Header/ Footer in Latex

Creating headers and footers in Latex can be done by using the package “fancyhdr”. This is a short introduction, showing the most important features of the package. If you know “fancyhdr” and are looking for something particular, refer to the fancyhdr-documentation.

First of all, you need to tell Latex to use the package:

\usepackage{fancyhdr}

and change the style from “plain” to “fancy”:

\pagestyle{fancy}

You will now the get the default fancy pagestyle which adds a line at the top of every page, except for some exceptions (title-page, abstract, new chapter in report).

Default fancyhdr page style:

Above the line, Latex will print headings:

Book/report

Left-hand side: section
Right- hand side: chapter
Note: if you use the optional documentclass argument “twoside”, Latex will alter the position of the section and chapter. (e.g. \documentclass[twoside]{report}, also introducing non-symmetric margins).

Article

For acticles, Latex will print the section only (chapters cannot be used with articles).
The footer only includes the page number which is centered by default.

Custom fancyhdr page style:

Even though fancyhdr has a default page style, you are free to define headers/footers yourself , which is not too difficult after all.

First you need to clear the default layout:

\fancyhead{}
\fancyfoot{}

There are seven letters you need to know before you can define your own header/footer:

E: Even page
O: Odd page
L: Left field
C: Center field
R: Right field
H: Header
F: Footer

Now you can start to define your own layout. The definitions are added before the document starts, usually after the “usepackages”:

\fancyhead[CO,CE]{---Draft---}
\fancyfoot[CO,CE]{Confidential}
\fancyfoot[RO, LE] {\thepage}

The decorative lines can be changed by increasing/decreasing their thickness (0pt means no line):

\renewcommand{\headrulewidth}{0.4pt}
\renewcommand{\footrulewidth}{0.4pt}

Note: After the fancyhdr-documentation, the default layout is produced by the following commands:

\fancyhead[LE,RO]{\slshape \rightmark}
\fancyhead[LO,RE]{\slshape \leftmark}
\fancyfoot[C]{\thepage}
\headrulewidth 0.4pt
\footrulewidth 0 pt

Reference: http://texblog.wordpress.com


Glossary in Latex

A glossary is a nice thing to have in a report and usually very helpful. As you probably can imaging, it is very easy to create in Latex. Nevertheless, there are a few things to be done, especially generating the glossary-files.

First you have to tell Latex to use the glossary package and to create the glo-file containing all the glossar-entries in your document:

\usepackage{glossary}
\makeglossary

Next you have to add glossary entries to your document. They are of the following form:

\glossary{name={entry name}, description={entry description}}

Note: Usually, the glossary-entry should be added to keywords where they first appear.

A glossary-entry produces by default the following format:

“entry name” “entry description (on multiple lines if necessary)”, “page number”

And finally you have to tell Latex where to place you glossary inside the document which is done by the following command at the location you want to produce the glossary:

\printglossary

Optinally, you can reference to the glossary in the index (toc-file) by adding the following command after “\printglossary”:

\addcontentsline{toc}{chapter}{Glossary}

What you do first is generate your PDF once. An ist-file as well as a glossary file (*.glo) are generated. The glossary-file contains all the glossary entries found in the document in plain text. Next you type the following command in the command-line:

makeindex document.glo -s document.ist -t document.glg -o document.gls

generating the two files with the extensions *.gls and *.glg. If entries are ignored or rejected, which can be seen either in the glg-file or directly in the output of the makeindex-command, you have to check your glossary entries. The important of the two files is the *.gls-file, as it is used by Latex for the actual glossary. You now need to re-generate the PDF and if everything works fine, your glossary should appear where you wanted it.

Reference: http://texblog.wordpress.com

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function.^[1] It also includes the theories of differentiation, integration and measure, infinite series,^[2] and analytic functions. These theories are often studied in the context of real numbers, complex numbers, and real and complex functions. However, they can also be defined and studied in any space of mathematical objects that has a definition of nearness (a topological space) or, more specifically, distance (a metric space).

Subdivisions

Mathematical analysis includes the following subfields.

• Real analysis, the rigorous study of derivatives and integrals of functions of real variables. This includes the study of sequences and their limits, series, and measures.
• Functional analysis^[7] studies spaces of functions and introduces concepts such as Banach spaces and Hilbert spaces.
• Harmonic analysis deals with Fourier series and their abstractions.
• Complex analysis, the study of functions from the complex plane to itself which are complex differentiable (that is, holomorphic).
• Differential geometry, the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally.
• p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
• Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers. It is normally classed as model theory.
• Numerical analysis, the study of algorithms for approximating the problems of continuous mathematics.

Classical analysis would normally be understood as any work not using functional analysis techniques, and is sometimes also called hard analysis; it also naturally refers to the more traditional topics. The study of differential equations is now shared with other fields such as dynamical systems, though the overlap with conventional analysis is large.

Vahid Damanafshan

Reference: Wikipedia

Lebesgue integration

In mathematics, Lebesgue integration refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a sub-domain of the real line or a higher dimensional Euclidean space with respect to the Lebesgue measure. For more information, Please go to this page.
Vahid Damanafshan
Reference: Wikipedia

Measure

In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area and volume of Euclidean geometry to suitable subsets of Rⁿ, n=1,2,3,.... For instance, the Lebesgue measure of [0,1] in the real numbers is its length in the everyday sense of the word, specifically 1.

To qualify as a measure (see Definition below), a function that assigns a non-negative real number or infinity to a set's subsets must satisfy a few conditions. One important condition is countable additivity. This condition states that the size of the union of a sequence of disjoint subsets is equal to the sum of the sizes of the subsets. However, it is in general impossible to consistently associate a size to each subset of a given set and also satisfy the other axioms of a measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the subsets on which the measure is to be defined are called measurable and they are required to form a sigma-algebra, meaning that unions, intersections and complements of sequences of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be consistently defined, are necessarily complex to the point of incomprehensibility, in a sense badly mixed up with their complement; indeed, their existence is a non-trivial consequence of the axiom of choice.

Measure theory was developed in successive stages during the late 19th and early 20th century by Emile Borel, Henri Lebesgue, Johann Radon and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.

Vahid Damanafshan

Reference: Wikipedia

Walter Rudin

Walter Rudin (born 1921) is an American mathematician, currently a Professor Emeritus of Mathematics at the University of Wisconsin–Madison.

He is known for three of his books on mathematical analysis: Functional Analysis, Principles of Mathematical Analysis, and Real and Complex Analysis. The second and third books are affectionately called Baby Rudin and Big Rudin (or sometimes Papa Rudin) by math students.

Rudin was born into a Jewish family in Austria in 1921. They fled to France after the Anschluss in 1938. When France surrendered to Germany in 1940, Rudin fled to England and served in the British navy for the rest of the war. After the war he left for the United States, and earned his Ph.D. from Duke University in North Carolina in 1949. After that he was a C.L.E. Moore Instructor at MIT before becoming a professor at the University of Wisconsin–Madison.

In 1953, he married fellow mathematician Mary Ellen Estill. The two now reside in Madison, Wisconsin, in a home built by architect Frank Lloyd Wright.

Vahid Damanafshan

Reference: Wikipedia

Bernhard Riemann

Georg Friedrich Bernhard Riemann (help·info) (German pronunciation: [ˈriːman]; September 17, 1826 – July 20, 1866) was an influential German mathematician who made contributions to analysis and differential geometry, some of them enabling the later development of general relativity.

Early life

Riemann was born in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is Germany today. His father, Friedrich Bernhard Riemann, was a poor Lutheranian pastor in Breselenz who fought in the Napoleonic Wars. His mother died before her children had reached adulthood. Riemann was the second of six children, shy, and suffered from numerous nervous breakdowns. Riemann exhibited exceptional mathematical skills, such as fantastic calculation abilities, from an early age, but suffered from timidity and a fear of speaking in public.

Middle life

In high school, Riemann studied the Bible intensively, but he was often distracted by mathematics. To this end, he even tried to prove mathematically the correctness of the Book of Genesis. His teachers were amazed by his adept ability to solve complicated mathematical operations, in which he often outstripped his instructor's knowledge. During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school). After the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg. In 1846, at the age of 19, he started studying philology and theology in order to become a priest and help with his family's finances.

During the spring of 1846, his father (Friedrich Riemann), after gathering enough money to send Riemann to university, allowed him to stop studying theology and start studying mathematics. He was sent to the renowned University of Göttingen, where he first met Carl Friedrich Gauss, and attended his lectures on the method of least squares.

In 1847, Riemann moved to Berlin, where Jacobi, Dirichlet, Steiner, and Eisenstein were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849.

Later life

Bernhard Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for Einstein's general theory of relativity. In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. Although this attempt failed, it did result in Riemann finally being granted a regular salary. In 1859, following Dirichlet's death, he was promoted to head the mathematics department at Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality^{[citation needed]}—an idea that was ultimately vindicated with Einstein's contribution in the early 20th century. In 1862 he married Elise Koch and had a daughter.

Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866.^[1] He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) where he was buried in the cemetery in Biganzolo (Verbania). Meanwhile, in Göttingen his housekeeper tidied up some of the mess in his office, including much unpublished work. Riemann refused to publish incomplete work and some deep insights may have been lost forever.^[1]

Vahid Damanafshan,

Reference: Wikipedia