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.     

Solvability Criterion for the Constrained Discrete Lyapunov and Riccati Equations

Conditions for the solvability of the discrete Lyapunov and the discrete Riccati equations subject to linear equality constraints are derived. These problems arise naturally in the context of output min-max robust control. It is shown that the following problems are equivalent to one another: (a) the solvability of the constrained discrete Riccati equation; and (b) the existence of a feedback gain that guarantees the solvability of the constrained discrete Lyapunov equation of the resulting closed loop. A simple criterion for the existence of a solution to both problems is presented. These problems are shown to be related to the discrete positive real property.     

Projective Generalizations of Lelieuvre's Formula

Generalizations of the classical affine Lelieuvre's formula to surfaces in projective three-dimensional space and to hypersurfaces in multi-dimensional projective space are given. A discrete version of the projective Lelieuvre's formula is presented too.     

Standing wave solutions for the discrete nonlinear Schrödinger equations with indefinite sign subquadratic potentials

We investigate the discrete nonlinear Schrödinger equations with indefinite sign subquadratic potentials. Making use of the critical point theory, we obtain a new result concerning the existence of a standing wave solution.     

Indefinite LQ optimal control with cross term for discrete‐time uncertain systems

Recently, there has been an increasing interest in the study on uncertain optimal control problems. In this paper, a linear quadratic (LQ) optimal control with cross term for discrete‐time uncertain systems is considered, whereas the weighting matrices in the cost function are allowed to be indefinite. Firstly, a recurrence equation for the problem is presented based on Bellman's principle of optimality in dynamic programming. Then, a necessary condition for the existence of an optimal linear state feedback control of the indefinite LQ problem is given by the recurrence equation. Moreover, a sufficient condition of well‐posedness for the indefinite LQ problem is presented by introducing a linear matrix inequality (LMI) condition. Furthermore, it is shown that the well‐posedness of the indefinite LQ problem, the solvability of the indefinite LQ problem, the LMI condition, and the solvability of the constrained difference equation are equivalent to each other. Finally, an example is presented to illustrate the results obtained.     

Exterior Difference System on Hypercubic Lattice

Based on the exterior differential calculus, we define the exterior difference system on the hypercubic lattice. The pairing formula, bracket, contract operator and Lie derivative operator for this system are also constructed. The discrete H. Cartan formulas are obtained too. After those preparations, we prove the discrete Frobenius Theorem, discrete retraction theorem and discrete Cartan–Kähler theorem.     

Isotropic affine spheres

In this paper, we study affine spheres which are isotropic and we obtain a complete classification. In particular, we show that all such affine spheres are hyperbolic affine spheres, isometric with SL(3 , R ) / SO(3), SL(3 , C ) / SU(3), SU*(6) / Sp(3) or E6 (-26) /F4 .     

A shorter proof for an explicit formula for discrete logarithms in finite fields

A shorter proof for an explicit formula for discrete logarithms in finite fields is given.     

Affine surfaces with constant affine curvatures

In this paper, we study affine non-degenerate Blaschke immersions from a surface M in ³. We will assume that M has constant affine curvature and constant affine mean curvature, i.e. both the determinant and the trace of the shape operator are constant. Clearly, affine spheres satisfy both these conditions. In this paper, we completely classify the affine surfaces with constant affine curvature and constant affine mean curvature, which are not affine spheres.Research Assistant of the National Fund for Scientific Research (Belgium).     

Weierstraß-type representation of affine spheres

Affine spheres are discussed in the context of loop groups. We show that every affine sphere can be obtained by solving two ordinary differential equations followed by an application of a generalized Birkhoff Decomposition Theorem (which we proof in the Appendix). A geometric interpretation of the coefficients of the ODE is given. Finally the method is applied to construct all ruled surfaces. Most of this work was done while the second named author was visiting the University of Kansas.     

A characterization of affine spheres in R 4

A surface immersed in R ⁴ is called a proper affine sphere if the position vector belongs to the affine normal plane. We classify proper affine spheres with ?^?? g ^??=0 whose affine mean curvature vector has constant length. Moreover, we find some concrete examples of affine spheres which are not affine umbilical.     

Détermination géométrique de la systole des groupes de triangles

In this Note we give an explicit formula for the length of the shortest geodesic loop for hyperbolic spheres with three singularities of order greater than 3.     

On locally strongly convex affine hypersurfaces with parallel cubic form. Part I

In this paper, we study locally strongly convex affine hypersurfaces of Rn+1 that have parallel cubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschke metric; it is known that they are affine spheres. In dimension n?7 we give a complete classification of such hypersurfaces; in particular, we present new examples of affine spheres.     

Discrete series characters for affine Hecke algebras and their formal degrees

We introduce the generic central character of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types E_n⁽¹⁾{E_{n}^{(1)}} , n=6, 7, 8) with arbitrary positive parameters and we prove an explicit product formula for their formal degrees (in all cases).     

An Extremal Class of Three-Dimensional Hyperbolic Affine Spheres

In analogy to an inequality of Chen [2], Scharlach and co-workers [7] have found a new, optimal inequality for (equi-) affine spheres. We classify those three-dimensional hyperbolic affine spheres for which the corresponding equality is assumed. This complements the classification of the elliptic case [3].     

Infinitely many stationary solutions of discrete vector nonlinear Schrödinger equation with symmetry

In this paper we study the existence of stationary solutions for the following discrete vector nonlinear Schrödinger equation

     

A Dissipation-Preserving Integrator for Damped Oscillatory Hamiltonian Systems

In this paper, based on discrete gradient, a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established. The solution of this system is a damped nonlinear oscillator. Basically, lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach. The new integrator gives a discrete analogue of the dissipation property of the original system. Meanwhile, since the integrator is based on the variation-of-constants formula for oscillatory systems, it preserves the oscillatory structure of the system. Some properties of the new integrator are derived. The convergence is analyzed for the implicit iterations based on the discrete gradient integrator, and it turns out that the convergence of the implicit iterations based on the new integrator is independent of $\|M\|$, where $M$ governs the main oscillation of the system and usually $\|M\|\gg1$. This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system. Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature.     

Characters of Simple Bounded Modules over an Untwisted Affine Lie Algebra

After V. Chari and A. Pressley, a simple integrable module with finite-dimensional weight spaces over an affine Lie algebra is either a standard module (highest or lowest weight), in which case its formal character is given by the famous Weyl–Kac formula, or a subquotient of a tensor product of loop modules. In this paper we compute formal characters of generic simple integrable modules of the latter type.     

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Pseudoframes for subspaces have been recently introduced by Li and Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence. In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for discrete subsets of ℝ ^d by employing a certain generalized Beurling density. We present several properties of Beurling dimension including a comparison with other notions of dimension showing, for instance, that our notion includes the mass dimension as a special case. Then we prove that Gabor pseudoframes for affine subspaces satisfy a certain Homogeneous Approximation Property, which implies invariance under time–frequency shifts of an approximation by elements from the pseudoframe. The main result of this paper is a classification of Gabor pseudoframes for affine subspaces by means of the Beurling dimension of their sets of parameters. This provides us, in particular, with a Nyquist dimension which separates sets of parameters of pseudoframes from those of non-pseudoframes and which links a fixed value to sets of parameters of pseudo-Riesz sequences. These results are even new for the special case of Gabor frames for an affine subspace.