Denjoy integral and Henstock-Kurzweil integral in vector lattices,I

In this paper we define the derivative and the Denjoy integral of mappings from a vector lattice to a complete vector lattice and show the fundamental theorem of calculus.      

Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral II

This paper is a continuation of the paper [T.Y. Lee, Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral, J. Math. Anal. Appl. 298 (2004) 677-692], in which we proved several Fubini-Tonelli type theorems for the Henstock-Kurzweil integral. Let f be Henstock-Kurzweil integrable on a compact interval . For a given compact interval , set

     

The Distributional Henstock-Kurzweil Integral and Applications: A Survey

This study presents a summary of the current state of research on the distributional Henstock-Kurzweil integral. Basic properties such as integration by parts, Hake theorem, inner product, Hölder inequality, second mean value theorem, orderings, Banach lattice, convergence theorems, fixed point theorems, are shown. This study also summarizes its applications in integral and differential equations.     

Differential and integral equations with Henstock-Kurzweil integrable functions

In this paper we apply fixed point theorems for increasing mappings in ordered normed spaces to prove existence and comparison results for solutions of discontinuous functional differential and integral equations containing Henstock-Kurzweil integrable functions.     

The Itô integral for Brownian motion in vector lattices: Part 1

In this paper the Itô integral for Brownian motion is constructed in a vector lattice and some of its properties are derived. The assumption is that there exists a conditional expectation operator on the vector lattice and the construction does not depend on a probability measure space. The classical case of the Itô integral is a special case of the constructed integral in the vector lattice.     

Estimates of the Remainder in Taylor's Theorem Using the Henstock-Kurzweil Integral

When a real-valued function of one variable is approximated by its nth degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue p-norms in cases where f ⁽ⁿ⁾ or f ⁽ⁿ⁺¹⁾ are Henstock-Kurzweil integrable. When the only assumption is that f ⁽ⁿ⁾ is Henstock-Kurzweil integrable then a modified form of the nth degree Taylor polynomial is used. When the only assumption is that f ⁽ⁿ⁾ ∈ C ⁰ then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1. Research partially supported by the Natural Sciences and Engineering Research Council of Canada. An adjunct appointment in the Department of Mathematical and Statistical Sciences, University of Alberta, made valuable library and computer resources available.     

Some Kurzweil-Henstock-type integrals and the wide Denjoy integral

Kurzweil-Henstock integrals related to local systems and the wide Denjoy integral are discussed in the frame of their comparability and compatibility. Dedicated to the memory of Ralph Henstock (1923–2007)     

Normed lattices with discrete ultrapowers

The paper is devoted to investigations of the order structure of ultrapowers. A characterization of discrete normed lattices, all of whose ultrapowers are discrete, is presented (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Monotone convergence theorems for Henstock-Kurzweil integrable functions and applications

In this paper we prove monotone convergence theorems for Henstock-Kurzweil integrable functions from a compact real interval to an ordered Banach space. These theorems are then applied to prove existence results for solutions of a discontinuous functional integral equation containing Henstock-Kurzweil integrable functions.     

The Itô integral for Brownian motion in vector lattices: Part 2

The Itô integral for Brownian motion in a vector lattice, as constructed in Part 1 of this paper, is extended to accommodate a larger class of integrands. This extension provides an analogue of the indefinite Itô integral in the classical setting which yields a local martingale. The assumption is that there exists a conditional expectation operator on the vector lattice and the construction does not depend on a probability measure space. The classical case of the extended Itô integral is a special case of the constructed integral in the vector lattice.     

Mathematical features of the Calculus

The fundamental theorems of the calculus describe the relationships between derivatives and integrals of functions. The value of any function at a particular location is the definite derivative of its integral and the definite integral of its derivative. Thus, any value is the magnitude of the slope of the tangent of its integral at that position, and any two subtracted values are the area under its derivative. The slope formula of secant lines actually is the mean value theorem for the derivative function in addition to representing the well-known Fermat definition of the derivative. The sine and other functions are discussed.     

A type of Strassen's theorem for positive vector measures with values in dual spaces

In this paper, we extend a type of Strassen's theorem for the existence of probability measures with given marginals to positive vector measures with values in the dual of a barreled locally convex space which has certain order conditions. In this process of the extension we also give some useful properties for vector measures with values in dual spaces.

     

Classification of integral lattices with large class number

A detailed exposition of Kneser's neighbour method for quadratic lattices over totally real number fields, and of the sub-procedures needed for its implementation, is given. Using an actual computer program which automatically generates representatives for all isomorphism classes in one genus of rational lattices, various results about genera of -elementary lattices, for small prime level are obtained. For instance, the class number of -dimensional -elementary even lattices of determinant is ; no extremal lattice in the sense of Quebbemann exists. The implementation incorporates as essential parts previous programs of W. Plesken and B. Souvignier.

     

《例说循环论证之谬误》注

指出《例说循环论证之谬误》中的第二个例子并非循环论证,并给出例子说明相关问题中更有可能产生循环论证之处。     

On the Radon-Nikodym theorem for measures with values in vector lattices

Measures with values in a countably order-complete vector lattice are considered. The underlying σ-algebra is assumed to be σ-isomorphic to the Borel sets of the real line. Given one such measure, densities are searched which are not necessarily scalar-valued for smaller measures. The results can be used to prove the existence of a least upper bound for two such measures.     

A transfer principle for inequalities in vector lattices

A transfer principle from inequalities with inner products to inequalities containing positive semidefinite symmetric bilinear operators with values in a vector lattices is proved. Some applications are also given.     

Intersecting antichains and shadows in linear lattices

We establish a homomorphism of finite linear lattices onto the Boolean lattices via a group acting on linear lattices. By using this homomorphism we prove the intersecting antichains in finite linear lattices satisfy an LYM-type inequality, as conjectured by Erd?s, Faigle and Kern, and we state a Kruskal-Katona type theorem for the linear lattices.     

Exploring students' mathematical performance,metacognitive experiences and skills in relation to fundamental theorem of calculus

Several studies have explored students’ understanding of the relationships between definite integrals and areas under curve(s). So far, however, there has been less attention to students’ understanding of the Fundamental Theorem of Calculus (FTC). In addition, students’ metacognitive experiences and skills whilst solving FTC questions have not previously been explored. This paper explored students’ mathematical performance, metacognitive experiences and metacognitive skills in relation to FTC questions by interviewing nine university and eight Year 13 students. The findings show that several students had difficulty solving questions related to the FTC and that students’ metacognitive experiences and skills could be further developed.     

The Fueter mapping theorem in integral form and the ℱ‐functional calculus

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n‐tuples of commuting operators (called ?‐functional calculus) based on a new notion of spectrum, called ?‐spectrum, for the n‐tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ?‐functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal average approximations for functions mapping in quasi-Banach spaces

In 1994, M.M. Popov [6] showed that the fundamental theorem of calculus fails, in general, for functions mapping from a compact interval of the real line into the _?p${?}_p$-spaces for 0<p<1$0<p<1$, and the question arose whether such a significant result might hold in some other non-Banach spaces. In this article we completely settle the problem by proving that the fundamental theorem of calculus breaks down in the context of any non-locally convex quasi-Banach space. Our approach introduces the tool of Riemann-integral averages of continuous functions, and uses it to bring out to light the differences in behavior of their approximates in the lack of local convexity. As a by-product of our work we solve a problem raised in [1] on the different types of spaces of differentiable functions with values on a quasi-Banach space.