arXiv research

Defines horocyclic evolutes, parallels, and involutes of spacelike frontals in hyperbolic 2-space.

Classifies self-similar curve shortening flows in hyperbolic 2-space.

In this paper we give the characterization of Fuchsian groups acting on quaternionic hyperbolic 2-space.

Study on closed $p$-elastic curves in hyperbolic and de Sitter planes.

HLoOP detects outliers in hyperbolic 2-space.

In this article, we prove that the equation of the Schrödinger maps from ${\bf R}^2$ to the hyperbolic 2-space ${\bf H}^2$ is SU(1,1)-gauge equivalent to the following 1+2 dimensional nonlinear Schrödinger-type system of unknown three complex functions $p,q,r$ and a real function $u$: {c} iq_t+q_{z{\bar z}}-2u q+2({\ba…

The paper studies singularities of pedal curves of hyperbolic frontals.

In this paper, we partially settle down the long standing open problem of the finite time blow-up property about the nonlinear Schr$\ddot{o}$dinger equations on some Riemannian manifolds like the standard 2-sphere $S^2$ and the hyperbolic 2-space $H^{2}(-1)$. Using the similar idea, we establish such blow-up results on…

In the present paper, we define the notions of Lorentzian Sabban frames and de Sitter evolutes of the unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$. In addition, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-…

Let n be aninteger>4. There is a smoothly knotted n-dimensional sphere in (n+2)-space such that the singular point set of its projection in (n+1)-space consists of double points and that the components of the singular point set are two. (The sphere is knotted in the sense that it does not bound any embedded (n+1)-ball …

Rigidity of $L^2$-spaces on manifolds is characterized.

The paper classifies 2D complete Lagrangian self-expanders in complex 2-space.

We explain how to define the quantization of q-Hamiltonian SU(2)-spaces as push-forwards in twisted K-homology, and prove a `quantization commutes with reduction' theorem for this setting. As applications, we show how the Verlinde formulas for flat SU(2) or SO(3) bundles are obtained by localization in twisted K-homolo…

The $N$-body problem with a $1/r^2$ potential has, in addition to translation and rotational symmetry, an effective scale symmetry which allows its zero energy flow to be reduced to a geodesic flow on complex projective $N-2$-space, minus a hyperplane arrangement. When $N=3$ we get a geodesic flow on the two-sphere min…

Study identifies unique minimizers for interaction kernels in particle systems.

We develop the Lorentzian geometry of a crooked halfspace in 2+1-dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types, giving an explicit slice for the action of the automorphism group. The set of parall…

Researchers reinterpret complex hyperbolic orbifolds using line arrangements.

The paper examines conditions for stochastic invariance of cones in SPDEs with jumps.

The paper classifies certain types of Lagrangian translators and self-expanders in complex 2-space.

The study proves a discrete Blaschke theorem for convex polygons in 2-dimensional space forms.

The study finds conditions for free boundary Hamiltonian stationary discs in complex 2-space.

A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper we show that if a generic immersion of a planar g…

Constructs Lorentzian harmonic maps and associated timelike surfaces.

Survey on the topology of singular foliations in complex 2-space.

Novel method learns memory kernels in Langevin equations.

Bayesian layer improves image segmentation and out-of-distribution detection.

This paper is concerned with the MAXVAR risk measure on L^2 space. We present an elementary and direct proof of its coherency and averseness. Based on the observation that the MAXVAR measure is a continuous convex combination of the CVaR measure, we provide an explicit formula for the risk envelope of MAXVAR.

We study the symplectic geometry of the moduli spaces $M_r=M_r(\s^3)$ of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of $n$ conjugacy classes in SU(2), denoted $C_r^n$, by the diagonal conjugation action …

We study the wave analog of the Liouville equation and the constant mean curvature equations in 2 space dimensions, which are energy critical. We exhibit a blow-up criteria for the former using tools from conformal geometry, and we exhibit finite time blow-up for the latter under suitable assumptions on the initial dat…

Jeffrey and Kirwan suggested expressions for intersection pairings on the reduced space of a Hamiltonian G-space in terms of multiple residues. In this paper we prove a residue formula for symplectic volumes of reduced spaces of a quasi-Hamiltonian SU(2)-space. The definition of quasi-Hamiltonian G-spaces was recently …

We prove a maximum principle for mild solutions to stochastic evolution equations with (locally) Lipschitz coefficients and Wiener noise on weighted $L^2$ spaces. As an application, we provide sufficient conditions for the positivity of forward rates in the Heath-Jarrow-Morton model, considering the associated Musiela …

Efficiently models learning curves using Gaussian processes with latent Kronecker structure.

In this remark, we show how the monopole Frøyshov invariant, as well as the analogues of the Involutive Heegaard Floer correction terms $\underline{d},\overline{d}$, are related to the $\mathrm{Pin}(2)$-equivariant Floer homology $\mathit{SWFH}^G_*$. We show that the only interesting correction terms of a $\mathrm{Pin}…

We show that Tolman's example (of a six dimensional Hamiltonian $T^2$-space with isolated fixed points and no compatible Kähler structure) can be constructed from the flag variety $U(3)/U(1)^3$ by $U(2)$-equivariant symplectic surgery. This implies that Tolman's space has a ``transversal multiplicity-free'' action of $…

Bayesian Gaussian Processes layer detects out-of-distribution data in medical imaging.

An oriented link L in a 3-sphere S in complex 2-space is a C-boundary if it bounds a piece of algebraic curve in the 4-ball bounded by S. Using Kronheimer and Mrowka's proof of the Thom Conjecture, we construct many oriented knots which are not concordant to a C-boundary. We use the two-variable HOMFLY polynomial to gi…

Carrying further work of T.A. Crawford, we show that each component of the space of harmonic maps from the $2$-sphere to complex projective $2$-space of degree $d$ and energy $4 πE$ is a smooth closed submanifold of the space of all $C^j$ maps $(j \geq 2)$. We achieve this by showing that the Gauss transform which rela…

We consider the generalized Segal-Bargmann transform, defined in terms of the heat operator, for a noncompact symmetric space of the complex type. For radial functions, we show that the Segal-Bargmann transform is a unitary map onto a certain L^2 space of meromorphic functions. For general functions, we give an inversi…

We study paracontact metric $(κ,μ)$-spaces with $κ=-1$, equivalent to $h^2=0$ but not $h=0$. In particular, we will give an alternative proof of Theorem 3.2 of [11] and present examples of paracontact metric $(-1,2)$-spaces and $(-1,0)$-spaces of arbitrary dimension with tensor $h$ of every possible constant rank. We w…

SQUEAK reduces space complexity for Nystrom approximations in KRR.

Paper tackles measure estimation in barycentric coding model.

Kernel online convex optimization (KOCO) is a framework combining the expressiveness of non-parametric kernel models with the regret guarantees of online learning. First-order KOCO methods such as functional gradient descent require only $\mathcal{O}(t)$ time and space per iteration, and, when the only information on t…

Paper proves weak unique continuation for harmonic functions on RCD spaces but finds counterexample for strong uniqueness.

Study Dirichlet-to-Neumann maps on manifolds, focusing on covering and total spaces.

This paper shows moduli spaces of RCD(0,2) structures are contractible.

We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the $L^2$-space produces the same evolution as the gradient flow of the relative entropy in the $L^2$-Wasserstein space. This means that the heat flow is well defined by either one of the two gradient fl…

We provide a deterministic space-efficient algorithm for estimating ridge regression. For $n$ data points with $d$ features and a large enough regularization parameter, we provide a solution within $\varepsilon$ L$_2$ error using only $O(d/\varepsilon)$ space. This is the first $o(d^2)$ space deterministic streaming al…

This paper develops a weighted $L^2$-method for the (half) Dirac equation. For Dirac bundles over closed Riemann surfaces, we give a sufficient condition for the solvability of the (half) Dirac equation in terms of a curvature integral. Applying this to the Dolbeault-Dirac operator, we establish an automatic transversa…