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Type of internal set in nonstandard analysis
a hyperfinite set or *-finite set is a type of internal set. An internal set H of internal cardinality g ∈ *N (the hypernaturals) is hyperfinite if and
Hyperfinite_set
Topics referred to by the same term
Look up hyperfinite in Wiktionary, the free dictionary. Hyperfinite may refer to: Hyperfinite set, a type of internal set in non-standard analysis Hyperfinite
Hyperfinite
Extremely small quantity in calculus; thing so small that there is no way to measure it
> 1 + 1, |x| > 1 + 1 + 1, ..., and infinitesimal if x ≠ 0 and a similar set of conditions holds for x and the reciprocals of the positive integers. A
Infinitesimal
Calculus using a logically rigorous notion of infinitesimal numbers
Infinitesimal Approach Hyperfinite set Hyperinteger Hyperreal number Influence of nonstandard analysis Infinitesimal Internal set theory Nonstandard model
Nonstandard_analysis
Swiss mathematician (1707–1783)
represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve
Leonhard_Euler
French mathematician and lawyer (1601–1665)
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
Pierre_de_Fermat
Real numbers adjoined with a nil-squaring element
determine the set of z such that the difference in slopes ("Galilean angle") between the lines from z to p and q is constant. This set is a cycle in the
Dual_number
Mathematical notion of infinitesimal difference
approach is popular in algebraic geometry. Differentials in smooth models of set theory. This approach is known as synthetic differential geometry or smooth
Differential_(mathematics)
Modern application of infinitesimals
f(x_{i})} for all i = 0, …, N (an alternative explanation is that every hyperfinite set admits a maximum). Consider the real point c = s t ( x i 0 ) {\displaystyle
Nonstandard_calculus
Mathematical symbol used to denote integrals and antiderivatives
(decimal) and ∫ (named entity). The original IBM PC code page 437 character set included a couple of characters ⌠,⎮ and ⌡ (codes 244 and 245 respectively)
Integral_symbol
Type of set in mathematical logic
theory and nonstandard analysis, an internal set is a set that is a member of a model. The concept of internal sets is a tool in formulating the transfer principle
Internal_set
Unique von Neumann algebra
mathematics, there are up to isomorphism exactly two separably acting hyperfinite type II factors; one infinite and one finite. Murray and von Neumann
Hyperfinite_type_II_factor
Mathematical notation used for calculus
to gain the precision of these concepts is that the set of real numbers must be extended to the set of hyperreal numbers. Leibniz experimented with many
Leibniz's_notation
Generalization of the real numbers
they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that
Surreal_number
Geometrical concept relating area and volume
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
Cavalieri's_principle
Textbook by Augustin-Louis Cauchy (1821)
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
Cours_d'analyse
Element of a nonstandard model of the reals, which can be infinite or infinitesimal
quantification over sets, or other higher-level structures such as functions and relations, which are typically constructed out of sets. Each real set, function
Hyperreal_number
German polymath (1646–1716)
we now call conjunction, disjunction, negation, identity, set inclusion, and the empty set. The principles of Leibniz's logic and, arguably, of his whole
Gottfried_Wilhelm_Leibniz
Concept in model theory
because at the set-theoretic level, the propositions in such a language are interpreted to apply only to internal sets rather than to all sets. As Robinson
Transfer_principle
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
Constructive nonstandard analysis
Constructive_nonstandard_analysis
Heuristic principle enunciated by Gottfried Wilhelm Leibniz
{\displaystyle a} is finite and d x {\displaystyle dx} is infinitesimal, then one sets a + d x = a . {\displaystyle a+dx=a.} Similarly, u d v + v d u + d u d v
Transcendental law of homogeneity
Transcendental_law_of_homogeneity
Function from the limited hyperreal to the real numbers
{\displaystyle \Delta x} is taken to be infinitesimal, exploiting a hyperfinite partition of the interval [a,b]. Given a sequence ( u n ) {\displaystyle
Standard_part_function
Principle that whatever succeeds for the finite also succeeds for the infinite
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
Law_of_continuity
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
Increment_theorem
1734 book by George Berkeley
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
The_Analyst
Branch of mathematics
differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their
Calculus
Mathematical model for describing material deformation under stress
function u i {\displaystyle u_{i}} . If the elastic medium is visualised as a set of infinitesimal cubes in the unstrained state, after the medium is strained
Infinitesimal_strain_theory
System of mathematical set theory
Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard
Internal_set_theory
set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of
Glossary_of_calculus
by noting that it allows one to rigorously manipulate things such as "the set of all small numbers", or to rigorously say things like "η1 is smaller than
Criticism of nonstandard analysis
Criticism_of_nonstandard_analysis
American mathematician
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
Abraham_Robinson
Calculus textbook by Guillaume de l'Hôpital (1696)
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
Analyse des infiniment petits pour l'intelligence des lignes courbes
Analyse_des_infiniment_petits_pour_l'intelligence_des_lignes_courbes
Mathematical term
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
Microcontinuity
Mathematical treatise by Archimedes
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
The Method of Mechanical Theorems
The_Method_of_Mechanical_Theorems
French mathematician (1789–1857)
dislike of mathematics. Cauchy was named a baron, a title by which Cauchy set great store. In 1834, his wife and two daughters moved to Prague, and Cauchy
Augustin-Louis_Cauchy
System of numbers with non-finite quantities
{Q} }a_{q}\varepsilon ^{q},} where Q {\displaystyle \mathbb {Q} } is the set of rational numbers, the coefficients a q {\displaystyle a_{q}} are real
Levi-Civita_field
In descriptive set theory and related areas of mathematics, a hyperfinite equivalence relation on a standard Borel space X is a Borel equivalence relation
Hyperfinite equivalence relation
Hyperfinite_equivalence_relation
1976 mathematics textbook by H. Jerome Keisler
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
Elementary Calculus: An Infinitesimal Approach
Elementary_Calculus:_An_Infinitesimal_Approach
Mathematical procedure equivalent to differential calculus
formulas. Heinrich Wieleitner (1929) wrote: Fermat replaces A with A+E. Then he sets the new expression roughly equal (angenähert gleich) to the old one, cancels
Adequality
Proof technique in nonstandard analysis
It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N of hypernatural numbers. Overspill
Overspill
Hyperreal number that is equal to its own integer part
hyperintegers are the image of the integer part function on the hyperreals. The set ∗ Z {\displaystyle ^{*}\mathbb {Z} } of all hyperintegers is an internal
Hyperinteger
Named set of points in nonstandard analysis
also a halo is the set of points infinitesimally close to a given point. Given a hyperreal number x in R∗, the monad of x is the set monad ( x ) = { y
Monad_(nonstandard_analysis)
Formalization in mathematical topos theory
Modelling the Universe: Synthetic Differential Geometry and Frame-Valued Sets (PDF file) F.W. Lawvere, Outline of synthetic differential geometry (PDF
Synthetic differential geometry
Synthetic_differential_geometry
Ordered field that does not satisfy the Archimedean property
principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental
Non-Archimedean_ordered_field
*-algebra of bounded operators on a Hilbert space
are the hyperfinite type II1 factor and the hyperfinite type II∞ factor, found by Murray & von Neumann (1936). These are the unique hyperfinite factors
Von_Neumann_algebra
Mathematical concept for comparing objects
Property of segments that have the same length and the same direction Hyperfinite equivalence relation Quotient by an equivalence relation – Generalization
Equivalence_relation
Concept in mathematics
is Fermion Walsh system in non-commutative Lp spaces associated with hyperfinite type II factor. The Fermion Walsh system is a non-commutative, or "quantum"
Walsh_function
C*-algebra
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
Approximately finite-dimensional C*-algebra
Approximately_finite-dimensional_C*-algebra
two standard Borel spaces X and Y are Borel-isomorphic iff |X| = |Y|. Hyperfinite equivalence relation Wadge hierarchy Entourage (topology) – Topological
Borel_equivalence_relation
Mathematics award
theorem of factors of type III, classification of automorphisms of the hyperfinite factor, classification of injective factors, and applications of the
Fields_Medal
finite set of elements of G {\displaystyle G} generates a finite subgroup (or more generally if G is amenable) then the factor is the hyperfinite factor
Crossed_product
Descriptive set theory relation
such as that of a hyperfinite equivalence relation, but is of interest in and of itself. A main area of study in invariant descriptive set theory is the relative
Countable_Borel_relation
Hungarian and American mathematician and physicist (1903–1957)
continuous geometry other than projective space was the projections of the hyperfinite type II factor. In more pure lattice theoretical work, he solved the
John_von_Neumann
Application of Fourier analysis to non-abelian topological groups
infinite symmetric group, where the von Neumann group algebra is the hyperfinite type II1 factor. The further theory divides up the Plancherel measure
Noncommutative harmonic analysis
Noncommutative_harmonic_analysis
continuous geometry other than projective space was the projections of the hyperfinite type II factor. Menger and Birkhoff gave axioms for projective geometry
Continuous_geometry
Locally compact topological group with an invariant averaging operation
is hyperfinite (A. Connes). Note that A. Connes also proved that the von Neumann group algebra of any connected locally compact group is hyperfinite, so
Amenable_group
together with Loeb measure theory allows one to define Brownian motion as a hyperfinite random walk, obviating the need for cumbersome measure-theoretic developments
Influence of nonstandard analysis
Influence_of_nonstandard_analysis
Lie group of complex numbers of unit modulus; topologically a circle
non-standard analysis, the most natural replacement of the circle group is the hyperfinite cyclic group C N = ∗ Z / N ∗ Z , {\displaystyle C_{N}={}^{*}\mathbb {Z}
Circle_group
Partially unsolved problem in mathematics
original idea of embedding the infinite-dimensional Hilbert space in a hyperfinite-dimensional Hilbert space (see Non-standard analysis#Invariant subspace
Invariant_subspace_problem
Topological algebra associated to continuous groups
consists only of complex multiples of the identity. NG is isomorphic to the hyperfinite type II1 factor if and only if G is countable, amenable, and has the
Group algebra of a locally compact group
Group_algebra_of_a_locally_compact_group
2 *-anti-automorphism of a von Neumann factor of the same type. For hyperfinite factors, the class of von Neumann factors completely classified by Connes
Jordan_operator_algebra
American mathematician (born 1947)
Papers, 1980 'Notes on *finite cooperative games', RAND Papers, 1981 'Hyperfinite Von Neumann games', Mathematical Social Sciences, Vol. 9, No. 2 (1985)
Alain_A._Lewis
A finite depth hyperfinite subfactor is amenable. About the non-amenable case: there are unclassifiably many irreducible hyperfinite subfactors of index
Planar_algebra
Quantum computational theorem on problem complexity
H.; Hastings, Matthew B. (January 2014). "Quantum Systems on Non-$k$-Hyperfinite Complexes: a generalization of classical statistical mechanics on expander
NLTS_conjecture
HYPERFINITE SET
HYPERFINITE SET
Surname or Lastname
English
English : habitational name from a place in North Yorkshire, so named from Old English setl ‘seat’, ‘dwelling’.
Surname or Lastname
English and Scottish
English and Scottish : habitational name from any of the numerous and widespread places so called. The majority of these are named with Old English middel ‘middle’ + tūn ‘enclosure’, ‘settlement’; a smaller group, with examples in Cumbria, Kent, Northamptonshire, Northumbria, Nottinghamshire, and Staffordshire, have as their first element Old English mylen ‘mill’.
Surname or Lastname
English
English : habitational name from places called Merton in London, Devon, Norfolk, and Oxfordshire, named in Old English with mere ‘lake’, ‘pool’ + tūn ‘enclosure’, ‘settlement’. Compare Marton, Martin 2.
Female
Japanese
(節å) Japanese name SETSUKO means "temperate child."
Surname or Lastname
English and Scottish
English and Scottish : habitational name from any of the places so called. In over thirty instances from many different areas, the name is from Old English midel ‘middle’ + tūn ‘enclosure’, ‘settlement’. However, Middleton on the Hill near Leominster in Herefordshire appears in Domesday Book as Miceltune, the first element clearly being Old English micel ‘large’, ‘great’. Middleton Baggot and Middleton Priors in Shropshire have early spellings that suggest gem̄ðhyll (from gem̄ð ‘confluence’ + hyll ‘hill’) + tūn as the origin.A Scottish family of this name derives it from lands at Middleto(u)n near Kincardine. The Scottish physician Peter Middleton practiced in New York City after 1752 and was one of the founders of the medical school at King's College (now Columbia University) in 1767. One of the earliest of the Charleston, SC, Middleton family of prominent legislators was Arthur Middleton, born in Charleston in 1681.
Male
Greek
(Σήθι) Greek form of Egyptian Seti, SETHI means "of Seth."Â
Male
Greek
(Σήθος) Greek form of Egyptian Sutekh, possibly SETHOS means "one who dazzles." In mythology, this is the name of an ancient evil god of Chaos, storms, and the desert, who slew Osiris.Â
Surname or Lastname
English
English : topographic name for someone who lived in the center of a village, from Middle English midde ‘mid’ + toun ‘village’, ‘town’.English : habitational name from places in Lancashire, Worcestershire, and West Yorkshire, so named in Old English as ‘farmstead at a river confluence’, from (ge)m̄ðe ‘river confluence’ + tūn ‘farmstead’, ‘settlement’.
Surname or Lastname
English
English : habitational name from a place in Kent named Meopham, from an Old English personal name MÄ“apa + Old English hÄm ‘homestead’, ‘settlement’.
Surname or Lastname
English
English : occupational name for a stone- or bricklayer, from Middle English setter ‘one who lays stones or bricks in building’ (agent derivative of setten ‘to set’).English : occupational name from Old French saietier ‘silk weaver’ (an agent derivative of sayete, a kind of silk).English : from an agent derivative of Middle English setten ‘to place (decoration, on a garment or metal surface)’, probably an occupational name for an embroiderer.German : unexplained.Norwegian : unexplained.
Surname or Lastname
English and Irish
English and Irish : variant of Mayhew.Variant of French Mailhot.A William Mayo born in Wiltshire, England, c. 1684 was a surveyor who settled in VA about 1623 and helped survey the VA-NC boundary and found Richmond and Petersburg, VA. [newpara]The Mayo Clinic in Rochester, MN, was founded by William Worrall Mayo (1819–1911), who immigrated to the U.S. from England, in 1845, and his sons, all gifted and innovative physicians and surgeons.
Surname or Lastname
English
English : habitational name from Mitcham in Surrey, so named from Old English micel ‘big’ + hÄm ‘homestead’, ‘settlement’.
Male
English
Anglicized form of Hebrew Sheth, SETH means "buttocks." In the bible, this is the name of the third son of Adam and Eve. Compare with other forms of Seth.
Male
Hindi/Indian
(सेठ) Hindi name derived from the Sanskrit word setu, SETH means "bridge." Compare with other forms of Seth.
Surname or Lastname
Scottish and English
Scottish and English : topographic name for someone who lived near a mill, Middle English mille, milne (Old English myl(e)n, from Latin molina, a derivative of molere ‘to grind’). It was usually in effect an occupational name for a worker at a mill or for the miller himself. The mill, whether powered by water, wind, or (occasionally) animals, was an important center in every medieval settlement; it was normally operated by an agent of the local landowner, and individual peasants were compelled to come to him to have their grain ground into flour, a proportion of the ground grain being kept by the miller by way of payment.English : from a short form of a personal name, probably female, as for example Millicent.
Surname or Lastname
English
English : habitational name from places in Cheshire and East Yorkshire, so named from Old English mylen ‘mill’ + tūn ‘enclosure’, ‘settlement’.
Surname or Lastname
English
English : patronymic from Setter.
Surname or Lastname
English
English : habitational name from a place in Shropshire, so named from Welsh mynydd ‘hill’ + Old English tūn ‘enclosure’, ‘settlement’.
Male
Italian
Italian form of Roman Latin Septimus, SETTIMIO means "seventh."
Surname or Lastname
English
English : habitational name from Milwich in Staffordshire, so named from Old English myln ‘mill’ + wīc ‘dairy farm’; ‘(trading) settlement’.
HYPERFINITE SET
HYPERFINITE SET
Girl/Female
Hindu, Indian, Marathi, Tamil
Glowing; Iridescent
Girl/Female
Greek French
Light.
Boy/Male
Indian, Punjabi, Sikh
Love of Fame
Male
French
French form of Anglo-Saxon Eadmund, EDMOND means "protector of prosperity."
Male
English
Anglicized form of Hebrew Bethuw'el, BETHUEL means "God destroys" or "man of God." In the bible, this is the name of a town and also the name of the father of Rebecca.
Girl/Female
Arabic, Indian, Kannada, Muslim
Pleasant
Male
Basque
, son of the raven.
Boy/Male
Tamil
Agnikumara | அகà¯à®¨à®¿à®•à¯à®®à®¾à®°à®¾
Son of Agni (Son of Agni)
Female
German
Low German form of German Swanhild, SWANHILDA means "swan battle."
Male
Italian
Italian form of Roman Tertius, TERZO means "third."
HYPERFINITE SET
HYPERFINITE SET
HYPERFINITE SET
HYPERFINITE SET
HYPERFINITE SET
a.
Properly or firmly set.
n.
A setula.
n.
An instrument used to set or turn the teeth of a saw a little sidewise, that they may make a kerf somewhat wider than the thickness of the blade, to prevent friction; -- called also saw-wrest.
pl.
of Setula
n.
A settled place of abode; residence; a right growing out of residence; legal residence or establishment of a person in a particular parish or town, which entitles him to maintenance if a pauper, and subjects the parish or town to his support.
n.
A vessel, as a tub, in which something, as pulverized ore suspended in a liquid, is allowed to settle.
n.
The act of peopling, or state of being peopled; act of planting, as a colony; colonization; occupation by settlers; as, the settlement of a new country.
n.
That which settles, or is settled, established, or fixed.
n.
The act of one who, or that which, settles; the act of establishing one's self, of colonizing, subsiding, adjusting, etc.
n.
The act or process of adjusting or determining; composure of doubts or differences; pacification; liquidation of accounts; arrangement; adjustment; as, settlement of a controversy, of accounts, etc.
n.
One who settles, becomes fixed, established, etc.
n.
A colony newly established; a place or region newly settled; as, settlement in the West.
n.
Especially, one who establishes himself in a new region or a colony; a colonist; a planter; as, the first settlers of New England.
n.
That which settles or finishes; hence, a blow, etc., which settles or decides a contest.
a.
Eager in appetite or desire of gratification; affected by keen hunger; ravenous; as, an eagle or a lion sharp-set.
n.
Fractures or dislocations caused by settlement.
n.
Matter that subsides; settlings; sediment; lees; dregs.
n.
That which settles at the bottom of a liquid; lees; dregs; sediment.
n.
A small, short hair or bristle; a small seta.
a.
Having small bristles or setae.