The paper proves isoparametric functions on Finsler space forms under specific conditions.
problem Understanding isoparametric functions in Finsler space forms.
method Proving transnormal functions as isoparametric functions and constructing global and local isoparametric functions using the distance function.
result Generalization of Theorem B to Finsler space forms.
The paper proves unboundedness of Donaldson-Hitchin functionals on G2 and tG2 forms.
problem Proving unboundedness of Donaldson-Hitchin functionals on G2 and tG2 forms.
method Explicit local calculations combined with covering arguments.
result Proves unboundedness above and below of the Donaldson-Hitchin functionals on G2 and tG2 forms.
New variational principle found for non-variational differential equations.
problem Non-variational differential equations without variational multipliers.
method Connecting functional forms with antiexact differential forms to identify obstructions.
result Formulation of variational problem for non-variational equations.
We obtained that any 2-form and any smooth function on 2-manifolds with boundary can be realized as the curvature form and the gaussian curvature function of some Riemmanian metric, respectively.
Differential forms on the Fréchet manifold F(S,M) of smooth functions on a compact k-dimensional manifold S can be obtained in a natural way from pairs of differential forms on M and S by the hat pairing. Special cases are the transgression map associating (p-k)-forms on F(S,M) to p-forms on M (hat pairing with a const…
The study calculates harmonic functions and 1-forms on specific 4D spaces.
problem Computing harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
method Computed the expansion of harmonic functions and 1-forms.
result Computed the expansion of harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called "mean curvature of the second fundamental form". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs…
Defines a filtration on variational bicomplex for concise functional form conditions.
problem Expressing functional form vanishing conditions concisely.
method Introduces a filtration on the variational bicomplex and studies its properties.
result Graded components of the filtration inherit module structures, simplifying functional form conditions.
Develops a framework for potentials on Lauritzen manifolds using bi-forms.
problem Developing a framework for potentials on Lauritzen manifolds using bi-forms.
method Developing a framework for potentials on Lauritzen manifolds using bi-forms.
result Constructing a canonical contrast bi-form on dually curvature-free Lauritzen manifolds.
The study simplifies complex functions on surfaces using a special transformation.
problem Understanding functions with degenerate singularities on various surfaces.
method Established a 'normal form' for functions using a specific transformation.
result Any function in the class can be simplified to a 'simplest' Morse function through a transformation.
Study on polynomial growth functions and forms on gradient Ricci solitons.
problem Estimating dimensions of polynomial growth holomorphic functions and forms.
method Relating to spectral data of the f-Laplacian, proving estimates under curvature assumptions. result Sharp dimension estimates and almost sharp frequency estimates for polynomial growth holomorphic functions.
Survey on smooth function and form density in Riemannian Sobolev spaces.
problem Density of smooth functions and forms in Sobolev spaces on Riemannian manifolds.
method Careful examination of weak covariant derivatives and partial derivatives.
result Equivalence of weak covariant derivatives to weak partial derivatives.
Study of tautological forms on curve moduli spaces.
problem Understanding tautological forms on moduli spaces of curves.
method Defined and studied a system of tautological rings on moduli spaces of marked curves, showing certain 2-forms are tautological and rings are finite dimensional.
result Characterized the Kawazumi-Zhang invariant as a tautological form.
Paper introduces magnetic Hodge Laplacian for differential forms.
problem No specific problem stated; general spectral analysis of differential forms.
method Introduced magnetic Hodge Laplacian, discussed spectral results.
result Similarities and differences with magnetic Laplacian on functions.
Topological complexity for closed 1-forms
problem Topological complexity for closed 1-forms
method Introduce and study a corresponding version of topological complexity
result Establish analogues of basic properties of ordinary topological complexity
The paper simplifies complex 2D functions near their critical points.
problem Simplifying smooth functions on 2-manifolds near critical points.
method Explicit construction of coordinate changes to canonical form.
result Estimates the radius of required neighbourhoods for specific singularity types.
Functional for Spin(7) forms defined on compact manifolds.
problem Characterizing conformal Spin(7) forms on Riemannian manifolds.
method Algebraic equation of degree two for self-dual four-forms, construction of a functional.
result Construction of a functional whose critical points are conformally Ricci-flat Spin(7) structures.
The paper investigates Liouville type theorems for various harmonic forms on Riemannian manifolds.
problem Investigating Liouville type properties of harmonic forms on Riemannian manifolds.
method Normalized integral Ricci curvature and BiRic curvature.
result Established Liouville theorems for p-harmonic function, p-harmonic 1 form, and harmonic q form (with q≥2). We introduce a new statistical test of the hypothesis that a balanced panel of firms have the same growth rate distribution or, more generally, that they share the same functional form of growth rate distribution. We applied the test to European Union and US publicly quoted manufacturing firms data, considering functio…
This paper generalizes beta divergence beyond its classical form associated with power variance functions of Tweedie models. Generalized form is represented by a compact definite integral as a function of variance function of the exponential dispersion model. This compact integral form simplifies derivations of many pr…
We present two range characterizations for the attenuated geodesic X-ray transform defined on pairs of functions and one-forms on simple surfaces. Such characterizations are based on first isolating the range over sums of functions and one-forms, then separating each sub-range in two ways, first by implicit conditions,…
Paper introduces a new multilinear functional for spectral triples and computes its properties.
problem Computing properties of spectral triples and their associated Hodge operators.
method Introduces a new multilinear functional for spectral triples and computes its properties using noncommutative residue and perturbed de-Rham Hodge operators.
result Recover two forms, torsion of the linear connection, and four forms by the noncommutative residue and perturbed de-Rham Hodge Dirac triple.
Study Z2 harmonic functions with singularities on flat space.
problem Understanding Z2 harmonic functions with point singularities.
method Analyzes Z2 harmonic functions on R2 with point singularities. result Characterizes Z2 harmonic functions on R2 with point singularities. New conformally invariant forms help identify Einstein metrics.
problem Identifying Einstein metrics in conformal classes.
method Constructing new conformally invariant one-forms.
result Global obstructions to the existence of Einstein metrics.
The paper proves optimal estimates and inequalities for spectral functions on certain manifolds.
problem Optimal estimates and inequalities for spectral functions on weakly 1-complete manifolds.
method Establishes optimal fundamental estimates and weak Morse inequalities for lower energy forms.
result Optimal fundamental estimates and weak Morse inequalities are proven for lower energy forms on weakly 1-complete manifolds.
Self-test loss functions improve data-driven modeling of weak-form operators and gradient flows.
problem Challenges in selecting test functions for data-driven modeling involving weak-form operators and gradient flows.
method Introducing self-test loss functions that depend on unknown parameters and are quadratic.
result Self-test loss functions conserve energy for gradient flows and coincide with log-likelihood ratios for stochastic differential equations.
Introduces a new Yang-Mills functional for connections and scalars over circle bundles.
problem Yang-Mills functional over circle bundles with two-forms.
method Dimensional reduction and Euler-Lagrange equations.
result Special three-dimensional solutions satisfy a duality condition.
The study characterizes and studies stability of biharmonic hypersurfaces in complex space forms.
problem Characterizing and studying biharmonic hypersurfaces in complex space forms.
method Characterizing hypersurfaces as critical points of a higher order energy functional.
result Existence and non-existence results for CPn and CHn. Study Hamiltonian stationary Lagrangian surfaces in complex space forms.
problem Characterize Lagrangian surfaces with harmonic mean curvature in complex space forms.
method Analyze surfaces with constant and harmonic mean curvature, using second fundamental form parallelism and Gaussian curvature constancy.
result Complete classification of Lagrangian surfaces with harmonic mean curvature and constant Gaussian curvature.
One way to recognise an object is to study how the echo has been shaped during the interaction with the target. Wideband sonar allows the study of the energy distribution for a large range of frequencies. The frequency distribution contains information about an object, including its inner structure. This information is…
Extends plate problems to differential forms on manifolds.
problem Eigenvalue problems for buckling and clamped plates on differential forms.
method Characterizes smallest eigenvalues, proves spectra equivalence, obtains estimates.
result Spectra of plate problems on forms coincide with functions in bounded domains.
Bi-forms extend contrast functions to handle torsion in information geometry.
problem Insufficient contrast-based approaches for geometric structures with torsion.
method Introducing contrast bi-forms, a generalization of contrast functions.
result Bi-forms provide a unified framework for statistical potentials.
In this paper, we discuss anisotropic submanifolds and isoparametric hypersurfaces in a Randers space form (N,F) with the navigation datum (h,W). We find that (N, F) with respect to the BH-volume and (N,h) have the same isoparametric hypersurfaces although, in general, their isoparametric functions are different. This …
Regularized empirical risk minimization with constrained labels (in contrast to fixed labels) is a remarkably general abstraction of learning. For common loss and regularization functions, this optimization problem assumes the form of a mixed integer program (MIP) whose objective function is non-convex. In this form, t…
A new method for analyzing shapes and forms using additive models on manifolds.
problem Analyzing shapes and forms under geometric transformations.
method Extending generalized additive regression to models for shapes/forms using squared geodesic distance and Riemannian L2-Boosting algorithm. result Automated model selection and intuitive visualization of covariate effects in shape/form space.
In reinforcement learning, Return, which is the weighted accumulated future rewards, and Value, which is the expected return, serve as the objective that guides the learning of the policy. In classic RL, return is defined as the exponentially discounted sum of future rewards. One key insight is that there could be many…
Due to the success of deep learning to solving a variety of challenging machine learning tasks, there is a rising interest in understanding loss functions for training neural networks from a theoretical aspect. Particularly, the properties of critical points and the landscape around them are of importance to determine …
We give an explicit description of the spectrum of the Hodge--Laplace operator on p-forms of an arbitrary lens space for any p. We write the two generating functions encoding the p-spectrum as rational functions. As a consequence, we prove a geometric characterization of lens spaces that are p-isospectral for e…
A formula for triangle area in Deep Sets form.
problem Finding a polynomial formula for triangle area in Deep Sets form.
method Expressing area as a permutation-invariant function and finding a suitable Deep Sets form.
result Explicit polynomial formula for triangle area in Deep Sets form.
Extends spectral Einstein functionals computation to 4D spin manifolds with boundary.
problem Computing spectral Einstein functionals for 4D spin manifolds with boundary.
method Generalizes Dabrowski's results to 4D spin manifolds with boundary using noncommutative residue.
result Generalized spectral Einstein functionals computation for 4D spin manifolds with boundary.
Study of degenerate contrast functions on Lie groupoids and their geometric structures.
problem Understanding geometric structures on Lie groupoids with degenerate metrics.
method Using Lie groupoids and algebroids, analyze contrast functions and degenerate two-forms.
result Reduction of degenerate two-forms to pseudometric structures under regular conditions.
Study asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces.
problem Asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces with ADE singularities.
method Investigates a function on the unit disc defined by fiber integrals of the forms with a smooth test function, showing a lower bound of Hölder exponent at the origin for both cscK-metrics and Ricci-flat metrics.
result Shows bounds of Hölder exponent for both cscK-metrics and Ricci-flat metrics.
Functionals involving surface curvature are important across a range of scientific disciplines, and their extrema are representative of physically meaningful objects such as atomic lattices and biomembranes. Inspired in particular by the relationship of the Willmore energy to lipid bilayers, we consider a general funct…
New integration theory on topological spaces, including fractals.
problem Developing a universal integration theory for arbitrary topological spaces.
method Introducing a new integration framework using unital magma valued functions and measures.
result Integration, differentiation, and orientation defined for arbitrary topological spaces.
Geometrically describes the linear and quadratic forms for rational links.
problem Predicting generating functions for colored HOMFLY-PT polynomials of rational links.
method Direct geometric description of linear and quadratic forms in terms of configuration spaces.
result Direct geometric description of forms for rational links.
Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.
problem Understanding shared properties of geodesically equivalent Finsler metrics.
method Computing first integrals as coefficients of a characteristic polynomial.
result Geodesically invariant functions are first integrals of geodesically equivalent Finsler metrics.
The aim of this paper is to study the fast computation of the lower and upper bounds on the value function for utility maximization under the Heston stochastic volatility model with general utility functions. It is well known there is a closed form solution of the HJB equation for power utility due to its homothetic pr…
The paper proves unboundedness of a functional on G2 forms and describes manifold limits.
problem Proving unboundedness of a functional on G2 forms and describing manifold limits.
method Scaling arguments, geometric estimates, collapsing theorem for orbifolds.
result Explicit descriptions of large volume limits of two G2 manifolds.