The paper classifies solutions to a specific Toda system around a singular source.
problem Characterizing solutions to a particular Toda system around a singular point.
method Utilizing an ordinary differential equation and properties of SU(n+1) Toda system, the paper precisely characterizes solutions.
result Characterization of solutions to the SU(n+1) Toda system around a singular source.
New method solves PDEs with singular sources efficiently.
problem Solving PDEs with delta distributions and their derivatives.
method Particle-without-Particle method using pseudospectral grids.
result Improved convergence rates for various PDE types.
Study of symplectic manifolds degenerating into singular spaces.
problem Understanding degenerations of symplectic manifolds into singular spaces.
method Topological framework focusing on Lagrangian manifolds and holomorphic membranes.
result Degenerations into singular toric varieties yield exotic Lagrangian tori.
This study calculates total masses for Toda systems with singular sources.
problem Establishing a priori bounds for solutions to Toda systems on compact surfaces.
method Lie-theoretic approach using DPW method, Drinfeld-Sokolov gauge, and W-invariants.
result Total masses are related to the longest element in the Weyl group.
We simplify D4+-front singularities and apply to geometric invariants.
problem Simplifying D4+-front singularities in R3. method Coordinate transformation on source and isometry on target.
result Gauss-Bonnet type theorem for fronts with D4+-singularity. Study geometric properties of S1 singularities and their deformations.
problem Understanding differential geometric properties of S1 singularities and deformations.
method Representing deformation using diffeomorphisms and isometries, studying geometric properties.
result Differential geometric properties of S1 singularities and Whitney umbrellas in deformations.
We simplify swallowtail singularity and classify nearby curves.
problem Understanding swallowtail singularity and its nearby curves.
method Coordinate transformation on source and isometry on target.
result Classification of asymptotic and characteristic curves near swallowtail.
New form of D4−-singularities for fronts in 3D space.
problem Understanding singularities of fronts in 3D space.
method Coordinate transformation on source and isometry on target.
result Computed differential geometric invariants near D4−-singularity. Study D4−-front singularities, compute invariants, and derive a Gauss-Bonnet theorem.
problem Characterize and analyze D4−-front singularities in 3D space. method Develop coordinate transformations and isometries, compute differential invariants.
result Derive a Gauss-Bonnet type theorem for D4−-fronts. We show that for a C^infty stable map of an oriented 4-manifold into a 3-manifold, the algebraic number of singular fibers of a specific type coincides with the signature of the source 4-manifold.
We consider the Toda systems of VHS type with singular sources and provide a criterion for the existence of solutions with prescribed asymptotic behaviour near singularities. We also prove the uniqueness of solution. Our approach uses Simpson's theory of constructing Higgs-Hermitian-Yang-Mills metrics from stability.
Transfer knowledge from multiple sources to improve matrix completion.
problem Matrix completion with noisy data.
method Aggregating singular subspaces information from multiple sources to solve a two-way PCA problem and transform into a low-dimensional linear regression.
result Guaranteed statistical efficiency in transforming the high-dimensional target matrix completion problem.
Geometric study of cuspidal S1 singularities using diffeomorphisms and isometries.
problem Understanding geometric properties of cuspidal S1 singularities. method Form representing deformation using diffeomorphisms and isometries, necessary and sufficient condition for frontal maps.
result Investigation of geometric properties and cuspidal cross caps in deformations.
The flat geometry of the I1 singularity: (x,y)↦(x,xy,y2,y3)math.DG Study the flat geometry of a specific singularity in 4D space.
problem Understanding the flat geometry of a least degenerate singularity in 4D space.
method Obtained a generic normal form invariant under diffeomorphisms and isometries, classified submersions preserving the image, studied the height function.
result Obtained a generic normal form for the I1 singularity invariant under diffeomorphisms and isometries. The paper solves conditions for non-singular extensions of fold maps.
problem Conditions for the existence of non-singular extensions of horizontal stable fold maps.
method Defined a combinatorial object called a pairing map to prove equivalence of existence.
result Existence of non-singular extensions is equivalent to the existence of a pairing map.
Paper classifies SU(n) Toda systems related to hypergeometric equations.
problem Existence and classification of SU(n) Toda systems with singular sources.
method Determining necessary conditions, constructing solutions, classifying under reality assumption.
result All SU(3) Toda systems satisfying conditions are related to hypergeometric equations.
SMART transfers knowledge across related studies for multi-task learning.
problem Deterioration of multi-task learning performance with small target sample size.
method SMART assumes spectral similarity between source and target models, estimating target coefficients through structured regularization.
result SMART achieves near-minimax error rates, improving estimation accuracy and robustness to negative transfer.
Geometric invariants of fold maps and their signatures.
problem Characterizing cobordism groups of fold maps.
method Combining immersion data and stable partial framings.
result A Poincare-Hopf type formula for the signature of manifolds.
This study examines the topology of singularities in optimal semicouplings between unequal spaces.
problem Topology of singularities in optimal semicouplings between unequal spaces.
method Continuous strong deformation retracts and Uniform Halfspace condition.
result Homotopy-reductions from a source space onto singularities of c-optimal semicouplings. Maps with boundary definite fold points restrict manifold structure.
problem Restricting the global structure of manifolds with boundary.
method Introducing boundary special generic maps and deriving differential-topological restrictions.
result New results on non-singular extensions of special generic maps.
For a germ of a smooth map f and a subgroup G_V of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form V in the source or in the target we study the G_V-moduli space of f that parameterizes the G_V-orbits inside the G-orbit of f. We find, for example, that this modu…
Hybrid model improves music source separation by 1.4 dB.
problem Improving music source separation accuracy.
method End-to-end hybrid spectrogram and waveform model, using model decision for domain choice.
result 1.4 dB improvement in Signal-to-Distortion (SDR) on MusDB HQ dataset.
Nearly all field theories suffer from singularities when particles are introduced. This is true in both classical and quantum physics. Classical field singularities result in the notorious self-force problem, where it is unknown how the dynamics of a particle change when the particle interacts with its own (self) field…
Introduces quasi-holomorphic maps and their properties.
problem Understanding singularities and stratifications in non-complex manifolds.
method Pontryagin--Thom construction, cobordism groups, Thom polynomials.
result Thom polynomials determine cohomology classes of quasi-holomorphic maps.
Develops a method for extracting sources in multi-subject fMRI data.
problem Analyzing brain imaging datasets from multiple subjects with varying levels of jointness.
method Deflation-based algorithm using higher order cumulants and thin-SVD factorization.
result The algorithm accurately identifies joint, partially-joint, and individual sources with high precision.
Fold maps are higher dimensional versions of Morse functions, which play important roles in the studies of smooth manifolds, and such general maps also have been fundamental tools in the studies of smooth manifolds by using generic maps. In this paper, we study {\it simple} fold maps, which are fold maps such that any …
The study connects complex surface singularities to numerical semigroups and their properties.
problem Understanding the geometry and properties of strongly flat semigroups and their generalizations.
method Analyzing complex surface singularities and their associated semigroups, proving properties of Frobenius numbers.
result Strongly flat semigroups associated with negative definite Seifert homology spheres are numerical semigroups.
A generic smooth map of a closed 2k-manifold into (3k−1)-space has a finite number of cusps (Σ1,1-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities (Σ1,0-singularities). Two fold maps are fold bordant if the…
Weak Bianchi identity found for spacetimes with timelike singularities.
problem Weak Bianchi identity for spacetimes with curvature singularities.
method Found sufficient conditions for a weak version of the second Bianchi identity.
result Identified a class of spherically symmetric static spacetimes with timelike singularities for which the identity holds.
New minimax results show how target labels benefit under covariate-shift.
problem Understanding the relative benefits of source and target labeled data under covariate-shift.
method Developed new minimax results and showed how a semi-supervised procedure can adapt to unknown transfer-exponent γ.
result Target labels can dramatically improve classification in certain regimes of covariate-shift.
We provide a geometric explanation for the existence of magnification relations for the A, D, E family of caustic singularities, which were established in recent work. In particular, it was shown that for families of general mappings between planes exhibiting any of these caustic singularities, and for any non-caustic …
Study on the parity of fold map singular points, showing non-invariance for odd-dimensional manifolds.
problem Parity of connected components of fold map singular points for odd-dimensional manifolds.
method Constructive proofs using open book decompositions, round fold maps, and allowable moves.
result Parity of connected components is not a homotopy invariant for odd-dimensional manifolds.
Proposes a regularization method for unsupervised domain adaptation that aligns predictions with target data's top singular vectors.
problem Domain adaptation challenges in high joint error scenarios.
method Regularizes classifier to align with unsupervised target data guided by label alignment property (LAP).
result The method improves performance in MNIST-USPS domain adaptation and cross-lingual sentiment analysis.
The miltifractal properties and scaling behaviour of the exchange rate variations of the Iranian rial against the US dollar from a daily perspective is numerically investigated. For this purpose the multifractal detrended fluctuation analysis (MF-DFA) is used. Through multifractal analysis, the scaling exponents, gener…
Optimal transport aligns rotated linear regression models across domains.
problem Aligning rotated linear regression models across domains with differing statistical properties.
method Combines K-means clustering, OT, and SVD to estimate rotation angle and adapt regression model.
result Optimal transport map recovers underlying rotation in R2. We design non-singular cloaks enabling objects to scatter waves like objects with smaller size and very different shapes. We consider the Schrodinger equation which is valid e.g. in the contexts of geometrical and quantum optics. More precisely, we introduce a generalized non-singular transformation for star domains, a…
The paper proves local Lipschitz regularity for harmonic map heat flows in RCD spaces into CAT(0) spaces.
problem Understanding the regularity of harmonic map heat flows in nonsmooth spaces.
method Combines metric Hamilton-Jacobi argument with elliptic contact estimates.
result Local Lipschitz regularity for the flow from RCD spaces into CAT(0) spaces.
In this article, we consider a generalized Radon transform that comes up in ultrasound reflection tomography. In our model, the ultrasound emitter and receiver move at a constant distance apart along a circle. We analyze the microlocal properties of the transform R that arises from this model. As a consequence, we sh…
Study homotopy equivalence of spaces of gradient-like flows and Morse functions on surfaces.
problem Understanding the topology of spaces of smooth functions and flows on surfaces.
method Proves homotopy equivalence of spaces of gradient-like flows and Morse functions on surfaces, with detailed decomposition into orbits.
result Spaces of gradient-like flows and Morse functions on surfaces are homotopy equivalent to manifolds.
Non-compact manifolds prevent C0-stable mappings from being dense.
problem Density of C0-stable mappings on non-compact manifolds. method Using topologically critical points to show non-density.
result The set of C0-stable mappings is never dense on non-compact manifolds. Study submersions with definite folds on manifolds with boundary into Euclidean spaces.
problem Understanding differential-topological properties of manifolds with boundary under submersions with definite folds.
method Analyzing submersions with definite folds on manifolds with boundary into Euclidean spaces, focusing on restrictions to the boundary and using results for m-functions.
result Obtained restrictions on the diffeomorphism types of the source manifolds and studied the diffeomorphism types and Euler characteristics of manifolds admitting such maps.
Continuity of roots of hyperbolic polynomials with smooth coefficients.
problem Continuity of the solution map for hyperbolic polynomials.
method Proving continuity of the solution map from hyperbolic polynomials of degree d with C^d coefficients to their increasingly ordered roots.
result Continuity of the solution map for hyperbolic polynomials with C^d coefficients.
This work classifies monodromy in vineyards using singularity theory.
problem Understanding and predicting monodromy in vineyards for topological data analysis.
method Using a connection with singularity theory, the study classifies monodromy in vineyards of 1-manifolds in R^2.
result Monodromy in vineyards occurs only if they contain a specific singularity of the distance function.
We introduce a wide category of superspaces, called locally finitely generated, which properly includes supermanifolds but enjoys much stronger permanence properties, as are prompted by applications. Namely, it is closed under taking finite fibre products (i.e. is finitely complete) and thickenings by spectra of Weil s…
Study on Sturm-Liouville problems with zero potential and Neumann boundary conditions.
problem Understanding properties of Sturm-Liouville problems with zero potential.
method Developed simple criteria for assessing properties of regular Sturm-Liouville problems in terms of coefficient functions.
result Proved various properties of Sturm-Liouville problems with zero potential under Neumann boundary conditions.
Almost regular Poisson structures are characterized and their holonomy groupoids are shown to be Poisson groupoids.
problem Understanding the behavior of Poisson structures with singularities.
method Analyzing Hamiltonian vector fields and their submodules, focusing on holonomy groupoids.
result Holonomy groupoids of almost regular Poisson structures are Poisson groupoids, providing a desingularization.
For a given multicusp f=c(θ0,...,θi) (1≤i), we present a direct sum decomposition theorem of the source space of iωˉf, where iωˉf is a higher version of the reduced Kodaira-Spencer-Mather map ωˉf. As a corollary of our direct sum decomposition theorem, we show that for any $i\in \mathb…
We study the problem of recovery both the attenuation a and the source f in the attenuated X-ray transform in the plane. We study the linearization as well. It turns out that there are natural Hamiltonian flow that determines which singularities we can recover. If the perturbations δa, δf are supported in a com…