On algebraic points in the plane near smooth curves∗

In the paper, we solve the problem on the number of points with algebraic real coordinates near smooth curve. This question is a natural extension of the problems of number theory connected with integer points in the domains and the rational numbers near curves. The main idea of the proof is based on the metric theory on Diophantine approximations.     

Semibounded local hamiltonians in ℝ4 for a laplacian perturbed on curves with corner points

The Laplace operator in ⁴, under perturbations with support on curves having corner points, is considered in the context of self-adjoint extensions. Some classes of local and semibounded extensions that generate semigroups preserving the positivity are constructed. The relation between these extensions and local Dirichlet forms containing an additional energy form on the curve is established.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 2, pp. 179–199, February, 1996.     

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 linear complexity of the Naor–Reingold sequence with elliptic curves

The Naor–Reingold sequences with elliptic curves are used in cryptography due to their nice construction and good theoretical properties. Here we provide a new bound on the linear complexity of these sequences. Our result improves the previous one obtained by I.E. Shparlinski and J.H. Silverman and holds in more cases.     

On the asymptotics of the rows of the Padé table of analytic functions with logarithmic branch points

For the functions $ f(z) = \sum\nolimits_{n = 0}^\infty {z^{l_n } } /a_n $ , where l _n and a _n are arithmetic progressions and their Padé approximants π _n,m (z; f), we establish an asymptotics of the decrease of the difference f(z) ? π _n,m (z; f) for the case in which z ∈ D = {z: |z| < 1}, m is fixed, and n → ∞. In particular, we obtain proximate orders of decrease of best uniform rational approximations to the functions ln(1 ? z) and arctan z in the disk D _q = {z: |z| ≤ q < 1}.     

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 the α-invariants of cubic surfaces with Eckardt points

In this paper, we show that the αm,2-invariant (introduced by Tian (1991) [27] and (1997) [29]) of a smooth cubic surface with Eckardt points is strictly bigger than . This can be used to simplify Tian's original proof of the existence of Kähler-Einstein metrics on such manifolds. We also sketch the computations on cubic surfaces with one ordinary double points, and outline the analytic difficulties to prove the existence of orbifold Kähler-Einstein metrics.     

On the spectrum of a discrete Laplacian on ℤ with finitely supported potential

We study spectral properties of the discrete Laplacian L?=??Δ?+?V on ? with finitely supported potential V. We give sufficient and necessary conditions for L to satisfy that the number of negative (resp. positive) eigenvalues is equal to one of the points x on which V(x) is negative (resp. positive). In addition, we prove that L has at least one discrete eigenvalue. If ∑ _x∈? V(x)?=?0, then L has both negative and positive discrete eigenvalues.     

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 Apéry sets of monomial curves

In this paper, we use the Apéry table of the numerical semigroup associated to an affine monomial curve in order to characterize arithmetic properties and invariants of its tangent cone. In particular, we precise the shape of the Apéry table of a numerical semigroup of embedding dimension 3, when the tangent cone of its monomial curve is Buchsbaum or 2-Buchsbaum, and give new proofs for two conjectures raised by Sapko (Commun. Algebra 29:4759–4773, 2001) and Shen (Commun. Algebra 39:1922–1940, 2001). We also provide a new simple proof in the case of monomial curves for Sally’s conjecture (Numbers of Generators of Ideals in Local Rings, 1978) that the Hilbert function of a one-dimensional Cohen-Macaulay ring with embedding dimension three is non-decreasing. Finally, we obtain that monomial curves of embedding dimension 4 whose tangent cones are Buchsbaum, and also monomial curves of any embedding dimensions whose numerical semigroups are balanced, have non-decreasing Hilbert functions. Numerous examples are provided to illustrate the results, most of them computed by using the NumericalSgps package of GAP (Delgado et al., NumericalSgps-a GAP package, 2006).     

On the Nyström discretization of integral equations on planar curves with corners

The Nyström method can produce ill-conditioned systems of linear equations when applied to integral equations on domains with corners. This defect can already be seen in the simple case of the integral equations arising from the Neumann problem for Laplace?s equation. We explain the origin of this instability and show that a straightforward modification to the Nyström scheme, which renders it mathematically equivalent to Galerkin discretization, corrects the difficulty without incurring the computational penalty associated with Galerkin methods. We also present the results of numerical experiments showing that highly-accurate solutions of integral equations on domains with corners can be obtained, irrespective of whether their solutions exhibit bounded or unbounded singularities, assuming that proper discretizations are used.     

The Chen–Ruan cohomology of moduli of curves of genus 2 with marked points

In this work we describe the Chen–Ruan cohomology of the moduli stack of smooth and stable genus 2 curves with marked points. In the first half of the paper we compute the additive structure of the Chen–Ruan cohomology ring for the moduli stack of stable n-pointed genus 2 curves, describing it as a rationally graded vector space. In the second part we give generators for the even Chen–Ruan cohomology ring as an algebra on the ordinary cohomology.     

On the μ-invariant of p-adic L-functions attached to elliptic curves with complex multiplication

The main result of this paper proves that the μ-invariant is zero for the Iwasawa module which arises naturally in the study of p-power descent on an elliptic curve with complex multiplication and good ordinary reduction at the prime p.     

On KKT points of Celis-Dennis-Tapia subproblem

The Celis-Dennis-Tapia(CDT) problem is a subproblem of the trust region algorithms for the constrained optimization. CDT subproblem is studied in this paper. It is shown that there exists the KKT point such that the Hessian matrix of the Lagrangian is positive semidefinite, if the multipliers at the global solution are not unique. Next the second order optimality conditions are also given, when the Hessian matrix of Lagrange at the solution has one negative eigenvalue. And furthermore, it is proved that all feasible KKT points satisfying that the corresponding Hessian matrices of Lagrange have one negative eigenvalue are the local optimal solutions of the CDT subproblem.     

On a form of the scattering matrix for ρ-perturbations of an abstract wave equations

We present the definition of ρ-perturbations of an abstract wave equation. As a special case, this definition involves perturbations with compact support for the classical wave equation. We construct the scattering matrix for equations of such a type. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 445–457, April, 1999.     

On the points of order 3 of the period parallelogram of the Weierstrass elliptic function

We examine the significance of the values of the Weierstrass ℘ function at the points of order three in the period parallelogram. Based upon the well-known duplication formulae and the differential equation of the Weierstrass ℘ function, we derive a set of identities involving the values of ℘ at these points.      

On the convergence of continued fractions at Runckel’s points

We consider the limit periodic continued fractions of Stieltjes type

On the conditions for the coincidence of two cubic Bézier curves

In a recent article, Wang et al. [2] derive a necessary and sufficient condition for the coincidence of two cubic Bézier curves with non-collinear control points. The condition reads that their control points must be either coincident or in reverse order. We point out that this uniqueness of the control points for polynomial cubics is a straightforward consequence of a previous and more general result of Barry and Patterson, namely the uniqueness of the control points for rational Bézier curves. Moreover, this uniqueness applies to properly parameterized polynomial curves of arbitrary degree.