arXiv research

New method constructs axial vector fields and defines quasi-local spin-angular momentum.

Stationary and axially symmetric space-times play an important role in astrophysics, particularly in the theory of neutron stars and black holes. The static vacuum sub-class of these space-times is known as Weyl's class, and contains the Schwarzschild space-time as its most prominent example. This paper is going to stu…

Here are described the axiumbilic points that appear in generic one parameter families of surfaces immersed in R4. At these points the ellipse of curvature of the immersion, Little, Garcia - Sotomayor has equal axes. A review is made on the basic preliminaries on axial curvature lines and the associated axiumbilic poin…

For singular corank 1 surfaces in $\mathbb R^3$ we introduce a distinguished normal vector called the axial vector. Using this vector and the curvature parabola we define a new type of curvature called the axial curvature, which generalizes the singular curvature for frontal type singularities. We then study contact pr…

We show the existence of a Hawking vector field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth Einstein-Maxwell space-time without assuming the underlying space-time is analytic. It extends one result of Friedrich, Rácz and Wald, which was limited to the interior of th…

Improved Beckner's inequality for axially symmetric functions on S^4.

In this paper are studied the simplest patterns of axial curvature lines (along which the normal curvature vector is at a vertex of the ellipse of curvature) near a critical point of a surface mapped into R4. These critical points, where the rank of the mapping drops from 2 to 1, occur isolated in generic one parameter…

Defines axial curvatures for corank 1 singular manifolds in higher dimensions.

We analyze quantum Yang-Mills theory on $\mathbb{R}^2$ using a novel discretization method based on an algebraic analogue of stochastic calculus. Such an analogue involves working with "Gaussian" free fields whose covariance matrix is indefinite rather than positive definite. Specifically, we work with Lie-algebra valu…

We prove positive mass theorem with angular momentum and charges for axially symmetric, simply connected, maximal, complete initial data sets with two ends, one designated asymptotically flat and the other either (Kaluza-Klein) asymptotically flat or asymptotically cylindrical, for 4-dimensional Einstein-Maxwell theory…

In this paper, we prove that the even solution of the mean field equation $Δu=λ(1-e^u)$ on $S^2$ must be axially symmetric when $4<λ\leq 8$. In particular, zero is the only even solution for $λ=6$. This implies the rigidity of Hawking mass for stable constant mean curvature(CMC) sphere with even symmetry.

The article improves Beckner's inequality for axially symmetric functions on the n-dimensional sphere.

We prove that Wilson loop expectation values for arbitrary simple closed contours obey an area law up to second order in perturbative two-dimensional Yang-Mills theory. Our analysis occurs within a general family of axial-like gauges, which include and interpolate between holomorphic gauge and the Wu-Mandelstam-Liebran…

New surfaces described that are symmetric and solve a specific equation.

Finite index solutions to Bernoulli problem are always axially symmetric.

Sharp inequality proven for symmetric functions on a 4D sphere.

Smooth minimizers found for Willmore energy surfaces.

Axial-LOB predicts stock prices from LOB data using attention layers.

This paper tackles gauge fixing and regularity for perturbations around spherical backgrounds.

We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume…

For any n>1 we give an explicit example of an n-axially symmetric Cartesian current in B^3 x S^2 with non-trivial vertical part and non-constant graph part minimizing the relaxed Dirichlet energy among the n-axially symmetric Cartesian currents with the same boundary. This stands in sharp contrast with a results of Har…

3D Axial-Attention improves lung nodule classification accuracy.

We study mean curvature flow of smooth, axially symmetric surfaces in $\mathbb{R}^3$ with Neumann boundary data. We show that all singularities at the first singular time must be of type I.

Based on the Hamiltonian dimensional reduction of $3+1$ axially symmetric, Ricci-flat Lorentzian spacetimes to a $2+1$ Einstein-wave map system with the (negatively curved) hyperbolic 2-plane target, we construct a positive-definite, (spacetime) gauge-invariant energy functional for linear axially symmetric perturbatio…

New minimal hypersurfaces found via transformations.

We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows - surfaces of revolution - in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additio…

Derives a Hamiltonian model for 3D axially symmetric magnetohydrodynamics.

This work proves Kerr black holes are dynamically stable under certain perturbations.

In this paper, we illustrated one scenario to modify the Ivanenko-Landau-Kähler equation. Since Ivanenko and Landau introduced the equation in 1928, the equation has been regarded as having a certain role as a fermion in particular in the discrete Lattice. Also, although it correctly is formulated as an alternative cla…

Researchers found a way to measure energy in black hole perturbations.

For Riemannian metrics of constant positive curvature on a punctured sphere with conic singularities at the punctures and co-axial monodromy of the developing map, possible angles at the singularities are completely described. This completes the recent result of Mondello and Panov. The related problem of describing pos…

We study the provenance of singularity formation under mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that if the mean curvature is uniformly bounded on any finite time interval, then no singularities can develop during that time under both mean curvature flow an…

The Einstein/Maxwell equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities phi: R^3Σ-> H^2_C, where Sigma is a subset of the axis of symmetry, and H^2_C is the complex hyperbolic plane. Motivated by this problem, we prove the existence and uniqueness of harmonic m…

New theory shows how membranes can break symmetry.

We discuss the existence of Killing tensors for certain (physically motivated) stationary and axially symmetric vacuum space-times. We show nonexistence of a nontrivial Killing tensor for a Tomimatsu-Sato metric (up to valence 7), for a C-metric (up to valence 9) and for a Zipoy-Voorhees metric (up to valence 11). The …

We study the convergence of an axially symmetric hypersurface evolving by volume preserving mean curvature flow. Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as for example on the curvature, we prove that it converges to a hemisphere, when t…

We make a detailed study of the moduli space of winding number two (k=2) axially symmetric vortices (or equivalently, of co-axial composite of two fundamental vortices), occurring in U(2) gauge theory with two flavors in the Higgs phase, recently discussed by Hashimoto-Tong (hep-th/0506022) and Auzzi-Shifman-Yung (hep-…

We suggest an alternative mathematical model for the electron in which the dynamical variables are a coframe (field of orthonormal bases) and a density. The electron mass and external electromagnetic field are incorporated into our model by means of a Kaluza-Klein extension. Our Lagrangian density is proportional to ax…

Constructs a triple on an Atiyah algebroid with connection.

Solves Christoffel-Minkowski problem for axially symmetric bodies.

Sharp uniqueness result for Q-curvature type equation on S^6.

Study on surface configurations with curvature and elasticity.

The possibility of the global Lagrangian reduction of a mechanical system with symmetry is shown to be connected with the characteristic class of a principal fiber bundle of the configuration space over the factor manifold. It is proved that the reduced system is globally Lagrangian if and only if the product of the mo…

We prove stability and exponential convergence of the Perfectly Matched Layer (PML) method for acoustic scattering on manifolds with axial analytic quasicylindrical ends. These manifolds model long-range geometric perturbations (e.g. bending or stretching) of tubular waveguides filled with homogeneous or inhomogeneous …

Study electric field and potential of torus knots, focusing on z-axis.

In the paper, some concepts of modern differential geometry are used as a basis to develop an invariant theory of mechanical systems, including systems with gyroscopic forces. An interpretation of systems with gyroscopic forces in the form of flows of a given geodesic curvature is proposed. For illustration, the proble…

We relate Gaussian curvature to the gyroscopic force, thus giving a mechanical interpretation of the former and a geometrical interpretation of the latter. We do so by considering the motion of a spinning disk constrained to be tangent to a curved surface. It is shown that the spin gives rise to a force on the disk whi…

Very recently Ben Andrews and Haizhong Li showed that every embedded cmc torus in the three dimensional sphere is axially symmetric. There is a two-parametric family of axially symmetric cmc surfaces; more precisely, for every real number H and every C > 2 (H+\sqrt{1+H^2}) there is an axially symmetry surface Σ_{H,C} w…