It is shown that in a tower of coverings the regularized determinant of a generalized Laplacian converges to the -determinant. This shows generic nontriviality of analytic torsion or regularized determinants since the -counterparts are easier to compute. We further have an "Euler product expansion" for regula…
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48 results for “ζ-regularized determinant”
Regularized determinants of Laplace type operators, analytic surgery and relative determinantsmath.DG
We study the asymptotic behaviour of regularized determinants of certain Laplace type operators with respect to singular deformations of the underlying manifold which are obtained by stretching a tubular neighborhood of an embedded separating hypersurface to a cylinder of infinite length. Using the asymptotic expansion…
Research determines criteria for semi-regular tilings in hyperbolic space.
problem Finding combinatorial criteria for semi-regular tilings in hyperbolic geometry.
method Combinatorial analysis of vertex-types and geodesic polygons.
result Determined criteria for existence and uniqueness of semi-regular tilings.
Determines regular homotopy classes for link immersions of simple singularities.
problem Classifying immersions of link singularities.
method Computing complete invariants of immersions and comparing with Dynkin diagrams.
result Inclusion map of link into 5-sphere is regularly homotopic to immersion associated with Dynkin diagram.
Study finds determinant of Laplacian on a special surface with a conical point.
problem Determining the determinant of the Laplacian on a specific type of surface.
method Explicit expression for the zeta-regularized determinant of the Laplacian on a compact Riemann surface with a conical singularity.
result An explicit expression for the determinant of the Laplacian is derived.
Researchers compute determinants and torsions of Rumin complex in specific Lie group representations.
problem Computing determinants and torsions of Rumin complex in specific Lie group representations.
method Analyzing Schrodinger and generic representations of the (2,3,5) nilpotent Lie group.
result Computed the spectrum and zeta regularized determinant of Rumin differentials in Schrodinger representations and evaluated their alternating product in generic representations.
Determinants remain constant along specific families of differential operators.
problem Local constancy of regularized determinants for differential operators.
method Analyzing families of operators , showing flat-regularized determinant's constancy.
result The flat-regularized determinant is constant in when restricted to under suitable assumptions.
Adaptive regularization improves deep learning model performance.
problem Improving generalization in deep learning models.
method Adaptive regularization via residual smoothing based on the heat equation.
result Our algorithm outperforms other optimization methods in generalization.
Study pseudo-laplacians and ζ(1) for spinor bundles over Riemann surfaces.
problem Analyzing self-adjoint extensions of Dolbeault Laplacians on Riemann surfaces.
method Defined -regularized determinants, introduced Robin mass, derived comparison formulas.
result Explicit expressions for Robin mass in spinor bundles and scalar cases.
The paper calculates determinants for Laplacians on spinor bundles over surfaces with flat metrics.
problem Calculating determinants for Laplacians on spinor bundles over surfaces with flat metrics.
method Explicit expressions for determinants of self-adjoint extensions of Laplacians using Bergman tau-function and theta-constants.
result An explicit expression for the determinant of the Szegö extension and comparison formulas for different extensions.
Formula for Laplacian determinants on polygonal domains with slits.
problem Determining the -regularized determinant of the Laplacian on polygonal domains with slits.
method Patchwork method for heat trace asymptotics, comparison formula for smooth conformal metrics.
result Polyakov-Alvarez type formula for Laplacian determinants on polygonal domains with slits.
Flat conical Laplacian in the square of the canonical bundle and its regularized determinantsmath.DG
Study on conical Laplacian operators on Riemann surfaces, focusing on determinants and moduli spaces.
problem Analyzing determinants of Laplacian operators on Riemann surfaces with conical metrics.
method Introduce and compare two methods to regularize determinants of the conical Laplacian acting in the bundle .
result Explicit expressions for the regularized determinants of the conical Laplacian in .
We consider a regular singular Sturm-Liouville operator on the line segment . We impose certain boundary conditions such that we obtain a semi-bounded self-adjoint operator. It is known that the -function of this operator $ζ_L(s)=\sum_{λ\in\spec(L)\setminus\{0…
Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cuspsmath.DG
We study the eta invariants of Dirac operators and the regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner M"uller and use relative traces to define these spectral invariants. We show the regularity of eta and zeta functions at s=0. The Selberg trace formula and the detai…
Paper proves DN map determination for simple surfaces with low regularity metrics.
problem Determining DN map from scattering relation for surfaces with low regularity metrics.
method Modified technical results and used microlocal analysis for metrics with finite regularity.
result Scattering relation determines DN map for surfaces, and for metrics using Lipschitz distance function.
We discuss the regularized determinant of elliptic boundary value problems on a line segment. Our framework is applicable for separated and non-separated boundary conditions.
The paper calculates the Euler characteristic of regular spherical polygon spaces.
problem Determining the Euler characteristic of regular spherical polygon spaces.
method Constructing a manifold and a function such that , determining the index of critical points, and using Morse surgeries.
result Calculating the Euler characteristic for all and odd .
The Residue Determinantmath.AP
The purpose of this paper is to present the construction of a canonical determinant functional on elliptic pseudodifferential operators associated to the Guillemin-Wodzicki residue trace. The resulting functional is multiplicative, a local invariant, and not defined by a regularization procedure. The residue determinan…
Sub-Riemannian Selberg trace formulae for compact quotients of SL(2, R)
problem Computing zeta-regularized determinants of sub-Laplacians
method Using Fourier decomposition and Selberg trace formulae
result Compact determinant formula expressed in terms of base hyperbolic surface and relative Selberg product
Paper optimizes Laplacian regularization for sparse network clustering.
problem Improving spectral clustering in sparse networks.
method Formally determines optimal Laplacian regularization.
result Proper regularization is closely tied to state-of-the-art techniques.
This is the first of a series of articles in which we are going to study the regularized determinants of the Laplacians of Calabi Yau metrics acting on (0,q) forms on the moduli space of CY manifolds with a fixed polarization. It is well known that in case of the elliptic curves the Kronecker limit formula gives an exp…
Given a flag in each of the vertex-transitive tessellations of the Euclidean plane by regular polygons, we determine the flag stabilizer under the action of the automorphism group of a regular cover. In so doing we give a presentation of these tilings as quotients of regular (infinite) polyhedra.
The study of semi-regular biperiodic alternating links and their geometric properties.
problem Understanding the geometric properties of biperiodic alternating links.
method Analyzing semi-regular links in the thickened torus, determining volumes, and relating commensurability and arithmeticity to Euclidean tilings.
result There exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field.
The paper discusses fractional Sobolev immersions of flat domains into 3D space.
problem Developing regularity and isometric immersions of flat domains with fractional Sobolev regularity.
method Analysis of weak Codazzi-Mainardi equations, study of $W^{2, rac2s}$ planar deformations, and properties of the distributional Jacobian determinant.
result Generalization of isometric immersions with local fractional Sobolev regularity.
Study flat conical surfaces' Laplacian determinants, showing differences in holonomy.
problem Comparing Laplacian determinants on flat conical surfaces with different holonomy.
method Analysis of self-adjoint extensions of the Laplacian on flat conical surfaces of varying holonomy.
result Significant differences in -regularized determinants for surfaces with trivial and non-trivial holonomy.
Study shows exponential growth of Laplacian determinant on random hyperbolic surfaces.
problem Understanding the behavior of Laplacian determinants on random hyperbolic surfaces.
method Investigated various models of random hyperbolic surfaces and their Laplacian determinants as genus increases.
result For all popular models, the determinant grows exponentially with a universal exponent as the genus goes to infinity.
We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy exampl…
Paper finds essential regularity in singular connections.
problem Determining if singularities in connections are removable or essential.
method Introduces RT-equations and a procedure to lift connections to essential regularity.
result A computable procedure to lift connections to essential regularity.
In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.
The regular genus of certain 4-manifolds is determined, providing new insights.
problem Determining the regular genus of higher-dimensional closed PL manifolds.
method Using crystallization graphs and combinatorial topology, the regular genus is calculated for specific manifolds.
result The regular genus of is 6, and is 16.
Formula derived for spectral determinant of sphere with conical singularities.
problem Calculating the spectral determinant of a sphere with conical singularities.
method Explicit closed formula derived using zeta regularization and Liouville action.
result Metrics with equal conical angles are a stationary point of the determinant, and a minimum if surface area is small.
Compact polyhedral surfaces (or, equivalently, compact Riemann surfaces with conformal flat conical metrics) of an arbitrary genus are considered. After giving a short self-contained survey of their basic spectral properties, we study the zeta-regularized determinant of the Laplacian as a functional on the moduli space…
In this note we specialize and illustrate the ideas developed in the paper math.DG/0201112 of the first author ("Index theory, eta forms, and Deligne cohomology ") in the case of the determinant line bundle. We discuss the surgery formula in the adiabatic limit using the adiabatic decomposition formula of the zeta regu…
Adaptive sparseness for correntropy-based robust regression via automatic relevance determinationcs.LG
Adaptive sparseness enhances robust regression using MCC and ARD.
problem Developing a robust regression method with adaptive sparseness.
method Integrating MCC with ARD in a Bayesian framework using variational Bayesian inference.
result MCC-ARD regression outperforms existing methods in prediction and feature selection.
Let be a holomorphic vector bundle on a compact Kähler manifold . If we fix a metric on , we get a Laplace operator acting upon smooth sections of over . Using the zeta function of , one defines its regularized determinant . We conjectured elsewhere that, when varies, this deter…
The Quillen-Bismut-Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators. We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut-Freed connection on…
New proof for convex solutions of Monge-Ampère equation.
problem Interior regularity of strictly convex solutions
method Doubling inequality for Hessian in extrinsic distance function
result Interior regularity established
We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, …
We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using …
Study on conical singularities in 2D surfaces, deriving Polyakov formulas.
problem Analyzing zeta-regularized determinants in surfaces with conical singularities.
method Demonstrated variational and integrated Polyakov formulas for conical singularities, circular sectors, and cones.
result Explicit formulas for the determinant of conical sectors and cones derived.
The paper analyzes methods for sparse Bayesian regression in nonlinear system identification.
problem Learning sparse models in Bayesian regression with nonlinear applications.
method Two classes of methods: regularization and thresholding based, built on automatic relevance determination (ARD).
result Analytical demonstration of favorable performance with sparse solutions in linear problems.
Classifies tilings of hyperbolic plane by regular polygons.
problem Decidability of tiling patterns in hyperbolic plane.
method Finite set of local and inductive combinatorial constraints.
result First known weakly aperiodic protosets of regular polygons in hyperbolic plane.
We propose an adaptive optimization method for deep learning that dynamically adjusts batch size.
problem Optimizing deep learning models with varying sensitivity to batch size selection.
method Adaptive regularization with dynamically determined stochastic batch size based on gradient norms.
result Our method outperforms state-of-the-art optimization algorithms in generalization and robustness.
Formula derived for zeta functions of 3D foliated systems.
problem Analyzing zeta functions of 3D Riemannian foliated dynamical systems.
method Relating dynamical spectral -functions to zeta functions using the distributional dynamical Lefschetz trace formula.
result Proved a regularized determinant formula for zeta functions.
In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent bounds for that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in for all…
We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for st…
Researchers determine all possible representations of monodromy for Schwarzian equations on punctured surfaces.
problem Determine representations of monodromy for Schwarzian equations on punctured surfaces.
method Explicit constructions of complex affine structures on punctured surfaces, with prescribed holonomy.
result All possible representations of monodromy for Schwarzian equations on punctured surfaces are determined.
Study on geometric Jensen-Shannon divergence for Gaussian measures in Hilbert space.
problem Computing divergence between Gaussian measures in infinite-dimensional Hilbert space.
method Closed form expression and regularization for divergence calculation.
result Closed form expression and regularization for Geometric Jensen-Shannon divergence.