scholarly journals Riemann-Stieltjes Integral

2016 ◽  
Vol 24 (3) ◽  
pp. 199-204 ◽  
Author(s):  
Keiko Narita ◽  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Total Variation ◽  
Bounded Variation ◽  
Stieltjes Integral ◽  
Real Numbers ◽  
Basic Properties ◽  
Finite Sequences ◽  
The Way

Abstract In this article, the definitions and basic properties of Riemann-Stieltjes integral are formalized in Mizar [1]. In the first section, we showed the preliminary definition. We proved also some properties of finite sequences of real numbers. In Sec. 2, we defined variation. Using the definition, we also defined bounded variation and total variation, and proved theorems about related properties. In Sec. 3, we defined Riemann-Stieltjes integral. Referring to the way of the article [7], we described the definitions. In the last section, we proved theorems about linearity of Riemann-Stieltjes integral. Because there are two types of linearity in Riemann-Stieltjes integral, we proved linearity in two ways. We showed the proof of theorems based on the description of the article [7]. These formalizations are based on [8], [5], [3], and [4].

scholarly journals The Basic Existence Theorem of Riemann-Stieltjes Integral

2016 ◽  
Vol 24 (4) ◽  
pp. 253-259 ◽  
Author(s):  
Kazuhisa Nakasho ◽  
Keiko Narita ◽  
Yasunari Shidama
Keyword(s):  
Continuous Function ◽  
Existence Theorem ◽  
Bounded Variation ◽  
Real Line ◽  
Closed Interval ◽  
Stieltjes Integral ◽  
Function Of Bounded Variation ◽  
Basic Properties ◽  
Finite Sequences

Summary In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15], [12], [10], and [11].

scholarly journals The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (3) ◽  
pp. 185-191
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Riemann Integral ◽  
Basic Properties ◽  
Bounded Functions ◽  
Definition Of

Summary In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

scholarly journals Silencing the city?

2013 ◽  
Vol 3 (3) ◽  
pp. 61-70 ◽  
Author(s):  
Jean-Paul Thibaud ◽  
Pascal Amphoux
Keyword(s):  
Urban Environment ◽  
Social Life ◽  
Research Topic ◽  
Basic Properties ◽  
Current State ◽  
Design Issue ◽  
Sonic Environment ◽  
The City ◽  
The Way

The notion of silence must be handled very carefully. In addition, its use reveals the way we deal with the urban environment as well as social life. What does the notion of silence convey about the current state of the urban sonic environment? How can we clarify the various meanings and the stakes involved in silence? Three themes are developed in order to answer these questions: silence as a research topic presents three complementary perspectives (acoustic, sociocultural, technological); silence as a polysemous notion emphasises the ideas of keeping quiet, tranquillity and pause; silence as a design issue relies on basic properties and principles in order to orient the design of the urban sonic environment.

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).

Distribution of variance associated with measurement of post-thaw function in ram sperm

1995 ◽  
Vol 7 (1) ◽  
pp. 129 ◽  
Author(s):  
AY Edward ◽  
DP Windsor ◽  
IW Purvis ◽  
LG Sanchez-Partida ◽  
WM Maxwell
Keyword(s):  
Total Variation ◽  
Semen Analysis ◽  
Amidase Activity ◽  
Ram Sperm ◽  
Semen Characteristic ◽  
The Way

Post-thaw characteristics of ram semen frozen as pellets were assessed using biochemically (amidase activity) or motility-based (Hamilton Thorn Motility Analyzer) techniques. The total variation associated with each semen characteristic measured was partitioned between rams (5), ejaculates within rams (5), pellets within ejaculates (5) and within pellets (2). A variety of variance distributions were observed for the characteristics measured. Of the 18 post-thaw characteristics examined, 10 had > 50% of variance distributed between within-ejaculate components. This has important implications for the way in which such measurements may be used in post-thaw semen analysis.

A minimal counterexample to universal baireness

1999 ◽  
Vol 64 (4) ◽  
pp. 1601-1627 ◽  
Author(s):  
Kai Hauser
Keyword(s):  
Fine Structure ◽  
Set Theory ◽  
Canonical Model ◽  
Core Model ◽  
Real Numbers ◽  
The Real ◽  
Minimal Counterexample ◽  
General Method ◽  
The Way

AbstractFor a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.

scholarly journals Cayley-Dickson Construction

2012 ◽  
Vol 20 (4) ◽  
pp. 281-290
Author(s):  
Artur Korniłowicz
Keyword(s):  
Complex Numbers ◽  
Real Numbers ◽  
Basic Properties

Summary Cayley-Dickson construction produces a sequence of normed algebras over real numbers. Its consequent applications result in complex numbers, quaternions, octonions, etc. In this paper we formalize the construction and prove its basic properties.

On the Topologies of R∞

1981 ◽  
Vol 24 (1) ◽  
pp. 47-51
Author(s):  
Chung-Wu Ho
Keyword(s):  
Homotopy Type ◽  
Type A ◽  
Real Numbers ◽  
Finite Sequences ◽  
Usual Topology

AbstractLet R∞ be the set of all the finite sequences of real numbers. The author shows that there are uncountably many distinct topologies on the set R∞, each of which coincides with the usual topology when restricted to an Rn. However, under each of these topologies, R∞ is always of the same homotopy type. A generalization to some other spaces is mentioned.

Sign in / Sign up

Export Citation Format

Share Document