arXiv research

New complete hypersurfaces found in affine geometry.

In this paper, we study the problem of finding the affine factorable surfaces in a 3-dimensional isotropic space with prescribed Gaussian (K) and mean (H) curvature. Because the absolute figure two different types of these surfaces appear by permutation of coordinates. We firstly classify the affine factorable surfaces…

Extends Choi-Wang inequality to Li-Xia affine connections.

New invariants found for mappings between non-symmetric affine spaces.

New non-quadratic hypersurfaces found for higher dimensions.

The paper proves there are many Lagrangian fillings for Legendrian links of affine type.

The paper studies a new class of affine maximal surfaces with singularities.

We find that for any n-dimensional, compact, convex subset K of R^{n+1} there is an affinely-spherical hypersurface M in R^{n+1} with center at the relative interior of K, such that the disjoint union of M and K is the boundary of an (n+1)-dimensional, compact, convex set. This so-called affine hemisphere M is uniquely…

We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new in…

We give characterizations of affine transformations and affine vector fields in terms of the spray. By utilizing the Jacobi type equation that characterizes affine vector fields, we prove some rigidity theorems of affine vector fields on compact or forward complete non-compact Finsler manifolds with non-positive total …

Paper finds inequalities for convex domains in hyperbolic space.

Dual Bayesian Affine Estimators for Wiener-type state-space models

Using the Blaschke-Berwald metric and the affine shape operator of a hypersurface M in the (n+1)-dimensional real affine space we can define some generalized curvature tensor named the Opozda-Verstraelen affine curvature tensor. In this paper we determine curvature conditions of pseudosymmetry type expressed by this te…

New non-quadratic Euclidean complete affine maximal type hypersurfaces found for N≥2, θ∈(0,(N-1)/N].

Uniform proof of property R_infinity for specific Artin-Tits groups.

We consider stochastic partial differential equations appearing as Markovian lifts of matrix valued (affine) Volterra type processes from the point of view of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce in particular Volterra Wishart processes with fractional kernels and values in the con…

Geometric models for algebraic suspensions using affine deformation spaces.

We consider equitorsion second type almost geodesic mappings of a non-symmetric affine connection space in this article. Using different computational methods, we obtained some invariants of these mappings. Last generalized Thomas projective parameter and Weyl projective tensor as invariants of a second type almost geo…

As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and various lifting procedures, and to the geometrical construction of Lagrangian-type dyn…

Since the work of Henri Cartan finite dimensional Riemannian symmetric spaces are an important subject of mathematical interest. They are related in a natural way to semisimple Lie groups. In this work we introduce and study their infinite dimensional generalization: Affine Kac-Moody symmetric spaces. Affine Kac-Moody …

In this paper we study time-inhomogeneous affine processes beyond the common assumption of stochastic continuity. In this setting times of jumps can be both inaccessible and predictable. To this end we develop a general theory of finite dimensional affine semimartingales under very weak assumptions. We show that the co…

Constructs positive energy representations from Toda equations Stokes data.

The study classifies certain types of incomplete surfaces with low curvature.

Study finds a measure for sponge components of Lalley-Gatzouras type.

The paper establishes inequalities for convex curves and applies them to lattice point estimates.

We investigate in detail the class of Euclidean affine Kac-Moody symmetric spaces and their orthogonal symmetric affine Kac-Moody algebras (OSAKAs). These spaces are the only class of Kac-Moody symmetric spaces, that is not directly derived from affine Kac-Moody algebras in the classical sense.

We study affine Jacobi structures on an affine bundle $π:A\to M$, i.e. Jacobi brackets that close on affine functions. We prove that there is a one-to-one correspondence between affine Jacobi structures on $A$ and Lie algebroid structures on the vector bundle $A^+=\bigcup_{p\in M}Aff(A_p,\R)$ of affine functionals. Som…

We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type $\mathcal{A}$ surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and examine the structure of the associated moduli space. For Type $\mathcal{…

We examine the local geometry of affine surfaces which are locally symmetric. There are 6 non-isomorphic local geometries. We realize these examples as Type A, Type B, and Type C geometries using a result of Opozda and classify the relevant geometries up to linear isomorphism. We examine the geodesic structures in this…

This paper aims to develop basic theory for the dual Orlicz $L_φ$ affine and geominimal surface areas for star bodies, which belong to the recent dual Orlicz-Brunn-Minkowski theory for star bodies. Basic properties for these new affine invariants will be provided. Moreover, related Orlicz affine isoperimetric inequalit…

We classify tube domains in $\mathbb C^{n+1}$ ($n\ge 1$) with affinely homogeneous base of their boundary and a.) with positive definite Levi form and b.) with Lorentzian type Levi form and affine isotropy of dimension at least $\frac{(n-2)(n-3)}2$.

Abstract mathematical formulas for statistical structures and curvatures.

Motivated by control-affine systems in optimal control theory, we introduce the notion of a point-affine distribution on a manifold X - i.e., an affine distribution F together with a distinguished vector field contained in F. We compute local invariants for point-affine distributions of constant type when dim(X)=n, ran…

Constructs generalized Frobenius manifolds for specific Weyl groups.

Affine hamiltonians are defined in the paper and their study is based especially on the fact that in the hyperregular case they are dual objects of lagrangians defined on affine bundles, by mean of natural Legendre maps. The variational problems for affine hamiltonians and lagrangians of order $k\geq 2$ are studied, re…

If $\mathcal{M}=(M,\nabla)$ is an affine surface, let $\mathcal{Q}(\mathcal{M}):=\ker(\mathcal{H}+\frac1{m-1}ρ_s)$ be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let $\tilde{\mathcal{M}}=(M,\tilde\nabla)$ be another affine structure on $M$ which is strongly projectively flat. We sh…

Book explores infinite translation surfaces, challenging traditional geometry.

New Poisson manifolds created over 2-tori.

Study decomposes geometric surfaces, finding special curves.

For a closed real algebraic plane affine curve dividing its complexification and equipped with a complex orientation, the Whitney number is expressed in terms of behavior of its complexification at infinity.

Study on faithfulness of Burau representation for Artin-Tits groups.

Using a quiver algebra of a cyclic quiver, we construct a faithful categorical action of the extended braid group of affine type A on its bounded homotopy category of finitely generated projective modules. The algebra is trigraded and we identify the trigraded dimensions of the space of morphisms of this category with …

It is shown that the diffeomorphism type of the complement to a real space line arrangement in any dimensional affine ambient space is determined only by the number of lines and the data on multiple points.

Holomorphic structures on Oeljeklaus-Toma manifolds are shown to be locally homogeneous.

New method constructs translationally equivariant hyperbolic affine spheres.

Extend classical theory of affine processes to path-dependent setting

This is the third companion paper of arXiv:1601.03586. When a gauge theory has a flavor symmetry group, we construct a partial resolution of the Coulomb branch as a variant of the definition. We identify the partial resolution with a partial resolution of a generalized slice in the affine Grassmannian, Hilbert scheme o…

Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimensions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures for systems with 2 s…