1.
Mathematics
–
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
2.
Complex number
–
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
3.
Regular Polytopes (book)
–
Regular Polytopes is a mathematical geometry book written by Canadian mathematician Harold Scott MacDonald Coxeter. Originally published in 1947, the book was updated and republished in 1963 and 1973, the book is a comprehensive survey of the geometry of regular polytopes, the generalisation of regular polygons and regular polyhedra to higher dimensions. Originating with an essay entitled Dimensional Analogy written in 1923, the first edition of the book took Coxeter twenty-four years to complete, regular Polytopes is a standard reference work on regular polygons, polyhedra and their higher dimensional analogues. It is unusual in the breadth of its coverage, its combination of mathematical rigour with geometric insight, Coxeter starts by introducing two-dimensional polygons and three-dimensional polyhedra. He then gives a rigorous definition of regularity and uses it to show that there are no other convex regular polyhedra apart from the five Platonic solids. The concept of regularity is extended to non-convex shapes such as star polygons and star polyhedra, to tessellations and honeycombs, Coxeter introduces and uses the groups generated by reflections that became known as Coxeter groups. The book combines algebraic rigour with clear explanations, many of which are illustrated with diagrams, the black and white plates in the book show solid models of three-dimensional polyhedra, and wire-frame models of projections of some higher-dimensional polytopes. At the end of each chapter Coxeter includes an Historical remarks section which provides a perspective of the development of the subject. The first two have been expounded by Sommerville and Neville, and we shall presuppose some familiarity with such treatises. Concerning the third, Poincaré wrote, A man who pursues it, will end up holding on to the fourth dimension. ”The original 1948 edition received a more complete review by M. Goldberg in MR0027148
4.
Fisher's inequality
–
Let, v be the number of varieties of plants, b be the number of blocks. Fishers inequality states simply that b ≥ v, let the incidence matrix M be a v × b matrix defined so that Mi, j is 1 if element i is in block j and 0 otherwise. Then B = MMT is a v × v matrix such that Bi, i = r and Bi, since r ≠ λ, det ≠0, so rank = v, on the other hand, rank = rank ≤ b, so v ≤ b. Fishers inequality is valid for more classes of designs. A pairwise balanced design is a set X together with a family of subsets of X such that every pair of elements of X is contained in exactly λ subsets. The set X is allowed to be one of the subsets, and if all the subsets are copies of X, the size of X is v and the number of subsets in the family is b. Theorem, For any non-trivial PBD, v ≤ b, in another direction, Ray-Chaudhuri and Wilson proved in 1975 that in a 2s- design, the number of blocks is at least. Bose, A Note on Fishers Inequality for Balanced Incomplete Block Designs, Annals of Mathematical Statistics,1949, R. A. Fisher, An examination of the different possible solutions of a problem in incomplete blocks, Annals of Eugenics, volume 10,1940, pages 52–75. Stinson, Douglas R. Combinatorial Designs, Constructions and Analysis, New York, Springer, ISBN 0-387-95487-2 Street, Anne Penfold, Street, Deborah J. Combinatorics of Experimental Design
5.
Harold Scott MacDonald Coxeter
–
Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
6.
Graph drawing
–
A drawing of a graph or network diagram is a pictorial representation of the vertices and edges of a graph. This drawing should not be confused with the graph itself, very different layouts can correspond to the same graph, in the abstract, all that matters is which pairs of vertices are connected by edges. In the concrete, however, the arrangement of vertices and edges within a drawing affects its understandability, usability, fabrication cost. The problem gets worse, if the changes over time by adding and deleting edges. Upward planar drawing uses the convention that every edge is oriented from a vertex to a higher vertex. Many different quality measures have been defined for graph drawings, in an attempt to find means of evaluating their aesthetics. In addition to guiding the choice between different layout methods for the graph, some layout methods attempt to directly optimize these measures. The crossing number of a drawing is the number of pairs of edges cross each other. If the graph is planar, then it is convenient to draw it without any edge intersections, that is, in this case. However, nonplanar graphs frequently arise in applications, so graph drawing algorithms must generally allow for edge crossings, the area of a drawing is the size of its smallest bounding box, relative to the closest distance between any two vertices. Drawings with smaller area are generally preferable to those with area, because they allow the features of the drawing to be shown at greater size. The aspect ratio of the box may also be important. Symmetry display is the problem of finding symmetry groups within a given graph, some layout methods automatically lead to symmetric drawings, alternatively, some drawing methods start by finding symmetries in the input graph and using them to construct a drawing. It is important that edges have shapes that are as simple as possible, in polyline drawings, the complexity of an edge may be measured by its number of bends, and many methods aim to provide drawings with few total bends or few bends per edge. Similarly for spline curves the complexity of an edge may be measured by the number of points on the edge. Several commonly used quality measures concern lengths of edges, it is desirable to minimize the total length of the edges as well as the maximum length of any edge. Additionally, it may be preferable for the lengths of edges to be rather than highly varied. Angular resolution is a measure of the sharpest angles in a graph drawing, if a graph has vertices with high degree then it necessarily will have small angular resolution, but the angular resolution can be bounded below by a function of the degree
7.
Graph theory
–
In mathematics graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, Graphs are one of the prime objects of study in discrete mathematics. Refer to the glossary of graph theory for basic definitions in graph theory, the following are some of the more basic ways of defining graphs and related mathematical structures. To avoid ambiguity, this type of graph may be described precisely as undirected, other senses of graph stem from different conceptions of the edge set. In one more generalized notion, V is a set together with a relation of incidence that associates with each two vertices. In another generalized notion, E is a multiset of unordered pairs of vertices, Many authors call this type of object a multigraph or pseudograph. All of these variants and others are described more fully below, the vertices belonging to an edge are called the ends or end vertices of the edge. A vertex may exist in a graph and not belong to an edge, V and E are usually taken to be finite, and many of the well-known results are not true for infinite graphs because many of the arguments fail in the infinite case. The order of a graph is |V|, its number of vertices, the size of a graph is |E|, its number of edges. The degree or valency of a vertex is the number of edges that connect to it, for an edge, graph theorists usually use the somewhat shorter notation xy. Graphs can be used to model many types of relations and processes in physical, biological, social, Many practical problems can be represented by graphs. Emphasizing their application to real-world systems, the network is sometimes defined to mean a graph in which attributes are associated with the nodes and/or edges. In computer science, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the structure of a website can be represented by a directed graph, in which the vertices represent web pages. A similar approach can be taken to problems in media, travel, biology, computer chip design. The development of algorithms to handle graphs is therefore of major interest in computer science, the transformation of graphs is often formalized and represented by graph rewrite systems. Graph-theoretic methods, in forms, have proven particularly useful in linguistics. Traditionally, syntax and compositional semantics follow tree-based structures, whose power lies in the principle of compositionality
8.
Field (mathematics)
–
In mathematics, a field is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus an algebraic structure, which is widely used in algebra, number theory. The best known fields are the field of numbers. In addition, the field of numbers is widely used, not only in mathematics. Finite fields are used in most cryptographic protocols used for computer security, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Formally, a field is a set together with two operations the addition and the multiplication, which have the properties, called axioms of fields. An operation is a mapping that associates an element of the set to every pair of its elements, the result of the addition of a and b is called the sum of a and b and denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, associativity of addition and multiplication For all a, b and c in F, one has a + = + c and a · = · c. Commutativity of addition and multiplication For all a and b in F one has a + b = b + a and a · b = b · a. Existence of additive and multiplicative identity elements There exists an element 0 in F, called the identity, such that for all a in F. There is an element 1, different from 0 and called the identity, such that for all a in F. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element in F, denoted −a, such that a + =0. For every a ≠0 in F, there exists an element in F, denoted a−1, 1/a, or 1/a, distributivity of multiplication over addition For all a, b and c in F, one has a · = +. The elements 0 and 1 being required to be distinct, a field has, at least, for every a in F, one has − a = ⋅ a. Thus, the inverse of every element is known as soon as one knows the additive inverse of 1. A subtraction and a division are defined in every field by a − b = a +, a subfield E of a field F is a subset of F that contains 1, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. It is straightforward to verify that a subfield is indeed a field, two groups are associated to every field. The field itself is a group under addition, when considering this group structure rather the field structure, one talks of the additive group of the field
9.
Finite geometry
–
A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points, a geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are systems that could be called finite geometries, attention is mostly paid to the finite projective. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field, Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of three or greater is isomorphic to a projective space over a finite field. However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes, similar results hold for other kinds of finite geometries. The following remarks apply only to finite planes, There are two main kinds of finite plane geometry, affine and projective. In an affine plane, the sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms. An affine plane geometry is a nonempty set X, along with a nonempty collection L of subsets of X, such that, For every two distinct points, there is exactly one line that contains both points. Playfairs axiom, Given a line ℓ and a point p not on ℓ, There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not trivial, while the first two specify the nature of the geometry, the simplest affine plane contains only four points, it is called the affine plane of order 2. Since no three are collinear, any pair of points determines a line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered parallel, or a square where not only opposite sides, but also diagonals are considered parallel. More generally, an affine plane of order n has n2 points and n2 + n lines, each line contains n points. The affine plane of order 3 is known as the Hesse configuration. A projective plane geometry is a nonempty set X, along with a nonempty collection L of subsets of X, such that, the intersection of any two distinct lines contains exactly one point
10.
Up to
–
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