arXiv research

The paper classifies solutions to a specific Toda system around a singular source.

Study of symplectic manifolds degenerating into singular spaces.

We simplify $D_4^+$-front singularities and apply to geometric invariants.

Study geometric properties of S1 singularities and their deformations.

New form of $D_4^-$-singularities for fronts in 3D space.

Study $D_4^-$-front singularities, compute invariants, and derive a Gauss-Bonnet theorem.

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.

Geometric study of cuspidal $S_1$ singularities using diffeomorphisms and isometries.

We study the flat geometry of the least degenerate singularity of a singular surface in $\mathbb R^4$, the $I_{1}$ singularity parametrised by $(x,y)\mapsto(x,xy,y^{2},y^{3})$. This singularity appears generically when projecting a regular surface in $\mathbb R^5$ orthogonally to $\mathbb R^4$ along a tangent direction…

The paper solves conditions for non-singular extensions of fold maps.

SMART transfers knowledge across related studies for multi-task learning.

This study examines the topology of singularities in optimal semicouplings between unequal spaces.

We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the normal bundle of the singular set in the source manifold. These invariants were in…

Maps with boundary definite fold points restrict manifold structure.

We construct a form of swallowtail singularity in R^3 which uses coordinate transformation on the source and isometry on the target. As an application, we classify configurations of asymptotic curves and characteristic curves near swallowtail.

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.

Partial differential equations with distributional sources---in particular, involving (derivatives of) delta distributions---have become increasingly ubiquitous in numerous areas of physics and applied mathematics. It is often of considerable interest to obtain numerical solutions for such equations, but any singular (…

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.

Objective: Joint analysis of multi-subject brain imaging datasets has wide applications in biomedical engineering. In these datasets, some sources belong to all subjects (joint), a subset of subjects (partially-joint), or a single subject (individual). In this paper, this source model is referred to as joint/partially-…

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 …

A generic smooth map of a closed $2k$-manifold into $(3k-1)$-space has a finite number of cusps ($Σ^{1,1}$-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 ($Σ^{1,0}$-singularities). Two fold maps are fold bordant if the…

In this article we obtain total masses of solutions to the Toda system associated to a general simple Lie algebra with singular sources at the origin. The determination of such total masses is one of the important steps towards establishing the a priori bound for solutions to the mean field type of Toda system on compa…

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 …

The (twice-contracted) second Bianchi identity is a differential curvature identity that holds on any smooth manifold with a metric. In the case when such a metric is Lorentzian and solves Einstein's equations with an (in this case inevitably smooth) energy-momentum-stress tensor of a "matter field" as the source of sp…

We present new minimax results that concisely capture the relative benefits of source and target labeled data, under covariate-shift. Namely, we show that the benefits of target labels are controlled by a transfer-exponent $γ$ that encodes how singular Q is locally w.r.t. P, and interestingly allows situations where tr…

Study on the parity of fold map singular points, showing non-invariance for odd-dimensional manifolds.

Proposes a regularization method for unsupervised domain adaptation that aligns predictions with target data's top singular vectors.

Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical semigroups. More precisely, we prove that the strongly flat semigroups, which satisfy the…

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.

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…

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…

Study homotopy equivalence of spaces of gradient-like flows and Morse functions on surfaces.

Non-compact manifolds prevent $C^0$-stable mappings from being dense.

Study submersions with definite folds on manifolds with boundary into Euclidean spaces.

Continuity of roots of hyperbolic polynomials with smooth coefficients.

This work classifies monodromy in vineyards using singularity theory.

This paper establishes certain existence and classification results for solutions to $SU(n)$ Toda systems with three singular sources at 0, 1, and $\infty$. First, we determine the necessary conditions for such an $SU(n)$ Toda system to be related to an $n$th order hypergeometric equation. Then, we construct solutions …

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…

A special generic map is a smooth map regarded as a natural generalization of Morse functions with just 2 singular points on homotopy spheres. Canonical projections of unit spheres are simplest examples of such maps and manifolds admitting special generic maps into the plane are completely determined by Saeki in 1993 a…

Study on Sturm-Liouville problems with zero potential and Neumann boundary conditions.

For a given multicusp $f=c_{(θ_0,..., θ_i)}$ $(1\le i)$, we present a direct sum decomposition theorem of the source space of ${}_i\barωf$, where ${}_i\barωf$ is a higher version of the reduced Kodaira-Spencer-Mather map $\barωf$. 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…

In this paper, we construct round fold maps or stable fold maps with concentric singular value sets introduced by the author on smooth bundles over spheres or bundles over more general manifolds. The class of round fold maps includes special generic maps on spheres and such maps have been constructed on smooth bundles …