Multiple brake orbits in bounded convex symmetric domains

In this paper, we prove that there exist at least two geometrically distinct brake orbits in every bounded convex symmetric domain in .     

Multiple brake orbits of even Hamiltonian systems on torus

In this paper,we establish a relationship between the Morse index at rest points in the saddle point reduction and the brake-orbit-type Maslov index at corresponding brake orbits.As an application,we give a criterion to find brake orbits which are contractible and start at {0}×T~n■T~(2 n) for even Hamiltonian on T~(2 n) by the methods of the Maslov-index theory and a critical point theorem formulated by Bartsch and Wang(1997).Explicitly,if all trivial solutions of a Hamiltonian are nondegenerate in the brake orbit boundary case,there are at least max{i_(L_0)(z_0)} pairs of nontrivial 1-periodic brake orbits if i_(L_0)(z_0) 0 or at least max{-i_(L_0)(z_0)-n}pairs of nontrivial 1-periodic brake orbits if i_(L_0)(z_0) -n.In the end,we give an example to find brake orbits for certain Hamiltonian via this criterion.     

Shortest curves on convex hypersurfaces

We give a survey of results of the author on the geometry of geodesics and shortest curves on general convex hypersurfaces in spaces with constant curvature. We present qualitatively new theorems, which unite and generalize the main classical results; we give applications to the solution of a number of actual problems of geometry of convex surfaces in the large. We give generalizations of some well-known theorems on geodesics and shortest curves to Riemannian manifolds and nonregular multidimensional convex metrics.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 16, pp. 155–194, 1984.     

Multiple periodic orbits of asymptotically linear autonomous convex Hamiltonian systems

We investigate multiple periodic solutions of convex asymptotically linear autonomous Hamiltonian systems. Our theorem generalizes a result by Mawhin and Willem.     

Periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces

This paper discusses the existence and multiplicity of periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces. We prove that each such energy hypersurface carries at least one symmetric periodic orbit. Under some suitable pinching conditions, we also get an existence result of multiple symmetric periodic orbits.     

Stable closed characteristics on partially symmetric convex hypersurfaces

Let n be a positive integer and P=diag(−I_n−κ,I_κ,−I_n−κ,I_κ) for some integer κ∈[0,n]. In this paper, we prove that for any convex compact smooth hypersurface Σ in R²ⁿ with n?2 there always exists at least one closed characteristic on Σ which possesses at least 2n−4κ Floquet multipliers on the unit circle of the complex plane, provided Σ is P-symmetric, i.e., x∈Σ implies Px∈Σ.     

Shortest paths on convex hypersurfaces of Riemannian spaces

A convex hypersurface in a Riemannian space M^m is part of the boundary of an m-dimensional locally convex set. It is established that there exists an intrinsic metric of such a hypersurface and it has curvature which is bounded below in the sense of A. D. Aleksandrov; curves with bounded variation of rotation in are shortest paths in M^m. For surfaces in R^m these facts are well known; however, the constructions leading to them are in large part inapplicable to spaces M^m. Hence approximations to by smooth equidistant (not necessarily convex) ones and normal polygonal paths, introduced (in the case of R³) by Yu. F. Borisov are used.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 66, pp. 114–132, 1976.     

Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in {\varvec{R}}^{2n}

In this paper, let $n$ be a positive integer and $P=diag(-I_{n-\kappa },I_\kappa ,-I_{n-\kappa },I_\kappa )$ for some integer $\kappa \in [0, n]$ , we prove that for any compact convex hypersurface $\Sigma $ in $\mathbf{R}^{2n}$ with $n\ge 2$ there exist at least two geometrically distinct P-invariant closed characteristics on $\Sigma $ , provided that $\Sigma $ is P-symmetric, i.e., $x\in \Sigma $ implies $Px\in \Sigma $ . This work is shown to extend and unify several earlier works on this subject.     

Resonance identity for symmetric closed characteristics on symmetric convex Hamiltonian energy hypersurfaces and its applications

In this paper, we establish a new resonance identity for symmetric closed characteristics on symmetric compact convex hypersurface Σ   in R²ⁿ${\mathbf{R}}^{2n}$ when there exist only finitely many geometrically distinct symmetric closed characteristics. As its applications, some interesting results about the stability and multiplicity of symmetric closed characteristics are obtained, and also we prove that if Σ   is C^∞$C^{\infty }$-generic, it carries infinitely many symmetric closed characteristics.     

Providing support for decisions based on time series information under conditions of asymmetric loss

In an experiment decision-makers used time series information on the past demand for products to decide on production levels to meet the next period's demand. Either shortages cost more per unit than surpluses or the asymmetry of losses was in the opposite direction. The decision-makers were either: (i) unsupported, (ii) provided with statistical point forecasts of the next period's demand or (iii) asked to estimate probability distributions of next period's demand using the fractile method––the decisions were inferred from these distributions (Decomposition). Providing statistical forecasts led to decisions incurring significantly lower expected costs than those achieved by the unsupported decision makers. However, the decomposition procedure did not significantly reduce expected costs because, contrary to earlier evidence, the fractile method generally led to distributions that were too wide and flat. Decision-makers in both treatments (i) and (ii) also performed significantly better when shortages were more costly than surpluses.     

Affine Kähler hypersurfaces satisfying certain conditions on the curvature tensor

The notion of affine Kähler immersions has been recently introduced by Nomizu-Pinkall-Podestà ([N-Pi-Po]). This work is aimed at giving some results towards the classification of non degenerate affine Kähler hypersurfaces with symmetric and parallel Ricci tensor; this problem generalizes the classical results due to Nomizu-Smyth ([N-S]) in the theory of Kählerian hypersurfaces. In a second section we deal with the case of “semisymmetric” affine Kähler immersions, when the curvature tensor R satisfies R · R = 0 and the Ricci tensor is symmetric, providing a complete classification; for affine Kähler curves we prove that the conditions above are actually equivalent to saying that the immersion is isometric for a suitable Kähler metric in C².     

Elliptic equations on convex domains with nonhomogeneous Dirichlet boundary conditions

In this paper, we will study the differentiability on the boundary of solutions of elliptic non-divergence differential equations on convex domains. The results are divided into two cases: (i) at the boundary points where the blow-up of the domain is not the half-space, if the boundary function is differentiable then the solution is differentiable; (ii) at the boundary points where the blow-up of the domain is the half-space, the differentiability of the solution needs an extra Dini condition for the boundary function. Counterexample is given to show that our results are optimal.     

On the number of P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in R~(2n)

In this paper, let n 2 be an integer, P = diag(-In-κ, Iκ,-In-κ, Iκ) for some integer κ∈ [0, n),and Σ∈ R2 nbe a partially symmetric compact convex hypersurface, i.e., x ∈Σ implies P x ∈Σ. We prove that if Σ is(r, R)-pinched withRr2~(1/2), then there exist at least n- κ geometrically distinct P-symmetric closed characteristics on Σ, as a consequence, Σ carry at least n geometrically distinct P-invariant closed characteristics.