On statistical limit points

The set of all statistical limit points of a given sequence is characterized as an -set. It is also characterized in terms of discontinuity points of distribution functions of .

     

Remote limit points on surfaces

A flow (continuous real action) on a compact orientable surface M of genus greater than one (a sphere with at least two handles) has sufficient room for orbits to wrap around one of the handles in an exotic fashion. Specifically, an orbit that is wrapping around one handle can, between wraps, spend increasing amounts of time wrapping and unwrapping around a second handle before returning to the first for the next wrap around it. As a result the omega limit set of such an orbit can contain a simple closed curve of fixed points around the second handle in spite of wrapping around the first handle. In an earlier paper (Colloq. Math. 84/85 (2000) 235), the authors constructed such a flow from this perspective and studied its lift to the universal covering space of the surface. In this paper it is shown that many of the properties of the example are consequences of a general theory that extends classical limit cycle theory.     

Universal distribution of limit points

We consider sequences of functions that have in some sense a universal distribution of limit points of zeros in the complex plane. In particular, we prove that functions having universal approximation properties on compact sets with connected complement automatically have such a universal distribution of limit points. Moreover, in the case of sequences of derivatives, we show connections between this kind of universality and some rather old results of Edrei/MacLane and Pólya. Finally, we show the lineability of the set what we call Jentzsch-universal power series.     

Some remarks about the limit point and limit circle theory

LetL be a formally selfadjoint differential operator andp a real-valued function, both ona≤x<∞. The deficiency indices are the numbers of solutions ofLu=λpu for Im λ>0 and for Im λ<0 which have a certain regularity atx=∞. (A) Ifp(x)≥0 this regularity means that the integral ofp(x)│u│² converges at infinity. (B) Ifp changes its sign for arbitrarily large values ofx butL has a positive definite Dirichlet integral it is natural to relate the regularity to this integral. Weyl’s classical study of the deficiency indices is reviewed for (A) with the help of elementary theory of quadratic forms. Individual bounds are found for the deficiency indices also whenL is of odd order. It is then indicated how the method carriers over to (B).     

Complementary remarks about the limit point and limit circle theory

A method to obtain bounds for the number of solutions with regular behaviour at infinity of selfadjoint ordinary differential equations is given. In this method, which is presented for a polar problem, the sequence of nested ellipsoids in a previous paper of mine is replaced by the use of certain projections. At the same time I take the opportunity to complete the unsatisfactory bibliography of my earlier paper.     

On the limit points of the Kaplan-Meier estimator

In this paper the limit points of the Kaplan-Meier estimator is discussed. We use the method of strong approximation to get the unit ball of the reproducing kernel Hilbert space.     

Continued fractions with three limit points

On page 45 in his lost notebook, Ramanujan asserts that a certain q-continued fraction has three limit points. More precisely, if A_n/B_n denotes its nth partial quotient, and n tends to ∞ in each of three residue classes modulo 3, then each of the three limits of A_n/B_n exists and is explicitly given by Ramanujan. Ramanujan's assertion is proved in this paper. Moreover, general classes of continued fractions with three limit points are established.     

Extremal A-statistical limit points via ideals

In this paper, following the line of recent work of Savaş et al. [20] we apply the notion of ideals to A-statistical limit superior and inferior for a sequence of real numbers.     

On statistical limit points of double sequences

We investigate the structure of the set of all statistical limit points of a double sequence and prove certain results, mainly showing that this set can be characterized as a F_σ-set.     

Non-metricity in the continuum limit of randomly-distributed point defects

We present a homogenization theorem for isotropically-distributed point defects, by considering a sequence of manifolds with increasingly dense point defects. The loci of the defects are chosen randomly according to a weighted Poisson point process, making it a continuous version of the first passage percolation model. We show that the sequence of manifolds converges to a smooth Riemannian manifold, while the Levi-Civita connections converge to a non-metric connection on the limit manifold. Thus, we obtain rigorously the emergence of a non-metricity tensor, which was postulated in the literature to represent continuous distribution of point defects.     

On the limit points of the smallest eigenvalues of regular graphs

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval ${[-1-\sqrt2, -2)}$ and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval ${[\alpha_1, -1-\sqrt2)}$ where α ₁ is the smallest root ${(\approx -2.4812)}$ of the polynomial x ³?+?2x ² ? 2x ? 2. From these results, we determine the largest and second largest limit points of smallest eigenvalues of regular graphs less than ?2. Moreover we determine the supremum of the smallest eigenvalue among all connected 3-regular graphs with smallest eigenvalue less than ?2 and we give the unique graph with this supremum value as its smallest eigenvalue.     

Cascade search: Stability of reachable limit points

This paper is a continuation of the series of author’s previous works focused on the cascade search method. The stability of the subset of limit points of a given searching multi-cascade reachable from x is considered with respect to a small variation of the initial point x and to a small perturbation of the searching functional φ generating the multi-cascade. The results are applied to the cascade search problem for common preimages of a closed subspace under n (n ≥ 1) multi-valued mappings.     

On limit points of spectra of the random graph first-order properties

New bounds on the minimum and maximum limit points of spectra of first-order properties of the Erdös–Rényi random graph are obtained. These results are used to improve bounds on the minimal quantifier depths of first-order formulas with infinite spectra. Moreover, we prove that there are no limit points of the spectra in the interval (1–2^1–k , 1).     

On the hyperbolic limit points of groups acting on hyperbolic spaces

We study the hyperbolic limit points of a groupG acting on a hyperbolic metric space, and consider the question of whether any attractive limit point corresponds to a unique repulsive limit point. In the special case whereG is a (non-elementary) finitely generated hyperbolic group acting on its Cayley graph, the answer is affirmative, and the resulting mapg ⁺↦g ⁻, is discontinuous everywhere on the hyperbolic boundary. We also provide a direct, combinatorial proof in the special case whereG is a (non-abelian) free group of finite type, by characterizing algebraically the hyperbolic ends ofG. Partially supported by a grant from M.U.R.S.T., Italy.