12,081 research outputs found
Generalized Alpha-Close-to-Convex Functions
We define the classes GÎČ(α,k,Îł) as follows: fâGÎČ(α,k,Îł)
if and only if, for zâE={zââ:|z|<1}, |arg{(1-α2z2)fâČ(z)/eâiÎČÏâČ(z)}|â€ÎłÏ/2, 0<Îłâ€1; αâ[0,1]; ÎČâ(âÏ/2,Ï/2),
where Ï
is a function of bounded boundary rotation. Coefficient estimates, an inclusion
result, arclength problem, and some other properties of these classes are studied
Topological Data Analysis with Bregman Divergences
Given a finite set in a metric space, the topological analysis generalizes
hierarchical clustering using a 1-parameter family of homology groups to
quantify connectivity in all dimensions. The connectivity is compactly
described by the persistence diagram. One limitation of the current framework
is the reliance on metric distances, whereas in many practical applications
objects are compared by non-metric dissimilarity measures. Examples are the
Kullback-Leibler divergence, which is commonly used for comparing text and
images, and the Itakura-Saito divergence, popular for speech and sound. These
are two members of the broad family of dissimilarities called Bregman
divergences.
We show that the framework of topological data analysis can be extended to
general Bregman divergences, widening the scope of possible applications. In
particular, we prove that appropriately generalized Cech and Delaunay (alpha)
complexes capture the correct homotopy type, namely that of the corresponding
union of Bregman balls. Consequently, their filtrations give the correct
persistence diagram, namely the one generated by the uniformly growing Bregman
balls. Moreover, we show that unlike the metric setting, the filtration of
Vietoris-Rips complexes may fail to approximate the persistence diagram. We
propose algorithms to compute the thus generalized Cech, Vietoris-Rips and
Delaunay complexes and experimentally test their efficiency. Lastly, we explain
their surprisingly good performance by making a connection with discrete Morse
theory
Centroid-Based Clustering with ab-Divergences
Centroid-based clustering is a widely used technique within unsupervised learning
algorithms in many research fields. The success of any centroid-based clustering relies on the
choice of the similarity measure under use. In recent years, most studies focused on including several
divergence measures in the traditional hard k-means algorithm. In this article, we consider the
problem of centroid-based clustering using the family of ab-divergences, which is governed by two
parameters, a and b. We propose a new iterative algorithm, ab-k-means, giving closed-form solutions
for the computation of the sided centroids. The algorithm can be fine-tuned by means of this pair of
values, yielding a wide range of the most frequently used divergences. Moreover, it is guaranteed to
converge to local minima for a wide range of values of the pair (a, b). Our theoretical contribution
has been validated by several experiments performed with synthetic and real data and exploring the
(a, b) plane. The numerical results obtained confirm the quality of the algorithm and its suitability to
be used in several practical applications.MINECO TEC2017-82807-
A generalization of starlike functions of order alpha
For every and we define a class of analytic
functions, the so-called -starlike functions of order , on the open
unit disk. We study this class of functions and explore some inclusion
properties with the well-known class of starlike functions of order .
The paper is also devoted to the discussion on the Herglotz representation
formula for analytic functions when is -starlike of
order . As an application we also discuss the Bieberbach conjecture
problem for the -starlike functions of order . Further application
includes the study of the order of -starlikeness of the well-known basic
hypergeometric functions introduced by Heine.Comment: 13 pages, 4 figures, submitted to a journa
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