1.
Anagram
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The original word or phrase is known as the subject of the anagram. Any word or phrase that exactly reproduces the letters in order is an anagram. Someone who creates anagrams may be called an anagrammatist, and the goal of a serious or skilled anagrammatist is to produce anagrams that in some way reflect or comment on their subject, Anagrams are often used as a form of mnemonic device. Such an anagram may be a synonym or antonym of its subject, a parody and it sometimes changes a proper noun or personal name into a sentence, such as with William Shakespeare = I am a weakish speller or Madam Curie = Radium came. It can change parts of speech, such as the adjective silent to the verb listen, Anagrams itself can be anagrammatized as Ars magna. Anagrams can be traced back to the time of Moses, as Themuru or changing and they were popular throughout Europe during the Middle Ages, for example with the poet and composer Guillaume de Machaut. They are said to go back at least to the Greek poet Lycophron, in the third century BCE, Anagrams in Latin were considered witty over many centuries. Est vir qui adest, explained below, was cited as the example in Samuel Johnsons A Dictionary of the English Language, any historical material on anagrams must always be interpreted in terms of the assumptions and spellings that were current for the language in question. In particular, spelling in English only slowly became fixed, there were attempts to regulate anagram formation, an important one in English being that of George Puttenhams Of the Anagram or Posy Transposed in The Art of English Poesie. The origins of these are not documented, Latin continued to influence letter values. There was a tradition of allowing anagrams to be perfect if the letters were all used once. This can be seen in a popular Latin anagram against the Jesuits, Societas Jesu turned into Vitiosa seces, the rules were not completely fixed in the 17th century. When it comes to the 17th century and anagrams in English or other languages, the lawyer Thomas Egerton was praised through the anagram gestat honorem, the physician George Ent took the anagrammatic motto genio surget, which requires his first name as Georgius. James Is courtiers discovered in James Stuart a just master, walter Quin, tutor to the future Charles I, worked hard on multilingual anagrams on the name of father James. Dryden in MacFlecknoe disdainfully called the pastime the torturing of one poor word ten thousand ways, an example from France was a flattering anagram for Cardinal Richelieu, comparing him to Hercules or at least one of his hands, where Armand de Richelieu became Ardue main dHercule. Examples from the century are the transposition of Horatio Nelson into Honor est a Nilo. With the advent of surrealism as a movement, anagrams regained the artistic respect they had had in the Baroque period. The German poet Unica Zürn, who made use of anagram techniques, came to regard obsession with anagrams as a dangerous fever
2.
Multiset
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In mathematics, a multiset is a generalization of the concept of a set that, unlike a set, allows multiple instances of the multisets elements. For example, and are different multisets although they are the same set, however, order does not matter, so and are the same multiset. The multiplicity of an element is the number of instances of the element in a specific multiset, however, the use of multisets predates the word multiset by many centuries. Knuth attributes the first study of multisets to the Indian mathematician Bhāskarāchārya, knuth also lists other names that were proposed or used for multisets, including list, bunch, bag, heap, sample, weighted set, collection, and suite. The number of times an element belongs to the multiset is the multiplicity of that member, the total number of elements in a multiset, including repeated memberships, is the cardinality of the multiset. For example, in the multiset the multiplicities of the members a, b, and c are respectively 2,3, and 1, to distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used, the multiset can be represented as. In multisets, as in sets and in contrast to tuples, the order of elements is irrelevant, The multisets and are equal. Wayne Blizard traced multisets back to the origin of numbers, arguing that “in ancient times. This shows that people implicitly used multisets even before mathematics emerged and this shows that necessity in this structure has been always so urgent that multisets have been several times rediscovered and appeared in literature under different names. For instance, they were referred to as bags by James Lyle Peterson in 1981, a multiset has been also called an aggregate, heap, bunch, sample, weighted set, occurrence set, and fireset. Although multisets were implicitly utilized from ancient times, their explicit exploration happened much later, the first known study of multisets is attributed to the Indian mathematician Bhāskarāchārya circa 1150, who described permutations of multisets. The work of Marius Nizolius contains another early reference to the concept of multisets, athanasius Kircher found the number of multiset permutations when one element can be repeated. Jean Prestet published a rule for multiset permutations in 1675. John Wallis explained this rule in detail in 1685. In the explicit form, multisets appeared in the work of Richard Dedekind, other mathematicians formalized multisets and began to study them as a precise mathematical object in the 20th century. One of the simplest and most natural examples is the multiset of prime factors of a number n, here the underlying set of elements is the set of prime divisors of n. For example, the number 120 has the prime factorization 120 =233151 which gives the multiset, a related example is the multiset of solutions of an algebraic equation. A quadratic equation, for example, has two solutions, however, in some cases they are both the same number
3.
C (programming language)
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C was originally developed by Dennis Ritchie between 1969 and 1973 at Bell Labs, and used to re-implement the Unix operating system. C has been standardized by the American National Standards Institute since 1989, C is an imperative procedural language. Therefore, C was useful for applications that had formerly been coded in assembly language. Despite its low-level capabilities, the language was designed to encourage cross-platform programming, a standards-compliant and portably written C program can be compiled for a very wide variety of computer platforms and operating systems with few changes to its source code. The language has become available on a wide range of platforms. In C, all code is contained within subroutines, which are called functions. Function parameters are passed by value. Pass-by-reference is simulated in C by explicitly passing pointer values, C program source text is free-format, using the semicolon as a statement terminator and curly braces for grouping blocks of statements. The C language also exhibits the characteristics, There is a small, fixed number of keywords, including a full set of flow of control primitives, for, if/else, while, switch. User-defined names are not distinguished from keywords by any kind of sigil, There are a large number of arithmetical and logical operators, such as +, +=, ++, &, ~, etc. More than one assignment may be performed in a single statement, function return values can be ignored when not needed. Typing is static, but weakly enforced, all data has a type, C has no define keyword, instead, a statement beginning with the name of a type is taken as a declaration. There is no function keyword, instead, a function is indicated by the parentheses of an argument list, user-defined and compound types are possible. Heterogeneous aggregate data types allow related data elements to be accessed and assigned as a unit, array indexing is a secondary notation, defined in terms of pointer arithmetic. Unlike structs, arrays are not first-class objects, they cannot be assigned or compared using single built-in operators, There is no array keyword, in use or definition, instead, square brackets indicate arrays syntactically, for example month. Enumerated types are possible with the enum keyword and they are not tagged, and are freely interconvertible with integers. Strings are not a data type, but are conventionally implemented as null-terminated arrays of characters. Low-level access to memory is possible by converting machine addresses to typed pointers
4.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
5.
Computer science
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Computer science is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. An alternate, more succinct definition of science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems and its fields can be divided into a variety of theoretical and practical disciplines. Some fields, such as computational complexity theory, are highly abstract, other fields still focus on challenges in implementing computation. Human–computer interaction considers the challenges in making computers and computations useful, usable, the earliest foundations of what would become computer science predate the invention of the modern digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, further, algorithms for performing computations have existed since antiquity, even before the development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623, in 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner. He may be considered the first computer scientist and information theorist, for, among other reasons and he started developing this machine in 1834, and in less than two years, he had sketched out many of the salient features of the modern computer. A crucial step was the adoption of a card system derived from the Jacquard loom making it infinitely programmable. Around 1885, Herman Hollerith invented the tabulator, which used punched cards to process statistical information, when the machine was finished, some hailed it as Babbages dream come true. During the 1940s, as new and more powerful computing machines were developed, as it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. Computer science began to be established as an academic discipline in the 1950s. The worlds first computer science program, the Cambridge Diploma in Computer Science. The first computer science program in the United States was formed at Purdue University in 1962. Since practical computers became available, many applications of computing have become distinct areas of study in their own rights and it is the now well-known IBM brand that formed part of the computer science revolution during this time. IBM released the IBM704 and later the IBM709 computers, still, working with the IBM was frustrating if you had misplaced as much as one letter in one instruction, the program would crash, and you would have to start the whole process over again. During the late 1950s, the science discipline was very much in its developmental stages. Time has seen significant improvements in the usability and effectiveness of computing technology, modern society has seen a significant shift in the users of computer technology, from usage only by experts and professionals, to a near-ubiquitous user base
6.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
7.
Equivalence relation
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In mathematics, an equivalence relation is a binary relation that is at the same time a reflexive relation, a symmetric relation and a transitive relation. As a consequence of these properties an equivalence relation provides a partition of a set into equivalence classes, a given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. That is, for all a, b and c in X, a ~ b if and only if b ~ a. if a ~ b and b ~ c then a ~ c. X together with the relation ~ is called a setoid, the equivalence class of a under ~, denoted, is defined as =. Let the set have the equivalence relation, the following sets are equivalence classes of this relation, =, = =. The set of all classes for this relation is. The following are all equivalence relations, Has the same birthday as on the set of all people, is similar to on the set of all triangles. Is congruent to on the set of all triangles, is congruent to, modulo n on the integers. Has the same image under a function on the elements of the domain of the function, has the same absolute value on the set of real numbers Has the same cosine on the set of all angles. The relation ≥ between real numbers is reflexive and transitive, but not symmetric, for example,7 ≥5 does not imply that 5 ≥7. It is, however, a total order, the relation has a common factor greater than 1 with between natural numbers greater than 1, is reflexive and symmetric, but not transitive. The empty relation R on a non-empty set X is vacuously symmetric and transitive, a partial order is a relation that is reflexive, antisymmetric, and transitive. Equality is both a relation and a partial order. Equality is also the relation on a set that is reflexive. In algebraic expressions, equal variables may be substituted for one another, the equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class. A strict partial order is irreflexive, transitive, and asymmetric, a partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reflexive if and only if for all a ∈ X, a reflexive and symmetric relation is a dependency relation, if finite, and a tolerance relation if infinite. A preorder is reflexive and transitive, a congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraic structure, and which respects the additional structure
8.
Jordan normal form
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Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal, and with identical diagonal entries to the left and below them. This condition is satisfied if K is algebraically closed. The diagonal entries of the form are the eigenvalues. If the operator is given by a square matrix M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix, the Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for normal matrices, is a special case of the Jordan normal form. The Jordan normal form is named after Camille Jordan, some textbooks have the ones on the subdiagonal, i. e. immediately below the main diagonal instead of on the superdiagonal. The eigenvalues are still on the main diagonal, an n × n matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces is n. Or, equivalently, if and only if A has n linearly independent eigenvectors, consider the following matrix, A =. Including multiplicity, the eigenvalues of A are λ =1,2,4,4, the dimension of the eigenspace corresponding to the eigenvalue 4 is 1, so A is not diagonalizable. However, there is an invertible matrix P such that A = PJP−1, the matrix J is almost diagonal. This is the Jordan normal form of A, the section Example below fills in the details of the computation. In general, a complex matrix A is similar to a block diagonal matrix J = where each block Ji is a square matrix of the form J i =. So there exists an invertible matrix P such that P−1AP = J is such that the only non-zero entries of J are on the diagonal, J is called the Jordan normal form of A. Each Ji is called a Jordan block of A, in a given Jordan block, every entry on the superdiagonal is 1. Assuming this result, we can deduce the properties, Counting multiplicity. Given an eigenvalue λi, its geometric multiplicity is the dimension of Ker, the sum of the sizes of all Jordan blocks corresponding to an eigenvalue λi is its algebraic multiplicity. A is diagonalizable if and only if, for every eigenvalue λ of A, the Jordan block corresponding to λ is of the form λ I + N, where N is a nilpotent matrix defined as Nij = δi, j−1
9.
Modular arithmetic
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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, a familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7,00 now, then 8 hours later it will be 3,00. Usual addition would suggest that the time should be 7 +8 =15. Likewise, if the clock starts at 12,00 and 21 hours elapse, then the time will be 9,00 the next day, because the hour number starts over after it reaches 12, this is arithmetic modulo 12. According to the definition below,12 is congruent not only to 12 itself, Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations on integers, addition, subtraction, and multiplication. For a positive n, two integers a and b are said to be congruent modulo n, written, a ≡ b. The number n is called the modulus of the congruence, for example,38 ≡14 because 38 −14 =24, which is a multiple of 12. The same rule holds for negative values, −8 ≡72 ≡ −3 −3 ≡ −8. Equivalently, a ≡ b mod n can also be thought of as asserting that the remainders of the division of both a and b by n are the same, for instance,38 ≡14 because both 38 and 14 have the same remainder 2 when divided by 12. It is also the case that 38 −14 =24 is a multiple of 12. A remark on the notation, Because it is common to consider several congruence relations for different moduli at the same time, in spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been if the notation a ≡n b had been used. The properties that make this relation a congruence relation are the following, if a 1 ≡ b 1 and a 2 ≡ b 2, then, a 1 + a 2 ≡ b 1 + b 2 a 1 − a 2 ≡ b 1 − b 2. The above two properties would still hold if the theory were expanded to all real numbers, that is if a1, a2, b1, b2. The next property, however, would fail if these variables were not all integers, the notion of modular arithmetic is related to that of the remainder in Euclidean division. The operation of finding the remainder is referred to as the modulo operation. For example, the remainder of the division of 14 by 12 is denoted by 14 mod 12, as this remainder is 2, we have 14 mod 12 =2
10.
Up to
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In mathematics, the phrase up to appears in discussions about the elements of a set, and the conditions under which subsets of those elements may be considered equivalent. The statement elements a and b of set S are equivalent up to X means that a and b are equivalent if criterion X is ignored and that is, a and b can be transformed into one another if a transform corresponding to X is applied. Looking at the entire set S, when X is ignored the elements can be arranged in subsets whose elements are equivalent, such subsets are called equivalence classes. If X is some property or process, the phrase up to X means disregarding a possible difference in X, for instance the statement an integers prime factorization is unique up to ordering, means that the prime factorization is unique if we disregard the order of the factors. Further examples concerning up to isomorphism, up to permutations and up to rotations are described below, in informal contexts, mathematicians often use the word modulo for similar purposes, as in modulo isomorphism. The Tetris game does not allow reflections, so the former notation is likely to more natural. To add in the count, there is no formal notation. However, it is common to write there are seven reflecting tetrominos up to rotations, in this, Tetris provides an excellent example, as a reader might simply count 7 pieces ×4 rotations as 28, where some pieces have fewer than four rotation states. In the eight queens puzzle, if the eight queens are considered to be distinct, the regular n-gon, for given n, is unique up to similarity. In other words, if all similar n-gons are considered instances of the same n-gon, then there is only one regular n-gon. In group theory, for example, we may have a group G acting on a set X, another typical example is the statement that there are two different groups of order 4 up to isomorphism, or modulo isomorphism, there are two groups of order 4. This means that there are two classes of groups of order 4, assuming we consider groups to be equivalent if they are isomorphic. A hyperreal x and its standard part st are equal up to an infinitesimal difference, adequality All other things being equal Modulo Quotient set Quotient group Synecdoche Abuse of notation Up-to Techniques for Weak Bisimulation
11.
Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
12.
Singular-value decomposition
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In linear algebra, the singular value decomposition is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix to any m × n matrix via an extension of polar decomposition and it has many useful applications in signal processing and statistics. The diagonal entries σ i of Σ are known as the values of M. The columns of U and the columns of V are called the left-singular vectors and right-singular vectors of M, the singular value decomposition can be computed using the following observations, The left-singular vectors of M are a set of orthonormal eigenvectors of MM∗. The right-singular vectors of M are a set of eigenvectors of M∗M. The non-zero singular values of M are the roots of the non-zero eigenvalues of both M∗M and MM∗. Suppose M is a m × n matrix whose entries come from the field K, V∗ is the conjugate transpose of the n × n unitary matrix, V, thus also unitary. The diagonal entries σi of Σ are known as the values of M. A common convention is to list the singular values in descending order, in this case, the diagonal matrix, Σ, is uniquely determined by M. Thus the expression UΣV∗ can be interpreted as a composition of three geometrical transformations, a rotation or reflection, a scaling, and another rotation or reflection. For instance, the figure above explains how a matrix can be described as such a sequence. If the rotation is done first, M = PR, then R is the same and P = UΣU∗ has the same eigenvalues and this shows that the SVD is a generalization of the eigenvalue decomposition of pure stretches in orthogonal directions to arbitrary matrices which both stretch and rotate. As shown in the figure, the values can be interpreted as the semiaxes of an ellipse in 2D. This concept can be generalized to n-dimensional Euclidean space, with the values of any n × n square matrix being viewed as the semiaxes of an n-dimensional ellipsoid. Since U and V∗ are unitary, the columns of each of them form a set of orthonormal vectors, the matrix M maps the basis vector Vi to the stretched unit vector σi Ui. By the definition of a matrix, the same is true for their conjugate transposes U∗ and V. In short, the columns of U, U∗, V, and V∗ are orthonormal bases. Consider the 4 ×5 matrix M = A singular value decomposition of this matrix is given by UΣV∗ U = Σ = V ∗ = Notice Σ is zero outside of the diagonal and one diagonal element is zero
13.
Weyr canonical form
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In mathematics, in linear algebra, a Weyr canonical form is a square matrix satisfying certain conditions. A square matrix is said to be in the Weyr canonical form if the matrix satisfies the conditions defining the Weyr canonical form, the Weyr form was discovered by the Czech mathematician Eduard Weyr in 1885. The Weyr form did not become popular among mathematicians and it was overshadowed by the closely related, the Weyr form has been rediscovered several times since Weyr’s original discovery in 1885. This form has variously called as modified Jordan form, reordered Jordan form, second Jordan form. The current terminology is credited to Shapiro who introduced it in a paper published in the American Mathematical Monthly in 1999, recently several applications have been found for the Weyr matrix. Of particular interest is an application of the Weyr matrix in the study of invariants in biomathematics. The first superdiagonal blocks W i, i +1 are full column rank n i × n i +1 matrices in reduced form for i =1, …, r −1. All other blocks of W are zero, in this case, we say that W has Weyr structure. The following is an example of a basic Weyr matrix, W = = In this matrix, n =9 and n 1 =4, n 2 =2, n 3 =2, n 4 =1. So W has the Weyr structure. Also, W11 = = λ I4, W22 = = λ I2, W33 = = λ I2, W44 = = λ I1 and W12 =, W23 =, W34 =. Let W be a matrix and let λ1, …, λ k be the distinct eigenvalues of W. We say that W is in Weyr form if W has the following form, the following image shows an example of a general Weyr matrix consisting of three basic Weyr matrix blocks. The matrix W is called the Weyr form of A, Let A be a square matrix of order n over an algebraically closed field and let the distinct eigenvalues of A be λ1, λ2, …, λ k. So the problem of reducing A to the Weyr form reduces to the problem of reducing the nilpotent matrices N i to the Weyr form. Given a nilpotent square matrix A of order n over a closed field F, the following algorithm produces an invertible matrix C. Step 1 Let A1 = A Step 2 Compute a basis for the space of A1. Extend the basis for the space of A1 to a basis for the n -dimensional vector space F n
14.
Vector space
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A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers. The operations of addition and scalar multiplication must satisfy certain requirements, called axioms. Euclidean vectors are an example of a vector space and they represent physical quantities such as forces, any two forces can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces and these vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are commonly used. This is particularly the case of Banach spaces and Hilbert spaces, historically, the first ideas leading to vector spaces can be traced back as far as the 17th centurys analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Today, vector spaces are applied throughout mathematics, science and engineering, furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques, Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra. The concept of space will first be explained by describing two particular examples, The first example of a vector space consists of arrows in a fixed plane. This is used in physics to describe forces or velocities, given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w, when a is negative, av is defined as the arrow pointing in the opposite direction, instead. Such a pair is written as, the sum of two such pairs and multiplication of a pair with a number is defined as follows, + = and a =. The first example above reduces to one if the arrows are represented by the pair of Cartesian coordinates of their end points. A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below, elements of V are commonly called vectors. Elements of F are commonly called scalars, the second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are represented in boldface to distinguish them from scalars
15.
Hilbert space
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The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of algebra and calculus from the two-dimensional Euclidean plane. A Hilbert space is a vector space possessing the structure of an inner product that allows length. Furthermore, Hilbert spaces are complete, there are limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces, the earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis —and ergodic theory, john von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis, geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space, at a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be specified by its coordinates with respect to a set of coordinate axes. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of the space of sequences that are square-summable. The latter space is often in the literature referred to as the Hilbert space. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of vectors, denoted by ℝ3. The dot product takes two vectors x and y, and produces a real number x·y, If x and y are represented in Cartesian coordinates, then the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3. The dot product satisfies the properties, It is symmetric in x and y, x · y = y · x. It is linear in its first argument, · y = ax1 · y + bx2 · y for any scalars a, b, and vectors x1, x2, and y. It is positive definite, for all x, x · x ≥0, with equality if. An operation on pairs of vectors that, like the dot product, a vector space equipped with such an inner product is known as a inner product space. Every finite-dimensional inner product space is also a Hilbert space, multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist
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Cardinality
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In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = contains 3 elements, there are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted | A |, with a bar on each side, this is the same notation as absolute value. Alternatively, the cardinality of a set A may be denoted by n, A, card, while the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets. Two sets A and B have the same cardinality if there exists a bijection, that is, such sets are said to be equipotent, equipollent, or equinumerous. This relationship can also be denoted A≈B or A~B, for example, the set E = of non-negative even numbers has the same cardinality as the set N = of natural numbers, since the function f = 2n is a bijection from N to E. A has cardinality less than or equal to the cardinality of B if there exists a function from A into B. A has cardinality less than the cardinality of B if there is an injective function. If | A | ≤ | B | and | B | ≤ | A | then | A | = | B |, the axiom of choice is equivalent to the statement that | A | ≤ | B | or | B | ≤ | A | for every A, B. That is, the cardinality of a set was not defined as an object itself. However, such an object can be defined as follows, the relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation then consists of all sets which have the same cardinality as A. There are two ways to define the cardinality of a set, The cardinality of a set A is defined as its class under equinumerosity. A representative set is designated for each equivalence class, the most common choice is the initial ordinal in that class. This is usually taken as the definition of number in axiomatic set theory. Assuming AC, the cardinalities of the sets are denoted ℵ0 < ℵ1 < ℵ2 < …. For each ordinal α, ℵ α +1 is the least cardinal number greater than ℵ α
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Compact space
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In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a set of points. This notion is defined for general topological spaces than Euclidean space in various ways. One such generalization is that a space is compact if any infinite sequence of points sampled from the space must frequently get arbitrarily close to some point of the space. An equivalent definition is that every sequence of points must have an infinite subsequence that converges to some point of the space, the Heine-Borel theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses a number of points in the closed unit interval some of those points must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, the same set of points would not accumulate to any point of the open unit interval, so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded, in particular, the sequence of points 0, 1, 2, 3, … has no subsequence that converges to any given real number. Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points, the term compact was introduced into mathematics by Maurice Fréchet in 1904 as a distillation of this concept. Various equivalent notions of compactness, including sequential compactness and limit point compactness, in general topological spaces, however, different notions of compactness are not necessarily equivalent. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, the term compact set is sometimes a synonym for compact space, but usually refers to a compact subspace of a topological space. In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano had been aware that any bounded sequence of points has a subsequence that must eventually get close to some other point. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts until it closes down on the limit point. The full significance of Bolzanos theorem, and its method of proof, in the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points. The idea of regarding functions as points of a generalized space dates back to the investigations of Giulio Ascoli. The uniform limit of this sequence then played precisely the same role as Bolzanos limit point and this ultimately led to the notion of a compact operator as an offshoot of the general notion of a compact space. It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property, in 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous
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Hausdorff space
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In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space and it implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology, hausdorffs original definition of a topological space included the Hausdorff condition as an axiom. Points x and y in a topological space X can be separated by neighbourhoods if there exists a neighbourhood U of x, X is a Hausdorff space if all distinct points in X are pairwise neighborhood-separable. This condition is the separation axiom, which is why Hausdorff spaces are also called T2 spaces. The name separated space is also used, a related, but weaker, notion is that of a preregular space. X is a space if any two topologically distinguishable points can be separated by neighbourhoods. Preregular spaces are also called R1 spaces, the relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is both preregular and Kolmogorov, a topological space is preregular if and only if its Kolmogorov quotient is Hausdorff. For a topological space X, the following are equivalent, X is a Hausdorff space, limits of nets in X are unique. Limits of filters on X are unique, any singleton set ⊂ X is equal to the intersection of all closed neighbourhoods of x. The diagonal Δ = is closed as a subset of the product space X × X, almost all spaces encountered in analysis are Hausdorff, most importantly, the real numbers are a Hausdorff space. More generally, all spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups, a simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set. Pseudometric spaces typically are not Hausdorff, but they are preregular, indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff. The related concept of Scott domain also consists of non-preregular spaces, while the existence of unique limits for convergent nets and filters implies that a space is Hausdorff, there are non-Hausdorff T1 spaces in which every convergent sequence has a unique limit. Subspaces and products of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff, in fact, every topological space can be realized as the quotient of some Hausdorff space. Hausdorff spaces are T1, meaning that all singletons are closed, another nice property of Hausdorff spaces is that compact sets are always closed
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Homeomorphism
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In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος = similar and μορφή = shape, roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. A function f, X → Y between two spaces and is called a homeomorphism if it has the following properties, f is a bijection, f is continuous. A function with three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic, a self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form a relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes, the open interval is homeomorphic to the real numbers R for any a < b. The unit 2-disc D2 and the square in R2 are homeomorphic. An example of a mapping from the square to the disc is, in polar coordinates. The graph of a function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is an homeomorphism between the domain of the parametrization and the curve, a chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space. The stereographic projection is a homeomorphism between the sphere in R3 with a single point removed and the set of all points in R2. If G is a group, its inversion map x ↦ x −1 is a homeomorphism. Also, for any x ∈ G, the left translation y ↦ x y, the right translation y ↦ y x, rm and Rn are not homeomorphic for m ≠ n. The Euclidean real line is not homeomorphic to the circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2. The third requirement, that f −1 be continuous, is essential, consider for instance the function f, [0, 2π) → S1 defined by f =
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Canonical representation of a positive integer
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For example,1200 =24 ×31 ×52 =3 ×2 ×2 ×2 ×2 ×5 ×5 =5 ×2 ×3 ×2 ×5 ×2 ×2 = etc. The requirement that the factors be prime is necessary, factorizations containing composite numbers may not be unique. This theorem is one of the reasons why 1 is not considered a prime number, if 1 were prime. Book VII, propositions 30,31 and 32, and Book IX, proposition 14 of Euclids Elements are essentially the statement, proposition 30 is referred to as Euclids lemma. And it is the key in the proof of the theorem of arithmetic. Proposition 31 is proved directly by infinite descent, proposition 32 is derived from proposition 31, and prove that the decomposition is possible. Book IX, proposition 14 is derived from Book VII, proposition 30, indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. Article 16 of Gauss Disquisitiones Arithmeticae is a modern statement. < pk are primes and the αi are positive integers and this representation is commonly extended to all positive integers, including one, by the convention that the empty product is equal to 1. This representation is called the representation of n, or the standard form of n. For example 999 = 33×37,1000 = 23×53,1001 = 7×11×13 Note that factors p0 =1 may be inserted without changing the value of n, allowing negative exponents provides a canonical form for positive rational numbers. However, as Integer factorization of large integers is much harder than computing their product, gcd or lcm, these formulas have, in practice, many arithmetical functions are defined using the canonical representation. In particular, the values of additive and multiplicative functions are determined by their values on the powers of prime numbers, the proof uses Euclids lemma, if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b. We need to show that every integer greater than 1 is either prime or a product of primes, for the base case, note that 2 is prime. By induction, assume true for all numbers between 1 and n, if n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = ab and 1 < a ≤ b < n, by the induction hypothesis, a = p1p2. pj and b = q1q2. qk are products of primes. But then n = ab = p1p2. pjq1q2. qk is a product of primes, assume that s >1 is the product of prime numbers in two different ways, s = p 1 p 2 ⋯ p m = q 1 q 2 ⋯ q n. We must show m = n and that the qj are a rearrangement of the pi, by Euclids lemma, p1 must divide one of the qj, relabeling the qj if necessary, say that p1 divides q1