On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves

We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.     

Regular simplices and lower volume bounds for hyperbolicn-manifolds

For an-dimensional compact hyperbolic manifoldM ⁿ a new lower volume bound is presented. The estimate depends on the volume of a hyperbolic regularn-simplex of edge length equal to twice the in-radius ofM ⁿ. Its proof relies upon local density bounds for hyperbolic sphere packings.     

O the volume growth of the hyperbolic regular $$\varvec{}$$-simplex

In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the n-dimensional volume of a regular simplex and the \((n-1)\)-dimensional volume of its facets. In addition to the methods of U. Haagerup and M. Munkholm we use a third volume form based on the hyperbolic orthogonal coordinates of a body. In the case of the ideal, regular simplex our upper bound gives the best known upper bound. On the other hand, also in the ideal case our general lower bound, improved the best known one for \(n=3\).     

Eigenvalue estimates for minimal hypersurfaces in hyperbolic space

Recently Candel [A. Candel, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc. 359 (2007) 3567-3575] proved that if M is a simply-connected stable minimal surface isometrically immersed in H³, then the first eigenvalue of M satisfies 1/4?λ(M)?4/3 and he asked whether the bound is sharp and gave an example such that the lower bound is attained. In this note, we prove that the upper bound can never be attained. Also we extend the result by proving that if M is compact stable minimal hypersurface isometrically immersed in Hn+1 where n?3 such that its smooth Yamabe invariant is negative, then (n−1)/4?λ(M)?n²(n−2)/(7n−6).     

A remark on the mean curvature of a graph-like hypersurface in hyperbolic space

In this paper we investigate the mean curvature H of a radial graph in hyperbolic space Hn+1. We obtain an integral inequality for H, and find that the lower limit of H at infinity is less than or equal to 1 and the upper limit of H at infinity is more than or equal to −1. As a byproduct we get a relation between the n-dimensional volume of a bounded domain in an n-dimensional hyperbolic space and the (n−1)-dimensional volume of its boundary. We also sharpen the main result of a paper by P.-A. Nitsche dealing with the existence and uniqueness of graph-like prescribed mean curvature hypersurfaces in hyperbolic space.     

Upper and Lower Bounds on Resonances for Manifolds Hyperbolic Near Infinity

For a conformally compact manifold that is hyperbolic near infinity and of dimension n + 1, we complete the proof of the optimal O(r ⁿ⁺¹) upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an r ⁿ⁺¹ lower bound on the counting function for scattering poles.     

Hyperbolic Volume of Manifolds with Geodesic Boundary and Orthospectra

In this paper we describe a function F _n : R ₊ → R ₊ such that for any hyperbolic n-manifold M with totally geodesic boundary ${\partial M \neq \emptyset}In this paper we describe a function F _n : R ₊ → R ₊ such that for any hyperbolic n-manifold M with totally geodesic boundary ?M 1 ?{\partial M \neq \emptyset} , the volume of M is equal to the sum of the values of F _n on the orthospectrum of M. We derive an integral formula for F _n in terms of elementary functions. We use this to give a lower bound for the volume of a hyperbolic n-manifold with totally geodesic boundary in terms of the area of the boundary.     

Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré-Mel’nikov method

We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic-hyperbolic type, having a homoclinic orbit. More precisely, we consider an (n + 2)-degree-of-freedom near integrable Hamiltonian with n centers and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the center manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel’nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound) and can be approximated by a trigonometric polynomial (which gives an upper bound).     

Asymptotics of convex sets in Euclidean and hyperbolic spaces

We study convex sets C of finite (but non-zero) volume in Hn and En. We show that the intersection C_∞ of any such set with the ideal boundary of Hn has Minkowski (and thus Hausdorff) dimension of at most (n−1)/2, and this bound is sharp, at least in some dimensions n. We also show a sharp bound when C_∞ is a smooth submanifold of _∂∞Hn. In the hyperbolic case, we show that for any k?(n−1)/2 there is a bounded section S of C through any prescribed point p, and we show an upper bound on the radius of the ball centered at p containing such a section. We show similar bounds for sections through the origin of a convex body in En, and give asymptotic estimates as 1?k?n.     

A lower bound for the volumes of complex hyperbolic orbifolds

In this paper, an explicit formula for a lower bound for the volumes of complex hyperbolic orbifolds depending on the dimension and the maximal order of torsion elements in their fundamental groups is obtained. This generalizes Adeboye (Pac. J. Math. 237:1?C19 2008, Theorem 1) to the setting of complex hyperbolic spaces.     

Thickness of a Simplex Whose Edge Lengths Fall Within a Prescribed Range

The existence of a simplex in hyperbolic, Euclidean, or spherical space whose edge lengths lie within a prescribed range has been studied in a few papers. The question considered there can be put this way. Let ? be a positive number (with a natural upper bound in the spherical case). What is the minimum number λ = λ _n (?) such that an n-simplex exists in the n-dimensional space whenever its edge lengths fall within [λ, ?]? The papers derive exact expressions for λ _n (?) in each of the three geometries. They do not tell, however, how 'thick' at least the simplex will be depending on ? and a lower bound µ ε [ λ, ?] of the edge lengths. Such a dependence clearly exists. One can easily guess for instance that, when µ is close to ?, the simplex cannot be too thin compared to the regular simplex for which µ =?. We define here a suitable thickness of a convex body and then estimate from below this thickness of a simplex in terms of n, ?, and µ for each of the three geometries.     

Computing the determinantal representations of hyperbolic forms

The numerical range of an n × n matrix is determined by an n degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an n degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus g = 1. We reformulate the Fiedler-Helton-Vinnikov formulae for the genus g = 0, 1, and present an elementary computation of the reformulation. Several examples are provided for computing the real symmetric matrices using the reformulation.     

Designs having the parameters of projective and affine spaces

Two constructions are described that yield an improved lower bound for the number of 2-designs with the parameters of PG _d (n, q), and a lower bound for the number of resolved 2-designs with the parameters of AG _d (n, q).     

Discreteness criteria based on a test map in PU(n, 1)

The discreteness of isometry groups in complex hyperbolic space is a fundamental problem. In this paper, the discreteness criteria of a n-dimensional subgroup G of SU(n,1) are investigated by using a test map which may not be in G.     

Bounds on the number of numerical semigroups of a given genus

Lower and upper bounds are given for the number n_g of numerical semigroups of genus g. The lower bound is the first known lower bound while the upper bound significantly improves the only known bound given by the Catalan numbers. In a previous work the sequence n_g is conjectured to behave asymptotically as the Fibonacci numbers. The lower bound proved in this work is related to the Fibonacci numbers and so the result seems to be in the direction to prove the conjecture. The method used is based on an accurate analysis of the tree of numerical semigroups and of the number of descendants of the descendants of each node depending on the number of descendants of the node itself.     

The equivariant LS-category of polar actions

We will provide a lower bound for arbitrary proper actions in terms of the stratification by orbit types, and an upper bound for proper polar actions in terms of the equivariant LS-category of its generalized Weyl group. As an application we reprove a theorem of Singhof that determines the classical Lusternik-Schnirelmann category for U(n) and SU(n).     

Aufbau der hyperbolischen Geometrie aus dem Geradenorthogonalitätsbegriff

We present an axiom system for plane hyperbolic geometry in a language with lines as the only individual variables and the binary relation of line-perpendicularity as the only primitive notion. It was made possible by results obtained by K. List and H.L. Skala. A similar axiomatization is possible for n-dimensional hyperbolic geometry with n≥4. We also point out that plane hyperbolic geometry admits a AE-axiomatization in terms of line-perpendicularity alone, an axiomatization we could not find.     

On the moduli space of quadruples of points in the boundary of complex hyperbolic space

We consider the space $ \mathcal{M} $ of ordered quadruples of distinct points in the boundary of complex hyperbolic n-space, $ H_\mathbb{C}^n $ , up to its holomorphic isometry group PU(n, 1): One of the important problems in complex hyperbolic geometry is to construct and describe a moduli space for $ \mathcal{M} $ . For n = 2, this problem was considered by Falbel, Parker, and Platis. The main purpose of this paper is to construct a moduli space for $ \mathcal{M} $ for any dimension n ≥ 1.     

Two-dimensional minimax Latin hypercube designs

We investigate minimax Latin hypercube designs in two dimensions for several distance measures. For the ?^∞-distance we are able to construct minimax Latin hypercube designs of n points, and to determine the minimal covering radius, for all n. For the ?¹-distance we have a lower bound for the covering radius, and a construction of minimax Latin hypercube designs for (infinitely) many values of n. We conjecture that the obtained lower bound is attained, except for a few small (known) values of n. For the ?²-distance we have generated minimax solutions up to n=27 by an exhaustive search method. The latter Latin hypercube designs are included in the website www.spacefillingdesigns.nl.     

Asymptotic Estimates on the Time Derivative of Φ-entropy on Riemannian Manifolds

In this note,we obtain an asymptotic estimate for the time derivative of the Φ-entropy in terms of the lower bound of the Bakry–Emery Γ2 curvature.In the cases of hyperbolic space and the Heisenberg group(more generally,the nilpotent Lie group of rank two),we show that the time derivative of the Φ-entropy is non-increasing and concave in time t,also we get a sharp asymptotic bound for the time derivative of the entropy in these cases.