arXiv research

The paper explains geometric correspondences for homothetic navigation.

Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.

We generalize the notion of Zermelo navigation to arbitrary pseudo-Finsler metrics possibly defined in conic subsets. The translation of a pseudo-Finsler metric $F$ is a new pseudo-Finsler metric whose indicatrix is the translation of the indicatrix of $F$ by a vector field $W$ at each point, where $W$ is an arbitrary …

The paper studies constant mean curvature hypersurfaces in Finsler manifolds.

In this paper, a characteristic condition of Einstein Kropina metrics is given. By the characteristic condition, we prove that a non-Riemannian Kropina metric $F=\frac{α^2}β$ with constant Killing form $β$ on an n-dimensional manifold $M$, $n\geq 2$, is an Einstein metric if and only if $α$ is also an Einstein metric. …

Proves uniqueness of catenoid-like shapes in a ball.

Solves geodesic completeness on pseudo-homothetic Lie group.

In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of $\mathcal{F}$-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabil…

Mathematically, a homothetic function is a function of the form $f({\bf x})=F(h(x_1,...,x_n))$, where $h$ is a homogeneous function of any degree $d\ne 0$ and $F$ is a monotonically increasing function. In economics homothetic functions are production functions whose marginal technical rate of substitution is homogeneo…

Paper classifies ruled surfaces in Lorentz-Minkowski space for a specific flow.

We show that any horizontally homothetic submersion from a compact manifold of nonnegative sectional curvature is a Riemannian submersion.

The paper explores conditions for homothetic Killing vectors on spacetime hypersurfaces.

In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane $\mathbb{E}^2$ were obtained. Moreover, the relations between the Complex velocities one-parameter motion in the Complex plane were provided by \cite…

The aim of this paper is to classify compact, simply connected Kähler manifolds which admit totally geodesic, holomorphic complex homothetic foliation by curves.

In this article we derive a complete classification of all submanifolds in space forms with codimension two for which the Gauss map is homothetic.

We obtain a local classification of complex homothetic foliations on Kaehler manifolds by complex curves. This is used to construct almost Kaehler, Ricci-flat metrics subject to additional curvature properties.

In this paper, we first give two fundamental principles under a technique to characterize conformal vector fields of $(α,β)$ spaces to be homothetic and determine the local structure of those homothetic fields. Then we use the principles to study conformal vector fields of some classes of $(α,β)$ spaces under certain c…

We classify homothetical surfaces with constant mean curvature in hyperbolic space.

A homothetical surface arises as a graph of a function $z = \varphi_1(v_1) \varphi_2(v_2)$. In this paper, we study the homothetical surfaces in three dimensional psuedo-Galilean space$\left(\mathbb{G}_3^1\right)$ satisfying the conditions $Δ^{II}\textbf{x}_i=λ_i\textbf{x}_i,$ where $Δ^{II}$ is the Laplacian with respe…

In an incomplete market, with incompleteness stemming from stochastic factors imperfectly correlated with the underlying stocks, we derive representations of homothetic (power, exponential and logarithmic) forward performance processes in factor-form using ergodic BSDE. We also develop a connection between the forward …

In this paper, we completely classify the homothetical hypersurfaces having null Gauss-Kronocker curvature in a Euclidean (n+1)-space. Several applications to the production functions in economics are also given.

Study on minimal surfaces in a 3D space with 2m-norm.

A selfsimiar manifold is a Riemannian manifold $\left(M,g\right)$ endowed with a homothetic vector field $ξ$. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian ma…

Study on solitons in deformed Kenmotsu manifolds with specific vector fields.

The paper classifies vertices in planar polygons formed by convex domains.

We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large…

Flat holonomies imply homotheticity in certain curved spaces.

In this paper, it is proved that any conformal vector field is homothetic on a locally projectively flat $(α,β)$-space of non-Randers type in dimension $n\ge 3$, and the local solutions of such a vector field are determined. While on a locally projectively flat Randers space, examples showthat the conformal vector fiel…

The paper explores almost Ricci solitons on Finsler spaces, proving conditions for their existence.

This paper studies rapidly forming singularities in the Yang-Mills flow. It is shown that a sequence of blow-ups near the singular point converges, modulo the gauge group, to a homothetically shrinking soliton with non-zero curvature. The proof uses Hamilton's monotonicity formula. Examples of homothetically shrinking …

Proves stability of cone-volume measure with nearly constant density.

Symmetry groups help define solitons in curved spaces.

HRL4IN tackles interactive navigation tasks with mobile manipulators, improving efficiency and performance.

In the study of the curve shortening flow on general closed curves, Abresch and Langer posed a conjecture that the homothetic curves can be regarded as saddle points between multi-folded circles and some singular curves. In other words, these homothetic curves are the watershed between curves with a nonsingular future …

We study the curve diffusion flow for closed curves immersed in the Minkowski plane $\mathcal{M}$, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in $\mathcal{M}$ depending on its length. The indiactrix $\partial\mathcal{…

Introduces a natural parallel translation for navigation data.

In this paper we study the set of balanced metrics (in Donaldson's terminology) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kahler metrics. We prove that this set is finite when $M$ admits…

The paper analyzes self-similar solutions for mean curvature flow in 3D.

Challenge to separate Earth's magnetic field from vehicle's magnetic field for accurate navigation.

In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions of conformal diffeomorphisms and vector fields that preserve a GQE structure. W…

VALAN is a framework for navigation agents in photo-realistic environments.

The paper solves navigation problems on conic Kropina manifolds and establishes curvature relationships.

Robotic navigation improves with RL and ultrasound images.

Improved robot navigation using multi-head attention for natural language instructions.

The problem of pursuing a moving target is always one of the main topics in navigation. In the literatures, there are two well-known algorithms called Pure Pursuit and Pure Rendezvous navigation in the 3-dimensional space $\mathbb{R}^3$. In this paper, these two methods are combined to introduce a novel family of pursu…

Mobile robot navigation in complex and dynamic environments is a challenging but important problem. Reinforcement learning approaches fail to solve these tasks efficiently due to reward sparsities, temporal complexities and high-dimensionality of sensorimotor spaces which are inherent in such problems. We present a nov…

SLAM-net learns to navigate visually in challenging indoor environments.

Study on CPSRM from/to Kähler manifolds, deriving integrability and geodesic results.