Refined theorem on linear perturbations with applications in singularity theory and optimization.
problem Linear perturbations and their implications in singularity theory and optimization.
method New perspective of Hausdorff measures for refined transversality theorem.
result Applications in singularity theory and optimization.
New theorem applies Poincaré-Bendixson to non-linear Cauchy-Riemann equations.
problem Applying Poincaré-Bendixson theorem to non-linear equations.
method Abstract theorem for flows with discrete Lyapunov function.
result Similar result holds for bounded solutions of non-linear Cauchy-Riemann equations.
Gradient bounds and Liouville theorems for quasi-linear equations on manifolds with nonnegative Ricci curvature.
problem Establishing bounds and theorems for solutions to quasi-linear elliptic equations on compact manifolds with nonnegative Ricci curvature.
method Gradient bounds, Liouville-type theorems, local splitting theorem, Harnack-type inequality, ABP estimate.
result Gradient bounds and Liouville-type theorems for solutions to quasi-linear equations on compact manifolds with nonnegative Ricci curvature.
Study Loday algebroids, prove splitting theorem, and linearize problems.
problem Splitting and linearization of Loday algebroids.
method Local splitting-type results, Euler-like derivations.
result Established a general linearization principle.
Paper proves linked simplices in odd dimensions and disjoint tuples in even dimensions.
problem Proving linked simplices and disjoint tuples in various dimensions.
method Algebraic proofs based on Radon theorem.
result Linear Conway--Gordon--Sachs and van Kampen--Flores theorems proved.
The Cauchy-Kowalewski theorem helps count geometric structures.
problem Counting linear connections and statistical structures.
method Applying the Cauchy-Kowalewski theorem in the analytic case.
result Analytic solutions for geometric structures are found.
The paper strengthens a theorem on crossings under linear perturbations with Hausdorff measure estimates.
problem Understanding multiple-point crossings under linear perturbations.
method Establishes a transversality theorem with Hausdorff measure estimates for exceptional parameter sets.
result Explicit upper bounds on the Hausdorff dimension of the exceptional set.
Generalizes theorem for topological G-manifolds with linear Lie groups G.
problem Understanding topological G-manifolds with linear Lie groups G. method Using countable CW complexes and Palais-proper actions.
result Topological G-manifolds have G-homotopy type of countable G-CW complexes. Study noncompact RCD(0,N) spaces with linear volume growth, proving diameter bounds and a splitting theorem.
problem Understanding non-compact RCD(0, N) spaces with linear volume growth.
method Analyzing properties of level sets and applying geometric inequalities.
result Diameter of level sets of a Busemann function grows at most linearly.
Extended Gauss-Markov theorem for linear estimation with bounded bias.
problem Linear estimation with bounded bias operator.
method Derive optimal estimator formulas for Nuclear and Spectral norms, analyze generalization error.
result Cross-validated Nuclear and Spectral regressors outperform Ridge regression in simulations.
Geometric invariant theory for real Lie groups proved.
problem Closed orbits and null cone stratification in real reductive Lie groups.
method Completely self-contained proof focusing on geometric and analytic methods.
result Applies to non-rational linear actions.
The paper proves a flat torus theorem for certain groups acting on convex domains.
problem Establishing a flat torus theorem for specific groups acting on convex domains.
method Analyzing discrete groups in mPGLd(R) acting convex co-compactly on a properly convex domain. result An analogue of the flat torus theorem for mCAT(0) spaces is proven for these groups. We revisit the linearization theorems for proper Lie groupoids around general orbits (statements and proofs). In the the fixed point case (known as Zung's theorem) we give a shorter and more geometric proof, based on a Moser deformation argument. The passing to general orbits (Weinstein) is given a more conceptual inte…
Hadwiger's Theorem states that Euclidean-invariant convex-continuous valuations of definable sets are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable real-valued functions on n-dimensional Euclidean space. This generalizes intrinsic vol…
In these lectures notes I discuss the Linearization Theorem for Lie groupoids, and its relation to the various classical linearization theorems for submersions, foliations and group actions. In particular, I explain in some detail the recent metric approach to this problem.
New proof of generalized Chow-Rashevskii theorem for non-linear systems.
problem Generalized Chow-Rashevskii Theorem for non-linear systems.
method Independent proof structure allowing generalizations to orbits of compositions of flows.
result Proof structure applicable to applications in Control Theory and controllability criteria.
Study CR immersions into Kähler manifolds, proving new theorems.
problem Understanding CR immersions and umbilical points in complex geometry.
method Analysis of second fundamental forms and eigenvalues of the Kohn Laplacian.
result Extension of Webster's theorem and new gap theorem.
The paper finds commutator formulas for Gradient Ricci Shrinker metrics and applies them to linear stability.
problem Linear stability of Gradient Ricci Shrinker metrics.
method Found commutator formulas and generalized a stability theorem.
result Generalized a necessary condition for linear stability.
Paper proves almost rigidity theorem and applies it to study RCD(0,N) spaces.
problem Understanding the structure of noncompact RCD(0,N) spaces with linear volume growth.
method Developed an almost rigidity theorem and applied it to study RCD(0, N) spaces.
result Obtained sublinear growth of diameter of geodesic spheres and non-existence of harmonic functions with polynomial growth.
Simplified proof of foliation closure theorem for linear foliations.
problem Proving the closure of linear foliations on Riemannian manifolds.
method Direct geometric approach, focusing on projectable foliations and compatible connections.
result Smoothness of the closure of linear foliations directly proven.
The Ambrose-Singer theorem is extended to cohomogeneity one Riemannian manifolds.
problem Characterizing isometric actions with specific orbit properties.
method Using a linear connection with covariant equations similar to the Ambrose-Singer theorem.
result Isometric cohomogeneity one foliations described in terms of such connections.
Theoretical validation of linear PCA and ICA for accurate nonlinear BSS.
problem Blind source separation for high-dimensional nonlinear source mixtures.
method Theoretical validation of a cascade of linear PCA and ICA.
result Zero-element-wise-error nonlinear BSS is achieved under certain conditions.
Minimal graph theorem proven for convex domains.
problem Characterizing minimal graphs over convex domains.
method Analyzing minimal surface equation solutions on convex domains.
result Minimal graphs over convex domains are linear.
Second part of proving linearization theorem for sl2(C).
problem Proving linearization theorem for sl2(C).
method Developed Nash-Moser method for functions flat at a point.
result Linearization result for a more general class of Lie algebras.
The author presents the generalized Stokes theorem for R-linear forms on Lie algebroids (which can be non-local). We apply the Stokes formula on forms to prove that two homotopic homomorphisms of Lie algebroids implies the existence of a chain operator joining their pullback operators.
Optimal Liouville theorem for minimal disks in any codimension.
problem Characterizing harmonic functions on minimal disks in high-dimensional spaces.
method Analyzing harmonic functions and using Liouville's theorem.
result Optimal Liouville theorem for minimal disks in any codimension.
Solves symplectic and conformal symplectic group actions equivalence problem.
problem Equivalence problem for symplectic and conformal symplectic group actions.
method Computing differential invariants via the Lie-Tresse theorem.
result Solves equivalence problem for symplectic and conformal symplectic group actions.
In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The first states that if such a manifold has small linear diameter growth then its fund…
Paper proves flatness of anisotropic minimal graphs in half-spaces.
problem Anisotropic minimal graphs with free boundaries in half-spaces.
method Proves flatness using linear growth conditions.
result Anisotropic minimal graphs in half-spaces are flat if they have at most one-sided linear growth.
Local model for Poisson manifolds around submanifolds.
problem Linearization of Poisson manifolds around submanifolds.
method Constructing a first order local model and giving conditions for it to be a normal form.
result Includes known linearization theorems for fixed points and symplectic leaves.
Lectures on linearized Kapustin-Witten equations on half-line.
problem Analyzing differential operator from Kapustin-Witten equations.
method General theorems of R. Mazzeo and E. Witten applied.
result Instances of asymptotic solutions found.
Improved non-squeezing theorem for calibrated geometries proved.
problem Proving an improved non-squeezing theorem for calibrated geometries.
method Two proofs: direct and reduction to classical case.
result Established an improved non-squeezing theorem for calibrated geometries.
We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry). The result is also a generalization of Conn's linearization theorem from one-point leaves to arbitrary symplectic leaves (however, we do not make use of Conn's theorem).
The paper proves sphere theorems for charged bodies in linear potential theory.
problem Analyzing the capacitary potential of a charged body to deduce geometric inequalities.
method Analyzing the mean curvature and applying inequalities to domains with spherical symmetry.
result Domains with spherical symmetry are the only ones satisfying the given curvature condition.
The paper develops theory for holomorphic null curves in SL2(C).
problem Developing theory for holomorphic null curves in SL2(C).
method Establish Runge, Mergelyan, Mittag-Leffler, and Carleman type theorems for holomorphic null immersions.
result Proves every open Riemann surface admits a proper holomorphic null embedding into SL2(C).
New curvature measure defined for graphs, with bounds on diameter and spectral gap.
problem Defining curvature for graphs and proving its properties.
method Solving linear systems to compute curvature; applying minimax theorem.
result Graphs with positive curvature have bounded diameter and spectral gap.
In this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth non-linearizable Hamiltoni…
The paper classifies involutions on S^4, proving linearities under certain conditions.
problem Classifying involutions on S^4 with specific fixed-point sets.
method Combining surgery theory, Schoenflies theorem, and equivariant topology.
result Linear involutions on S^4 with 1-dimensional fixed-point sets are proven.
Solves low-rank approximation problems in Hilbert spaces.
problem Low-rank approximation in Hilbert spaces.
method Closed-form solutions and error bounds for bounded linear operators.
result Generalization to bounded linear operators from finite dimensions.
The A^-genus is characterized as a linear combination of Pontrjagin numbers.
problem Characterizing the A^-genus using Pontrjagin numbers. method Combining Atiyah-Hirzebruch idea and Calabi-Yau theorem.
result A rational linear combination of Pontrjagin numbers vanishing on certain manifolds implies it is a multiple of the A^-genus. The paper extends a theorem about momentum maps to singular symplectic spaces.
problem Extending a theorem about momentum maps to singular symplectic spaces.
method Using integral affine stratification and equivariant locally trivial fibrations, the paper extends the linear variation theorem to singular values of the momentum map.
result Cohomology classes of symplectic forms on reduced spaces vary linearly within strata.
We give a short topological proof for Rubermans Theorem about mutation and volume, using the Maskit combination theorem and the homology of the linear group.
In 1960, J. Peetre proved the finiteness of the order of linear local operators. Later on, J. Slovák vastly generalized this theorem, proving the finiteness of the order of a broad class of (non-linear) local operators. In this paper, we use the language of sheaves and ringed spaces to prove a simpler version of Slovák…
Universal approximation theorem for differentiable maps on infinite-dimensional manifolds
problem Approximation of differentiable maps on infinite-dimensional manifolds
method Weighted universal approximation theorem
result Universal approximation theorem for differentiable maps
We prove a Liouville theorem for the plurisubharmonic functions on complete Kaelher manifolds. As the applications, we prove a splitting theorem for complete Kaehler manifolds with nonnegative biscetional curvature in terms of the linear growth harmonic functions and a optomal gap theorem for such manifolds.
The paper proposes new cross-correlators using Price's Theorem and piecewise-linear decomposition.
problem Optimal method for estimating cross-correlations using finite samples.
method General mathematical framework using Price's Theorem and piecewise-linear decomposition.
result Some cross-correlators based on Huber's loss functions, MP functions, and LSE functions have higher SNR.
Representer theorem conditions extended to reflexive Banach spaces.
problem Conditions for representer theorem in reflexive Banach spaces.
method Proved necessary and sufficient conditions for representer theorems in reflexive Banach spaces.
result Solution independence of regularizer in reflexive Banach spaces.
We prove an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah-Singer index theorem for selfadjoint elliptic operators. For the special case of one-parameter families, we compare our theorem with a classical result of Salamon and Zehnder. Finally, we use the index theo…