arXiv research

Using the method of Nehari manifold, we prove the existence of at least two distinct weak solutions to elliptic equation of four order with singulatities and with critical Sobolev growth.

Researchers find multiple ways to deform manifolds with specific curvature properties.

In this paper we construct explicit smooth solutions to the Strominger system on generalized Calabi-Gray manifolds, which are compact non-Kähler Calabi-Yau 3-folds with infinitely many distinct topological types and sets of Hodge numbers.

This paper deals with the existence of solutions to a class of fourth order nonlinear elliptic equations. The technique used relies on critical points theory. The solutions appeared as critical points of a functional restricted to a suitable manifold.In the case of constant coefficients we obtain the existence of tree …

We address the problem of determining the hypersurfaces $f\colon M^{n} \to \mathbb{Q}_s^{n+1}(c)$ with dimension $n\geq 3$ of a pseudo-Riemannian space form of dimension $n+1$, constant curvature $c$ and index $s\in \{0, 1\}$ for which there exists another isometric immersion $\tilde{f}\colon M^{n} \to \mathbb{Q}^{n+1}…

The paper studies inverse mean curvature flow on hypersurfaces in space forms.

New method distinguishes feature relevance in non-linear contexts.

Ancient solutions and translators identified for Lagrangian flow.

Page's Einstein metric on CP_2 # (-CP_2) is conformally related to an extremal Kaehler metric. Here we construct a family of conformally Kähler solutions of the Einstein-Maxwell equations that deforms the Page metric, while sweeping out the entire Kaehler cone of CP_2 # (-CP_2).The same method also yields analogous sol…

It is conjectured that for each knot $K$ in $S^3$, the fundamental group of its complement surjects onto only finitely many distinct knot groups. Applying character variety theory we obtain an affirmative solution of the conjecture for a class of small knots that includes 2-bridge knots.

Study finds methods to learn multiple solutions from single task in offline RL.

GEMSS discovers multiple sparse solutions in high-dimensional data.

In this paper, partially invariant solutions (PISs) method is applied in order to obtain new four-dimensional Einstein Walker manifolds. This method is based on subgroup classification for the symmetry group of partial differential equations (PDEs) and can be regarded as the generalization of the similarity reduction m…

In this article, we study connections between representation theory and efficient solutions to the conjugacy problem on finitely generated groups. The main focus is on the conjugacy problem in conjugacy separable groups, where we measure efficiency in terms of the size of the quotients required to distinguish a distinc…

In this paper we present explicit formulas for the fundamental solution to the Klein-Gordon operator on some higher dimensional generalizations of the Möbius strip and the Klein bottle with values in distinct pinor bundles. The fundamental solution is described in terms of generalizations of the Weierstraß $\wp$-functi…

We examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension. It is shown that, generically, singular data can give rise to two distinct solutions which are both stable, and satisfy the local energy inequality. We also discuss how uniqueness can…

In this article, we extend Anderson's higher-dimensional Dehn filling construction to a large class of infinite-volume hyperbolic manifolds. This gives an infinite family of topologically distinct asymptotically hyperbolic Einstein manifolds with the same conformal infinity. The construction involves finding a sequence…

We introduce a notion of Ricci flow in generalized geometry, extending a previous definition by Gualtieri on exact Courant algebroids. Special stationary points of the flow are given by solutions to first-order differential equations, the Killing spinor equations, which encompass special holonomy metrics with solutions…

Study shows different behaviors of noncompact hypersurfaces under mean curvature flow.

New isometric solutions found for heterotic G2-system.

This article is about applications of linear algebra to knot theory. For example, for odd prime p, there is a rule (given in the article) for coloring the arcs of a knot or link diagram from the residues mod p. This is a knot invariant in the sense that if a diagram of the knot under study admits such a coloring, then …

We investigate active learning with access to two distinct oracles: Label (which is standard) and Search (which is not). The Search oracle models the situation where a human searches a database to seed or counterexample an existing solution. Search is stronger than Label while being natural to implement in many situati…

We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…

Unique optimal strategy identified for state-dependent risk aversion.

We study supergravity solutions corresponding to fivebranes wrapped on a three-sphere inside a G_2 holonomy manifold. By changing a parameter the solutions interpolate between a G_2 manifold X_i \cong S^3 x R^4 with flux on a three-sphere and a distinct G_2 manifold X_j \cong S^3 x R^4 with branes on another three-sphe…

This work includes a number of novel contributions for the multiple-source adaptation problem. We present new normalized solutions with strong theoretical guarantees for the cross-entropy loss and other similar losses. We also provide new guarantees that hold in the case where the conditional probabilities for the sour…

For a variety of regularized optimization problems in machine learning, algorithms computing the entire solution path have been developed recently. Most of these methods are quadratic programs that are parameterized by a single parameter, as for example the Support Vector Machine (SVM). Solution path algorithms do not …

In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time $t \in [0, \infty)$ with unif…

We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If the associated roots of the ODEs are real and distinct, we give a universal uppe…

A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors, we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on 4-manifolds with non-trivial fu…

This article studies quadratic semimartingale BSDEs arising in power utility maximization when the market price of risk is of BMO type. In a Brownian setting we provide a necessary and sufficient condition for the existence of a solution but show that uniqueness fails to hold in the sense that there exists a continuum …

The paper studies special surfaces in pseudo-Euclidean space.

Developing a range-aware Bayesian optimization framework for discovering diverse designs within target property windows.

In this paper we construct an end of a self-similar shrinking solution of the mean curvature flow asymptotic to an isoparametric cone C and lying outside of C. We call a cone C in $R^{n+1}$ an isoparametric cone if C is the cone over a compact embedded isoparametric hypersurface $Γ\subset S^n$. The theory of isoparamet…

In this article we propose a model for stochastic delay differential equation with jumps (SDDEJ) in a differentiable manifold $M$ endowed with a connection $\nabla$. In our model, the continuous part is driven by vector fields with a fixed delay and the jumps are assumed to come from a distinct source of (càdlàg) noise…

The Allen-Cahn system on manifolds yields multiple phase distributions.

Study on $m$-positivity in Kähler manifolds with new Monge-Ampère-type equation.

We introduce exact macroscopic on-line learning dynamics of two-layer neural networks with ReLU units in the form of a system of differential equations, using techniques borrowed from statistical physics. For the first experiments, numerical solutions reveal similar behavior compared to sigmoidal activation researched …

This work analyzes two methods for combining multiple binary labels in bipartite ranking.

The Ward equation, also called the modified 2+1 chiral model, is obtained by a dimension reduction and a gauge fixing from the self-dual Yang-Mills field equation on $R^{2,2}$. It has a Lax pair and is an integrable system. Ward constructed solitons whose extended solutions have distinct simple poles. He also used a li…

In this article we consider nonholonomic deformations of disk solutions in general relativity to generic off-diagonal metrics defining knew classes of exact solutions in 4D and 5D gravity. These solutions possess Lie algebroid symmetries and local anisotropy and define certain generalizations of manifolds with Killing …

We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this two part treatise, we present our developments in the context of solving two main classes …

CPRA efficiently finds diverse solutions in CO problems using UL and parallelization.

We construct infinitely many Legendrian links in the standard contact $\mathbb{R}^3$ with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in $S^3$ that bound topologically distinct pieces of algebraic curves in $B^4 \subset \mathbb{C}^2$, is applied to find contact 3-…

Typically clustering algorithms provide clustering solutions with prespecified number of clusters. The lack of a priori knowledge on the true number of underlying clusters in the dataset makes it important to have a metric to compare the clustering solutions with different number of clusters. This article quantifies a …

We study an equation lying `mid-way' between the periodic Hunter-Saxton and Camassa-Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural r…

MaxSketch improves distinct counting in high-dimensional, noisy data streams.

Double InfoGAN improves CA by generating clearer, more accurate images.