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 defines and studies the geometric mean for tensors and its associated Riemannian geometry.
Paper defines multiplicities for quaternion eigenvalues without complex matrix concepts.
A new geometric definition of integration for differential forms.
Defines new representations for hyperbolic groups, unifying existing definitions.
Study of J-Hermitian matrices and geometric mean definition.
Paper defines and proves geometric uniqueness of Einstein field equations.
Study of metrics on positive-definite matrices from power potential, linking to power means.
New geometric structures defined on SPD matrices for better understanding.
Farrell and Hsiang noticed that the geometric surgery groups defined By Wall, Chapter 9, do not have the naturality Wall claims for them. They were able to fix the problem by augmenting Wall's definitions to keep track of a line bundle. The definition of geometric Wall groups involves homology with local coefficients a…
The search for a geometric interpretation of the constrained brackets of Dirac led to the definition of the Courant bracket. The search for the right notion of a "double" for Lie bialgebroids led to the definition of Courant algebroids. We recount the emergence of these concepts.
I describe three geometric approaches to resolving variants of P v. NP, present several results that illustrate the role of group actions in complexity theory, and make a first step towards completely geometric definitions of complexity classes.
We give a geometric construction of the multivariable Conway potential function for colored links. In the case of a single color, it is Kauffman's definition of the Conway polynomial in terms of a Seifert matrix.
Improved method for computing Fréchet means on SPD matrices.
Study elliptic isometries on a matrix manifold with specific metrics.
We analyze sub-Riemannian and lightlike metrics from the point of view of their rigidity as geometric structures. Following Cartan's and Gromov's formal definitions, they are never rigid, yet, in generic cases, they naturally give rise to rigid geometric structures!?
Positive definite matrices abound in a dazzling variety of applications. This ubiquity can be in part attributed to their rich geometric structure: positive definite matrices form a self-dual convex cone whose strict interior is a Riemannian manifold. The manifold view is endowed with a "natural" distance function whil…
We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate joint diagonalization (AJD). Today there is a considerable interest in estimating t…
Geometrically interprets frequency in electric circuits.
Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume compari…
In 1978 Brakke introduced the mean curvature flow in the setting of geometric measure theory. There exist multiple variants of the original definition. Here we prove that most of them are indeed equal. One central point is to correct the proof of Brakke's §3.5, where he develops an estimate for the evolution of the mea…
New geometric mechanism solves four envelope problems.
With the renewed and growing interest in geometric continuity in mind, this article gives a general definition of geometrically continuous polygonal surfaces and geometrically continuous spline functions on them. Polynomial splines defined by G1 gluing data in terms of rational functions are analyzed further. A general…
As the third of our series of papers on differential geometry of microlinear Frolicher spaces, this paper is devoted to the Frolicher-Nijenhuis calculus of their named bracket. The main result is that the Frolicher-Nijenhuis bracket satisfies the graded Jacobi identity. It is also shown that the Lie derivation preserve…
We define analytic indices which involve the eta form and the analytic torsion form. We show that these indices are independent of the geometric choices made in their definitions, and hence are topological in nature.
We give a differential geometric description of the Cartan (or tractor) bundle and its canonical connection in CR geometry, thus offering a direct, alternative, definition to the usual abstract approach.
We study the differential geometric properties of the manifold of non-singular symmetric real matrices endowed with the trace metric; in case of positive definite matrices we describe the full group of isometries
This paper studies rectifiability in Carnot groups and proves geometric area formulas.
We point out a new view on slow invariant manifolds (SIM) in dynamical systems which departs from a purely geometric covariant characterization implying coordinate independency. The fundamental idea is to treat the SIM as a well-defined geometric object in phase space and elucidate characterizing geometric properties t…
We analyse the definition of quasi-local energy in GR based on a Hamiltonian analysis of the Einstein-Hilbert action initiated by Brown-York. The role of the constraint equations, in particular the Hamiltonian constraint on the timelike boundary, neglected in previous studies, is emphasized here. We argue that a consis…
This work revisits, from a geometric perspective, the notion of discrete connection on a principal bundle, introduced by M. Leok, J. Marsden and A. Weinstein. It provides precise definitions of discrete connection, discrete connection form and discrete horizontal lift and studies some of their basic properties and rela…
Survey of three geometric frameworks for action-dependent field theories.
Study explains Zipf's law using geometric mechanisms from a finite alphabet.
The closure conditions of the inexact exterior differential form and dual form (an equality to zero of differentials of these forms) can be treated as a definition of some differential-geometrical structure. Such a connection discloses the properties and specific features of the differential-geometrical structures. The…
New Riemannian optimization improves variance estimation in mixed models.
These notions in the title are of fundamental importance in any branch of physics. However, there have been great difficulties in finding physically acceptable definitions of them in general relativity since Einstein's time. I shall explain these difficulties and progresses that have been made. In particular, I shall i…
In this paper we extend Badzioch's, Dorabiala's, and Williams' definition of cohomological higher smooth torsion to a twisted cohomological higher torsion invariant. Additionally, we show that this still satisfies geometric additivity and transfer, and will also satisfy additivity and transfer for coefficients.
It is shown how the coherent states permit to find different geometrical objects as the geodesics, the conjugate locus, the cut locus, the Calabi's diastasis and its domain of definition, the Euler-Poincaré characteristic, the number of Borel-Morse cells, the Kodaira embedding theorem.
We prove that smooth 1-dimensional topological field theories over a manifold are equivalent to vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth version of Rezk'…
We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we int…
Study on geometric Jensen-Shannon divergence for Gaussian measures in Hilbert space.
A new definition of umbilic points at infinity for polynomial surfaces.
New geometric spine for Artin groups defined by cube complexes.
In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this…
The paper uses Frenet frame to unify electrical and geometric quantities.
We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also define generalizations of virtual link diagrams and Gauss codes to represent virtua…
This is an expository paper, which provides a first introduction to geometric structures on . The paper contains definitions and characteristic properties of generalized complex, generalized Kaehler, generalized (normal, almost) contact and generalized Sasakian structures. A few of these properties are n…
In this thesis we study the Seiberg-Witten theory of an oriented homology 3-sphere. The goal is to extract topological invariants - the Seiberg-Witten invariants - by counting the solutions to the Seiberg-Witten equations on the manifold. The first question we consider is whether the Seiberg-Witten invariants depend on…
The paper studies geometric structures of wormholes using a new connection.