Topology

Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure. Topology builds on set theory, considering both sets of points and families of sets.
The word topology is used both for the area of study and for a family of sets with certain properties described below that are used to define a topological space. Of particular importance in the study of topology are functions or maps that are homeomorphisms. Informally, these functions can be thought of as those that stretch space without tearing it apart or sticking distinct parts together.
When the discipline was first properly founded, toward the end of the 19th century, it was called geometria situs (Latin geometry of place) and analysis situs (Latin analysis of place). From around 1925 to 1975 it was an important growth area within mathematics.
Topology is a large branch of mathematics that includes many subfields. The most basic division within topology is point-set topology, which investigates such concepts as compactness, connectedness, and countability; algebraic topology, which investigates such concepts as homotopy and homology; and geometric topology, which studies manifolds and their embeddings, including knot theory.


The homotopy groups of spheres describe the different ways spheres of various dimensions can be wrapped around each other. They are studied as part of algebraic topology. The topic can be hard to understand because the most interesting and surprising results involve spheres in higher dimensions. These are defined as follows: an n-dimensional sphere, n-sphere, consists of all the points in a space of n+1 dimensions that are a fixed distance from a center point. This definition is a generalization of the familiar circle (1-sphere) and sphere (2-sphere).

A homotopy from a circle around a sphere down to a single point.

The goal of algebraic topology is to categorize or classify topological spaces. Homotopy groups were invented in the late 19th century as a tool for such classification, in effect using the set of mappings from an n-sphere in to a space as a way to probe the structure of that space. An obvious question was how this new tool would work on n-spheres themselves. No general solution to this question has been found to date, but many homotopy groups of spheres have been computed and the results are surprisingly rich and complicated. The study of the homotopy groups of spheres has led to the development of many powerful tools used in algebraic topology.

It is often suggested that a topologist cannot tell the difference between a coffee cup and a doughnut. This is because these objects when thought of as topological spaces are homeomorphic. The above picture depicts a continuous deformation of a coffee cup into a doughnut such that at each stage the object is homeomorphic to the original.

Did you know?

• ...that it is possible to turn a sphere inside out without tears or creases? (The sphere is allowed to pass through itself).
• ...that the Klein bottle gives a two-fold covering space of itself?
• ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined?
• ...that you cannot knot strings in 4-dimensions? You can, however, knot 2-dimensional surfaces like spheres.

Statistics

Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the natural and social sciences to the humanities, government and business.
Statistical methods are used to summarize and describe a collection of data; this is called descriptive statistics. In addition, patterns in the data may be modeled in a way that accounts for randomness and uncertainty in the observations, and then used to draw inferences about the process or population being studied; this is called inferential statistics.
Statistics arose no later than the 18th century from the need of states to collect data on their people and economies, in order to administer them. The meaning broadened in the early 19th century to include the collection and analysis of data in general.


The normal distribution or Gaussian distribution is a continuous probability distribution that describes data that clusters around a mean or average. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known as the Gaussian function or bell curve. The normal distribution can be used to describe, at least approximately, any variable that tends to cluster around the mean. For example, the heights of adult males in the United States are roughly normally distributed, with a mean of about 70 inches.

Anscombe's quartet comprises four datasets which have identical simple statistical properties (mean, standard deviation, correlation, etc), yet which are revealed to be very different when inspected graphically. Each dataset consists of eleven (x,y) points. They were constructed in 1973 by the statistician F.J. Anscombe to demonstrate the importance of graphing data before analyzing it, and of the effect of outliers on the statistical properties of a dataset.

William Edwards Deming (October 14, 1900—December 20, 1993) was an American statistician, professor, author, lecturer, and consultant. Deming is widely credited with improving production in the United States during World War II, although he is perhaps best known for his work in Japan. There, from 1950 onward he taught top management how to improve design (and thus service), product quality, testing and sales (the last through global markets). Deming made a significant contribution to Japan's later renown for innovative high-quality products and its economic power. He is regarded as having had more impact upon Japanese manufacturing and business than any other individual not of Japanese heritage. Despite being considered something of a hero in Japan, he was only beginning to win widespread recognition in the U.S. at the time of his death.

• ...that William Sealy Gosset published under the pseudonym Student in order to avoid detection by his employer, and so his most famous achievement is now referred to as Student's t-distribution, which might otherwise have been Gosset's t-distribution?
• ...that in 1747, by dividing 12 men suffering from scurvy into six pairs and giving each group different additions to their basic diet for a period of two weeks, the surgeon James Lind conducted one of the first controlled experiments?
• ...that the Cauchy distribution is an example of a distribution which has no mean, variance or higher moments defined?
• ...that according to Benford's law, the first digit from many real-life sources of data is 1 almost one third of the time?
• ...that the Law of Truly Large Numbers of Diaconis and Mosteller states that with a sample size large enough, any outrageous thing is likely to happen?

Set Theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of objects can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the most well known.
Set theory, formalized using first-order logic, is the most common foundational system for mathematics. The language of set theory is used in the definitions of nearly all mathematical objects, such as functions, and concepts of set theory are integrated throughout the mathematics curriculum. Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams, to study collections of commonplace physical objects

A set is a collection of distinct objects considered as a whole. Sets are one of the most fundamental concepts in mathematics and their formalization at the end of the 19th century was a major event in the history of mathematics and lead to the unification of a number of different areas. The idea of function comes along naturally, as "morphisms" between sets.
The study of the structure of sets, set theory, can be viewed as a foundational ground for most of mathematical theories. Sets are usually defined axiomatically using an axiomatic set theory, this way to study sets was introduced by Georg Cantor between 1874 and 1884 and deeply inspired later works in logic. Sets are representable in the form of Venn diagrams, for instance it can represent the idea of union, intersection and other operations on sets.

The union of two sets is the set that contains everything that belongs to any of the sets, but nothing else. It is possible to define the union of several sets, and even of an infinite family of sets.


Georg Cantor (March 3, 1845 – January 6, 1918) was a German mathematician. He is best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.

Did you know?

• ... that there exists a composition of binary relations consistent with the composition of functions ?
• ... that there is an ordinal arithmetic extending the arithmetic of integers to the ordinal numbers ?

Science

Science, in the broadest sense of the term, refers to any system of knowledge attained by verifiable means. In a more restricted sense, science refers to a system of acquiring knowledge based on empiricism, experimentation, and methodological naturalism, as well as to the organized body of knowledge humans have gained by such qualified research.
Scientists maintain that scientific investigation must adhere to the scientific method, a rigorous process for properly developing and evaluating natural explanations for observable phenomena based on reliable empirical evidence and neutral, unbiased independent verification, and not on arguments from authority or popular preferences. Science therefore bypasses supernatural explanations, it instead only considers natural explanations that may be falsifiable.
Fields of science are distinguished as pure science or applied science. Pure science is principally involved with the discovery of new truths with less or no regard to their practical applications. Applied science is principally involved with the application of existing knowledge in new ways.
Mathematics is the language in which scientific information is best presented, often it is the only way to formulate and present scientific knowledge. Therefore whether mathematics is a science in itself or the framework of science is a matter of perspective.

In physical cosmology, the Big Bang is the scientific theory that the universe emerged from a tremendously dense and hot state about 13.7 billion years ago. The theory is based on the observations indicating the expansion of space (in accord with the Robertson-Walker model of general relativity) as indicated by the Hubble redshift of distant galaxies taken together with the cosmological principle. Extrapolated into the past, these observations show that the universe has expanded from a state in which all the matter and energy in the universe was at an immense temperature and density. Physicists do not widely agree on what happened before this, although general relativity predicts a gravitational singularity (for reporting on some of the more notable speculation on this issue, see cosmogony).
The term Big Bang is used both in a narrow sense to refer to a point in time when the observed expansion of the universe (Hubble's law) began — calculated to be 13.7 billion (1.37 × 10¹⁰) years ago (±2%) — and in a more general sense to refer to the prevailing cosmological paradigm explaining the origin and expansion of the universe, as well as the composition of primordial matter through nucleosynthesis as predicted by the Alpher-Bethe-Gamow theory.


Carl Edward Sagan (November 9, 1934 – December 20, 1996) was an American astronomer, astrobiologist, and highly successful science popularizer. He pioneered exobiology and promoted the Search for Extra-Terrestrial Intelligence (SETI). He is world-famous for writing popular science books and for co-writing and presenting the award-winning 1980 television series Cosmos: A Personal Voyage, which was the most-watched PBS program until Ken Burns' The Civil War in 1990. A book to accompany the program was also published. He also wrote the novel Contact, the basis for the 1997 film of the same name starring Jodie Foster. During his lifetime, Sagan published more than 600 scientific papers and popular articles and was author, co-author, or editor of more than 20 books. In his works, he frequently advocated scientific skepticism, humanism, and the scientific method.

A Tesla coil is a category of disruptive discharge transformer coils, named after their inventor, Nikola Tesla. Tesla coils are composed of coupled resonant electric circuits. Nikola Tesla actually experimented with a large variety of coils and configurations, so it is difficult to describe a specific mode of construction that will meet the wants of those who ask about "Tesla" coils. "Early coils" and "later coils" vary in configuration and setup. Tesla coils in general are very popular devices among high-voltage enthusiasts.

Did you know...

• ...that silica aerogel (pictured) holds 15 entries in the Guinness Book of Records for material properties, including best insulator and lowest-density solid?
• ...that acoustic levitation is a method for suspending matter in a fluid by using acoustic radiation pressure from intense sound waves in the medium?
• ...that the genus Entomocorus includes a catfish species that lives only one year?
• ...that a successful experimental system must be stable and reproducible enough for scientists to make sense of the system's behavior, but unpredictable enough that it can produce useful results?
• ...that Abbott Lawrence Rotch established the Blue Hill Meteorological Observatory in 1885, which maintains the longest-running meteorological record of any observation site in the United States?

Physics

Physics (from Greek φυσική / physikê meaning science of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern the fundamental principles governing the universe. Physics deals with the elementary constituents of the universe and their interactions. Therefore, it can be thought of as a foundational science, upon which stands "the central science" of chemistry, and the earth sciences, biological sciences, and social sciences. Discoveries in basic physics have important ramifications for all of science.
Physics, like all the sciences, is a work in progress. Experimental and theoretical physics researchers continue to find new phenomena and to create and refine new models and theories.
As the fundamental science its biggest and even main goal is to bring one unified theory to the universe.

Lightning is an atmospheric discharge of electricity accompanied by thunder, which typically occurs during thunderstorms, and sometimes during volcanic eruptions or dust storms. In the atmospheric electrical discharge, a leader of a bolt of lightning can travel at speeds of 60,000 m/s (130,000 mph), and can reach temperatures approaching 30,000 °C (54,000 °F), hot enough to fuse silica sand into glass channels known as fulgurites which are normally hollow and can extend some distance into the ground. There are some 16 million lightning storms in the world every year.

• October 29, 2009 - Results from the Fermi Gamma-ray Space Telescope indicate that "either that space-time is smooth and continuous, as general relativity proposed, or that the grains of space-time are smaller than we ever thought possible, and are having only the most minuscule effect on light waves". Discovery
• October 28, 2009 - NASA's Ares I-X rocket is sucessfully test launched from Kennedy Space Center LC-39B.
• November 7, 2008 - Physicists revisited hydrogen production experiment, performed by Italian scientist G. D. Botto in 1833 (PhysOrg.com)
• September 10, 2008 - The first particle beam is successfully steered around the Large Hadron Collider accelerator ring at CERN near Geneva, Switzerland.
• May 1, 2008 - A team of nanoscientists have grown twisted nanowires that look like pine trees by twisting the original wire from which the trees were grown. (Science, [1] [2])
• April 13, 2008 - John Wheeler died aged 96. "For me, he was the last Titan, the only physics superhero still standing." (Max Tegmark, New York Times)
• December 11, 2007 - NASA's voyager 2 probe's recent measurements prove that the solar system is asymmetrical. Several scientists suggest that this is caused by galactic magnetic fields. (CNN)
• November 24, 2007 - Speculation about space based solar power plants continues, and with oil prices rising, they might be economical too. (New Scientist)

The heliospheric current sheet extends to the outer reaches of the Solar System, and results from the influence of the Sun's rotating magnetic field on the plasma in the interplanetary medium.

 

Did you know...

• ...that in the Large Hadron Collider (LHC) (pictured) protons, accelerated to the energy of 8 TeV, move at 99.999999% of light speed and become 7,000 times (on grounds of mathematical calculations) heavier than at rest?
• ...that, at a speed of 299,792,458 m/s, light can travel to the Moon in 1.2 seconds?
• ...that the submarine telescope ANTARES, intended to detect neutrinos, may also be used to observe bioluminescent plankton and fish?
• ...that lasers can be used to separate two isotopes very efficiently?
• ...that neutron stars are so dense that a teaspoonful (5 mL) would have ten times the mass of all human world population?
• ...that every year, the Moon moves 3.82 cm away from Earth?
• ...that gold leaf is about 500 atoms thick?

Number Theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated. (See the list of number theory topics.)
The term "arithmetic" is also used to refer to number theory. This is a somewhat older term, which is no longer as popular as it once was. Number theory used to be called the higher arithmetic, but this too is dropping out of use. Nevertheless, it still shows up in the names of mathematical fields (arithmetic functions, arithmetic of elliptic curves, arithmetic geometry). This sense of the term arithmetic should not be confused either with elementary arithmetic, or with the branch of logic which studies Peano arithmetic as a formal system. Mathematicians working in the field of number theory are called number theorists.

In number theory, Sylvester's sequence is a sequence of integers in which each member of the sequence is the product of the previous members, plus one. Sylvester's sequence is named after James Joseph Sylvester, who first investigated it in 1880.
Its values grow doubly exponentially, and the sum of its reciprocals forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions with the same sum. The recurrence by which it is defined allows the numbers in the sequence to be factored more easily than other numbers of the same magnitude, but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its members. Values derived from this sequence have also been used to construct finite Egyptian fraction representations of 1, Sasakian Einstein manifolds, and hard instances for online algorithms.

Time-keeping on a clock gives an example of modular arithmetic, the "clock group" is represented by the group Z/12Z for a 12-hour clock and Z/24Z for a 24-hour clock.

                                               Did you know?

• ...that every positive integer can be expressed as the sum of four squares of integers?
• ...that it is impossible to separate any power higher than the second into two like powers?
• ...that the only integer that can be written as reverse powers is 16, i.e. 16 = 2^4 = 4^2?

Mathematics

Mathematics from the Greek: μαθηματικά or mathēmatiká, is the study of patterns. Such patterns include quantities (numbers) and their operations, interrelations, combinations and abstractions; and of space configurations and their structure, measurement, transformations, and generalizations. Mathematics evolved through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the systematic study of positions, shapes and motions of physical objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.


The trigonometric functions are functions of an angle; they are most important when studying triangles and modeling periodic phenomena, among many other applications. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers.
The study of trigonometric functions dates back to Babylonian times, and a considerable amount of fundamental work was done by ancient Greek, Indian and Arab mathematicians.

Credit: Phil Tregoning A Bézier curve is a parametric curve important in computer graphics and related fields. Widely publicized in 1962 by the French engineer Pierre Bézier, who used them to design automobile bodies, the curves were first developed in 1959 by Paul de Casteljau using de Casteljau's algorithm.
In the animation above, a quartic Bézier curve is constructed using control points P₀ through P₄. The green line segments join points moving at a constant rate from one control point to the next; the parameter t shows the progress over time. Meanwhile, the blue line segments join points moving in a similar manner along the green segments, and the magenta line segment points along the blue segments. Finally, the black point moves at a constant rate along the magenta line segment, tracing out the final curve in red. The curve is a fourth-degree function of its parameter t.
Quadratic and cubic Bézier curves are most common since higher-degree curves are more computationally costly to evaluate. When more complex shapes are needed, low-order Bézier curves are patched together.

• ...that there are 6 unsolved mathematics problems whose solutions will earn you one million US dollars each?
• ...that there are different sizes of infinite sets in set theory? More precisely, not all infinite cardinal numbers are equal?
• ...that every natural number can be written as the sum of four squares?
• ...that the largest known prime number is over 12 million digits long?
• ...that the set of rational numbers is equal in size to the subset of integers; that is, they can be put in one-to-one correspondence?
• ...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
• ...that it is unknown whether π and e are algebraically independent?
• ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
• ...that it is possible for a three dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.

Mathematic Logic

Logic (from Classical Greek λόγος logos; meaning 'speech/word') is the study of the principles and criteria of valid inference and demonstration. The term "logos" was also believed by the Greeks to be the universal power by which all reality was sustained and made coherent and consistent. As a formal science, logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and through the study of arguments in natural language. The field of logic ranges from core topics such as the study of fallacies and paradoxes, to specialized analysis of reasoning using probability and to arguments involving causality. Logic is also commonly used today in argumentation theory. [1] Traditionally, logic is studied as a branch of philosophy, one part of the classical trivium, which consisted of grammar, logic, and rhetoric. Since the mid-nineteenth century formal logic has been studied in the context of the foundations of mathematics. In 1903 Bertrand Russell and Alfred North Whitehead attempted to establish logic as the cornerstone of mathematics formally with the publication of Principia Mathematica. However, the system of Principia is no longer much used, having been largely supplanted by set theory. The development of formal logic and its implementation in computing machinery is the foundation of computer science.  In logic and mathematics, or, also known as logical disjunction or inclusive disjunction is a logical operator that results in true whenever one or more of its operands are true. In grammar, or is a coordinating conjunction. Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false. More generally a disjunction is a logical formula that can have one or more literals separated only by ORs. A single literal is often considered to be a degenerate disjunction. Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician and advocate for social reform. A prolific writer, he was also a populariser of philosophy and a commentator on a large variety of topics, ranging from very serious issues to those much less so. Continuing a family tradition in political affairs, he was a prominent anti-war activist, championing free trade between nations and anti-imperialism. Did you know? ...that the Law of noncontradiction is the same as the Law of contradiction? ...that nand alone or nor alone make a functionally complete set of logical operators? ...that according to the continuum hypothesis, there is no cardinal number in between aleph null and the cardinality of the continuum? ...that all 24 valid syllogistic forms have names like 'Camestrop' and 'Felapton'? ...that if a statement P implies another statement Q, and a third statement R also implies Q, and either P or R is true; then Q has to be true? ...that the dunce hat was named after a logician? ...that the collective noun for a group of logicians is a "sequitur of logicians"

Mathematic Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry is one of the two fields of pre-modern mathematics, the other being the study of numbers.
In modern times, geometric concepts have been extended. They sometimes show a high level of abstraction and complexity. Geometry now uses methods of calculus and abstract algebra, so that many modern branches of the field are not easily recognizable as the descendants of early geometry. (See areas of mathematics.) A geometer is one who works or is specified in geometry.

The trigonometric functions are functions of an angle; they are most important when studying triangles and modeling periodic phenomena, among many other applications. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers.
The study of trigonometric functions dates back to Babylonian times, and a considerable amount of fundamental work was done by ancient Greek, Indian and Arab mathematicians.

The above shows an example of doubly ruled surface - the hyperboloid of one sheet. Although the wires are straight lines, they are lying within the surface. Through any point on this surface pass two straight lines, so it is doubly ruled.

Euclid (also referred to as Euclid of Alexandria) (Greek: Εὐκλείδης) (c. 325–c. 265 BC), a Greek mathematician, who lived in Alexandria, Hellenistic Egypt, almost certainly during the reign of Ptolemy I (323 BC–283 BC), is often considered to be the "father of geometry". His most popular work, Elements, is thought to be one of the most successful textbooks in the history of mathematics. Within it, the properties of geometrical objects are deduced from a small set of axioms, thereby founding the axiomatic method of mathematics.

Did you know?

• ...that the hyperboloid of one sheet is a doubly ruled surface?
• ...that as the dimension of a hypersphere tends to infinity, its "volume" (content) tends to 0?
• ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
• ...that a regular heptagon is the regular polygon with the fewest number of sides which is not constructible with a compass and straightedge?
• ...that it is possible for a three dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.

Discrete Mathematics

Discrete mathematics, also called finite mathematics or decision mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the notion of continuity. Objects studied in finite mathematics are largely countable sets such as integers, finite graphs, and formal languages.
Discrete mathematics has become popular in recent decades because of its applications to computer science. Concepts and notations from discrete mathematics are useful to study or describe objects or problems in computer algorithms and programming languages. In some mathematics curricula, finite mathematics courses cover discrete mathematical concepts for business, while discrete mathematics courses emphasize concepts for computer science majors.

Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics. Aspects of combinatorics include "counting" the objects satisfying certain criteria (enumerative combinatorics), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and finding algebraic structures these objects may have (algebraic combinatorics).
Combinatorics is as much about problem solving as theory building, though it has developed powerful theoretical methods, especially since the later twentieth century (see the page List of combinatorics topics for details of the more recent development of the subject). One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas.

A Penrose tiling, an example of a tiling that can completely cover an infinite plane, but only in a pattern which is non-repeating (aperiodic).

Did you know?

• ...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
• ...that the Catalan numbers solve a number of problems in combinatorics such as the number of ways to completely parenthesize an algebraic expression with n+1 factors?

Mathematic Crytopraphy

Cryptography (from Greek κρύπτω, "to conceal, to obscure", and γράφω, "to etch, to inscribe, to write down") is, traditionally, the study of means of converting information from its normal, comprehensible form into an incomprehensible format, rendering it unreadable without secret knowledge — the art of encryption. Cryptography is often used to replace or in combination with steganography. In the past, cryptography helped ensure secrecy in important communications, such as those of spies, military leaders, and diplomats. In recent decades, the field of cryptography has expanded its remit in two ways. Firstly, it provides mechanisms for more than just keeping secrets: schemes like digital signatures and digital cash, for example. Secondly, cryptography has come to be in widespread use by many civilians who do not have extraordinary needs for secrecy, although typically it is transparently built into the infrastructure for computing and telecommunications, and users are not aware of it.

The Enigma machine was a portable cipher machine used to encrypt and decrypt secret messages. More precisely, Enigma was a family of related electro-mechanical rotor machines — there are a variety of different models.The Enigma was used commercially from the early 1920s on, and was also adopted by military and governmental services of a number of nations — most famously, by Nazi Germany before and during World War II. The German military model, the Wehrmacht Enigma, is the version most commonly discussed. Allied codebreakers were, in many cases, able to decrypt messages protected by the machine (see cryptanalysis of the Enigma). The intelligence gained through this source — codenamed ULTRA — was a significant aid to the Allied war effort. Some historians have suggested that the end of the European war was hastened by up to a year or more because of the decryption of German ciphers.

• New Windows malware "Gpcode.AK" appears in early June 2008, using RC4-128 and RSA-1024 ciphers to take document files hostage for ransom.
• RSA-640 was factored on November 2, 2005.
• From 14 August 2005–18 August 2005 the 25th Annual International Cryptology Conference CRYPTO 2005 took place in Santa Barbara, California, USA. At the rump session, an improved collision attack on SHA-1 was announced.
• RSA-200 was factored on 9 May 2005. At 663 bits (200 decimal digits), the number is the largest of the RSA numbers yet factored.
• The US Secret Service is reported to be using 4,000 of its computers in a distributed dictionary attack to solve passwords used to protect encryption keys [1]. They report particular success in crafting custom dictionaries based on knowledge of a suspect's personal interests.
• In Australia, the Vigenère cipher is being used to communicate with an extortionist via the advertisements in a newspaper 

Did you know...

...that the Pigpen cipher was used by the Freemasons for correspondence and record keeping?
...that Polish mathematician Marian Rejewski (pictured) deduced the wiring of the German Enigma machine in 1932 using theorems about permutations?

...that acoustic cryptanalysis is a type of attack that exploits sound in order to compromise a system?
...that one scheme to defeat spam involves proving that the sender has performed a small amount of computation: a proof-of-work system?

Mathematic Computer Science

Computer science is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. Computer science encompasses many branches; some emphasize the computation of specific results (such as computer graphics), while others (such as computational complexity theory) relate to properties of computational problems. Still others focus on the challenges in implementing computations. For example, programming language theory studies approaches to describing a computation, while computer programming applies specific programming languages to craft a solution to some concrete computational problems.


In mathematics and computing, an algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. The computational complexity and efficient implementation of the algorithm are important in computing, and this depends on suitable data structures.
Informally, the concept of an algorithm is often illustrated by the example of a recipe, although many algorithms are much more complex; algorithms often have steps that repeat (iterate) or require decisions (such as logic or comparison). In most higher level programs, algorithms act in complex patterns, each using smaller and smaller sub-methods which are built up to the program as a whole. In many programming languages, algorithms are implemented as functions or procedures.
The concept of an algorithm originated as a means of recording procedures for solving mathematical problems such as finding the common divisor of two numbers. The concept was formalized in 1936 through Alan Turing's Turing machines and Alonzo Church's lambda calculus, which in turn formed the foundation of computer science.
Most algorithms can be directly implemented by computer programs; any other algorithms can at least in theory be simulated by computer programs.

Alan Mathison Turing (June 23, 1912 – June 7, 1954) was a British mathematician, logician, and cryptographer. Turing is often considered to be the father of modern computer science.
With the Turing test, Turing made a significant and characteristically provocative contribution to debates regarding artificial intelligence: whether it will ever be possible to say that a machine is conscious and can think. He provided an influential formalisation of concepts of algorithm and computation with the Turing machine, formulating the now widely accepted "Turing" version of the Church–Turing thesis: that any practical computing model has either the equivalent or a subset of capabilities of a Turing machine. During World War II, Turing worked at Bletchley Park, Britain's codebreaking centre and was for a time head of Hut 8, the section responsible for German Naval cryptanalysis. He devised techniques for breaking German ciphers, including the method of the bombe, an electromechanical machine which found settings for the Enigma machine.
After the war, he worked at the National Physical Laboratory, creating an early design for a stored-program computer, but never actually built. In 1947 he moved to the University of Manchester, to work mainly on software for the Manchester Mark I, one of the earliest true computers.
In 1952, Turing was convicted of acts of gross indecency after he admitted a relationship with a man in Manchester. He was placed on probation and required to undergo hormone therapy. When Turing died in 1954, an inquest found that he had committed suicide by eating an apple laced with cyanide.