On the Generalized Hurwitz Equation and the Baragar–Umeda Equation

We consider the generalized Hurwitz equation \({a_1x_1^2 + \cdots + a_nx_n^2 = dx_1 \cdots x_n - k}\) and the Baragar–Umeda equation \({ax^2 + by^2 + cz^2 = dxyz + e}\) for solvability in integers.     

The number of integer points on Vinogradov’s quadric

An asymptotic formula is obtained for the number of integer solutions of bounded height on Vinogradov’s quadric. Two leading terms are determined, and a strong estimate for the error term is given.     

Harmonic morphisms of graphs and the Riemann–Hurwitz theorem

Several versions of the Riemann–Hurwitz theorem for branched coverings of graphs are presented. A larger class of graphs which may have not only multiple edges but also loops and darts is considered. This makes it possible to render the class of graphs closed with respect to morphisms arising as quotient maps for actions of finite groups.     

On the existence of ɛ-fixed points

In this paper we prove some approximate fixed point theorems which extend, in a broad sense, analogous results obtained by Brânzei, Morgan, Scalzo and Tijs in 2003. By assuming also the weak demiclosedness property we state two fixed point theorems. Moreover, we study the existence of ?-Nash equilibria.     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

On the centre and the set of ω-limit points of continuous maps on dendrites

It is well known that for dynamical systems generated by continuous maps of a graph, the centre of the dynamical system is a subset of the set of ω-limit points.In this paper we provide an example of a continuous self-map f₁ of a dendrite such that ω(f₁) is a proper subset of C(f₁).The second example is a continuous self-map f₂ of a dendrite having a strictly increasing sequence of ω-limit sets which is not contained in any maximal one. Again, this is impossible for continuous maps on graphs.     

On the number of points of bounded height¶on arithmetic projective spaces

Let K be a number field and its ring of integers. Let be a Hermitian vector bundle over . In the first part of this paper we estimate the number of points of bounded height in (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d>1 in of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of ℙ ^N _K blown up at a linear subspace of codimension two. Received: 20 February 1998 / Revised version: 9 November 1998     

On the stability of Mañé critical hypersurfaces

We construct examples of Tonelli Hamiltonians on ${\mathbb{T}^n}$ (for any n ≥ 2) such that the hypersurfaces corresponding to the Mañé critical value are stable (i.e. geodesible). We also provide a criterion for instability in terms of closed orbits in free homotopy classes and we show that any stable energy level of a Tonelli Hamiltonian must contain a closed orbit.     

Asymptotic behavior of integer programming and the stability of the Castelnuovo–Mumford regularity

Mathematical Programming - The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer...     

On the non-round points of the boundary of the numerical range ∗

Let A be a bounded linear operator acting on a Hilbert space. It is well known (Donoghue, 1957) that comer points of the numerical range W(A) are eigenvalues of A. Recently (1995), this result was generalized by Hiibner who showed that points of infinite curvature on the boundary of W(A) lie in the spectrum of A. Hübner also conjectured that all such points are either corner points or lie in the essential spectrum of A. In this paper, we give a short proof of this conjecture.     

On fixed points of algebraic actions on ℂ n

It is shown that the action regarded for a rather long time by experts as a possible example disproving the conjecture on the existence of fixed points for reductive algebraic group actions on affine spaces is not an action on an affine variety, and therefore provides no example of this kind. Moreover, it is shown that the actions naturally related to the original one provide no examples of this kind as well. Supported by CRDF grant RM1-206 and INTAS grant INTAS-OPEN-97-1570. Moscow Independent University. Moscow Institute of Electronics and Mathematics. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 34, No. 1, pp. 41–50, January–March, 2000 Translated by V. L. Popov     

On the invariant density of the random β-transformation

We construct a Lebesgue measure preserving natural extension of a skew product system related to the random β-transformation K _β . This allows us to give a formula for the density of the absolutely continuous invariant probability measure of K _β , answering a question of Dajani and de Vries, and also to evaluate some estimates on the typical branching rate of the set of β-expansions of a real number.     

On the distribution of periodic points for β-shifts

It is shown that for -shifts the periodic points are uniformly distributed with respect to the unique measure of maximal entropy, and that the invariant measures with support on a single periodic orbit are dense in the space of all invariant measures. Dedicated to Prof. Dr. E. Hlawka on the occasion of his 60th birthday     

On the geometry of maximal spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime

In this paper, we establish new characterizations of totally geodesic spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime, which is supposed to obey the null convergence condition. As applications, we get nonparametric results concerning to entire maximal vertical graphs in a such ambient spacetime. Proceeding, we obtain a lower estimate of the index of relative nullity of complete r-maximal spacelike hypersurfaces immersed in Robertson–Walker spacetimes of constant sectional curvature. In particular, we prove a sort of weak extension of the classical Calabi–Bernstein theorem.     

On the multiplicity of eigenvalues of the Sturm–Liouville problem on graphs

Bounds for the multiplicity of the eigenvalues of the Sturm–Liouville problem on a graph, which are valid for a wide class of consistency (transmission) conditions at the vertices of the graph, are given. The multiplicities are estimated using the topological characteristics of the graph. In the framework of the notions that we use, the bounds turn out to be exact.