Real Analysis

This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of the interval techniques as they are now practised, provide a historical and epistemological background of interval mathematics and uncertainty in science and technology, and finally describe some typical applications that clarify the need for interval computations to cope with uncertainty in a wide variety of scientific disciplines.

Keywords. Interval mathematics, Uncertainty, Quantitative Knowledge, Reliability, Complex interval arithmetic, Machine interval arithmetic, Interval automatic differentiation, Computer graphics, Ray tracing, Interval root isolation.

Recommended Citation:

Hend Dawood. Interval Mathematics as a Potential Weapon against Uncertainty. In S. Chakraverty, editor, Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems. chapter 1, pages 1-38. IGI Global, Hershey, PA, 2014. ISBN 978-1-4666-4991-0.

The final publication is available at IGI Global via http://dx.doi.org/10.4018/978-1-4666-4991-0.ch001

In this paper we develop several inequalities concerning multivariate polynomials. In particular we obtain the following result:
Let $\vec{a}_{1},\vec{a}_{2}\ldots, \vec{a_n}$ be a linearly independent vectors in the space $\mathbb{R}^{l}$ and $\sigma:\{1,2,\ldots,l\}\longrightarrow \{1,2,\ldots,l\}$ be any permutation. If for each $g_k\in \mathbb{R}[x_1,x_2,\ldots,x_l]$ with $1\leq k\leq n$ and the norm of the cross products $\bigg | \bigg |\vec{a}_{i} \diamond \vec{a}_{j}\diamond \cdots \diamond \vec{a}_v\bigg |\bigg |\geq K$ for $K\in \mathbb{R}$ for all $1\leq i,j,\ldots,v\leq n$, then there exist some constant $C>0$ such that if
\begin{align}
    \int \limits_{a_{i_{\sigma(l)}}}^{a_{j_{\sigma(l)}}}\int \limits_{a_{i_{\sigma(l-1)}}}^{a_{j_{\sigma(l-1)}}}\cdots \int \limits_{a_{i_{\sigma(1)}}}^{a_{j_{\sigma(1)}}}g_kdx_{\sigma(1)}dx_{\sigma(2)}\cdots dx_{\sigma(l)}>0\nonumber
\end{align}then
\begin{align}
    \sum \limits_{k=1}^{n}\int \limits_{a_{i_{\sigma(l)}}}^{a_{j_{\sigma(l)}}}\int \limits_{a_{i_{\sigma(l-1)}}}^{a_{j_{\sigma(l-1)}}}\cdots \int \limits_{a_{i_{\sigma(1)}}}^{a_{j_{\sigma(1)}}}g_kdx_{\sigma(1)}dx_{\sigma(2)}\cdots dx_{\sigma(l)}&\leq C\times \frac{1}{K}\sqrt{n} \nonumber \\&\times \int \limits_{a_{i_{\sigma(l)}}}^{a_{j_{\sigma(l)}}}\int \limits_{a_{i_{\sigma(l-1)}}}^{a_{j_{\sigma(l-1)}}}\nonumber \\& \cdots \int \limits_{a_{i_{\sigma(1)}}}^{a_{j_{\sigma(1)}}}\sqrt{\bigg(\sum \limits_{k=1}^{n}(\mathrm{max}(g_k))^2\bigg)}\nonumber \\& dx_{\sigma(1)}dx_{\sigma(2)}\cdots dx_{\sigma(l)}\nonumber
\end{align}

Starting from a type of variation in the pattern of definiteness checking in Old Romanian (16th to 18th century), which has never been noticed before, and from the realization of genitive phrases in Old Romanian, we examine the significance of these phenomena for the evolution of the Romanian definite article. We claim that the existence of the A [ def]+ N[+def]+ GenDP/PP structure in Old Romanian is significant for understanding the rise of the enclitic article. Of the many proposed solutions to this problem, we have retained two; both relate the enclitic article to features of Romanian which are/were not present (to the same extent) in other Romance languages. First, the development of the enclitic article has been related to the creation of the inflectional genitive (Coteanu 1956, Giusti 1991), with the suffixed article being the speller of the Case feature; in contrast, the other Romance languages developed a prepositional genitive. Secondly, the enclitic article was related to the strong preference for the post-position of the adjective, including the demonstrative adjective in Romanian, a preference strengthened by the same N+A order in the local Dacian idioms (Brâncuş 2004). Graur (1967), Renzi (1993) agree that the enclitic article develops by re-analysis of examples like homo ille bonus: the demonstrative, required as a licenser of the modifier, is re-analyzed as a constituent-part of the head noun: homo ille bonus [homo-ille] bonus. Both hypotheses are correct, but incomplete. Through our study of definiteness checking in Old Romanian, we prove that the Romanian definite article is a post-posed demonstrative, re-analyzed as a definite suffix. Secondly we claim that the development of the definite article out of a constituent which had [locative, deictic] features allowed the specialization of the Genitive into a referential anchoring Genitive co-occurring with the definite article and a non-anchored non-referential Genitive. Romanian thus verifies the typological generalization that only languages that have articles may develop specialized forms for anchoring vs. non-anchoring Gen (Koptjevskaya-Tamm 2005: 168). It is the correlation between the anchoring role of the Gen and the presence of the post-posed demonstrative which is still visible in the lower N[+def] examples noticed above. At the time of the lower N[+def] construction, the article was still interpretable as [locative, deictic]. The post-nominal demonstrative was required to license an anchoring inflected genitive, and was re-analyzed as the mark of an individuated, anchored head.

This paper was a review about theory of Henstock integral. Riemann gave a definition of integral based on the sum of the partitions in Integration area (interval [a, b]). Those partitions is a -positive constant. Independently, Henstock and Kurzweil replaces -positive constant on construction Riemann integral into a positive function, ie (x)>0 for every x[a, b]. This function is a partition in interval [a, b]. From this partitions, we can defined a new integral called Henstock integral. Henstock integral is referred to as a complete Riemann integral, because the basic properties of the Henstock integral is more constructive than Riemann Integral.

An article was published some time ago by two idiot mainstream mathematics academics:

https://www.maa.org/book/export/html/117830

In my article, I share with you the correct approach which has worked without fail in my teaching career. Enjoy!

Once you've read the article, take this quiz.

What is the remarkable fact about the Mean Value Theorem?

The Mean Value Theorem identity is given by  f ' (c) = [ f(b)-f(a)]/ (b-a)

[A]  There is a c in (a,b) such that a tangent line to x=c is parallel to the secant line whose endpoints are (a,f(a)) and (b, f(b)).

[B]  It is the average rate of change between x=a and x=b.

[C]  There may be more than one c such that f ' (c) = [ f(b)-f(a)]/ (b-a).

[D]  It's not the reason calculus works.

[E]  The slope of the tangent line to f at x=c is equal to the level magnitude (aka arithmetic mean) of all the y ordinate lengths of f ' in the interval (a,b).

In this document we present the main theory of Continuity of real valued functions. We provide detailed proofs of the fundamental results on continuous functions. We treat also continuous functions on compact sets as well as the behavior of monotone continuous functions

We show that if n>1 then there exists a Lebesgue null set in R^n containing a point of differentiability of each Lipschitz function f:R^n→R^(n-1); in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of σ-porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof.

In the field of analysis, it is common to distinguish hard (quantitative) analysis from soft (qualitative) analysis. Hard analysis deals with finite quantities and their quantitative properties. On the other hand, soft analysis tends to deal with more infinitary objects and their qualitative properties. It is well known that various correspondence principles can connect the results obtained by hard and soft analysis. This report will develop some quantitative notions in hard analysis by refining the corresponding ones in soft analysis and then applying them to deduce the quantitative Lebesgue differentiation theorem and regularity lemma.

Scientists are, all the time, in a struggle with uncertainty which is always a threat to a trustworthy scientific knowledge. A very simple and natural idea, to defeat uncertainty, is that of enclosing uncertain measured values in real closed intervals. On the basis of this idea, interval arithmetic is constructed. The idea of calculating with intervals is not completely new in mathematics: the concept has been known since Archimedes, who used guaranteed lower and upper bounds to compute his constant Pi. Interval arithmetic is now a broad field in which rigorous mathematics is associated with scientific computing. This connection makes it possible to solve uncertainty problems that cannot be efficiently solved by floating-point arithmetic. Today, application areas of interval methods include electrical engineering, control theory, remote sensing, experimental and computational physics, chaotic systems, celestial mechanics, signal processing, computer graphics, robotics, and computer-assisted proofs. The purpose of this book is to be a concise but informative introduction to the theories of interval arithmetic as well as to some of their computational and scientific applications.

While Newton and Leibniz were in the dark about much of why calculus works, they took a brute force approach by approximating slopes of non-parallel secant lines and in their ignorance laid the foundations for much of the rot one finds in mainstream mathematics academia: theory of limits, infinity, infinitesimals and many other ill-formed concepts.

The Abel prize is awarded to unremarkable idiots in most cases and yet my work alone is worth many Abel prizes. I was the first to produce a rigorous formulation of calculus - the New Calculus:

https://www.academia.edu/41616655/An_Introduction_to_the_Single_Variable_New_Calculus

Thus far I have been met with stiff resistance from the incompetent, arrogant and incorrigibly stupid math academics of the mainstream establishment. They have labeled me a crank and attributed all sorts of mental diseases in order to discredit me. There are several web sites dedicated to libeling my good name and presenting a false picture of me and my work.

https://youtu.be/XBjLuj5LQPo

We prove the sequential criterion for limits, the sequential criterion for continuity, the sequential criterion for absence of uniform continuity, and the preservation of Cauchy sequences by the uniformly continuous functions. Further, we give applications and solve several exercises that demonstrate the use of the criteria and highlight their importance.

The Heine-Borel theorem, which states that a subset of R is compact if and only if it is closed and bounded, is a fundamental theorem of real analysis, as it is equivalent to the completeness property of R. Herein, we give a proof of the Heine-Borel theorem that can be characterized as sequential, in the sense that in order to prove that a closed and bounded interval is compact and then to generalize this result to any closed and bounded subset of R, we construct a monotone sequence and use the monotone convergence theorem. All statements we use in the proof are explained extensively and in details, but at the cost of making the proof lengthy.

Στο παρόν έγγραφο θα παρουσιάσουμε την βασική θεωρία και μεθοδολογίες πάνω στο θέμα της σύνθεσης πραγματικών συναρτήσεων μιας μεταβλητής. Στην αρχή θα παρουσιάζεται το θέμα που μελετάμε με απλό τρόπο και έπειτα θα παραθέτουμε τον φορμαλιστικό μαθηματικό τρόπο με τον οποίο περιγράφεται η κάθε έννοια.

No mainstream math professor or math educator I've ever known, could tell the difference between C : 2r and C/2r.  In fact, most people mistakenly believe these are the same thing. The difference is both significant and important. One can't understand the concept of number without understanding this difference.

We prove a sequential if-and-only-if criterion for a real-valued function of one real variable to be Riemann integrable. The criterion follows from Riemann's definition of the integral and relates directly the integral to the limit of a sequence of Riemann sums, thus allowing us to make direct use of properties of the sequential limit to prove respective properties of the integral.

In this article, we offer a new polynomial or polynomial-exponential bounds for the exponential function. Its main interest is to be both simple and sharp, under some clear conditions on the parameters involved. Applications are given for a probability function and the Kummer beta function.

Starting from the comparison test for the convergence of real series, we prove the Cauchy-Hadamard theorem that determines the interval of convergence of a real power series. The comparison test results from the fact that every increasing and bounded above real sequence converges, which in turn results from the axiom of completeness of R.

In standard real analysis textbooks, the theorem that differentiability implies continuity is proved by means of the product rule for limits. Herein, we prove the theorem using only the epsilon-delta definitions of differentiability and continuity.

Defining the limit superior and the limit inferior of a real sequence as the supremum and the infimum, respectively, of the limit set of the sequence, we prove the basic properties of the two limits demonstrating the advantages of this definition.

This work aims to give meaning to the expression: P = p1 × p2 × p3 × .....p4 × ...... which is known as the infinite product of real terms. For its study, we will start from the notion of finite products of a set of real factors, and then we will proceed to define the infinite version using the concept of limit. Below, we will discuss some theorems and various important applications of the infinite products in mathematics, such as expressing some important functions.

Heine-Borel theorem: A set in R (and in any euclidean space)  is Compact if and only if its is Closed and Bounded. In this document we provide the necessary material for a proof of the Heine-Borel theorem, that we eventually prove using Lebesque Covering Lemma.