arXiv research

This paper studies the critical dynamics of random surfaces, focusing on area and genus evolution.

We relate the total curvature and the isoperimetric deficit of a curve $γ$ in a two-dimensional space of constant curvature with the area enclosed by the evolute of $γ$. We provide also a Gauss-Bonnet theorem for a special class of evolutes.

In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a 3-manifold under the Ricci flow. This estimate generalizes an area estimate of Hamilton, which we will recall in the first part of t…

Study proves existence and properties of shrinkers in area-preserving curve-shortening flow.

The paper studies how curves evolve under area constraints and converges to a critical point.

We describe some relations between the long-time asymptotic behavior of the vacuum Einstein evolution equations and the geometrization of 3-manifolds. These relations are expressed in terms of evolution of CMC hypersurfaces in the vacuum space-time.Some results are also obtained on the singularity avoidance of CMC foli…

New curve flow preserves area and converges to a circle.

In this paper we consider the evolution of a graph-like hypersurface by anisotropic mean curvature flow, under some restrictions on the anisotropic area integrand. We find interior estimates (in both time and space) on the gradient of such hypersurfaces, depending only on the height of the graph and the anisotropic are…

In this article we give a complete description of the evolution of an area decreasing map $f:M\to N$ induced by its mean curvature in the situation where $M$ and $N$ are complete Riemann surfaces with bounded geometry, $M$ being compact, for which their sectional curvatures $σ_M$, $σ_N$ satisfy $\minσ_M\ge\supσ_N$.

We consider the mean curvature flow of the graph of a smooth map $f:\mathbb{R}^2\to\mathbb{R}^2$ between two-dimensional Euclidean spaces. If $f$ satisfies an area-decreasing property, the solution exists for all times and the evolving submanifold stays the graph of an area-decreasing map $f_t$. Further, we prove unifo…

In this paper we consider planar polygons with parallel opposite sides. This type of polygons can be regarded as discretizations of closed convex planar curves by taking tangent lines at samples with pairwise parallel tangents. For this class of polygons, we define discrete versions of the area evolute, central symmetr…

In this paper we study properties of the area evolute (AE) and the center symmetry set (CSS) of a convex planar curve $γ$. The main tool is to define a Minkowski plane where $γ$ becomes a constant width curve. In this Minkowski plane, the CSS is the evolute of $γ$ and the AE is an involute of the CSS. We prove that the…

We consider a regular embedded network composed by two curves, one of them closed, in a convex domain $Ω$. The two curves meet only in one point, forming angle of $120$ degrees. The non-closed curve has a fixed end point on $\partialΩ$. We study the evolution by curvature of this network. We show that the maximal exist…

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygo…

EvoNet predicts the evolution of dynamic graphs using a graph neural network and recurrent architecture.

Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefini…

We consider an evolving plane curve with two endpoints that can move freely on the $x$-axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is governed by area-preserving curvature flow equation. The main result shows that an…

New method estimates Schrödinger bridges using ML techniques.

We analyze the sectoral dynamics of startup venture financing. Based on a dataset of 52000 start-ups and 110000 funding rounds in the United States from 2000 to 2017, and by applying both Principal Component Analysis (PCA) and Tensor Component Analysis (TCA) in sector space, we visualize and measure the evolution of th…

One dimensional stylized model taking into account spatial activity of firms with uniformly distributed customers is proposed. The spatial selling area of each firm is defined by a short interval cut out from selling space (large interval). In this representation, the firm size is directly associated with the size of i…

Some optimization problems coming from the Differential Geometry, as for example, the minimal submanifolds problem and the harmonic maps problem are solved here via interior solutions of appropriate multitime optimal control problems. Section 1 underlines some science domains where appear multitime optimal control prob…

The paper studies area-preserving and length-preserving inverse curvature flow for planar curves with singularities.

Analyzed geometric and diffusion properties of a coupled system.

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof o…

This paper gives two methods for constructing associative 3-folds in R^7, based around the fundamental idea of evolution equations, and uses these methods to construct examples of these geometric objects. The paper is a generalisation of the work by Joyce in math.DG/0008021, math.DG/0008155, math.DG/0010036 and math.DG…

AR model forecasts partially observed dynamical time series by estimating evolution function and imputing missing variables.

Paper proves a generalized Penrose conjecture for flat initial data.

Small bubbles sliding on a boundary maintain half-spherical shape.

The paper proves reverse inequalities in various geometric settings using curvature radius data.

New algorithm improves signal reconstruction from noisy measurements with side information.

Two flows for convex curves converge to circles smoothly.

Accounting frameworks follow stipulations of existing Accounting Theories. This exploratory research sets out to trace the evolution of accounting theories of Charge and Discharge Syndrome and the Corollary of Double Entry. Furthermore, it dives into the theories of Income Determination, garnishing it with areas of div…

Studying the impact of climate change on precipitation is constrained by finding a way to evaluate the evolution of precipitation variability over time. Classical approaches (feature-based) have shown their limitations for this issue due to the intermittent and irregular nature of precipitation. In this study, we prese…

Develops a machine learning method for parameter estimation in branching processes models.

Study shows how curved surfaces evolve smoothly to spherical shapes.

In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric difference. We also study a new nonlocal flow of convex plane curves and discuss its e…

This paper studies the large time existence for the motion of closed hypersurfaces in a radially symmetric potential. In physical, this surface can be considered as an electrically charged membrane with a constant charge per area in a radially symmetric potential. The evolution of such surface has been investigated by …

Given a simple closed plane curve $Γ$ of length $L$ enclosing a compact convex set $K$ of area $F$, Hurwitz found an upper bound for the isoperimetric deficit, namely $L^2-4πF\leq π|F_{e}|$, where $F_{e}$ is the algebraic area enclosed by the evolute of $Γ$. In this note we improve this inequality finding strictly posi…

Phase segregation, the process by which the components of a binary mixture spontaneously separate, is a key process in the evolution and design of many chemical, mechanical, and biological systems. In this work, we present a data-driven approach for the learning, modeling, and prediction of phase segregation. A direct …

The RG-2 flow is the two-loop approximation for the world-sheet non-linear sigma model renormalization group flow. The first truncation of the flow is the well known Ricci flow, at two loops higher order curvature terms appear, changing almost completely the behaviour of the evolution equation. In this article we study…

We analyze the stability properties of equilibrium solutions and periodicity of orbits in a two-dimensional dynamical system whose orbits mimic the evolution of the price of an asset and the excess demand for that asset. The construction of the system is grounded upon a heterogeneous interacting agent model for a singl…

Language is dynamic, constantly evolving and adapting with respect to time, domain or topic. The adaptability of language is an active research area, where researchers discover social, cultural and domain-specific changes in language using distributional tools such as word embeddings. In this paper, we introduce the gl…

Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.

Paper reviews the evolution of alpha from human insight to AI-powered systems.

A new framework reduces traffic congestion by 36%.

Develops adiabatic theory for ACW flow on surfaces.

With the breakthrough of computational power and deep neural networks, many areas that we haven't explore with various techniques that was researched rigorously in past is feasible. In this paper, we will walk through possible concepts to achieve robo-like trading or advising. In order to accomplish similar level of pe…

This review explores Ricci soliton inequalities in Riemannian geometry.