arXiv research

Solves four problems related to circle families in the plane.

Research on the least complex surface in certain 4D shapes.

Solves four problems related to sphere families in 3D space.

In this paper we deal with symplectic Lie algebras. All symplectic structures are determined for dimension four and the corresponding Lie algebras are classified up to equivalence. Symplectic four dimensional Lie algebras are described either as solutions of the cotangent extension problem or as symplectic double exten…

We study a non-concave optimization problem in which a financial company maximizes the expected utility of the surplus under a risk-based regulatory constraint. For this problem, we consider four different prevalent risk constraints (Expected Shortfall, Expected Discounted Shortfall, Value-at-Risk, and Average Value-at…

New geometric mechanism solves four envelope problems.

Paper solves four problems of pseudo-circle envelopes in Minkowski plane.

A local classification of locally conformal flat Riemannian Einstein-like four-manifolds as well as a local classification of all locally conformal flat Riemannian four-manifolds for which all Jacobi operators have parallel eigenspaces along every geodesic is given. Non-trivial explicit examples are presented. The prob…

In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature to a metric conformal to the standard one. Our proof involves the study of crit…

In this paper we consider three deeply connected classificational problems on four-dimensional manifolds. First we consider and describe locally regular distributions. Second we give a classification of almost complex structures of general position in terms of distributions. Finally we classify nondegenerate Monge-Ampe…

We review the relations between (twisted) supersymmetric gauge theories in four dimensions and moduli problems in four-dimensional topology, and we study in detail the non-abelian monopole equations from this point of view. The relevance of exact results in N=1 and N=2 supersymmetric gauge theories to the computation o…

We study the problem of finding the minimal (maximal) genus for a surface where a given four-valent graph with fixed opposite edge structure can be embedded into. We find several partial relations and give new reformulations in combinatorial and knot theoretic languages.

Study classifies certain Einstein 4-manifolds with twistorial properties.

We show vanishing results about the infimum of the topological entropy of the geodesic flow of homogeneous smooth four manifolds. We prove that any closed oriented geometric four manifold has zero minimal entropy if and only if it has zero simplicial volume. We also show that if a four manifold M admits a geometric dec…

The problem of conformal transformation and conformal flatness of Finsler spaces has been studied by so many researchers $\left[ 6,16,17,20,21\right] .$ Recently, Prasad et. al $\left[ 19\right]$ have studied three dimensional conformally flat Landsberg and Berwald spaces and have given some important results. The pur…

The notion of type of a differential 2-form in four variables is introduced and for 2-forms of type < 4, local normal models are given. If the type of a 2-form $Ω$ is 4, then the equivalence under diffeomorphisms of $Ω$ is reduced to the equivalence of a symplectic linear frame functorially attached to $Ω$. As the equi…

We solve a portfolio selection problem with four objectives, finding convex scalarizations for part of the Pareto front.

We survey different classification results for surfaces with parallel mean curvature immersed into some Riemannian homogeneous four-manifolds, including real and complex space forms, and product spaces. We provide a common framework for this problem, with special attention to the existence of holomorphic quadratic diff…

In this review article we discuss four recent methods for computing Maurer-Cartan structure equations of symmetry groups of differential equations. Examples include solution of the contact equivalence problem for linear hyperbolic equations and finding a contact transformation between the generalized Hunter-Saxton equa…

Proves uniqueness of certain spacetime solutions with extremal horizons.

Study shows strict inequality for Dirac-type equation on spin manifolds, leading to existence results.

Reduces energy for 4D submanifolds in R^n.

The paper classifies hypersurfaces in a specific 4D geometry.

A generalized flag manifold is a homogeneous space of the form $G/K$, where $K$ is the centralizer of a torus in a compact connected semisimple Lie group $G$. We classify all flag manifolds with four isotropy summands and we study their geometry. We present new $G$-invariant Einstein metrics by solving explicity the Ei…

In this note, the geography problem in dimension four is reviewed and then its extension to dimension six for the symplectic case is explained. Finally some examples in dimension six are provided.

The paper tackles isotropy of symplectic forms using Hodge flows.

Most known four-dimensional cohomogeneity-one Einstein metrics are diagonal in the basis defined by the left-invariant one-forms, though some essentially non-diagonal ones are known. We consider the problem of explicitly seeking non-diagonal Einstein metrics, and we find solutions which in some cases exhaust the possib…

This paper is concerned with the study of the Monge optimal transport problem in sub-Riemannian manifolds where the cost is given by the square of the sub-Riemannian distance. Our aim is to extend previous results on existence and uniqueness of optimal transport maps to cases of sub-Riemannian structures which admit ma…

The economy globalization measure problem is discussed. Four macroeconomic indices of twenty among the "richest" countries are examined. Four types of "distances" are calculated.Two types of networks are next constructed for each distance measure definition. It is shown that the globalization process can be best charac…

Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.

Makeev proved that among centrally symmetric four-dimensional polytopes, with more than twenty facets and circumscribed about the Euclidean ball of diameter one, there is no universal cover for the family of unit diameter sets. In this paper we examine the converse problem, and prove that each centrally symmetric polyt…

Given an Einstein structure with positive scalar curvature on a four-dimensional Riemannian manifolds, that is $Ric=λg$ for some positive constant $λ$. For convenience, the Ricci curvature is always normalized to $Ric=1$. A basic problem is to classify four-dimensional Einstein manifolds with positive or nonnegative cu…

Study of parabolas in Funk metric on unit disk.

Explains peculiarities of 4D scalar curvature via Yamabe invariant.

We consider the problem of allocating samples to a finite set of discrete distributions in order to learn them uniformly well in terms of four common distance measures: $\ell_2^2$, $\ell_1$, $f$-divergence, and separation distance. To present a unified treatment of these distances, we first propose a general optimistic…

Novel bistable structures made from four-bar linkages, proving existence and construction.

Following an approach of the second author for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuit…

In this paper, we study the prescribed $Q$-curvature problem on closed four-dimensional Riemannian manifolds when the total integral of the $Q$-curvature is a positive integer multiple of the one of the four-dimensional round sphere. This problem has a variational structure with a lack of compactness. Using some topolo…

New method produces positive colored superpolynomials from four-point functions.

Survey on knot theory's impact on four-dimensional topology.

Geometric formulation of 4D supergravity for mathematicians.

Construct divide knots with specific genus properties.

We first provide an alternative proof of the classical Weitzneböck formula for Einstein four-manifolds using Berger curvature decomposition, motivated by which we establish a unified framework for a Weitzenböck formula for a large class of canonical metrics on four-manifolds (or a Weitzenböck formula for "Einstein metr…

Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the …

Gradient descent proves global convergence for 4-layer matrix factorization.

Four-dimensional Einstein Dehn filling is impossible.

Study finds all 4D Lie groups with harmonic curvature.

We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the m…