arXiv research

New proof for certain groups in higher dimensions.

The study examines higher-order modern portfolio theory with complex critical points and feasible portfolio variety.

Study critical exponents in normal subgroups of higher rank Lie groups.

Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.

A new method uses higher-order Langevin dynamics with critical damping for better generative modeling.

Extends Gauduchon's result to higher dimensions, showing balanced metrics.

Paper explores weak solutions' regularity in critical dimensions without conservation law.

We study the problem of prescribing the Paneitz curvature on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of suc…

Research examines coamenable subgroups in higher rank groups.

We consider a variational problem for submanifolds Q $\subset$ M with nonempty boundary $\partial$Q = K. We propose the definition that the boundary K of any critical point Q have constant mean curvature, which seems to be a new perspective when dim Q \textless{} dim M . We then construct small nearly-spherical solutio…

In this note I use cup-products and higher Massey products to find topological lower bounds on the number of geometrically distinct critical points of any closed 1-form in a given cohomology class.

We compute the minimum number of critical points of a small codimension smooth map between two manifolds. We give as well some partial results for the case of higher codimension when the manifolds are spheres.

The article studies critical points of a new energy functional in higher dimensions.

HC test measures word-frequency similarity for authorship attribution.

We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.

New functionals defined for free boundary minimal submanifolds in higher dimensions.

The goal of this paper is to study weakly Einstein critical metrics of the volume functional on a compact manifold $M$ with smooth boundary $\partial M$. Here, we will give the complete classification for an $n$-dimensional, $n=3$ or $4,$ weakly Einstein critical metric of the volume functional with nonnegative scalar …

We define functionals generalising the Seiberg-Witten functional on closed $spin^c$ manifolds, involving higher order derivatives of the curvature form and spinor field. We then consider their associated gradient flows and, using a gauge fixing technique, are able to prove short time existence for the flows. We then pr…

We investigate existence and stability of rotationally symmetric critical immersions of variational problems of higher order which were considered by Nitsche.

The paper explores optimization of higher Steklov eigenvalues in various dimensions.

Heat flow on lens spaces settles into Morse functions with four critical points.

New method generates critical points for complex functionals.

Paper introduces a meta-critic for accelerating off-policy actor-critic learning.

We introduce $Z$-critical connections for holomorphic vector bundles and prove their existence under stability conditions.

This paper compares uncertainty estimation methods for deep learning in autonomous vehicles.

A new sampler tackles critical phenomena by leveraging scale invariance.

In this paper I suggest an alternative approach (using generic flat bundles and higher Massey products) to a Lusternik-Schnirelman type theory for closed 1-forms (cf. also math.DG/9811113)

Paper develops estimates for Lagrangian phase changes in 2D.

In this study, we investigate the use of global information to speed up the learning process and increase the cumulative rewards of reinforcement learning (RL) in competition tasks. Within the actor-critic RL, we introduce multiple cooperative critics from two levels of the hierarchy and propose a reinforcement learnin…

The study characterizes geometries of hypersurfaces in warped product and conformal manifolds.

We prove that the number of critical points of a Li-Tam Green's function on a complete open Riemannian surface of finite type admits a topological upper bound, given by the first Betti number of the surface. In higher dimensions, we show that there are no topological upper bounds on the number of critical points by con…

The usual Gromoll-Meyer's generalized Morse lemma near degenerate critical points on Hilbert spaces, so called splitting lemma, is stated for at least $C^2$-smooth functionals. In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower…

Study reveals structural differences in financial networks near and far from crises using balance theory.

Investigates multifractal scaling in critical dynamics of random surfaces.

We show that for three dimensional gravity with higher genus boundary conditions, if the theory possesses a sufficiently light scalar, there is a second order phase transition where the scalar field condenses. This three dimensional version of the holographic superconducting phase transition occurs even though the pure…

We generalize Cohen & Jones & Segal's flow category whose objects are the critical points of a Morse function and whose morphisms are the Morse moduli spaces between the critical points to an n-category. The n-category construction involves repeatedly doing Morse theory on Morse moduli spaces for which we have to const…

Anosov groups' measures on limit sets are uniquely determined by their dimension.

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second varia…

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

We study the regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler--Lagrange equations and consider the regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under…

Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total $(2k)$-th Gauss-Bonnet curvature function, called $(2k)$-minimal su…

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe…

Study on metrics maximizing eigenvalues of Paneitz operator on 4-manifolds.

The higher-power derivative terms involved in both Faddeev and Skyrme energy functionals correspond to $σ_2$-energy, introduced by Eells and Sampson. The paper provides a detailed study of the first and second variation formulae associated to this energy. Some classes of (stable) critical maps are outlined.

In this paper, we introduce the notion of motif closure and describe higher-order ranking and link prediction methods based on the notion of closing higher-order network motifs. The methods are fast and efficient for real-time ranking and link prediction-based applications such as web search, online advertising, and re…

New existence results for curvature problem on balls with specific conditions.

CSAC enables cooperative reinforcement learning for multi-stage tasks.

Study of polynomial strata using braid groups and translation surfaces.