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 .

     

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.     

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.     

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.     

Cascade search for preimages and coincidences: Global and local versions

In previous papers of the author, the cascade search principle was proposed, which makes it possible to construct a set-valued self-map of a metric spaceX from a set-valued functional or a collection of set-valued maps of X so that the new map generates a multicascade, i.e., a set-valued discrete dynamical system whose limit set coincides with the zero set of the given functional, with the coincidence set of the given collection, or with the common preimage of a closed subspace under the maps from this collection. Stability issues of cascade search were studied. This paper is devoted to a generalization and local modifications of the cascade search principle and their applications to problems concerning local search and approximation of common preimages of subspaces and coincidence sets for finite collections of set-valued maps of metric spaces.     

Ak-extreme point is the limit ofk-exposed points

It is proved that the relative boundary of ak-dimensional intersection of a hyperplane and a compact convex body is contained in the closure of the union of all intersections of dimension lower thanp that the same convex body makes with different hyperplanes.     

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.     

On limit points of Laplacian spectral radii of graphs

The study of limit points of eigenvalues of adjacency matrices of graphs was initiated by Hoffman [A.J. Hoffman, On limit points of spectral radii of non-negative symmetric integral matrices, in: Y. Alavi et al. (Eds.), Lecture Notes Math., vol. 303, Springer-Verlag, Berlin, Heidelberg, New York, 1972, pp. 165-172]. There he described all of the limit points of the largest eigenvalue of adjacency matrices of graphs that are no more than . In this paper, we investigate limit points of Laplacian spectral radii of graphs. The result is obtained: Let , β₀=1 and be the largest positive root of

     

Cascade search of the coincidence set of collections of multivalued mappings

Mathematical Notes - The present paper is a continuation of the previous papers of the author dealing with this subject. We study the cascade search of a given subset A, i.e., the construction on...     

On limit distributions of first crossing points of Gaussian sequences

Let {X_k, k?Z} be a stationary Gaussian sequence with EX₁ – 0, EX²_k = 1 and EX₀X_k = r_k. Define τ_x = inf{k: X_k >– βk} the first crossing point of the Gaussian sequence with the function – βt (β > 0). We consider limit distributions of τ_x as β→0, depending on the correlation function r_k. We generalize the results for crossing points τ_x = inf{k: X_k >β?(k)} with ?(– t)?t^γL(t) for t→∞, where γ > 0 and L(t) varies slowly.     

Almost sure limit points of independent copies of sample maxima

In this paper, we derive the almost sure limit sets of the random vector consisting of properly normalised independent copies of sample maximum of an i.i.d. sequence. The sets are derived under two different conditions on the common distribution function.     

Euclidean distance degree and limit points in a Morsification

Motivated by finding an effective way to compute the algebraic complexity of the nearest point problem for algebraic models, we introduce an efficient method for detecting the limit points of the stratified Morse trajectories in a small perturbation of any polynomial function on a complex affine variety. We compute the multiplicities of these limit points in terms of vanishing cycles. In the case of functions with only isolated stratified singularities, we express the local multiplicities in terms of polar intersection numbers.     

Recurrent critical points and typical limit sets of rational maps

We consider a rational map of the Riemann sphere with normalized Lebesgue measure and show that if there is a subset of the Julia set of positive -measure whose points have limit sets not contained in the union of the limit sets of recurrent critical points, then for -a.e. point and is conservative, ergodic and exact.