Paper extends Schur's theorem to spherical curves via monotonicity.
problem Comparing chord lengths of convex and spherical curves.
method Monotonicity and expansion module approach.
result Schur's Theorem extended to spherical curves.
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48 results for “Schur convexity”
The paper finds new inequalities for convex polygons.
problem Finding precise inequalities for convex polygons.
method Analytic isoperimetric inequalities based on Schur convex functions, followed by Bonnesen-style and inverse Bonnesen-style inequalities.
result Sharp discrete isoperimetric inequalities for planar convex polygons.
Convex hypersurfaces in curved spaces bound convex regions.
problem Characterizing convex hypersurfaces in curved spaces.
method Gauss-Codazzi equations, Schur comparison theorem, Alexandrov geometry.
result Closed convex hypersurfaces bound convex regions in curved spaces.
Closed surfaces minimize total curvature in curved spaces.
problem Minimizing total curvature in curved spaces.
method Isometric embedding via holonomy and Pogorelov's theory.
result Closed surfaces bound flat convex bodies.
Extends results for law-invariant functionals to random variable spaces.
problem Establishing results for a broad class of random variable spaces.
method Using structural results for law-invariant functionals and extending to new spaces.
result Unified perspective on law-invariant functionals, including quantile-based representations.
The spaces of Sp(n)-, Sp(n)U(1)- and Sp(n)Sp(1)- invariant, translation invariant, continuous convex valuations on the quaternionic vector space H^n are studied. Combinatorial dimension formulas involving Young diagrams and Schur polynomials are proved.
Defines a map linking quandle homology to Schur multiplier.
problem Understanding the relationship between quandle homology and Schur multiplier.
method Defined a map from second quandle homology to Schur multiplier and expressed second homology of Alexander quandles in terms of exterior algebras.
result Expressed the second homology of Alexander quandles in terms of exterior algebras.
Paper proves a new volume comparison theorem for Riemannian manifolds.
problem Comparing volumes of boundaries in Riemannian manifolds.
method Inspired by Schur's theorem, applies to Riemannian manifolds with Ricci curvature.
result Provides a new Schur's type volume comparison theorem.
The paper defines new types of positivity and proves properties of Schur forms for vector bundles.
problem Defining and characterizing new types of positivity for vector bundles.
method Introducing and characterizing two types of strongly decomposable positivity, proving properties of Schur forms.
result Schur forms of strongly decomposable positive vector bundles are positive or weakly positive, answering a question of Griffiths.
In this paper, we study a general almost Schur Lemma on pseudo-Hermitian (2n+1)-manifolds for . When the equality of almost Schur inequality holds, we derive the contact form is pseudo-Einstein and the pseudo-Hermitian scalar curvature is constant.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.
problem Proving new theorems in Riemannian and Alexandrov geometries.
method Inspired by the proof of the Schur-Toponogov theorem, a new proof of Toponogov's theorem is provided.
result A new theorem in Riemannian geometry and a new proof of Toponogov's theorem in Alexandrov geometry.
New CR almost Schur Lemma estimates curvature on compact manifolds.
problem Estimating curvature on compact pseudohermitian manifolds.
method Established a new CR almost Schur Lemma with specific positivity conditions.
result Estimates pseudohermitian scalar curvature as a constant.
Proves a formula for Kontsevich-Witten tau-function using Schur Q-polynomials.
problem Proving the Kontsevich-Witten tau-function formula.
method Directly shows Q-polynomial expansion satisfies Virasoro constraints.
result Direct proof of the formula without matrix model.
New unoriented versions of Schur and Bogomolov multipliers for finite groups.
problem Defining and analyzing unoriented versions of Schur and Bogomolov multipliers.
method Using cohomology groups and quotient groups to define unoriented multipliers.
result Triviality of unoriented Bogomolov multiplier for certain groups, nontriviality for others.
Quantifies Schur's theorem for curves in CAT(k) spaces.
problem Quantifying Schur's comparison theorem for curves in CAT(k) spaces.
method Comparison formula for curves in model planes, curvature measures, moment arm, and Reshetnyak's theorem.
result Sharpens and extends classical arm and bow lemmas and Riemannian analogues.
We show that the Schur multiplier of is , when is divisible by 4.
Let X=G/P be cominuscule rational homogeneous variety. (Equivalently, X admits the structure of a compact Hermitian symmetric space.) We say a Schubert class [S] is Schur rigid if the only irreducible subvarieties Y of X with homology class [Y] = r [S], for an integer r, are Schubert varieties. Robles and The identifie…
Efficiently computes embeddings for large graphs using coarsening.
problem Inefficient computation of graph embeddings for large-scale graphs.
method Graph coarsening based on Schur complements and Gaussian elimination.
result Efficiently computed embeddings on coarsened graph match Schur complement embeddings in expectation.
Schur theorem proven for weakly Landsberg Finsler metrics.
problem Proving the Schur theorem for a specific class of Finsler metrics.
method Using the Ricci curvature and properties of the mean Landsberg tensor, the theorem is proven for weakly Landsberg metrics.
result For weakly Landsberg Finsler metrics, the Ricci scalar must be constant.
New basis and Schur-Weyl duality for loop Hecke algebra defined.
problem Define a new basis for the loop Hecke algebra.
method Use higher linear rewriting theory and combinatorics of Dyck paths.
result Yields a conjecture of Damiani-Martin-Rowell and provides a representation theoretic interpretation.
New method connects neural networks to diagrammatic algebra.
problem Constructing permutation equivariant neural networks.
method Schur-Weyl duality between symmetric group and partition algebra.
result Simple diagrammatic method for calculating weight matrices.
The paper proves new curvature estimates in quaternionic contact geometry.
problem Estimating the curvature of compact qc manifolds.
method Establishing quaternionic contact versions of the Almost Schur Lemma.
result Curvature is a constant in terms of specific norms and components.
We study the restless bandit associated with an extremely simple scalar Kalman filter model in discrete time. Under certain assumptions, we prove that the problem is indexable in the sense that the Whittle index is a non-decreasing function of the relevant belief state. In spite of the long history of this problem, thi…
Study framizations of algebras using Schur--Weyl duality and tied braids.
problem Understanding framizations of algebras and their connections to quantum groups.
method Developing a general setting for framizations of algebras, including Yokonuma--Hecke and tied braids.
result Obtained Schur--Weyl duality for various algebras, including new framizations.
In this paper, we prove almost Schur Lemma on closed smooth metric measure spaces, which implies the results of X. Cheng and De Lellis-Topping whenever the weighted function f is constant.
Let L be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of the exterior current algebra of the Lie algebra sl_2. When L is an m-framed n-cable of a knot K in the three-sphere, its sutured annular Khovanov homology carries a commuting action of the symmetric group S_n…
Optimizes embedding accuracy for data variance and error.
problem Efficiently embedding data while minimizing distortion.
method Uses Johnson-Lindenstrauss embeddings with orthogonal matrices and singular-value latent variables.
result Achieves best accuracy in variance, mean-squared error, and length distortion.
Researchers prove a stability result for a 3-sphere inequality, extending previous work.
problem Quantitative stability of nonlinear Yamabe-type inequalities on the 3-sphere.
method Proved a two-term refinement of the Schur lemma inequality in the conformal class of the 3-sphere.
result Deduced quantitative stability of an entire family of nonlinear Yamabe-type inequalities.
A new method for efficient portfolio optimization using graph structures.
problem Optimizing portfolio weights while reducing computational complexity.
method Hierarchical graph structures and Schur complement method.
result Optimal portfolio weights can be computed efficiently by inverting small submatrices.
The paper studies connectivity of Schur-Horn map images in real Grassmannians.
problem Connectivity of Schur-Horn map images in real Grassmannians.
method Criterion for pre-images of vectors in \(\mathbb{R}^n\) to be connected.
result Criterion for pre-images of vectors in \(\mathbb{R}^n\) to be connected.
The Schur's theorem of antiholomorphic type is proved for arbitrary almost Hermitian manifolds, namely: If a connected almost Hermitian manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then this curvature is a global constant.
Defines odd Khovanov homology via categorification of q-Schur algebra.
problem Constructing odd Khovanov homology via representation theory.
method Supercategorification of q-Schur algebra, odd foams, tensor product on chain complexes.
result Odd Khovanov homology defined via categorification.
New gauge invariants from framed 3-manifolds match Hopf algebra indicators.
problem Computing indicators for Hopf algebras.
method Using Kuperberg invariants from framed 3-manifolds.
result Kuperberg invariants match higher Frobenius-Schur indicators of Hopf algebras.
Constructs new topological theories in 2D not fitting standard axioms.
problem Developing new topological theories in 2D that don't conform to traditional axioms.
method Universal construction by Blanchet et al., Kronecker's characterization, field extension, Hankel matrices, Schur polynomials, and foam evaluation.
result Introduction of non-multiplicative theories and classification over finite-dimensional state spaces.
In this short note we establish an integral geometric inequality in a smooth metric measure space of the nonnegative Bakry-Émery Ricci curvature. This result can be regarded as a mild generalization of the almost Schur theorem due to De Lellis and Topping (Calc. Var., DOI: 10.1007/s00526-011-0413-z).
In this article, we first establish the main tool - an integral formula for Riemannian manifolds with multiple boundary components (or without boundary). This formula generalizes Reilly's original formula from \cite{Re2} and the recent result from \cite{QX}. It provides a robust tool for sub-static manifolds regardless…
Method for conditional sampling with pre-trained normalizing flows.
problem Conditional sampling for incomplete observations.
method Variational Schur conditional sampling with normalizing flows.
result Successfully applied to invertible residual networks for inference and classification.
New algorithm learns ReLU networks efficiently using Schur polynomials.
problem PAC learning a linear combination of ReLU activations under Gaussian distribution.
method Uses tensor decomposition and Schur polynomials to identify and analyze higher-order moments.
result Near-optimal sample and computational complexity for learning ReLU networks.
Almost-Schur lemmamath.DG
Schur's lemma states that every Einstein manifold of dimension has constant scalar curvature. Here is defined to be Einstein if its traceless Ricci tensor $$\Rico:=\Ric-\frac{R}{n}g$$ is identically zero. In this short note we ask to what extent the scalar curvature is constant if the traceless Ricci …
In our previous paper in \cite{C}, we generalized the almost-Schur lemma of De Lellis and Topping for closed manifolds with nonnegative Rcci curvature to any closed manifolds. In this paper, we generalize the above results to symmetric -tensors and give the applications including th mean curvatures of closed …
Spatial statisticians and quantitative investors use the same mathematical object: a Schur complement, damped by one parameter.
problem The Schur complement is used in both spatial modeling and portfolio allocation, but the parameters are different.
method The Schur complement is interpreted as reliability shrinkage of a conditional Gaussian.
result The Schur complement is the same in both applications.
This paper introduces Schur-constant equilibrium distribution models of dimension n for arithmetic non-negative random variables. Such a model is defined through the (several orders) equilibrium distributions of a univariate survival function. First, the bivariate case is considered and analyzed in depth, stressing the…
On characteristic forms of positive vector bundles, mixed discriminants and pushforward identitiesmath.AG
The paper proves positivity of characteristic forms for certain vector bundles.
problem Characterizing positivity conditions for vector bundles.
method Operator theory, pushforward identities, and differential forms.
result Schur polynomials in Chern forms of Nakano and Griffiths positive vector bundles are positive as differential forms.
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.
Preprint proves quantum coideal Schur-Weyl duality and generalizes Jones-Wenzl projectors.
problem Quantum coideal Schur-Weyl duality and Jones-Wenzl projectors in type B/D.
method Combinatorial proofs and functional analytic arguments.
result Explicit proof of quantum coideal Schur-Weyl duality and generalization of Jones-Wenzl projectors.
Fix a finite group and a conjugacy invariant subset . Let be an oriented surface, possibly with punctures. We consider the question of when two homomorphisms taking punctures into are equivalent up to an orientation preserving diffeomorphism of . We provide an answer to this …
The paper optimizes policies constrained to Schur stabilizing controllers using a Newton-type algorithm.
problem Optimizing policies under linear constraints in control systems.
method Newton-type algorithm on a manifold of Schur stabilizing controllers with a Riemannian metric.
result Local convergence guarantees for the Newton-type algorithm without relying on exponential mapping or retractions.
Paper improves performance guarantees for Rademacher projections.
problem Improving statistical guarantees for Rademacher random projections.
method Algebraic framework for proving Schur-concavity properties.
result Novel Schur-concavity property of Rademacher projections with improved performance.