Paper proposes a new vanishing ideal for noisy data with improved algebraic structure.
problem Extracting nonlinear structures of noisy data with better algebraic properties.
method Simultaneously finds polynomials and data points for which polynomials approximately vanish on input data and almost exactly on discovered points.
result Our method discovers much fewer and lower-degree polynomials than state-of-the-art methods, accelerating runtime without degrading accuracy.
Paper tackles spurious vanishing problem in approximate vanishing ideals.
problem Capturing nonlinear structure of perturbed data points leads to spurious vanishing problem.
method Proposes a general method integrating coefficient normalization and iterative basis construction.
result Proposed method overcomes spurious vanishing problem, resulting in shorter feature vectors.
Geometrically interprets Atiyah class vanishing for Lie pairs.
problem Atiyah class vanishing for Lie pairs and Lie algebroids.
method New characterisation of ideal systems and geometric interpretation.
result Atiyah class vanishes for Lie pairs with kernel of fibration.
Gradient boosts monomial-order-free basis construction algorithms.
problem Lack of theoretical properties in monomial-order-free basis construction algorithms.
method Exploits gradient to sidestep spurious vanishing, achieve consistent output, and remove redundant bases.
result Proposes methods that equip monomial-order-free algorithms with theoretical properties.
In this expository article we first give an overview on multiplier ideal sheaves and geometric problems in Kählerian and Sasakian geometries. Then we review our recent results on the relationship between the support of the subschemes cut out by multiplier ideal sheaves and the invariant whose non-vanishing obstructs th…
The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a rela…
We formulate and establish a generalization of Kollár's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár's torsion-freeness, Kollár's vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge…
Defines Reidemeister torsion form on 3-manifold character varieties.
problem Character variety singularities and vanishing torsion.
method Differential form on character variety, Alexander polynomial, Culler-Shalen theory.
result Torsion vanishes only at singular points, related to reducibility and Euler characteristic.
Defines ideal simplicial volume for manifolds with boundary, showing it's bounded by classical volume.
problem Measuring the complexity of manifolds with boundary.
method Introduces ideal simplicial volume, compares it to classical volume, and computes specific examples.
result Ideal simplicial volume is bounded by classical volume and can be strictly smaller for certain manifolds.
The paper proves vanishing theorems for vector bundles with singular metrics.
problem Analyzing vector bundles with singular Hermitian metrics.
method Using a sheaf of locally square integrable holomorphic sections and a new Nadel-Nakano type vanishing theorem.
result Generalizations of vanishing theorems for vector bundles with singular metrics.
We consider the ideal-gas models of trading markets, where each agent is identified with a gas molecule and each trading as an elastic or money-conserving (two-body) collision. Unlike in the ideal gas, we introduce saving propensity λ of agents, such that each agent saves a fraction λ of its money and trades with t…
The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generali…
The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the h…
The paper proves injectivity and vanishing theorems on compact Kahler manifolds.
problem Injectivity and vanishing theorems on compact Kahler manifolds.
method Hodge theory, Bochner-Kodaira-Nakano identity, analytic method, transcendental method, Demailly-Peternell-Schneider equisingular approximation theorem, Hormander L2 estimates.
result The main injectivity theorem implies several Nadel type vanishing theorems.
New quantum invariant for framed 3-manifolds using ideal triangulations.
problem Quantum invariants of framed 3-manifolds with vanishing first Betti number.
method Based on ideal triangulations and Hopf algebras, using the pentagon equation and graphical representations.
result Construction of a new quantum invariant for closed framed 3-manifolds.
We describe how cross-kernel matrices, that is, kernel matrices between the data and a custom chosen set of `feature spanning points' can be used for learning. The main potential of cross-kernels lies in the fact that (a) only one side of the matrix scales with the number of data points, and (b) cross-kernels, as oppos…
The paper finds obstructions to Lie algebroid representations up to homotopy.
problem Obstacles to representations up to homotopy of Lie algebroids.
method Analyzes Pontryagin characters and vector bundles over Lie algebroids.
result Vanishing of Pontryagin characters implies no representation up to homotopy.
We consider the ideal-gas models of trading markets, where each agent is identified with a gas molecule and each trading as an elastic or money-conserving (two-body) collision. Unlike in the ideal gas, we introduce saving propensity λ of agents, such that each agent saves a fraction λ of its money and trades with t…
The ``classical BRST construction'' as developed by Batalin-Fradkin-Vilkovisky is a homological construction for the reduction of the Poisson algebra P=C∞(W) of smooth functions on a Poisson manifold W by the ideal I of functions which vanish on a constraint locus. This ideal is called first class if I…
The paper establishes conditions for Riemannian connections and semi-simplicity of Lie algebras using spray structures.
problem Conditions for Riemannian connections and semi-simplicity of Lie algebras.
method Using almost product structures and spray, the paper provides necessary and sufficient conditions for these properties.
result Equivalence of semi-simplicity of Lie algebras to derived ideal coincidence, interiority of derivations, and adjoint representation semi-simplicity.
Smooth groupoid algebras are H-unital, with implications for algebraic and homological properties.
problem Understanding the structure of convolution algebras on Lie groupoids.
method Analyzing smooth functions and invariant subsets to prove H-unitality.
result H-unitality of groupoid algebras and their quotients, leading to excision properties.
We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this int…
In this paper, we study generalized constant ratio (GCR) hypersurfaces in Euclidean spaces. We mainly focus on the hypersurfaces in E4. First, we deal with δ(2)-ideal GCR hypersurfaces. Then, we study on hypersurfaces with constant (first) mean curvature. Finally, we obtain the complete classification of G…
Proves new fixed point formulae for complex manifolds with boundary.
problem Fixed points on complex manifolds with boundary conditions.
method Logarithmic Lefschetz fixed point formulae, normal rescaling, relative duality.
result Resonant boundary terms record normal contact and tangential multiplicity.
The purpose of this survey is to present analytic versions of the injectivity theorem and their applications. The proof of our injectivity theorems is based on a combination of the L^2-method for the dbar-equation and the theory of harmonic integrals. As applications, we obtain Nadel type vanishing theorems and extensi…
We prove the ADO invariants are a q-holonomic family and establish recursion relations.
problem Understanding the q-holonomic properties of ADO link invariants. method Proving the ADO invariants are a q-holonomic family and establishing recursion relations. result The ADO invariants for r≥2 are a q-holonomic family, satisfying independent recursion relations. Study curvatures of submanifolds in space forms with topological obstructions.
problem Investigate intrinsic curvatures and their implications for submanifolds in space forms.
method Derive inequalities involving mean curvature and normal scalar curvature, derive topological obstructions.
result Prove existence of compact 3-dimensional minimal Wintgen ideal submanifolds in even-dimensional spheres.
Derives ideal train/test split for ridge regression in large data limit.
problem Finding optimal train/test split for ridge regression in large data scenarios.
method Mathematical derivation of optimal train/test split, considering ridge tuning parameter and asymptotic behavior.
result The optimal train/test split for ridge regression in the large data limit depends weakly on the ridge tuning parameter alpha.
The study explores Einstein hypersurfaces in warped product spaces and their properties.
problem Investigating Einstein hypersurfaces in warped product spaces.
method Analyzing constant curvature hypersurfaces and properties of gradient functions.
result Characterization of ideal Einstein hypersurfaces with specific curvature properties.
New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and …
The Hochschild and cyclic homology groups are computed for the algebra of `cusp' pseudodifferential operators on any compact manifold with boundary. The index functional for this algebra is interpreted as a Hochschild 1-cocycle and evaluated in terms of extensions of the trace functionals on the two natural ideals, cor…
Study shows L2-Betti numbers vanish for certain matrix groups over rings, proving non-acylindrical hyperbolicity.
problem Investigating L2-Betti numbers and acylindrical hyperbolicity for matrix groups over various rings. method Utilizing n-rigid rings, the study proves vanishing L2-Betti numbers and non-acylindrical hyperbolicity for specified groups. result Matrix groups over certain rings have vanishing L2-Betti numbers and are not acylindrically hyperbolic. We extend the Boutet de Monvel Toeplitz index theorem to complex manifold with isolated singularities following the relative K-homology theory of Baum, Douglas, and Taylor for manifold with boundary. We apply this index theorem to study the Arveson-Douglas conjecture. Let $\ball^m$ be the unit ball in Cm,…
We construct left invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven dimensional…
Paper uses algebraic signatures to identify probabilistic structures in empirical data.
problem Identifying probabilistic structure from observed binomials in empirical probability tensors.
method Treating vanishing binomials as algebraic signatures, matching signatures to identify models without parameter estimation.
result The method successfully identified rank-one structures in real language data, revealing interpretable sets of words.
Paper proves stability of multi-dimensional rarefaction waves in gas dynamics.
problem Challenges in constructing multi-dimensional rarefaction waves in gas dynamics.
method Geometric Weighted Energy Method (GWEM) to overcome derivative losses.
result Established nonlinear stability of multi-dimensional rarefaction waves for compressible Euler equations.
Study ideals in the Goldman Lie algebra S by mapping to a simpler algebra.
problem Identify and classify ideals in the Goldman Lie algebra S. method Construct an algebra homomorphism to a simpler structure and classify ideals.
result Found infinite classes of ideals in both Z-module and Q-module cases. The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
problem Finding minimal ideal triangulations for specific 3-manifolds.
method Analyzing properties of poor ideal three-edge triangulations and applying them to construct minimal triangulations.
result Poor ideal three-edge triangulations are proven to be minimal for certain 3-manifolds.
Smooth algebra analysis for one-dimensional singular foliations.
problem Analyzing smooth algebras of one-dimensional singular foliations.
method Analyzing natural ideals and using Dixmier-Malliavin theorem.
result Smooth algebras of one-dimensional singular foliations are pairwise nonisomorphic.
We give a simple method to find ideal points of the character variety of a 3-manifold from an ideal triangulation.
Paper provides new Alexander ideal-based obstruction to 0-concordance of knotted surfaces.
problem Tackles the 0-concordance problem for knotted surfaces in S4. method Uses Alexander ideals to induce a homomorphism and prove non-sliceness.
result Alexander ideal determines 0-concordance classes and non-sliceness.
Short proof for ideal polygons with near optimal orthogeodesic decomposition.
problem Decomposing ideal polygons into orthogeodesics.
method Short proof with orthogeodesic decomposition of length at most 2log(n). result Optimal orthogeodesic decomposition of ideal polygons with length 2log(n). A method uses CG to create efficient channels for ideal observers.
problem Computational intractability of ideal observers for high-dimensional image data.
method Conjugate gradient (CG) method for constructing efficient channels.
result CG-based channels approximate IO and HO performance efficiently.
The paper explores genericity of triples of antipodal chambers in affine buildings.
problem Investigating genericity of triples of antipodal ideal chambers in locally finite affine buildings.
method Defined ideal-genericity and affine-genericity at the ideal boundary and within the affine building, respectively. Established a barycenter map and showed its continuity.
result Affine-genericity implies ideal-genericity, but the latter is more suitable for constructing a barycenter map.
Study introduces dynamical ideals for non-commutative rings and classifies knots and links.
problem Classifying surface knots and links in smooth 4-manifolds.
method Introduced dynamical analog of prime ideals for non-commutative rings and proved a factorization theorem.
result Classified surface knots and links in smooth 4-manifolds.
We investigate the rigidity of hyperbolic cone metrics on 3-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to i…
Defines timelike ideal boundary for non-positively curved Lorentzian spaces.
problem Understanding the geometry of non-positively curved Lorentzian spaces.
method Introduces timelike ideal boundary as asymptotic classes of geodesic rays, endows with topology and metric, and studies upper curvature bounds.
result Established upper curvature bounds for the resulting metric space.