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Mathematical result on ordinals
The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points
Fixed-point lemma for normal functions
Fixed-point_lemma_for_normal_functions
Condition for a mathematical function to map some value to itself
technique of iterating a function to find a fixed point can also be used in set theory; the fixed-point lemma for normal functions states that any continuous
Fixed-point_theorem
Function of ordinals in mathematics
Every normal function f has arbitrarily large fixed points; see the fixed-point lemma for normal functions for a proof. One can create a normal function f ′ :
Normal_function
Infinite cardinal number
some limit ordinals that are fixed points of the omega function, because of the fixed-point lemma for normal functions. The first such is the limit of
Aleph_number
Fodor's lemma Fixed-point lemma for normal functions (axiomatic set theory) Moschovakis coding lemma Rasiowa–Sikorski lemma Bézout's lemma Dwork's lemma Euclid's
List_of_lemmas
Probability distribution
theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable
Normal_distribution
Type of transfinite numbers
arbitrarily large fixed points by the fixed-point lemma for normal functions. When α = ω {\displaystyle \alpha =\omega } , these fixed points are precisely
Epsilon_number
Special coordinate system in differential geometry
In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate
Normal_coordinates
Mathematical proposition equivalent to the axiom of choice
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every
Zorn's_lemma
American mathematician (1880–1960)
introduced the Veblen axioms for projective geometry and proved the Veblen–Young theorem. He introduced the Veblen functions of ordinals and used an extension
Oswald_Veblen
Function returning one of only two values
special case of this fact is the piling-up lemma for parity functions. The polynomial form of a Boolean function can also be used as its natural extension
Boolean_function
Computer approximation for real numbers
computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a significand (a signed sequence of a fixed number of digits
Floating-point_arithmetic
Hadamard space and x0 is a fixed point in X then the union of the space of Busemann functions vanishing at x0 and the space of functions hy(x) = d(x,y) − d(x0
Busemann_function
Probability distribution and special case of gamma distribution
null hypothesis (Neyman–Pearson lemma) and this leads also to optimality properties of generalised LRTs. However, the normal and chi-squared approximations
Chi-squared_distribution
Theorem in mathematics
complex-valued functions of a complex variable. It generalizes to functions from n-tuples (of real or complex numbers) to n-tuples, and to functions between
Inverse_function_theorem
Mathematical function with no sudden changes
where arguments and values of functions are real numbers and complex numbers. The concept has been generalized to functions between metric spaces and between
Continuous_function
Mathematical-logic system
the function space D → D, of functions on itself. However, no nontrivial such D can exist, by cardinality constraints because the set of all functions from
Lambda_calculus
Mathematics of smooth surfaces
Gauss's lemma and its generalisations. Roughly speaking this lemma states that geodesics starting at the base point must cut the spheres of fixed radius
Differential geometry of surfaces
Differential_geometry_of_surfaces
Order-preserving mathematical function
monotonic functions are invertible because they are guaranteed to have a one-to-one mapping from their range to their domain. However, functions that are
Monotonic_function
Gauss–Newton line – described in Journal for Geometry and Graphics, see also Newton line Gauss's area formula Gauss's lemma in Riemannian geometry Gauss map in
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
Branch of statistics
used decision theory with probability distributions and loss functions (or utility functions). The decision-theoretic approach to statistical inference
Mathematical_statistics
Second-order partial differential equation
and indeed real analytic. Harmonic functions also admit a variational characterization. Among functions with fixed boundary values, the solutions of Laplace's
Laplace's_equation
Mathematical models of strategic interactions
proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became
Game_theory
adjusted mortality rate Risk factor Risk function Risk perception Risk theory Risk–benefit analysis Robbins lemma Robust Bayesian analysis Robust confidence
List_of_statistics_articles
Distance from a point to the boundary of a set
applications, the signed distance function or signed distance field (SDF) is the orthogonal distance of a given point x to the boundary of a set Ω in a
Signed_distance_function
Mathematical function that can be computed by a program
recursive functions. Although these four are of a very different nature, they provide exactly the same class of computable functions, and, for every model
Computable_function
primes diverges Banach fixed-point theorem Banach–Tarski paradox Basel problem Bolzano–Weierstrass theorem Brouwer fixed-point theorem Buckingham π theorem
List_of_mathematical_proofs
Type of topological space
Fixed-point space – Space where all functions have fixed points, a Hausdorff space X such that every continuous function f : X → X has a fixed point.
Hausdorff_space
Thesis on the nature of computability
formalized the definition of the class of general recursive functions: the smallest class of functions (with arbitrarily many arguments) that is closed under
Church–Turing_thesis
Fundamental theorem in probability theory and statistics
number of density functions becomes close to the characteristic function of the normal density as the number of density functions increases without bound
Central_limit_theorem
Type of vector space in math
basis functions for a space such as L2([0, 1]). In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather
Hilbert_space
Branch of topology
is normal if and only if any two disjoint closed sets can be separated by a continuous function; this is Urysohn's lemma.) X is completely normal, or
General_topology
Function related to statistics and probability theory
parameter(s) or argument that maximizes the likelihood function serves as a point estimate for the unknown parameter, while the Fisher information (often
Likelihood_function
Branch of mathematical logic
coincide with a natural class of functions, such as the primitive recursive or polynomial-time computable functions. Functional interpretations have also
Proof_theory
Axiom of set theory
compactification. Urysohn's Lemma: For any two disjoint closed subsets A {\displaystyle A} and B {\displaystyle B} of a normal space X {\displaystyle X}
Axiom_of_choice
Natural moving frame in differential geometry of surfaces
each point p of an oriented surface, one may attach a unit normal vector u(p) in a unique way, as soon as an orientation has been chosen for the normal at
Darboux_frame
space Completely Hausdorff space Regular space Tychonoff space Normal space Urysohn's lemma Tietze extension theorem Paracompact Separated sets Direct sum
List of general topology topics
List_of_general_topology_topics
Existence and cardinality of models of logical theories
get a function from the first-order formulas φ {\displaystyle \varphi } to such functions f φ {\displaystyle f_{\varphi }} . The family of functions f φ
Löwenheim–Skolem_theorem
Combinatorial game theory theorem
the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite
Sprague–Grundy_theorem
Mathematical theorem
{\displaystyle f} at the point z 0 {\displaystyle z_{0}} is equal to ϕ {\displaystyle \phi } . This is an easy consequence of the Schwarz lemma. As a corollary
Riemann_mapping_theorem
Assignment of meaning to the symbols of a formal language
type: subsets of the domain, functions from the domain, functions that take a subset of the domain and return a function from the domain to subsets of
Interpretation_(logic)
Mathematical theorem
let u denote a Lipschitz-continuous function on Rn. The first step of the proof is to show that, for any fixed unit vector v, the v-directional derivative
Rademacher's_theorem
Notions of probabilistic convergence, applied to estimation and asymptotic analysis
convergence of the probability density functions implies convergence in distribution. The portmanteau lemma provides several equivalent definitions of
Convergence of random variables
Convergence_of_random_variables
Mathematical transform that expresses a function of time as a function of frequency
of integrable functions is that the Fourier-Stieltjes transform need not vanish at infinity, i.e., the Riemann–Lebesgue lemma fails for measures. Bochner's
Fourier_transform
Limitative results in mathematical logic
existence of at least one such statement follows from the diagonal lemma, which says that for any sufficiently strong formal system and any statement form F
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Area of mathematical logic
its domain, closed under all functions in its signature σ, which is regarded as a σ-structure by restricting all functions and relations in σ to the subset
Model_theory
Representation of a game in game theory
information, a normal-form representation of a game is a specification of players' strategy spaces and payoff functions. A strategy space for a player is
Normal-form_game
Generalised alphabetical order
for other related algorithms, such as the algorithms for the computation of the tangent cone. As Gröbner bases are defined for polynomials in a fixed
Lexicographic_order
Diagram that shows all possible logical relations between a collection of sets
of creature can be imagined as a point somewhere in the diagram. Living creatures that have two legs and can fly—for example, parrots—are then in both
Venn_diagram
Set-theoretic function
fixed point φ ( 2 , 0 , 0 ) {\displaystyle \varphi (2,0,0)} of the α ↦ φ ( 1 , α , 0 ) {\displaystyle \alpha \mapsto \varphi (1,\alpha ,0)} functions
Ordinal_collapsing_function
Branch of logic
in FO(LFP), first-order logic augmented with a least fixed point operator, and more generally for sentences in the infinitary logic L ∞ ω ω {\displaystyle
Finite_model_theory
Statistical estimator
synonymous to the notion of convergence in probability. As such, any theorem, lemma, or property which establishes convergence in probability may be used to
Consistent_estimator
In logic, a statement which is always true
component terms, with only the logical constants having a fixed meaning. It is a logical truth. For example, a formula that states "the ball is green or the
Tautology_(logic)
Hungarian and American mathematician and physicist (1903–1957)
topological fixed-point theorem, rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions. Von
John_von_Neumann
Concept in mathematical set theory
f(\alpha )=\gamma } for any α ∈ S 0 {\displaystyle \alpha \in S_{0}} . In modern parlance, the nonstationary ideal is normal. The lemma was first proved
Fodor's_lemma
Type of logical system
are predicates having predicates or functions as arguments, or in which quantification over predicates, functions, or both, are permitted. In first-order
First-order_logic
Paradox in set theory
set "normal" if it is not a member of itself, and "abnormal" if it is a member of itself. Clearly every set must be either normal or abnormal. For example
Russell's_paradox
Problem in computer science
effectively calculable function can be formalized by the general recursive functions or equivalently by the lambda-definable functions. He proves that the
Halting_problem
Mathematical result or axiom on order relations
Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914. It states that in any partially ordered
Hausdorff_maximal_principle
Logical operation
generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity (and vice versa). In intuitionistic logic,
Negation
Type of mathematical curve
form Isolated point y2 = x3 − x2 semicubical parabola y2 = x3 Double point y2 = x3 + x2 For the cubics that are not in Weierstrass normal form, the shape
Plane_cubic_curve
Standard system of axiomatic set theory
Zorn's lemma. Since the existence of a choice function when X {\displaystyle X} is a finite set is easily proved from axioms 1–8, AC only matters for certain
Zermelo–Fraenkel_set_theory
Branch of mathematics that studies algebraic structures
fixed ring) Morita equivalence, Morita duality Category of vector spaces Homological algebra Filtration (algebra) Exact sequence Functor Zorn's lemma
List of abstract algebra topics
List_of_abstract_algebra_topics
Branch of mathematics that studies sets
The latter was the starting point for a movement in real analysis of the study of "seriously" discontinuous functions. A young Georg Cantor entered
Set_theory
Discrete probability distribution
Poisson clumping Poisson point process Poisson regression Poisson sampling Poisson wavelet Queueing theory Renewal theory Robbins lemma Skellam distribution
Poisson_distribution
Theorem which asserts the existence of an object
(3 December 2014). From Sperner's Lemma to Differential Equations in Banach Spaces : An Introduction to Fixed Point Theorems and their Applications. KIT
Existence_theorem
Proof all ranked voting rules have spoilers
ISBN 978-3-319-23261-4 Quesada, Antonio (2002). "From social choice functions to dictatorial social welfare functions". Economics Bulletin. 4 (16): 1–7. Doron, Gideon;
Arrow's_impossibility_theorem
Method in probability theory
Stein operator. For the standard normal distribution, Stein's lemma yields such an operator: ( 2.2 ) E ( f ′ ( Y ) − Y f ( Y ) ) = 0 for all f ∈ C b 1
Stein's_method
Smallest convex set containing a given set
well as for finite point sets, convex hulls have also been studied for simple polygons, Brownian motion, space curves, and epigraphs of functions. Convex
Convex_hull
floating-point system Elementary functions (exponential, logarithm, trigonometric functions): Trigonometric tables — different methods for generating
List of numerical analysis topics
List_of_numerical_analysis_topics
Symbol representing a mathematical object
are used for denoting values of functions, such as the symbol y in the equation y = f(x), where x is the argument and f denotes the function itself. A
Variable_(mathematics)
Type of binary relation
length n for any n. The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded relations: for any set-like
Well-founded_relation
Solution concept of a non-cooperative game
Kakutani fixed-point theorem in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler Brouwer fixed-point theorem for the same
Nash_equilibrium
inconsistent. A stronger version of Fubini's theorem for positive functions, where the function is no longer assumed to be measurable but merely that
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Topology on Cartesian products of topological spaces
real-valued functions on I {\displaystyle I} , and convergence in the product topology is the same as pointwise convergence of functions. If the real
Product_topology
Symbolic description of a mathematical object
denote numbers, variables, operations, and functions. Other symbols include punctuation marks and brackets, used for grouping where there is not a well-defined
Expression_(mathematics)
Subfield of automated reasoning and mathematical logic
could then be checked for unsatisfiability using a number of methods. Gilmore's program used conversion to disjunctive normal form, a form in which the
Automated_theorem_proving
Concept in game theory
"reaction functions" since functions must only have one value per argument, and many reaction correspondences will be undefined, i.e., a vertical line, for some
Best_response
Impossible task in computing
no function symbols. Its S a t {\displaystyle {\rm {Sat}}} is NEXPTIME-complete (Theorem 3.22). Any first-order formula has a prenex normal form. For each
Entscheidungsproblem
Logical connective OR
evaluated. The logical disjunction operator thus usually constitutes a sequence point. In a parallel (concurrent) language, it is possible to short-circuit both
Logical_disjunction
Set theory concept
observation that Vω+1 is adequate for the integers, while Vω+2 is adequate for the real numbers, and most other normal mathematics can be built as relations
Von_Neumann_universe
Reasoning for mathematical statements
proof of the lemma." [1] "Whether constant π (i.e., pi) is normal is a confusing problem without any strict theoretical demonstration except for some statistical
Mathematical_proof
Mathematical function, inverse of an exponential function
a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10
Logarithm
Set of sentences in a formal language
and substitution axiom schemes for this symbol. Compactness theorem Consistent set Deduction theorem Lindenbaum's lemma Löwenheim–Skolem theorem A direct
Theory_(mathematical_logic)
Cryptography algorithm
cipher by itself is only suitable for the secure cryptographic transformation (encryption or decryption) of one fixed-length group of bits called a block
Block cipher mode of operation
Block_cipher_mode_of_operation
Isomorphism of an object to itself
automorphism. The inner automorphisms form a normal subgroup of Aut(G), denoted by Inn(G); this is called Goursat's lemma. The other automorphisms are called outer
Automorphism
functions from β to α → 1. Implies 2. f: X → Y means f is a function from X to Y. 3. The ordinary partition symbol, where κ→(λ)n m means that for every
Glossary_of_set_theory
Military strategy during the Cold War with regard to the use of nuclear weapons
attributed to disobedience, technical failures, the presence or lack of normal accidents, or other factors beyond strict human control; that "'lucky' nuclear
Deterrence_theory
Axiomatic set theories based on the principles of mathematical constructivism
_{0},0)} , which is the first common fixed point of the Veblen functions φ β {\displaystyle \varphi _{\beta }} for β < ε 0 {\displaystyle \beta <\varepsilon
Constructive_set_theory
Decimal representation of real numbers in computing
floating-point representation over decimal fixed-point and integer representation is that it supports a much wider range of values. For example, while a fixed-point
Decimal_floating_point
Standard example in game theory
loyalty to each other, and will have no opportunity for retribution or reward outside of the game. The normal game is shown below: Regardless of what the other
Prisoner's_dilemma
Language used to describe another language
An embedded metalanguage is a language formally, naturally and firmly fixed in an object language. This idea is found in Douglas Hofstadter's book,
Metalanguage
Representation of data of various types in lambda calculus
types by lambda terms, that is, by functions that are taking functions as their arguments and are returning functions as their results. The Church numerals
Church_encoding
South Korean mathematician (born 1957)
sets for the function algebra A−∞(Ω) . Complex Var. Elliptic Equ. 54 (2009), no. 9, 879–897. Kim, Kang-Tae; Poletsky, Evgeny; Schmalz, Gerd Functions holomorphic
Kang-Tae_Kim
re-expressed: first in terms of functions on G constant on K-cosets, and then in terms of projection operators (for example the averaging over K-cosets
System_of_imprimitivity
Particular class of sets which can be described entirely in terms of simpler sets
Axiomatic set theory Transitive set L(R) Ordinal definable Condensation lemma Gödel 1938. K. J. Devlin, "An introduction to the fine structure of the
Constructible_universe
Rules used for constructing, or transforming the symbols and words of a language
Ganesh, M.; Srivastava, Jaideep; Nerode, Anil (2001). "Normal forms and syntactic completeness proofs for functional independencies". Theoretical Computer Science
Syntax_(logic)
Kind of partial function between algebraic varieties
proof that this defines an equivalence relation relies on the following lemma: If two morphisms of varieties are equal on some non-empty open set, then
Rational_mapping
Properties linking logical conjunction and disjunction
{\displaystyle \varphi ^{D}\models \neg \psi ^{D}} . For a formula φ {\displaystyle \varphi } in disjunctive normal form, the formula φ ¯ D {\displaystyle {\overline
Conjunction/disjunction duality
Conjunction/disjunction_duality
Theorem in mathematical logic
ISBN 978-0-387-94258-2 Sebastian Enqvist, "A General Lindström Theorem for Some Normal Modal Logics", Logica Universalis 7, 2013, 233–264. doi:10.1007/s11787-013-0078-9
Lindström's_theorem
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FIXED POINT-LEMMA-FOR-NORMAL-FUNCTIONS
FIXED POINT-LEMMA-FOR-NORMAL-FUNCTIONS
FIXED POINT-LEMMA-FOR-NORMAL-FUNCTIONS
FIXED POINT-LEMMA-FOR-NORMAL-FUNCTIONS
FIXED POINT-LEMMA-FOR-NORMAL-FUNCTIONS
FIXED POINT-LEMMA-FOR-NORMAL-FUNCTIONS