arXiv research

The paper examines $HM$-tensional and $HS$-tensional maps between Riemannian manifolds.

We explore a computational model of an incompressible fluid with a multi-phase field in three-dimensional Euclidean space. By investigating an incompressible fluid with a two-phase field geometrically, we reformulate the expression of the surface tension for the two-phase field found by Lafaurie, Nardone, Scardovelli, …

Study Gauss maps of surfaces in Heisenberg group using hyperbolic geometry.

We consider maps into Riemannian manifolds of non-positive curvature and start developing a systematic PDE theory. We control the Sobolev $H^{2,2}$-norm of such a map in terms of its energy, the $L^2$-norm of its tension field and a topological term depending on the homotopy class. We also solve a Dirchlet problem with…

Let $M$ be a closed Riemannian surface and $u_n$ a sequence of maps from $M$ to Riemannian manifold $N$ satisfying $\sup_n(\|\nabla u_n\|_{L^2(M)}+\|τ(u_n)\|_{L^p(M)})\leq Λ$ for some $p>1$, where $τ(u_n)$ is the tension field of the mapping $u_n$. For the general target manifold $N$, if $p\geq \frac 65$, we prove th…

For the class of approximate harmonic maps $u\in W^{1,2}(Σ,N)$ from a closed Riemmanian surface $(Σ,g)$ to a compact Riemannian manifold $(N, h)$, we show that (i) the so-called energy identity holds for weakly convergent approximate harmonic maps $\{u_n\}:Σ\to N$, with tension fields $τ(u_n)$ bounded in the Morrey spa…

Study biharmonic submanifolds in warped product structures.

Let $u_n$ be a sequence of mappings from a closed Riemannian surface $M$ to a general Riemannian manifold $N$. If $u_n$ satisfies \beno \sup_{n}\big(\|\nabla u_n\|_{L^2(M)}+\|τ(u_n)\|_{L^{p}(M)}\big)\leq Λ\quad \text{for some}\,\,p>1, \eeno where $τ(u_n)$ is the tension field of $u_n$, then there hold the so called ene…

By using Moser's iteration technique, we show some removable singularity theorem of the tension field for biharmonic maps into manifolds of non-positive curvature, and the bubbling theorem of biharmonic maps and also harmonic maps.

Harmonic morphisms and p-harmonic functions constructed on symmetric spaces.

Minimal hypersurfaces are the only $H$-tensional in 4D space forms.

In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh-Taylor sign condition holds. We prove that as the surface tension tends to zero, when the Rayleigh-Taylor condition is satisfied, solutions converge to the Euler …

The absence of interesting harmonic sections for the Sasaki and Cheeger-Gromoll metrics has led to the consideration of alternatives, for example in the form of a two-parameter family of natural metrics shown to relax existence conditions for harmonicity. This article investigates harmonic Killing vector fields, proves…

We prove the removal singularity results for maps with bounded energy from the unit disk $B$ of $R^2$ centered at the origin to a closed Riemannian manifold whose tension field is unbounded in $L^2(B)$ but satisfies the following condition: {eqnarray*} (\int_{B_t\setminus B_{\frac{t}{2}}}|τ(u)|^2)^1/2\leq C_1(\frac{1}{…

Maps converge to simpler structures under certain tension conditions.

Surgeons normally need surgical scissors and tissue grippers to cut through a deformable surgical tissue. The cutting accuracy depends on the skills to manipulate these two tools. Such skills are part of basic surgical skills training as in the Fundamentals of Laparoscopic Surgery. The gripper is used to pinch a point …

We consider the regularity of an interface between two incompressible and inviscid fluids flows in the presence of surface tension. We obtain local in time estimates on the interface in $H^{\frac32k +1}$ and the velocity fields in $H^{\frac32k}$. These estimates are obtained using geometric considerations which show th…

We consdier in dimension four weakly convergent sequences of approximate biharmonic maos into sphere with bi-tension fields bounded in $L^p$ for some $p>1$. We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on $\mathbb…

New statistical manifolds derived from identity map biharmonicity.

The purpose of this paper is to study the harmonicity of maps to or from para-Sasakian manifolds. We derive the condition for the tension field of paraholomorphic map between almost para-Hermitian manifold and para-Sasakian manifold. The necessary and sufficient condition for a paraholomorphic map between para-Sasakian…

Estimates for plate eigenvalues with nonzero Poisson's ratio.

A triharmonic map is a critical point of the 3-energy in the space of smooth maps between two Riemannian manifolds. We study a triharmonic isometric immersion into a space form of non-positively constant curvature. We show that if the domain is complete and both the 4-enegy and the L^4-norm of the tension field are fin…

We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold (M, <,>) equipped with the Sasaki metric and we discuss the characterising condition for critical points. Likewise, we provide a useful method for computing the tension field in some particular situations. Such a me…

We consider in dimension four weakly convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in $L^p$ for $p>\frac43$. We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic ma…

Ambitwistor string matches superstring chiral integrands at zero tension.

The elastic net was introduced as a heuristic algorithm for combinatorial optimisation and has been applied, among other problems, to biological modelling. It has an energy function which trades off a fitness term against a tension term. In the original formulation of the algorithm the tension term was implicitly based…

We study the free boundary Euler equations in two spatial dimensions. We prove that if the boundary is sufficiently regular, then solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain when the coefficient of surface tension tends to infinity.

We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension \(n\) that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the $L^…

We develop a theory of axisymmetric surfaces minimizing a combination of surface tension and nematic elastic energies which may be suitable for describing simple film and bubble shapes. As a function of the elastic constant and the applied tension on the bubbles, we find the analogues of the unduloid, sphere, and nodoi…

Study of degenerating maps to Riemannian manifolds, proving asymptotic limits and existence of minimal cylinders.

Novel method separates astrophysical components from noisy data.

We study a generalized functional related to the pullback metrics (3). We derive the first variation formula which yield stationary maps. We introduce the stress-energy tensor which is naturally linked to conservation law and yield monotonicity formula via the coarea formula and comparison theorem in Riemannian geometr…

Characterizations of entire subsolutions for the 1-harmonic equation of a constant 1$-tension field are given with applications in geometry via transformation group theory. In particular, we prove that every level hypersurface of such a subsolution is calibrated and hence is area-minimizing over$\mathbb{R}$; and every…

In this note, we generalize biharmonic equation for rotationally symmetric maps ([4], [16], [10]) to equivariant maps between model spaces and use it to give a complete classification of rotationally symmetric conformal biharmonic maps from a $4$-dimensional space form into a $4$-dimensional model space. We also give a…

Networks, which represent agents and interactions between them, arise in myriad applications throughout the sciences, engineering, and even the humanities. To understand large-scale structure in a network, a common task is to cluster a network's nodes into sets called "communities", such that there are dense connection…

Model for material elasticity and plasticity using networks.

We consider biharmonic maps $φ:(M,g)\rightarrow (N,h)$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $α$ satisfies $1<α<\infty$. If for such an $α$, $\int_M|τ(φ)|^αdv_g<\infty$ and $\int_M|dφ|^2dv_g<\infty,$ where $τ(φ)$ is the tension field of $φ$, th…

New mechanism found for power laws including Zipf's law.

Let $u: (M, g)\to (N, h)$ be a map between Riemannian manifolds $(M, g)$ and $(N, h)$. The $p$-bienergy of $u$ is defined by $E_p(u)=\int_M|τ(u)|^pdν_g$, where $τ(u)$ is the tension field of $u$ and $p>1$. Critical points of $E_p(\cdot)$ are called $p$-biharmonic maps. In this paper we will prove nonexistence result of…

We investigate the Penrose limits of classical string and M-theory backgrounds. We prove that the number of (super)symmetries of a supergravity background never decreases in the limit. We classify all the possible Penrose limits of AdS x S spacetimes and of supergravity brane solutions. We also present the Penrose limi…

Compact gravity models yield tiny spin-two field masses.

In this paper we study the parabolic evolution equation $\partial_t u=(|Du|^{2}+2|\det Du|)^{-1} Δu$, where $u : M\times[0,\infty) \to N$ is an evolving map between compact flat surfaces. We use a tensor maximum principle for the induced metric to establish two-sided bounds on the singular values of Du, which shows tha…

We introduce and study generalized $1$-harmonic equations (1.1). Using some ideas and techniques in studying $1$-harmonic functions from [W1] (2007), and in studying nonhomogeneous $1$-harmonic functions on a cocompact set from [W2, (9.1)] (2008), we find an analytic quantity $w$ in the generalized $1$-harmonic equatio…

Let $\{u_n\}$ be a sequence of maps from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold $N$ with free boundary on a smooth submanifold $K\subset N$ satisfying \[ \sup_n \ \left(\|\nabla u_n\|_{L^2(M)}+\|τ(u_n)\|_{L^2(M)}\right)\leq Λ, \] where $τ(u_n)$ is the tension field o…

We explore the use of random forest and gradient boosting, two powerful tree-based machine learning algorithms, for the detection of cosmic strings in maps of the cosmic microwave background (CMB), through their unique Gott-Kaiser-Stebbins effect on the temperature anisotropies.The information in the maps is compressed…

TSCoNet forecasts correlated geophysical fields with uncertainty estimates.

New findings show privacy affects generalization error in a non-monotonic way.

Minimal biharmonic hypersurfaces in Euclidean spaces are ideal.