Study of zonal spherical functions on partial flag manifolds using Jacobi polynomials.
problem Understanding zonal spherical functions on partial flag manifolds.
method Matrix variate version of Koornwinder's method for constructing orthogonal polynomials, using Hermitian Jacobi polynomials and multivariate Schur polynomials.
result Conjecture that Hermitian Jacobi polynomials are elementary zonal spherical functions.
Deep networks better approximate functions with compositional structure.
problem Approximating functions with complex structures.
method Design deep networks with compositional structure, leveraging the blessing of compositionality.
result Deep networks can approximate functions better than shallow networks when the function has a compositional structure.
Improved bounds for eigenfunctions on hyperbolic surfaces found.
problem Establishing improved bounds for eigenfunctions of magnetic Laplacians.
method Using explicit eigenstates called magnetic zonal states.
result Explicit eigenstates called magnetic zonal states found.
Origami structures are enumerated and shown to be quantum modular.
problem Counting and understanding origami structures with real structures.
method Using combinatorics of zonal polynomials and Schur polynomials, and relating to quantum modular forms and double Hurwitz numbers.
result The generating functions of certain origami structures are quantum modular forms.
Identifies conjugate points in spherical harmonics solutions of quasi-geostrophic equations.
problem Locating conjugate points in spherical harmonics solutions.
method Utilizing structure constants and quasi-geostrophic equations on the sphere, identifying conjugate points.
result Existence and location of conjugate points along spherical harmonics solutions.
The Zeeman-Hamilton operators of free charged particles are identified with the Laplacians of certain Riemannian manifolds, called Zeeman manifolds. The quantum Hilbert space decomposes into subspaces (Zeeman zones) which are invariant under the actions both of the Zeeman operator and the natural Heisenberg group repre…
Adopting a zonal structure of electricity market requires specification of zones' borders. In this paper we use social welfare as the measure to assess quality of various zonal divisions. The social welfare is calculated by Market Coupling algorithm. The analyzed divisions are found by the usage of extended Locational …
Consider a family Z = { x i , y i Z=\{\boldsymbol{x_{i}},y_{i} Z = { x i , y i , 1 ≤ i ≤ N } 1\leq i\leq N\} 1 ≤ i ≤ N } of N N N pairs of vectors x i ∈ R d \boldsymbol{x_{i}} \in \mathbb{R}^d x i ∈ R d and scalars y i y_{i} y i that we aim to predict for a new sample vector x 0 \mathbf{x}_0 x 0 . Kriging models y y y as a sum of a deterministic function m m m , a drift which depends on the point $\boldsymbol…
Spatially-aware metrics improve uncertainty evaluation in segmentation.
problem Uncertainty evaluation metrics treat voxels independently, ignoring spatial context.
method Proposed three spatially aware metrics incorporating structural and boundary information.
result Improved alignment with clinically important factors and better discrimination between uncertainty patterns.
Paper proves existence of conjugate points on ellipsoids but not on spheres.
problem Existence of conjugate points in incompressible Euler flows.
method Formulated a differential-geometric criterion (M-criterion) and analyzed flows on spheres and ellipsoids.
result Zonal flows on ellipsoids can satisfy M-criterion, while not on spheres.
Study compares two market clearing methods for European power markets.
problem Optimizing market clearing for European power markets considering cost and social welfare.
method Introduces Cost Minimization and Social Welfare Maximization models, and four algorithms to solve the CM model.
result Cost Minimization reduces market power and decreases total procurement cost.
Refines Hurwitz numbers with a two-parameter theory.
problem Understanding polynomial structure and tropicalization of b b b -Hurwitz numbers. method Introducing CJT-refinement of symmetric functions on Fock space.
result Derives tropicalization of b b b -Hurwitz numbers and solves an open problem. Develops kernels for matchings, overcoming computational challenges.
problem Challenges in applying kernel methods to matchings due to their discrete, non-Euclidean nature.
method Characterizes stationary kernels, introduces heat and Matérn kernel families, and develops a sub-exponential algorithm for efficient evaluation.
result Establishes novel negative results and identifies an open problem in transferring the framework to trees.
Study spherical curves with curvature dependent on distance to a great circle.
problem Understanding spherical curves with curvature dependent on distance to a great circle.
method Introducing spherical angular momentum, characterizing known curves, finding new families, and obtaining arc length parametrizations.
result New families of spherical curves with intrinsic equations in elementary or Jacobi elliptic functions.
Researchers found new functions for spherical clothoids using special functions.
problem Developing new mathematical functions for spherical clothoids.
method Used confluent hypergeometric functions and Meixner-Pollaczek polynomials.
result Presented Cartesian coordinate functions and stereographic projections.
PGF kernels analyze spherical data using generalized RBF kernels.
problem Analysis of spherical data.
method Introduced PGF kernels and a semi-parametric learning algorithm.
result PGF kernels generalize RBF kernels for spherical data.
Study spherical Fourier transform on hypergeometric type harmonic manifolds.
problem Spherical Fourier transform on harmonic Hadamard manifolds.
method Representation of spherical functions by Gauss hypergeometric functions.
result Inversion formula, convolution rule, and Plancherel theorem are derived.
Spherical representations and functions are the building blocks for harmonic analysis on riemannian symmetric spaces. In this paper we consider spherical functions and spherical representations related to certain infinite dimensional symmetric spaces G ∞ / K ∞ = l i m → G n / K n G_\infty/K_\infty = \varinjlim G_n/K_n G ∞ / K ∞ = lim G n / K n . We use the representation t…
Defines spherical type surfaces via support function and classifies them.
problem Characterizing surfaces via support function.
method Weierstrass type representation for SS-surfaces with prescribed Gauss map.
result Every compact and connected SS-surface is the sphere.
The Bounded Spherical Functions are determined for a Cartan Motion Group
Study biharmonic functions on vector bundles with spherical symmetry.
problem Investigate biharmonic functions on vector bundles with spherically symmetric metrics.
method Analyze vertical lifts and radial functions of functions on vector bundle manifolds.
result Construct an infinite two-parameter family of proper biharmonic functions.
Estimates spherical functions on SL(3,R) improving previous results.
problem Estimating spherical functions on SL(3,R) with uniform decay.
method Estimates spherical functions using the method of stationary phase and classifies singularities.
result Improves previous results by removing restrictions on group parameter.
Paper explores neural network approximations on sphere domains.
problem Approximating functionals on sphere domains using neural networks.
method Encoder-decoder framework with spherical harmonics for infinite-dimensional domain.
result Approximation rates of neural networks with different encoder structures.
This paper deals with some simple results about spherical functions of type δ δ δ , namely new integral formulas, new results about behavior at infinity and some facts about the related C σ C_σ C σ functions.
Locally classifies 4D spherical symmetric Finsler spaces.
problem Classifying 4D spherical symmetric Finsler spaces.
method Local classification of pseudo-Finsler Berwald structures.
result Six classes of non-Riemannian SO(3)-symmetric pseudo-Finsler Berwald functions.
Existence and uniqueness of spherical helicoidal surfaces in 3-sphere via spherical curves.
problem Existence and uniqueness of spherical helicoidal surfaces in 3-sphere.
method Continuous function of distance to axis, spherical angular momentum of spherical curves.
result Existence and uniqueness theorem for spherical helicoidal surfaces in 3-sphere.
Paper studies generic dynamics of MCFs with spherical singularities.
problem Characterizing the generic behavior of mean curvature flow with spherical singularities.
method Level set formulation of mean curvature flow, analysis of arrival time function.
result Generically, the arrival time function has at most C 2 C^2 C 2 regularity. The paper classifies spherically symmetric Finsler metrics as Berwaldian or Riemannian.
problem Classifying spherically symmetric Finsler metrics.
method Proving all spherically symmetric Landsberg surfaces are Berwaldian and modifying the classification of spherically symmetric Finsler metrics.
result All Berwald spherically symmetric metrics of dimension n ≥ 3 n\geq 3 n ≥ 3 are Riemannian or given by a specific formula. Study null surfaces of pseudo-spherical curves in anti-de Sitter space.
problem Characterize null surfaces of pseudo-spherical spacelike framed curves in anti-de Sitter 3-space.
method Introduced nullcone fronts, classified singularities, defined Anti-de Sitter distance-squared functions.
result Relate singularities of nullcone fronts to those of framed curves.
Classifies spherical objects filled with water or air using acoustic echoes and Form Function.
problem Automatic recognition of spherical objects based on acoustic echoes.
method Form Function analysis of wideband sonar data, fed into a Multilayer Perceptron (MLP) classifier.
result Performance of the MLP classifier compared to SVM, highlighting the effectiveness of Form Function.
We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space E n = G / K E^n = G/K E n = G / K where G G G is the semidirect product R n ⋅ K R^n \cdot K R n ⋅ K of the translation group with a closed subgroup K K K of the orthogonal group O(n). We give exact parameterizations of the space of $(G,K…
The paper studies K-stability of spherical varieties and their degenerations.
problem Understanding K-stability and degenerations of polarized spherical varieties.
method Reduction to a variational problem on the moment polytope, convexity constraint, and solving the HMA equation.
result Determines strict semistability and polystable degenerations for Fano spherical varieties of rank two.
Develops a new theory for approximating functions on massive data.
problem Challenges in machine learning with massive data.
method eignets theory for local, stratified approximation.
result Solves inverse problems like finding data probability law and function smoothness.
Describes the space of spherical triangles on a smooth 3-manifold.
problem Understanding the geometric structure of spherical triangles.
method Analyzes the homotopy and analytic properties of the space.
result The space is a smooth 3-manifold embedded in R^6.
We present in this paper a \boundary version" for theorems about minimality of volume and energy functionals on a spherical domain of threedimensional Euclidean sphere.
Bayesian approach approximates probability functions of Gaussian mixtures.
problem Approximating probability functions of non-spherical Gaussian mixtures.
method Bayesian decomposition, spherical radial decomposition, random sampling.
result Established differentiability and integral representation of gradient for probability functions.
Proves Li-Yau inequality for Helfrich functional, ensuring embeddedness in spherical cases.
problem Ensuring embeddedness of minimizers in the Canham-Helfrich model.
method Proves Li-Yau inequality for Helfrich functional, converting singular volume integral to explicit energy threshold.
result Existence of smoothly embedded minimizers in physically relevant cases.
This paper is the second part of a study of the quantum free particle on spherical and hyperbolic spaces by making use of a curvature-dependent formalism. Here we study the analogues, on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ ( κ > 0 κ>0 κ > 0 ) and $H_\k^3$ ( κ < 0 κ<0 κ < 0 ), to the standard {\itshape spherical wav…
Unique metric found for discrete curvature on spherical cone-metrics.
problem Finding a unique metric with prescribed curvature on spherical cone-metrics.
method Discrete conformal approach to spherical cone-metrics.
result Existence of a unique metric realizing prescribed curvature in each conformal class.
The paper studies T-tensor of spherically symmetric Finsler metrics and characterizes metrics satisfying the T-condition.
problem Characterizing spherically symmetric Finsler metrics with vanishing T-tensor.
method Deriving a general expression for the T-tensor and characterizing metrics satisfying the T-condition.
result Characterization of spherically symmetric Finsler metrics with vanishing T-tensor.
The study finds the optimal metrics for free boundary minimal surfaces in spherical caps.
problem Optimizing metrics for free boundary minimal surfaces in spherical caps.
method Introducing functionals based on eigenvalues of Steklov-type problems and proving maximizers are induced by immersions.
result Maximizing metrics are induced by free boundary minimal immersions in geodesic balls of a round sphere.
We survey many of the important properties of spherically symmetric spacetimes as follows. We present several different ways of describing a spherically symmetric spacetime and the resulting metrics. We then focus our discussion on an especially useful form of the metric of a spherically symmetric spacetime in polar-ar…
The paper explores geometric properties of interception curves on planes and spheres.
problem Geometric properties of interception curves defined by differential equations.
method Parametric representation and spherical curve defined by Gudermannian function.
result Symmetry/asymmetry between spherical and planar cases, connections to lemniscate constants.
Conformal qc geometry of spherical qc manifolds are investigated. We construct the qc Yamabe operators on qc manifolds, which are covariant under the conformal qc transformations. A qc manifold is scalar positive, negative or vanishing if and only if its qc Yamabe invariant is positive, negative or zero, respectively. …
The paper verifies deep neural networks' ability to approximate functions on spheres.
problem Theoretical verification of deep neural networks' performance on spherical functions.
method Spherical analysis using reproducing kernels and convolutional factorizations.
result Rates of uniform approximation for functions in Sobolev spaces and additive ridge forms.
Researchers found multiple spherical Ricci metrics on tori with rotational symmetry.
problem Constructing and analyzing spherical Ricci metrics with rotational symmetry.
method Explicitly constructed a two-parameter family of metrics with rotational symmetry and showed their existence on tori.
result Infinitely many non-isometric spherical Ricci metrics can be realized on the same torus.
Paper computes stability of Q-Fano spherical varieties using test configurations and Futaki invariants.
problem Stability of Q-Fano spherical varieties.
method Test configurations, Futaki invariants, intersection numbers.
result Equivalence of stability criteria and existence of Kähler-Ricci g-solitons.
Analyzes smoothness and classification of maps between manifolds.
problem Analyzing interpolating sesqui-harmonic maps between Riemannian manifolds.
method Derives a conservation law and uses it to show smoothness of weak solutions; obtains classification results.
result Smoothness of weak solutions and classification results for interpolating sesqui-harmonic maps.