A new class of harmonic Hadamard manifolds, those spaces called of hypergeometric type, is defined in terms of Gauss hypergeometric equations. Spherical Fourier transform defined on a harmonic Hadamard manifold of hypergeometric type admits an inversion formula. A characterization of harmonic Hadamard manifold being of…
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48 results for “Hadamard differentiability”
Develops a new method for statistical optimal allocation problems.
problem Statistical optimal allocation problems with constraints.
method Functional differentiability approach and Hadamard differentiability of value functions.
result Validates margin assumption for fast convergence rate of plug-in methods.
We apply a suitable modification of the functional delta method to statistical functionals that arise from law-invariant coherent risk measures. To this end we establish differentiability of the statistical functional in a relaxed Hadamard sense, namely with respect to a suitably chosen norm and in the directions of a …
Paper proves existence of Hadamard states for Maxwell equations.
problem Proving existence of Hadamard states for Maxwell equations on spacetime.
method Introducing Cauchy radiation gauge and new Hodge decomposition.
result Existence of Hadamard states for Maxwell equations proven.
Paper discusses directional differentiability of interval-valued functions on Riemannian manifolds.
problem Equivalence of directional differentiability of interval-valued functions and their components.
method Analyzes directional differentiability of interval-valued functions on Riemannian manifolds.
result Directional differentiability of interval-valued functions is not equivalent to the directional differentiability of their components.
Any surface can be foliated into equipotential hypersurfaces of the level sets. A current result is that the contours are the progressing wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, as well as with a corresponding eikonal equation. The lev…
Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the actio…
The paper derives expansions for Green's operators and resolvents using Hadamard methods.
problem Analyzing normally hyperbolic operators and their Green's functions.
method Hadamard expansions for powers of Green's operators and resolvents.
result Derives expansions involving Hadamard coefficients for advanced/retarded Green's operators.
Harmonic maps between pinched Hadamard surfaces are quasi-conformal.
problem Characterizing harmonic maps between Hadamard surfaces.
method Proving harmonic quasi-isometries are quasi-conformal diffeomorphisms.
result Harmonic quasi-isometries of pinched Hadamard surfaces are injective.
New insights into identifying mixtures of product distributions using Hadamard extensions.
problem Identifying mixtures of product distributions on binary variables.
method Analysis of Hadamard extensions of matrix products.
result Conditions for full column rank of Hadamard extensions.
We prove a quantitative openness theorem for submersions under suitable assumptions on the differential. We then apply our result to a class of exponential maps appearing in Carnot-Carathéodory spaces and we improve a classical completeness result by Palais.
Generalized gradient flows in Hadamard manifolds and convex optimization on entanglement polytopesmath.OC
Paper addresses optimization on Hadamard manifolds, generalizing gradient flow.
problem Optimization of convex functions on Hadamard manifolds.
method Introduces a generalized gradient flow to minimize .
result Gradient flow attains infimum in limit for basic manifolds.
Cyclic projections in Hadamard spaces can be irregular, unlike in Hilbert spaces.
problem Understanding the behavior of cyclic projections in Hadamard spaces compared to Hilbert spaces.
method Constructing an example of convex subsets in a Hadamard space.
result Cyclic product of projections is not asymptotically regular in Hadamard spaces.
Study shows spectrum properties for specific Hadamard manifolds.
problem Spectrum properties of Hadamard manifolds.
method Absolute continuity and spectrum determination for two classes of Hadamard manifolds.
result Spectrum properties determined for specific Hadamard manifolds.
We consider the problem of finding sufficient conditions for a locally Lipschitz mapping between Finsler manifolds to be a global homeomorphism. For this purpose, we develop the notion of Clarke generalized differential in this context and, using this, we obtain a version of the Hadamard integral condition for invertib…
Sharp curvature inequality extends Euclidean isoperimetric inequality to Cartan-Hadamard manifolds.
problem Proving the Euclidean isoperimetric inequality on Cartan-Hadamard manifolds with nullity.
method Using the Chern-Gauss-Bonnet theorem to establish a sharp inequality for total curvature.
result The Euclidean isoperimetric inequality extends to Cartan-Hadamard manifolds with nullity.
The paper provides formulas for Hadamard coefficients using Green's operators.
problem Calculating Hadamard coefficients from Green's operators.
method Various methods including resolvents, powers of Green's operators, and product with the real line.
result Formulas for Hadamard coefficients in terms of Green's operators.
Formula for Hadamard coefficients from Green's operators on spacetimes.
problem Calculating Hadamard coefficients from Green's operators on spacetimes.
method Developed formulas for diagonal values and integrals over the diagonal of Hadamard coefficients.
result Formulated analogues of Hadamard expansions and resolvents for Green's operators.
Upper bounds for circumradius in Hadamard surfaces with curvature constraints.
problem Bounding circumradius in Hadamard surfaces with curvature constraints.
method Using curvature constraints to derive upper bounds for circumradius.
result Upper bounds for circumradius in terms of curvature bounds.
Solves Plateau problem for surfaces in pinched curvature manifolds.
problem Asymptotic Plateau problem for immersed surfaces in pinched curvature manifolds.
method Complete solution to asymptotic Plateau problem, providing dynamical stability of hypersurface laminations.
result Achieved complete solution to the asymptotic Plateau problem for immersed surfaces of constant extrinsic curvature in Cartan--Hadamard manifolds.
Proves Hadamard states for Dirac fields on manifolds with timelike boundaries.
problem Existence of Hadamard states for Dirac fields with MIT boundary conditions.
method Introducing geometric Møller operator to implement unitary isomorphism between spaces of initial data.
result Existence of Hadamard states for Dirac fields with MIT boundary conditions.
Various results based on some convexity assumptions (involving the exponential map along with affine maps, geodesics and convex hulls) have been recently established on Hadamard manifolds. In this paper we prove that these conditions are mutually equivalent and they hold if and only if the Hadamard manifold is isometri…
Paper computes link determinants using Fourier-Hadamard transforms.
problem Computing determinants of complex link structures.
method Fourier-Hadamard transforms of Boolean functions.
result Determinant of centrally symmetric links with even components equals zero.
New theorem extends Hadamard's to transversely affine geometry.
problem Transversely affine foliations and their properties.
method Introduced and investigated novel transversely affine foliations, extending Hadamard's theorem.
result Transversely affine version of Hadamard's theorem for foliations.
The paper proves Liouville-type theorems on Hadamard manifolds.
problem Non-existence of Killing-Yano tensors, Killing tensors, and harmonic symmetric tensors on Hadamard manifolds.
method Proofs use Liouville-type theorems on non-existence of subharmonic and harmonic functions on complete Riemannian manifolds, modified for Hadamard manifolds.
result Proves several Liouville-type theorems on Hadamard manifolds.
Maps between Hadamard manifolds are quasi-isometric to harmonic maps.
problem Understanding the relationship between quasi-isometric maps and harmonic maps on Hadamard manifolds.
method Extending a previous result to quotient spaces of Hadamard manifolds by convex cocompact discrete groups.
result Locally quasi-isometric maps to Hadamard manifolds are within bounded distance from a unique harmonic map.
Time-series Random Process Complexity Ranking Using a Bound on Conditional Differential Entropyeess.SP
This paper improves entropy bounds for ranking time-series complexity.
problem Ranking the complexity of time series processes.
method Building on information theoretic bounds, the paper improves the upper bound of conditional differential entropy using Hadamard's inequality and covariance matrix properties.
result The improved bounds can be used to rank the complexity of time series processes.
The study extends isoperimetric inequalities to non-positive curvature spaces.
problem Isoperimetric inequalities in spaces of non-positive curvature.
method Analyzes submanifolds and geodesics in Cartan-Hadamard manifolds.
result Extensions of isoperimetric inequalities to non-positive curvature spaces.
Efficient algorithm for Hadamard decomposition of matrices.
problem Decomposing matrices into low-rank factors efficiently.
method Alternating optimization with SVD-inspired initialization and momentum.
result Significantly improved performance compared to existing methods.
Study KKT conditions for multi-objective optimization on Hadamard manifolds.
problem Optimizing multi-objective interval-valued functions on Hadamard manifolds.
method Developed KKT conditions for Pareto optimal solutions under different ordering and convexity notions.
result Results are more general than on Euclidean spaces.
We use the combinatorial harmonic map theory to study the isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex and its isometric action on a Hadamard space, we formulate criterions for the actio…
Proves homological inequality for cycles in Hadamard spaces of asymptotic rank 2.
problem Establishing isoperimetric inequalities in Hadamard spaces of asymptotic rank two.
method Homological inequality for cycles in dimensions at least 2, assuming finite linearly controlled asymptotic dimension.
result Homological inequality for general cycles in Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes.
Linear inequality found for certain curved spaces.
problem Finding isoperimetric inequalities for curved spaces.
method Extending known result to homogeneous Hadamard manifolds.
result Linear isoperimetric inequality proven for higher-dimensional cycles.
The paper defines Fenchel conjugate and biconjugate on Hadamard manifolds.
problem Defining Fenchel conjugate and biconjugate on curved spaces.
method Introduced a new definition of Fenchel conjugate and biconjugate on Hadamard manifolds based on the tangent bundle.
result Developed a Fenchel-Moreau Theorem for geodesically convex functions on Hadamard manifolds.
Groups can act on spaces with non-trivial cohomology.
problem Understanding actions of groups on Hadamard spaces.
method Analyzing isometric actions on Hadamard manifolds and -spaces.
result Non-trivial -cohomology for certain groups.
Sharp inequality proved in 3D hyperbolic spaces using flow methods.
problem Proving a sharp Minkowski-type inequality in Cartan-Hadamard 3-spaces.
method Using harmonic mean curvature flow to prove the inequality.
result Sharpened estimates for total mean curvature in hyperbolic 3-space.
We study the Hadamard finite part of divergent integrals of differential forms with singularities on submanifolds. We give formulae for the dependence of the finite part on the choice of regularization and express them in terms of a suitable local residue map. The cases where the submanifold is a complex hypersurface i…
In this paper we extend a recent result of Collin-Rosenberg ({\it a solution to the minimal surface equation in the Euclidean disc has radial limits almost everywhere}) to a large class of differential operators in Divergence form. Moreover, we construct an example (in the spirit of \cite{CR2}) of a minimal graph in $\…
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.
Sharp Minkowski inequality for convex surfaces in curved spaces.
problem Establishing a precise lower bound for total mean curvature of convex surfaces.
method Harmonic mean curvature flow applied to Cartan-Hadamard manifolds.
result Improved Minkowski inequality for convex surfaces in nonpositively curved 3-spaces.
Study Hardy identities and inequalities on Cartan-Hadamard manifolds.
problem Existence and nonexistence of extremal functions in Hardy inequalities.
method Using the notion of a Bessel pair, we derive Hardy identities and inequalities.
result Established several Hardy type inequalities with improvements and understandings.
Answering a question by Margulis we prove that the conclusion of Selberg's Lemma fails for discrete isometry groups of negatively curved Hadamard manifolds.
In this paper we study a homological version of the higher-dimensional divergence invariants defined by Brady and Farb. We show that they are quasi-isometry invariants in the class of proper cocompact Hadamard spaces in the sense of Alexandrov and that they can moreover be used to detect the Euclidean rank of such spac…
Using the classical approach we show the existence of disc type solutions to the asymptotic Plateau problem in certain Hadamard manifolds which may have arbitrarily strong curvature and volume growth.
New entropy functionals for curved spaces help predict shape behavior.
problem Understanding entropy behavior in curved spaces.
method Introduced new entropy functionals for submanifolds of Cartan-Hadamard manifolds.
result Obtained sharp lower bounds on these entropies for certain closed hypersurfaces and observed a novel rigidity phenomenon.
Cartan-Hadamard manifold is a simply connected Riemannian manifold with non-positive sectional curvature. In this article, we have proved that a Cartan-Hadamard manifold satisfying steady gradient Ricci soliton with the integral condition of potential function is isometric to the Euclidean space. Next we have proved a …
Geometrically proves majorizing measure theorem on Hadamard manifolds.
problem Volume size relation between random process index space and its convex hull.
method Assumed Hadamard manifold, derived upper bound for volume ratio, applied to prove majorizing measure theorem.
result Upper bound for volume ratio between index space and convex hull.
A new depth measure and median defined on Hadamard manifolds.
problem Statistical depth and median on Hadamard manifolds.
method Horospherical depth and Busemann median defined using renormalized distance functions.
result The Busemann median exists for every Borel probability measure on Hadamard manifolds.