The Cauchy problem for indefinite improper affine spheres and their Hessian equation

We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we classify the helicoidal indefinite improper affine spheres and find a new family with geodesically complete non-flat affine metric. Moreover, we present interesting examples with singular curves and isolated singularities.     

A geometric representation of improper indefinite affine spheres with singularities

Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefinite improper affine sphere in ${\mathbb {R}^3}$ is the generalized distance of a pair of planar curves. Considering this representation, the singularity set of the improper affine sphere corresponds to the area evolute of the pair of curves, and this fact allows us to describe a clear geometric picture of the former. Other symmetry sets of the pair of curves, like the affine area symmetry set and the affine envelope symmetry set can be also used to describe geometric properties of the improper affine sphere.     

Representation formula for discrete indefinite affine spheres

We present a representation formula for discrete indefinite affine spheres via loop group factorizations. This formula is derived from the Birkhoff decomposition of loop groups associated with discrete indefinite affine spheres. In particular we show that a discrete indefinite improper affine sphere can be constructed from two discrete plane curves.     

Left-symmetric algebras and homogeneous improper affine spheres

The nonzero level sets in n-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the nth power of the function. The exponentials of the characteristic polynomials of certain left-symmetric algebras yield examples of such functions whose level sets are analogues of the generalized Cayley hypersurface of Eastwood–Ezhov. There are found purely algebraic conditions sufficient for the characteristic polynomial of the left-symmetric algebra to have the desired properties. Precisely, it suffices that the algebra has triangularizable left multiplication operators and the trace of the right multiplication is a Koszul form for which right multiplication by the dual idempotent is projection along its kernel, which equals the derived Lie subalgebra of the left-symmetric algebra.     

Singularities of convex improper affine maps

In this paper we consider convex improper affine maps of the three-dimensional affine space and classify their singularities. The main tool developed is a generating family with properties that closely resembles the area function for non-convex improper affine maps.     

An affine interpretation of Bäcklund transformations

The paper is devoted to an affine interpretation of Bäcklundmaps (Bäcklund transformations are a particular case of Bäcklund maps) for second order differential equations with unknown function of two arguments. Note that up to now there are no papers where Bäcklund transformations are interpreted as transformations of surfaces in a space other than Euclidean space. In this paper, we restrict our considerations to the case of so-called Bäcklund maps of class 1. The solutions of a differential equation are represented as surfaces of an affine space with induced connection determining a representation of zero curvature. We show that, in the case when a second order partial differential equation admits a Bäcklund map of class 1, for each solution of the equation there is a congruence of straight lines in an affine space formed by the tangents to the affine image of the solution. This congruence is an affine analog of a parabolic congruence in Euclidean space. The Bäcklund map can be interpreted as a transformation of surfaces of an affine space under which the affine image of a solution of the differential equation is mapped into a particular boundary surface of the congruence.     

The Wolff potential estimate for solutions to elliptic equations with signed data

In this note, we obtain sharp bounds for the Green’s function of the linearized Monge–Ampère operators associated to convex functions with either Hessian determinant bounded away from zero and infinity or Monge–Ampère measure satisfying a doubling condition. Our result is an affine invariant version of the classical result of Littman–Stampacchia–Weinberger for uniformly elliptic operators in divergence form. We also obtain the L ^p integrability for the gradient of the Green’s function in two dimensions. As an application, we obtain a removable singularity result for the linearized Monge–Ampère equation.     

Lyapunov exponents and stationary solutions for affine stochastic delay equations

A certain class of affine delay equations is considered. Two cases for the forcingfunction M are treated: M locally integrable deterministic, and M a random process with stationaryincrements. The Lyapunov spectrum of the homogeneous equation is used to decompose the state spaceinto finite-dimensional and finite-codimensional subspaces. Using a suitable variation of constants representation, formulas for the projection of the trajectories onto the above subspaces are obtained. If the homogeneous equation is hyperbolic and M has stationary increments, existence and uniqueness of a stationary solution for the affine stochastic delay equation is proved. The existence of Lyapunov exponents for the affine equation and their dependence on initial conditions is als studied.     

Improper affine spheres in ℝ3 and ℂ3

In this paper we determined all polynomial improper affine spheres in ?³ and ?³ without any curvature conditions. We have gotten the following result: A polynomial improper affine sphere in ?³ or ?³ is either a quadric or the ruled surface z = xy +?(x).     

Improper affine maps

We study a class of improper affine spheres with singularities called improper affine maps. New examples of genus 0 and 1 are described and the classification of complete improper affine maps with at most two embedded ends will be given.Mathematics Subject Classification (2000): 53A15, 53A35Research partially supported by MCYT-FEDER Grant No. BFM2001-3318 and Junta de Andalucía CEC: FQM0804.     

Modeling and passivity analysis of nonholonomic Hamiltonian systems with rheonomous affine constraints

This paper is devoted to modeling and theoretical analysis of Hamiltonian systems subject to nonholonomic rheonomous affine constraints. We first define rheonomous affine constraints and explain geometric representation of them. Next, a complete nonholonomicity condition for the rheonomous affine constraints is developed in terms of the rheonomous bracket. Then, the nonholonomic Hamiltonian system with rheonomous affine constraints (NHSRAC) is derived via a transformation and model reduction for the expanded Hamiltonian system defined on the expanded phase space. After that, we investigate passivity of the NHSRAC with the control input term and the output equation. Finally, in order to confirm the application potentiality of our new results, we show an example, a radius-variable ball on rotating table with a time-varying angular velocity.     

Higher order curvature information and its application in a modified diagonal Secant method

ABSTRACT

A secant equation (quasi-Newton) has one of the most important rule to find an optimal solution in nonlinear optimization. Curvature information must satisfy the usual secant equation to ensure positive definiteness of the Hessian approximation. In this work, we present a new diagonal updating to improve the Hessian approximation with a modifying weak secant equation for the diagonal quasi-Newton (DQN) method. The gradient and function evaluation are utilized to obtain a new weak secant equation and achieve a higher order accuracy in curvature information in the proposed method. Modified DQN methods based on the modified weak secant equation are globally convergent. Extended numerical results indicate the advantages of modified DQN methods over the usual ones and some classical conjugate gradient methods.     

Gevrey regularity of subelliptic Monge–Ampère equations in the plane

In this paper, we establish the Gevrey regularity of solutions for a class of degenerate Monge–Ampère equations in the plane. Under the assumptions that one principal entry of the Hessian is strictly positive and the coefficient of the equation is degenerate with appropriately finite type degeneracy, we prove that the solution of the degenerate Monge–Ampère equation will be smooth in Gevrey classes.     

Blaschke Dupin hypersurfaces and equiaffine tubes

We give constructions of Blaschke Dupin hypersurfaces and a Blaschke isoparametric ones in terms of the notion of an equiaffine tube. In particular, the construction of Blaschke isoparametric hypersurfaces includes the Calabi-type composition of improper affine spheres (or an improper one and a proper one).     

On the exterior Dirichlet problem for Hessian quotient equations

In this paper, we establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for Hessian quotient equations with prescribed asymptotic behavior at infinity. This extends the previous related results on the Monge–Ampère equations and on the Hessian equations, and rearranges them in a systematic way. Based on the Perron's method, the main ingredient of this paper is to construct some appropriate subsolutions of the Hessian quotient equation, which is realized by introducing some new quantities about the elementary symmetric polynomials and using them to analyze the corresponding ordinary differential equation related to the generalized radially symmetric subsolutions of the original equation.     

Graphical Calculus for the Double Affine Q-Dependent Braid Group

In this paper, we present a straightforward pictorial representation of the double affine Hecke algebra (DAHA) which enables us to translate the abstract algebraic structure of a DAHA into an intuitive graphical calculus suitable for a physics audience. Initially, we define the larger double affine Q-dependent braid group. This group is constructed by appending to the braid group a set of operators Q _i , before extending it to an affine Q-dependent braid group. We show specifically that the elliptic braid group and the DAHA can be obtained as quotient groups. Complementing this, we present a pictorial representation of the double affine Q-dependent braid group based on ribbons living in a toroid. We show that in this pictorial representation, we can fully describe any DAHA. Specifically, we graphically describe the parameter q upon which this algebra is dependent and show that in this particular representation q corresponds to a twist in the ribbon.     

A variational theory of the Hessian equation

By studying a negative gradient flow of certain Hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous Hessian equations. To achieve this we derive uniform, first‐ and second‐order a priori estimates for the elliptic and parabolic Hessian equations. Our results generalize well‐known results for semilinear elliptic equations and the Monge‐Ampère equation. © 2001 John Wiley & Sons, Inc.     

Complex Hessian Operator and Lelong Number for Unbounded m-subharmonic Functions

m-subharmonic functions are the right class of admissible solutions to the complex Hessian equation. In this paper, we generalize the definition of the complex Hessian operator to some unbounded m-subharmonic functions, and we prove that the complex Hessian operator is continuous on the monotonically decreasing sequences of m-subharmonic functions. Moreover we establish the Lelong-Jensen type formula and introduce the Lelong number for m-subharmonic functions. A useful inequality for the mixed Hessian operator is showed.     

Smooth Convex Approximation to the Maximum Eigenvalue Function

In this paper, we consider smooth convex approximations to the maximum eigenvalue function. To make it applicable to a wide class of applications, the study is conducted on the composite function of the maximum eigenvalue function and a linear operator mapping ^m to , the space of n-by-n symmetric matrices. The composite function in turn is the natural objective function of minimizing the maximum eigenvalue function over an affine space in . This leads to a sequence of smooth convex minimization problems governed by a smoothing parameter. As the parameter goes to zero, the original problem is recovered. We then develop a computable Hessian formula of the smooth convex functions, matrix representation of the Hessian, and study the regularity conditions which guarantee the nonsingularity of the Hessian matrices. The study on the well-posedness of the smooth convex function leads to a regularization method which is globally convergent.     

From Lie algebras of vector fields to algebraic group actions

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → Der_K(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.