On Ward's Perron-Stieltjes Integral

1957 ◽  
Vol 9 ◽  
pp. 96-109 ◽  
Author(s):  
Ralph Henstock
Keyword(s):  
Bounded Variation ◽  
Large Set ◽  
Stieltjes Integral ◽  
Countable Number ◽  
Finite Function ◽  
Set Of Points

In the paper (5), Ward defines an integral of Perron type of a finite function f with respect to another finite function g, where g need not be of bounded variation. There arise two problems, (a) and (b) below, that have not been dealt with in (5).If f = j at a countable number of points everywhere dense in (a, b), where f and j are both integrable with respect to g, then f — j can be nonzero on a large set of points of (a, b).

The existence of parametric surface integrals

1960 ◽  
Vol 1 (4) ◽  
pp. 419-427
Author(s):  
J. H. Michael
Keyword(s):  
Bounded Variation ◽  
Continuous Mapping ◽  
Measurable Subset ◽  
Signed Measure ◽  
Stieltjes Integral ◽  
Parametric Surface ◽  
Surface Integral ◽  
Linear Type ◽  
Definition Of ◽  
Image Position

In [2] we studied parametric n-surfaces (f, Mn), where Mn was a compact, oriented, topological n-manifold and f a continuous mapping of Mn into the real euclidean k-space Rn (k≧n). A definition of bounded variation was given and, for each surface with bounded variation and each projection P from Rk to Rn, a signed measure: Was constructed. This measure was used to define a linear type of surface integral: over a “measurable” subset A of Mn, as the Lebesgue-Stieltjes integral: .

Dynamics of $(e^{z} - 1)/z$: the Julia set and bifurcation

1998 ◽  
Vol 18 (6) ◽  
pp. 1363-1383 ◽  
Author(s):  
G. P. KAPOOR ◽  
M. GURU PREM PRASAD
Keyword(s):  
Real Root ◽  
Julia Set ◽  
Dynamical Behaviour ◽  
Positive Real ◽  
Finite Function ◽  
Positive Real Root ◽  
Transcendental Functions ◽  
Set Of Points

We describe the dynamical behaviour of the entire transcendental non-critically finite function $f_\lambda (z) = \lambda(e^z - 1)/z$, $\lambda > 0$. Our main result is to obtain a computationally useful characterization of the Julia set of $f_\lambda (z)$ as the closure of the set of points with orbits escaping to infinity under iteration, which in turn is applied to the generation of the pictures of the Julia set of $f_\lambda (z)$. Such a characterization was hitherto known only for critically finite entire transcendental functions [11]. We find that bifurcation in the dynamics of $f_\lambda (z)$ occurs at $\lambda = \lambda^{*}$ ($\approx 0.64761$) where $\lambda^\ast = {(x^{*})}^{2} /({e}^{x^{*}} -1)$ and $x^{*}$ is the unique positive real root of the equation $e^{x}(2 -x ) -2 = 0$.

scholarly journals On Monotonic and Nonnegative Solutions of a Nonlinear Volterra-Stieltjes Integral Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Tomasz Zając
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Functions Of Bounded Variation ◽  
Stieltjes Integral ◽  
Measures Of Noncompactness ◽  
Bounded Interval ◽  
Nonnegative Solutions ◽  
Real Functions

We study the existence of monotonic and nonnegative solutions of a nonlinear quadratic Volterra-Stieltjes integral equation in the space of real functions being continuous on a bounded interval. The main tools used in our considerations are the technique of measures of noncompactness in connection with the theory of functions of bounded variation and the theory of Riemann-Stieltjes integral. The obtained results can be easily applied to the class of fractional integral equations and Volterra-Chandrasekhar integral equations, among others.

scholarly journals ON THE POSITIVITY OF RIEMANN–STIELTJES INTEGRALS

2012 ◽  
Vol 87 (3) ◽  
pp. 400-405 ◽  
Author(s):  
JANI LUKKARINEN ◽  
MIKKO S. PAKKANEN
Keyword(s):  
Continuous Function ◽  
Bounded Variation ◽  
Nonnegative Function ◽  
Stieltjes Integral ◽  
Function Of Bounded Variation ◽  
Positive Continuous Function

AbstractWe study the question whether a Riemann–Stieltjes integral of a positive continuous function with respect to a nonnegative function of bounded variation is positive.

scholarly journals A sharp companion of Ostrowski’s inequality for the Riemann–Stieltjes integral and applications

2016 ◽  
Vol 15 (1) ◽  
pp. 69-78 ◽  
Author(s):  
Mohammad W. Alomari
Keyword(s):  
Bounded Variation ◽  
Approximation Problem ◽  
Stieltjes Integral ◽  
Ostrowski’S Inequality

AbstractA sharp companion of Ostrowski’s inequality for the Riemann-Stieltjes integral $\int_a^b {f(t)\;du(t)} $, where f is assumed to be of r-H-Hölder type on [a, b] and u is of bounded variation on [a, b], is proved. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.

scholarly journals Automatic detection of polyomavirus particles with local texture indicators

2013 ◽  
Vol 19 (S4) ◽  
pp. 67-68
Author(s):  
M.C. Proença ◽  
J.F. Moura Nunes ◽  
A.P. Alves de Matos
Keyword(s):  
Morphological Characteristics ◽  
Initial Guess ◽  
Large Set ◽  
False Negatives ◽  
Virus Family ◽  
Virus Particles ◽  
Intact Virus ◽  
Fully Automatic ◽  
Occurrence Matrix ◽  
Set Of Points

A fully automatic approach to locate polyomavirus particles in images of transmission electronic microscopy is presented, that can localize intact particles, many damaged capsids and an acceptable percentage of superposed ones; the performance is quantified in 25 electron micrographs containing nearly 390 particles, compared to the interpretation of two independent electron microscopy experts. As these particles often does not have a well-defined edge, in highly textured backgrounds (Figure 1) the initial guess of candidate points from local entropy proportion measures retains a large set of points, in order to avoid false negatives.Based in human criteria - focused on the pattern generated by the capsomeres typical of polyomavirus surface, this approach uses indicators calculated from the local co-occurrence matrix of grey levels to assess the textured pattern and prune the initial set of candidates; in some more complicated backgrounds about 2-10% of the elements will survive. A restricted set of the points accepted (Figure 2) is used to evaluate typical average and variance and reduce the set of survivors accordingly. These intermediate points are locally evaluated using i) a statistical index concerning the radiometric distribution of a circular neighborhood around the centroid of each candidate, and ii) a structural index resuming the expected morphological characteristics of eight radial intensity profiles encompassing the area of the possible particle. All the parameterization is based on the particle expected dimensions.This hierarchical approach attains 95% efficiency in the detection of intact virus particles, tolerating a certain lack of differentiation in the borders and a certain amount of shape alterations. Only 5 to 7% (according to the reference used) of the intact particles are missed by the algorithm – cases of particles with abnormal radial structures, particles with different degrees of agglutination or particles showing empty (or with different degrees of permeation) capsids. Superposed particles are detected with a success rate of 71% to 77%. In the case of distorted and/or doubtful particles, a correct detection of 72% is attained; these particles are the main source of differences between experts (a difference of 48% between both interpretations).This kind of algorithm is "tailored" for one virus family particle, in this case, the polyomavirus; it can be parameterized for different working magnifications, but cannot be applied to different virus, that has different texture, structure and typical radiometry characteristics. We believe that similar results can be obtained in other virus families with the necessary adaptations.

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