A report on Mathematics and Computer science

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

Charles Babbage, sometimes referred to as the "father of computing".

The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.

Ada Lovelace published the first algorithm intended for processing on a computer.

The quadratic formula expresses concisely the solutions of all quadratic equations

Rubik's cube: the study of its possible moves is a concrete application of group theory

The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.

Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.

The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.

Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.

Leonhard Euler created and popularized much of the mathematical notation used today.

Carl Friedrich Gauss, known as the prince of mathematicians

Mathematics is essential in many fields, including natural sciences, engineering, medicine, finance, computer science and social sciences.

- Mathematics

Computer science research also often intersects other disciplines, such as cognitive science, linguistics, mathematics, physics, biology, Earth science, statistics, philosophy, and logic.

- Computer science

9 related topics with Alpha

Logic

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Study of correct reasoning or good arguments.

A depiction from the 15th century of the square of opposition, which expresses the fundamental dualities of syllogistic.

Logic is studied in and applied to various fields, such as philosophy, mathematics, computer science, and linguistics.

Algorithm

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Ada Lovelace's diagram from "note G", the first published computer algorithm

Logical NAND algorithm implemented electronically in 7400 chip

Flowchart examples of the canonical Böhm-Jacopini structures: the SEQUENCE (rectangles descending the page), the WHILE-DO and the IF-THEN-ELSE. The three structures are made of the primitive conditional GOTO (IF test THEN GOTO step xxx, shown as diamond), the unconditional GOTO (rectangle), various assignment operators (rectangle), and HALT (rectangle). Nesting of these structures inside assignment-blocks result in complex diagrams (cf. Tausworthe 1977:100, 114).

The example-diagram of Euclid's algorithm from T.L. Heath (1908), with more detail added. Euclid does not go beyond a third measuring and gives no numerical examples. Nicomachus gives the example of 49 and 21: "I subtract the less from the greater; 28 is left; then again I subtract from this the same 21 (for this is possible); 7 is left; I subtract this from 21, 14 is left; from which I again subtract 7 (for this is possible); 7 is left, but 7 cannot be subtracted from 7." Heath comments that "The last phrase is curious, but the meaning of it is obvious enough, as also the meaning of the phrase about ending 'at one and the same number'."(Heath 1908:300).

A graphical expression of Euclid's algorithm to find the greatest common divisor for 1599 and 650.

"Inelegant" is a translation of Knuth's version of the algorithm with a subtraction-based remainder-loop replacing his use of division (or a "modulus" instruction). Derived from Knuth 1973:2–4. Depending on the two numbers "Inelegant" may compute the g.c.d. in fewer steps than "Elegant".

In mathematics and computer science, an algorithm is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation.

Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.

Many problems in mathematics have been shown to be undecidable after these initial examples were established.

Mathematical logic

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Mathematical logic is the study of formal logic within mathematics.

Computer scientists often focus on concrete programming languages and feasible computability, while researchers in mathematical logic often focus on computability as a theoretical concept and on noncomputability.

Lorenz cipher machine, used in World War II to encrypt communications of the German High Command

Cryptography

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Practice and study of techniques for secure communication in the presence of adversarial behavior.

Alphabet shift ciphers are believed to have been used by Julius Caesar over 2,000 years ago. This is an example with k = 3. In other words, the letters in the alphabet are shifted three in one direction to encrypt and three in the other direction to decrypt.

Reconstructed ancient Greek scytale, an early cipher device

First page of a book by Al-Kindi which discusses encryption of messages

16th-century book-shaped French cipher machine, with arms of Henri II of France

Enciphered letter from Gabriel de Luetz d'Aramon, French Ambassador to the Ottoman Empire, after 1546, with partial decipherment

Symmetric-key cryptography, where a single key is used for encryption and decryption

One round (out of 8.5) of the IDEA cipher, used in most versions of PGP and OpenPGP compatible software for time-efficient encryption of messages

Public-key cryptography, where different keys are used for encryption and decryption.

Whitfield Diffie and Martin Hellman, authors of the first published paper on public-key cryptography.

In this example the message is only signed and not encrypted.
1) Alice signs a message with her private key.
2) Bob can verify that Alice sent the message and that the message has not been modified.

Variants of the Enigma machine, used by Germany's military and civil authorities from the late 1920s through World War II, implemented a complex electro-mechanical polyalphabetic cipher. Breaking and reading of the Enigma cipher at Poland's Cipher Bureau, for 7 years before the war, and subsequent decryption at Bletchley Park, was important to Allied victory.

Poznań monument (center) to Polish cryptanalysts whose breaking of Germany's Enigma machine ciphers, beginning in 1932, altered the course of World War II

NSA headquarters in Fort Meade, Maryland

Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, electrical engineering, communication science, and physics.

Deductive reasoning

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Mental process of drawing deductive inferences.

The semantic approach is also referred to as the model-theoretic approach since the branch of mathematics known as model theory is often used to interpret these sentences.

But the subject of deductive reasoning is also pertinent to the computer sciences, for example, in the creation of artificial intelligence.

René Descartes, who is often credited as the father of modern philosophy, was often preoccupied with epistemological questions in his work.

Epistemology

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Branch of philosophy concerned with knowledge.

Bertrand Russell famously brought attention to the distinction between propositional knowledge and knowledge by acquaintance.

An Euler diagram representing a version of the traditional definition of knowledge that is adapted to the Gettier problem. This problem gives us reason to think that not all justified true beliefs constitute knowledge.

The analytic–synthetic distinction was first proposed by Immanuel Kant.

David Hume, one of the most staunch defenders of empiricism.

Certain forms exempt disciplines such as mathematics and logic from these requirements.

Work in this area spans several academic fields, including philosophy, computer science, economics, and statistics.

Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming.

Applied mathematics

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A numerical solution to the heat equation on a pump casing model using the finite element method.

Fluid mechanics is often considered a branch of applied mathematics and mechanical engineering.

Mathematical finance is concerned with the modelling of financial markets.

The Brown University Division of Applied Mathematics is the oldest applied math program in the U.S.

Applied mathematics has substantial overlap with statistics.

Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry.

Representation of the expression (8-6)*(3+1) as a Lisp tree, from a 1985 Master's Thesis.

Computer algebra

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RischIntegration.PNG of the algebraic function

In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects.