arXiv research

Novel bistable structures made from four-bar linkages, proving existence and construction.

We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of discrete nets on the smooth surface one can obtain second-order approximation g…

Approximates smooth surfaces using Laguerre geometry meshes.

Approximates surfaces using Laguerre geometry with spherical faces.

New discretizations of principal curvature lines discovered.

In this paper, we establish existence results for positive solutions to the Lichnerowicz equation of the following type in closed manifolds -Δu=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where $p>1, q>0$, and $A(x)>0$, $B(x)\geq0$ are given smooth functions. Our analysis is based on the global existence of positive solution…

We consider the following singularly perturbed Neumann problem \begin{eqnarray*} \ve^2 Δu -u +u^p = 0 \, \quad u>0 \quad {\mbox {in}} \quad Ω, \quad {\partial u \over \partial ν}=0 \quad {\mbox {on}} \quad \partial Ω, \end{eqnarray*} where $p>2$ and $Ω$ is a smooth and bounded domain in $\R^2$. We construct a new class…

Let $(\mathcal{M},g)$ be a compact Riemannian manifold of dimension $N$, $N\geq 2$. In this paper, we prove that there exists a family of domains $(Ω_\varepsilon)_{\varepsilon\in(0,\varepsilon_0)}$ and functions $u_\varepsilon$ such that $ -Δ_{g} u_\varepsilon=1 \quad \textrm{ in } Ω_\varepsilon, \quad u_\varepsilon=0 …

The purpose of this note is to attract attention to the following conjecture (metastable $r$-fold Whitney trick) by clarifying its status as not having a complete proof, in the sense described in the paper. Assume that $D=D_1\sqcup\ldots\sqcup D_r$ is disjoint union of $r$ disks of dimension $s$, $f:D\to B^d$ a proper …

A Lie group $G$ naturally acts on its Lie algebra $\gg$, called the adjoint action. In this paper, we determine the orbit types of the compact exceptional Lie group $G_2$ in its Lie algebra $\gg_2$. As results, the group $G_2$ has four orbit types in the Lie algebra $\gg_2$ as $$ G_2/G_2, \quad G_2/(U(1) \times U(1)), …

Study para-CR structures relaxing complex conjugation constraint, finding homogeneous models.

Researchers describe isometric deformations of T-hedra and T-surfaces.

We investigate length decreasing maps $f:M\to N$ between Riemannian manifolds $M$, $N$ of dimensions $m\ge 2$ and $n$, respectively. Assuming that $M$ is compact and $N$ is complete such that $$\sec_M>-σ\quad\text{and}\quad{\Ric}_M\ge(m-1)σ\ge(m-1)\sec_N\ge-μ,$$ where $σ$, $μ$ are positive constants, we show that the m…

The paper studies a new type of stochastic differential equations for financial claims.

New method characterizes surface quadrilateral layouts as special immersions.

The paper studies RBSDEs with arbitrary stopping times and their solutions.

Let $(M,g)$ be a two dimensional compact Riemannian manifold of genus $g(M)>1$. Let $f$ be a smooth function on $M$ such that $f \ge 0, \quad f\not\equiv 0, \quad \min_M f = 0.$ Let $p_1,\ldots,p_n$ be any set of points at which $f(p_i)=0$ and $D^2f(p_i)$ is non-singular. We prove that for all sufficiently small $λ>…

This paper develops a discrete theory of real Riemann surfaces using quad-graphs and linear discretization.

In this article we study the nonlocal equation \begin{align} (-Δ)^{\frac{n}{2}}u=(n-1)!e^{nu}\quad \text{in $\mathbb{R}^n$}, \quad\int_{\mathbb{R}^n}e^{nu}dx<\infty, \notag \end{align} which arises in the conformal geometry. Inspired by the previous work of C. S. Lin and L. Martinazzi in even dimension and T. Jin, A. M…

The paper proves symmetry and classification of solutions to an integral equation in the Heisenberg group.

Gradient estimates derived for solutions of a specific elliptic equation on Riemannian manifolds.

We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = Δu + |\nabla u|^2 u \quad \text{in } Ω\times(0,T) \\ u &= \varphi \quad \text{on } \partial Ω\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } Ω, \end{align*} where $Ω$ is a bounded, smooth do…

The paper studies 1-equivariant harmonic map flow behavior from R² to S².

We present some enumerative and structural results for flag homology spheres. For a flag homology sphere $Δ$, we show that its $γ$-vector $γ^Δ=(1,γ_1,γ_2,\ldots)$ satisfies: \begin{align*} γ_j=0,\text{ for all } j>γ_1, \quad γ_2\leq\binom{γ_1}{2}, \quad γ_{γ_1}\in\{0,1\}, \quad \text{ and }γ_{γ_1-1}\in\{0,1,2,γ_1\}, \e…

New inequality criterion for a mean field equation on spheres.

Let $σ>1$ and let $M$ be a complete Riemannian manifold. In a recent work [9], Grigor$^{\prime}$yan and Sun proved that a pointwise upper bound of volume growth is sufficient for uniqueness of nonnegative solutions of elliptic inequality $$(*)\quad\qquad\qquad\qquad Δu(x)+u^σ(x)\leq 0,\qquad x\in M.\quad\qquad \qquad\q…

The paper proves curvature rigidity for manifolds with specific scalar curvature bounds.

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = Δu + |\nabla u|^2 u \quad \text{in } Ω\times(0,T) \\ u &= u_b \quad \text{on } \partial Ω\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } Ω, \end{align*} with $u(x,t): \bar Ω\times [0,T) \to …

Solutions to a quadratic matrix equation are linked to strongly regular graphs and multiplicative characters.

Let $\nabla$ be a metric connection with totally skew-symmetric torsion $\T$ on a Riemannian manifold. Given a spinor field $Ψ$ and a dilaton function $Φ$, the basic equations in type II B string theory are \bdm \nabla Ψ= 0, \quad δ(\T) = a \cdot \big(d Φ\haken \T \big), \quad \T \cdot Ψ= b \cdot d Φ\cdot Ψ+ μ\cdot Ψ. …

Proves weakly non-collapsed RCD spaces are strongly non-collapsed.

Characterizes term structure models driven by Lévy processes.

The paper proves an equilibrium in a limited stock market participation model with power utilities.

The paper proves no positive weak solutions for a specific inequality on Riemannian manifolds.

Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n, [\hat{h}])$. The fractional Yamabe problem addresses to solve \[P^γ[g^+,\hat{h}] (u) = cu^{n+2γ\over n-2γ}, \quad u > 0 \quad \text{on } M\] where $c \in \mathbb{R}$ and $P^γ[g^+,\hat{h}]$ is the fractiona…

In this paper, we study the blow-up phenomena on the $α_k$-harmonic map sequences with bounded uniformly $α_k$-energy, denoted by $\{u_{α_k}: α_k>1 \quad \mbox{and} \quad α_k\searrow 1\}$, from a compact Riemann surface into a compact Riemannian manifold. If the Ricci curvature of the target manifold is of a positive l…

A discrete analog of the Tzitzeica equation is found in the form of quad-equation. Its continuous symmetry is an inhomogeneous Narita--Bogoyavlensky type lattice equation which defines a discretization of the Sawada--Kotera equation. The integrability of these discretizations is proven by construction of the Lax repres…

The paper constructs metrics with blow-up solutions for a curvature equation.

The study proves conditions for rigidity of Kleinian groups using measure theory and ergodic theory.

Algebraic method reveals criterion for quaternionic Möbius group reversibility.

Study solves overdetermined problem on compact surfaces.

The paper studies how norms of random vectors are preserved by random projections.

The paper explores discrete isothermic nets using checkerboard patterns in quadrilateral nets.

We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on larger parameter rectangles. For discrete planar quadrilateral nets, circular nets, $Q^*$-nets and conical…

In this paper, we introduce transformations of deep rectifier networks, enabling the conversion of deep rectifier networks into shallow rectifier networks. We subsequently prove that any rectifier net of any depth can be represented by a maximum of a number of functions that can be realized by a shallow network with a …

Discretizes special surfaces using Koenigs nets.

We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and their non-corresponding diagonals are parallel.…

Classifies nets with area-preserving transformations into two types.