Approximately finite-dimensional C*-algebra
In mathematics, an finite-dimensional C*-algebra is a C*-algebra, the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality was first described combinatorially by Ola Bratteli. George A. Elliott gave a complete classification of AF algebras using the K0 functor whose range consists of ordered abelian groups with sufficiently nice order structure; the classification theorem for AF-algebras serves as a prototype for classification results for larger classes of separable simple nuclear stably finite C*-algebras. Its proof divides into two parts; the invariant here is K0 with its natural order structure. First, one proves existence: a homomorphism between invariants must lift to a *-homomorphism of algebras. Second, one shows uniqueness: the lift must be unique up to approximate unitary equivalence. Classification follows from what is known as the intertwining argument. For unital AF algebras, both existence and uniqueness follow from the fact the Murray-von Neumann semigroup of projections in an AF algebra is cancellative.
The counterpart of simple AF C*-algebras in the von Neumann algebra world are the hyperfinite factors, which were classified by Connes and Haagerup. In the context of noncommutative geometry and topology, AF C*-algebras are noncommutative generalizations of C0, where X is a disconnected metrizable space. An arbitrary finite-dimensional C*-algebra A takes the following form, up to isomorphism: ⊕ k M n k, where Mi denotes the full matrix algebra of i × i matrices. Up to unitary equivalence, a unital *-homomorphism Φ: Mi → Mj is of the form Φ = a ⊗ I r, where r·i = j; the number r is said to be the multiplicity of Φ. In general, a unital homomorphism between finite-dimensional C*-algebras Φ: ⊕ 1 s M n k → ⊕ 1 t M m l is specified, up to unitary equivalence, by a t × s matrix of partial multiplicities satisfying, for all l ∑ k r l k n k = m l. In the non-unital case, the equality is replaced by ≤. Graphically, Φ, can be represented by its Bratteli diagram; the Bratteli diagram is a directed graph with nodes corresponding to each nk and ml and the number of arrows from nk to ml is the partial multiplicity rlk.
Consider the category whose objects are isomorphism classes of finite-dimensional C*-algebras and whose morphisms are *-homomorphisms modulo unitary equivalence. By the above discussion, the objects can be viewed as vectors with entries in N and morphisms are the partial multiplicity matrices. A C*-algebra is AF if it is the direct limit of a sequence of finite-dimensional C*-algebras: A = lim → ⋯ → A i → α i A i + 1 → ⋯, where each Ai is a finite-dimensional C*-algebra and the connecting maps αi are *-homomorphisms. We will assume; the inductive system specifying an AF algebra is not unique. One can always drop to a subsequence. Suppressing the connecting maps, A can be written as A = ∪ n A n ¯; the Bratteli diagram of A is formed by the Bratteli diagrams of in the obvious way. For instance, the Pascal triangle, with the nodes connected by appropriate downward arrows, is the Bratteli diagram of an AF algebra. A Bratteli diagram of the CAR algebra is given on the right; the two arrows between nodes means each connecting map is an embedding of multiplicity 2.
1 ⇉ 2 ⇉ 4 ⇉ 8 ⇉ … If an AF algebra A = − an ideal J in A takes the form ∪n −. In particular, J is itself an AF algebra. Given a Bratteli diagram of A and some subset S of nodes, the subdiagram generated by S gives inductive system that specifies an ideal of A. In fact, every ideal arises in this way. Due to the presence of matrix units in the inductive sequence, AF algebras have the following local characterization: a C*-algebra A is AF if and only if A is separable and any finite subset of A is "almost contained" in some finite-dimensional C*-subalgebra; the projections in ∪nAn in fact form an approximate unit of A. It is clear that the extension of a finite-dimensional C*-algebra by another finite-dimensional C*-algebra is again finite-dimensional. More the extension of an AF algebra by another AF algebra is again AF; the K-theoretic group K0 is an invariant of C*-algebras. It has its origins in topological K-theory and serves as the range of a kind of "dimension function." For an AF algebra A, K0 can be defined.
Let Mn be the C*-algebra of n × n matrices whose entries are elements of A. Mn can b
Script typeface
Script typefaces are based upon the varied and fluid stroke created by handwriting. They are used for display or trade printing, rather than for extended body text in the Latin alphabet; some Greek alphabet typefaces historically, have been a closer simulation of handwriting. Script typefaces are organized into regular formal types similar to cursive writing and looser, more casual scripts. A majority of formal scripts are based upon the letterforms of seventeenth and eighteenth century writing-masters like George Bickham, George Shelley and George Snell; the letters in their original form are generated by a quill or metal nib of a pen. Both are able to create thick strokes. Typefaces based upon their style of writing appear late in the eighteenth century and early nineteenth century. Contemporary revivals of formal script faces can be seen in Kuenstler Script and Matthew Carter's typeface Snell Roundhand; these typefaces are used for invitations and diplomas to effect an elevated and elegant feeling.
They may use typographic ligatures to have letters connect. Casual scripts show a more active hand; the strokes may vary in width but appear to have been created by wet brush rather than a pen nib. They appear in the early twentieth century and with the advent of photocomposition in the early-1950s their number increased, they were popularly used in advertising in North America into the 1970s. Examples of casual script types include Brush Script and Mistral; some may be non-connecting. Script typefaces place particular demands on printing technology if the letters are intended to join up and vary like handwriting. A typeface intended to mimic handwriting, such as Claude Garamond's grecs du roi typeface, will require many alternate characters. In digital type these can be substituted seamlessly through contextual ligature insertion in applications like InDesign, but this was complicated in metal. Another complexity in metal type was; this required careful design and casting for the sorts to fit together without gaps or the sorts breaking, or leaving gaps to be filled in by the natural spread of ink on paper.
Script typefaces have evolved in the second half of the 20th century due to developments in technology and the end of widespread use of metal type. Most signwriting on logos and shop frontages did not use fonts but was rather custom-designed lettering created by signpainters and engravers; as phototypesetting and computers have made printing text at a range of sizes far easier than in the metal type period, it has become common for businesses to use type for logos and signs rather than hand-drawn lettering. In addition, phototypesetting made overlap of characters simple, something complicated to achieve in metal type. Matthew Carter has cited his 1966 Snell Roundhand typeface as deliberately designed to replicate a style of calligraphy hard to simulate in metal. An additional development enabling more sophisticated script fonts has been the release of the OpenType format, which most fonts are now released in; this allows fonts to have a large character set, increasing the sophistication of design possible, contextual insertion, in which characters that match one another are inserted into a document automatically, so fonts can convincingly mimic handwriting without the user having to choose the correct substitute characters manually.
Many modern script typefaces emulate the styles of hand-drawn lettering from different historical periods. In Unicode there is a script Latin alphabet for mathematical use, with both capital and small letters. Few fonts provide support for all 52 characters, their presentations vary in style from roundhand to chancery hand and others; these characters are listed here: ℬℰℱℋℐℒℳℛ ℯℊℴ Antiqua Blackletter Chancery hand Record type Blackwell, Lewis. 20th Century Type. Yale University Press: 2004. ISBN 0-300-10073-6. Fiedl, Nicholas Ott and Bernard Stein. Typography: An Encyclopedic Survey of Type Design and Techniques Through History. Black Dog & Leventhal: 1998. ISBN 1-57912-023-7. Thi Truong, Mai-Linh. FontBook: Digital Typeface Compendium. FSI FontShop International: 2006, ISBN 978-3-930023-04-2 Macmillan, Neil. An A–Z of Type Designers. Yale University Press: 2006. ISBN 0-300-11151-7. Columbia University online facsimile of writing manuals including The Universal Penman Allan Haley article on using digital versions of script typefaces
Group (mathematics)
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
Monotonic function
In mathematics, a monotonic function is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, was generalized to the more abstract setting of order theory. In calculus, a function f defined on a subset of the real numbers with real values is called monotonic if and only if it is either non-increasing, or non-decreasing; that is, as per Fig. 1, a function that increases monotonically does not have to increase, it must not decrease. A function is called monotonically increasing, if for all x and y such that x ≤ y one has f ≤ f, so f preserves the order. A function is called monotonically decreasing if, whenever x ≤ y f ≥ f, so it reverses the order. If the order ≤ in the definition of monotonicity is replaced by the strict order < one obtains a stronger requirement. A function with this property is called increasing. Again, by inverting the order symbol, one finds a corresponding concept called decreasing. Functions that are increasing or decreasing are one-to-one If it is not clear that "increasing" and "decreasing" are taken to include the possibility of repeating the same value at successive arguments, one may use the terms weakly increasing and weakly decreasing to stress this possibility.
The terms "non-decreasing" and "non-increasing" should not be confused with the negative qualifications "not decreasing" and "not increasing". For example, the function of figure 3 first falls rises falls again, it is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing. A function f is said to be monotonic over an interval if the derivatives of all orders of f are nonnegative or all nonpositive at all points on the interval; the term monotonic transformation can possibly cause some confusion because it refers to a transformation by a increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform. In this context, what we are calling a "monotonic transformation" is, more called a "positive monotonic transformation", in order to distinguish it from a “negative monotonic transformation,” which reverses the order of the numbers; the following properties are true for a monotonic function f: R → R: f has limits from the right and from the left at every point of its domain.
F can only have jump discontinuities. The discontinuities, however, do not consist of isolated points and may be dense in an interval; these properties are the reason. Two facts about these functions are: if f is a monotonic function defined on an interval I f is differentiable everywhere on I, i.e. the set of numbers x in I such that f is not differentiable in x has Lebesgue measure zero. In addition, this result cannot be improved to countable: see Cantor function. If f is a m
Encyclopedia of Mathematics
The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM; the 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer; the CD-ROM contains three-dimensional objects. The encyclopedia has been translated from the Soviet Matematicheskaya entsiklopediya edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles; until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge online. This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online; this new wiki is a collaboration between the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the previous online version, but all entries can now be publicly updated to include the newest advancements in mathematics.
All entries will be monitored for content accuracy by members of an editorial board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Sov. Entsiklopediya, 1977. Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 1, Kluwer, 1987. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 2, Kluwer, 1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 3, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 4, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 5, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 6, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 7, Kluwer, 1991. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 8, Kluwer, 1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 9, Kluwer, 1993. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 10, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer, 1997. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer, 2000.
Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer, 2002. Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer, 1998. Encyclopedia of Mathematics, public wiki monitored by an editorial board under the management of the European Mathematical Society. List of online encyclopedias Official website Publications by M. Hazewinkel, at ResearchGate
