scholarly journals On Montel’s Theorem

1956 ◽  
Vol 10 ◽  
pp. 125-127 ◽  
Author(s):  
Yoshiro Kawakami
Keyword(s):  
Closed Set ◽  
Regular Functions ◽  
Measurable Set ◽  
Montel’S Theorem

In this note we shall prove a theorem which is related to Montel’s theorem [1] on bounded regular functions. Let E be a measurable set on the positive y-axis in the z( = x + iy)-plane, E(a, b) be its part contained in 0 ≦ a ≦ y ≦ b, and ∣E(a, b)∣ be its measure. We define the lower density of E at y = 0 byLEMMA, Let E be a set of positive lower density λ at y = 0. Then E contains a subset E1 of the same lower density at y = 0 such that E1 ∪ {0} is a closed set.

scholarly journals Remarks to the Paper “On Montel’s Theorem’ By Kawakami

1956 ◽  
Vol 10 ◽  
pp. 165-169
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Linear Measure ◽  
Measurable Set ◽  
Montel’S Theorem

We take a measurable set E on the positive η-axis and denote by μ(r) the linear measure of the part of E in the interval 0 < η < r. The lower density of E at η = 0 is defined byTheorem by Kawakami [1] asserts that if λ is positive, if a function f(ζ) = f(ξ + iη) is bounded analytic in ξ > 0 and continuous at E, and if f(ζ) → A as ζ → 0 along E, then f(ζ) → A as ζ → 0 in ∣η∣ ≦ kξ for any k > 0.

scholarly journals COMPUTATION OF THE PERIMETER OF MEASURABLE SETS VIA THEIR COVARIOGRAM. APPLICATIONS TO RANDOM SETS

2011 ◽  
Vol 30 (1) ◽  
pp. 39 ◽  
Author(s):  
Bruno Galerne
Keyword(s):  
Unit Volume ◽  
Lipschitz Constant ◽  
Directional Derivative ◽  
Random Sets ◽  
Directional Derivatives ◽  
Boolean Models ◽  
Closed Set ◽  
Finite Perimeter ◽  
Measurable Set ◽  
Grain Distribution

The covariogram of a measurable set A ⊂ Rd is the function gA which to each y ∈ Rd associates the Lebesgue measure of A ∩ (y + A). This paper proves two formulas. The first equates the directional derivatives at the origin of gA to the directional variations of A. The second equates the average directional derivative at the origin of gA to the perimeter of A. These formulas, previously known with restrictions, are proved for any measurable set. As a by-product, it is proved that the covariogram of a set A is Lipschitz if and only if A has finite perimeter, the Lipschitz constant being half the maximal directional variation. The two formulas have counterparts for mean covariogram of random sets. They also permit to compute the expected perimeter per unit volume of any stationary random closed set. As an illustration, the expected perimeter per unit volume of stationary Boolean models having any grain distribution is computed.

Loci of complex polynomials, part II: polar derivatives

2015 ◽  
Vol 159 (2) ◽  
pp. 253-273 ◽  
Author(s):  
BLAGOVEST SENDOV ◽  
HRISTO SENDOV
Keyword(s):  
Higher Order ◽  
Complex Polynomial ◽  
Complex Polynomials ◽  
Point Sets ◽  
Closed Set ◽  
Measurable Set ◽  
Closed Point ◽  
Polar Derivatives ◽  
Lebesgue Measurable Set ◽  
Derivatives Of

AbstractFor every complex polynomial p(z), closed point sets are defined, called loci of p(z). A closed set Ω ⊆ ${\mathbb C}$* is a locus of p(z) if it contains a zero of any of its apolar polynomials and is the smallest such set with respect to inclusion. Using the notion locus, some classical theorems in the geometry of polynomials can be refined. We show that each locus is a Lebesgue measurable set and investigate its intriguing connections with the higher-order polar derivatives of p.

scholarly journals On Measure of Sumsets

1961 ◽  
Vol 12 (4) ◽  
pp. 209-211
Author(s):  
A. M. MacBeath
Keyword(s):  
Abelian Groups ◽  
Present Problem ◽  
Finite Abelian Groups ◽  
Closed Set ◽  
Closed Sets ◽  
Alternative Method ◽  
Measurable Set ◽  
Continuous Analogue

In the second paper under this general title, it was shown how a theorem about the torus could be deduced by a limiting process from a theorem on finite abelian groups. The object of this paper is to prove a similar continuous analogue of H. B. Mann's (α+β)-theorem. It was found that the limiting process used in the second paper could not easily be modified to apply to the present problem, and an alternative method had to be found. The method is, roughly, to prove the result first for open sets satisfying certain conditions, then for closed sets by taking intersections of open sets, and finally for arbitrary measurable sets, since every measurable set contains a closed set of almost equal measure.

ON A NEW CLASS OF SEMI GENERALIZED CLOSED SETS IN STRONG GENERALIZED TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (11) ◽  
pp. 9353-9360
Author(s):  
G. Selvi ◽  
I. Rajasekaran
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Basic Properties ◽  
Closed Sets ◽  
New Class ◽  
Open Set ◽  
Continuous Maps

This paper deals with the concepts of semi generalized closed sets in strong generalized topological spaces such as $sg^{\star \star}_\mu$-closed set, $sg^{\star \star}_\mu$-open set, $g^{\star \star}_\mu$-closed set, $g^{\star \star}_\mu$-open set and studied some of its basic properties included with $sg^{\star \star}_\mu$-continuous maps, $sg^{\star \star}_\mu$-irresolute maps and $T_\frac{1}{2}$-space in strong generalized topological spaces.

scholarly journals Unavoidable subprojections in union-closed set systems of infinite breadth

2021 ◽  
Vol 94 ◽  
pp. 103311
Author(s):  
Yemon Choi ◽  
Mahya Ghandehari ◽  
Hung Le Pham
Keyword(s):  
Closed Set ◽  
Set Systems

REFLEXIVITY INDEX AND IRRATIONAL ROTATIONS

2021 ◽  
pp. 1-13
Author(s):  
BINGZHANG MA ◽  
K. J. HARRISON
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Irrational Rotations ◽  
Index 2

Abstract We determine the reflexivity index of some closed set lattices by constructing maps relative to irrational rotations. For example, various nests of closed balls and some topological spaces, such as even-dimensional spheres and a wedge of two circles, have reflexivity index 2. We also show that a connected double of spheres has reflexivity index at most 2.

scholarly journals Synthetic speech detection through short-term and long-term prediction traces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Clara Borrelli ◽  
Paolo Bestagini ◽  
Fabio Antonacci ◽  
Augusto Sarti ◽  
Stefano Tubaro
Keyword(s):  
Deep Learning ◽  
Speech Processing ◽  
Synthetic Speech ◽  
Opinion Formation ◽  
Closed Set ◽  
Speech Detection ◽  
Open Set ◽  
Technological Advances ◽  
Speech Generation ◽  
Long Term Prediction

AbstractSeveral methods for synthetic audio speech generation have been developed in the literature through the years. With the great technological advances brought by deep learning, many novel synthetic speech techniques achieving incredible realistic results have been recently proposed. As these methods generate convincing fake human voices, they can be used in a malicious way to negatively impact on today’s society (e.g., people impersonation, fake news spreading, opinion formation). For this reason, the ability of detecting whether a speech recording is synthetic or pristine is becoming an urgent necessity. In this work, we develop a synthetic speech detector. This takes as input an audio recording, extracts a series of hand-crafted features motivated by the speech-processing literature, and classify them in either closed-set or open-set. The proposed detector is validated on a publicly available dataset consisting of 17 synthetic speech generation algorithms ranging from old fashioned vocoders to modern deep learning solutions. Results show that the proposed method outperforms recently proposed detectors in the forensics literature.

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