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In computability theory, there are a number of basis theorems. These theorems show that particular kinds of sets always must have some members that are
Basis_theorem_(computability)
Topics referred to by the same term
Basis theorem can refer to: Basis theorem (computability), a type of theorem in computability theory showing that sets from particular classes must have
Basis_theorem
Polynomial ideals are finitely generated
mathematics, Hilbert's basis theorem asserts that every ideal of a polynomial ring over a field has a finite generating set (a finite basis in Hilbert's terminology)
Hilbert's_basis_theorem
The low basis theorem is one of several basis theorems in computability theory, each of which show that, given an infinite subtree of the binary tree
Low_basis_theorem
A prime p divides a^p–a for any integer a
Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after
Fermat's_little_theorem
Approximation of a function by a polynomial
Taylor's theorem also generalizes to multivariate and vector valued functions. It provided the mathematical basis for some landmark early computing machines:
Taylor's_theorem
Mathematical rule for inverting probabilities
formulation on an axiomatic basis, writing in a 1973 book that Bayes' theorem "is to the theory of probability what the Pythagorean theorem is to geometry". Stephen
Bayes'_theorem
In mathematics, a statement that has been proven
theory consists of some basis statements called axioms, and some deducing rules (sometimes included in the axioms). The theorems of the theory are the statements
Theorem
Subfield of automated reasoning and mathematical logic
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving
Automated_theorem_proving
Theorem that any three objects in space can be simultaneously bisected by a plane
offers a proof of the theorem. A more modern reference is Stone & Tukey (1942), which is the basis of the name "Stone–Tukey theorem". This paper proves
Ham_sandwich_theorem
Algebraic expansion of powers of a binomial
algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power ( x
Binomial_theorem
Theorem about a certain class of control-flow graphs
construction was based on Böhm's programming language P′′. The theorem forms the basis of structured programming, a programming paradigm which eschews
Structured_program_theorem
Theorem of quantum circuits
In quantum computing, the Gottesman–Knill theorem is a theoretical result by Daniel Gottesman and Emanuel Knill that states that stabilizer circuits—circuits
Gottesman–Knill_theorem
Theorem in dimensional analysis
Buckingham π theorem is a key theorem in dimensional analysis. It is a formalisation of Rayleigh's method of dimensional analysis. Loosely, the theorem states
Buckingham_pi_theorem
On polynomial rings over fields
invariant theory, and are at the basis of modern algebraic geometry. The two other theorems are Hilbert's basis theorem, which asserts that all ideals of
Hilbert's_syzygy_theorem
Property of artificial neural networks
In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate
Universal approximation theorem
Universal_approximation_theorem
Basic result in harmonic analysis on compact topological groups
} The final statement of the Peter–Weyl theorem (Knapp 1986, Theorem 1.12) gives an explicit orthonormal basis of L 2 ( G ) {\displaystyle L^{2}(G)} .
Peter–Weyl_theorem
Formula for area of a grid polygon
direction, using Pick's theorem (proved in a different way) as the basis for a proof of Euler's formula. Alternative proofs of Pick's theorem that do not use
Pick's_theorem
Yes-or-no question that cannot ever be solved by a computer
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct
Undecidable_problem
theorem Goodstein's theorem Green's theorem (to do) Green's theorem when D is a simple region Heine–Borel theorem Intermediate value theorem Itô's lemma Kőnig's
List_of_mathematical_proofs
Thesis on the nature of computability
theorems of computability theory. But because the computability theorist believes that Turing computability correctly captures what can be computed effectively
Church–Turing_thesis
Theorem in calculus relating line and double integrals
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in R
Green's_theorem
Theorem in quantum mechanics
In mathematical physics, Gleason's theorem shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from
Gleason's_theorem
Mathematical construct in computer algebra
polynomial rings are Noetherian (Hilbert's basis theorem). Condition 4 ensures that the result is a Gröbner basis, and the definitions of S-polynomials and
Gröbner_basis
Theory of stochastic processes
orthonormal basis of L2([a, b]) yields an expansion thereof in that form. The importance of the Karhunen–Loève theorem is that it yields the best such basis in
Kosambi–Karhunen–Loève theorem
Kosambi–Karhunen–Loève_theorem
Theorem in physics
Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with
Bell's_theorem
In additive number theory, a way to measure how dense a sequence of numbers is
an additive basis, and the least number of summands required is called the degree (sometimes order) of the basis. Thus, the last theorem states that any
Schnirelmann_density
Sufficiency theorem for reconstructing signals from samples
The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
Theorem in computational complexity theory
In computational complexity theory, the PCP theorem (also known as the PCP characterization theorem) states that every decision problem in the NP complexity
PCP_theorem
Coordinate change in linear algebra
of inertia is a theorem that asserts that the numbers of 1 and of –1 depend only on the bilinear form, and not on the change of basis. Symmetric bilinear
Change_of_basis
In the mathematical field of computability theory, a PA degree is a Turing degree that computes a complete extension of Peano arithmetic (Jockusch 1987)
PA_degree
Theorem
In mathematics, more specifically in differential geometry, the de Rham theorem says that the ring homomorphism from the de Rham cohomology to the singular
De_Rham_theorem
Theorem in topology
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f
Brouwer_fixed-point_theorem
Measure of algorithmic complexity
14words". It is also possible to show the non-computability of K by reduction from the non-computability of the halting problem H, since K and H are Turing-equivalent
Kolmogorov_complexity
On the dimension of vector space duals
The Erdős–Kaplansky theorem is a theorem from linear algebra. The theorem makes a fundamental statement about the dimension of the dual spaces of infinite-dimensional
Erdős–Kaplansky_theorem
Relates the homology of a fiber bundle with the homologies of its base and fiber
In mathematics, the Leray–Hirsch theorem is a basic result on the algebraic topology of fiber bundles. It is named after Jean Leray and Guy Hirsch, who
Leray–Hirsch_theorem
Mathematical result on infinite trees
The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic, especially in computability theory
Kőnig's_lemma
Fundamental theorem in probability theory and statistics
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample
Central_limit_theorem
Theorem in mathematics
In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following
Projection-slice_theorem
Mathematical proposition equivalent to the axiom of choice
several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's
Zorn's_lemma
theorem (computability) Cenzer, Douglas (1999), " Π 1 0 {\displaystyle \Pi _{1}^{0}} classes in computability theory", Handbook of computability theory
Π01_class
In mathematics, invariant of square matrices
some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if
Determinant
Branch of mathematical logic
arithmetic, results such as Post's theorem establish a close link between the complexity of a formula and the (non)computability of the set it defines. Another
Reverse_mathematics
American mathematician
with Carl Jockusch, the low basis theorem, and has done other work in mathematical logic, primarily in the area of computability theory. His doctoral students
Robert_I._Soare
Fundamental theorem in condensed matter physics
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves
Bloch's_theorem
Basic framework of mathematics
generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical
Foundations_of_mathematics
Finite extension of the rationals
that represents an element x has integer entries in some basis e. By the Cayley–Hamilton theorem, pA(A) = 0, and it follows that pA(x) = 0, so that x is
Algebraic_number_field
Describes the objects of a given type, up to some equivalence
In mathematics, a classification theorem answers the classification problem: "What are the objects of a given type, up to some equivalence?". It gives
Classification_theorem
Foundational law of electromagnetism relating electric field and charge distributions
as Gauss's flux theorem or sometimes Gauss's theorem, is one of Maxwell's equations. It is an application of the divergence theorem, and it relates the
Gauss's_law
Statement relating differentiable symmetries to conserved quantities
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law
Noether's_theorem
reason about simplicial sets and cubical sets. Synthetic computability theory develops computability theory in constructive mathematics by postulating, among
Synthetic_mathematics
Computation model defining an abstract machine
each producing output data from given input data. Computability theory, which studies computability of functions from inputs to outputs, and for which
Turing_machine
Number divisible only by 1 and itself
than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself
Prime_number
Concept in computability theory
In computability theory, a Turing reduction from a decision problem A {\displaystyle A} to a decision problem B {\displaystyle B} is an oracle machine
Turing_reduction
analyzing the proof-theoretic strength of Ramsey's theorem. High (computability) Low basis theorem R. Downey, R. A. Shore, Degree Theoretic Definitions
Low_(computability)
Category of mathematical proof
In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as
Proof_of_impossibility
Primality test for numbers of a certain form
In number theory, Proth's theorem is a theorem which forms the basis of a primality test for Proth numbers known as Proth's test. Proth numbers, sometimes
Proth's_theorem
Algorithm in computational number theory
(Hoffstein, Pipher & Silverman 2008, Theorem 6.68), with the corrections from the errata. INPUT a lattice basis b1, b2, ..., bn in Zm a parameter δ with
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Differential equation that is linear with respect to the unknown function
an ordinary differential operator of order n, Carathéodory's existence theorem implies that, under very mild conditions, the kernel of L is a vector space
Linear_differential_equation
Overview of and topical guide to computer science
Automata theory – Different logical structures for solving problems. Computability theory – What is calculable with the current models of computers. Proofs
Outline_of_computer_science
Square matrices satisfy their characteristic equation
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix
Cayley–Hamilton_theorem
Theorem used in quantum mechanics for angular momentum calculations
Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators in the basis of angular
Wigner–Eckart_theorem
In functional analysis, a Hilbert space
field of statistical learning theory because of the celebrated representer theorem which states that every function in an RKHS that minimises an empirical
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
1938 doctoral thesis by Alan Turing
systems after Gödel's theorem. Gödel showed that for any formal system S powerful enough to represent arithmetic, there is a theorem G that is true but the
Systems of Logic Based on Ordinals
Systems_of_Logic_Based_on_Ordinals
Theorem in linear algebra
In matrix theory, the Perron–Frobenius theorem, proved in its first part by Oskar Perron (1907) and extended by Georg Frobenius (1912), asserts that a
Perron–Frobenius_theorem
Multivariate functions can be written using univariate functions and summing
approximation theory, the Kolmogorov–Arnold representation theorem (or superposition theorem) states that every multivariate continuous function f : [
Kolmogorov–Arnold representation theorem
Kolmogorov–Arnold_representation_theorem
Type of artificial neural network architecture
architecture inspired by the Kolmogorov–Arnold representation theorem, also known as the superposition theorem. Unlike traditional multilayer perceptrons (MLPs),
Kolmogorov–Arnold_Networks
Mathematical theorem
of polynomials with rational coefficients. Even though Vincent's theorem is the basis of the fastest method for the isolation of the real roots of polynomials
Vincent's_theorem
Branch of mathematical logic
proof-theoretic semantics, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses on applications
Proof_theory
Counterintuitive result in probability
The infinite monkey theorem states that a monkey hitting keys independently and at random on a typewriter keyboard for an infinite amount of time will
Infinite_monkey_theorem
Inherent difficulty of computational problems
fields in theoretical computer science are analysis of algorithms and computability theory. A key distinction between analysis of algorithms and computational
Computational complexity theory
Computational_complexity_theory
Fixed-point theorem
mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed-point theorem for partially ordered sets
Bourbaki–Witt_theorem
Soviet-American mathematician
of NP-complete problems. This NP-completeness theorem, often called the Cook–Levin theorem, was a basis for one of the seven Millennium Prize Problems
Leonid_Levin
Result in algebra
In mathematics, Wedderburn's little theorem states that every finite division ring is a field; thus, every finite domain is a field. In other words, for
Wedderburn's_little_theorem
Type of vector space in math
space (the cardinality of a Hamel basis). Koashi, Masato, "Appendix: Linear algebra" (PDF) Hewitt & Stromberg (1965, Theorem 16.29) Prugovečki 1981, I, §4
Hilbert_space
System of arithmetic in proof theory
Type of mathematical function Grzegorczyk hierarchy – Functions in computability theory Reverse mathematics – Branch of mathematical logic Ordinal analysis –
Elementary function arithmetic
Elementary_function_arithmetic
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
pr}, {z1, ..., zt}, and {q1, ..., qs} forms a basis for the vector space. By the rank-nullity theorem, dim ( ker ( A − λ I ) ) ) = n − r {\displaystyle
Jordan_normal_form
Thermodynamic theorem
In classical statistical mechanics, the H-theorem, introduced by Ludwig Boltzmann in 1872, describes the tendency of the quantity H (defined below) to
H-theorem
Cycles in a graph that generate all cycles
be computed as the sum of the weights of its edges. The minimum weight basis of the cycle space is necessarily a cycle basis: by Veblen's theorem, every
Cycle_basis
Axiom of set theory
Hahn–Banach theorem in functional analysis, allowing the extension of linear functionals. The theorem that every Hilbert space has an orthonormal basis. The
Axiom_of_choice
Computer hardware technology that uses quantum mechanics
computers provide no additional power over classical computers in terms of computability. This means that quantum computers cannot solve undecidable problems
Quantum_computing
Formula relating lift on an airfoil to fluid speed, density, and circulation
The Kutta–Joukowski theorem is a fundamental theorem in aerodynamics that relates the lift per unit span of an airfoil (and any two-dimensional body, including
Kutta–Joukowski_theorem
Theorem in quantum mechanics
Koopmans' theorem states that in closed-shell Hartree–Fock theory (HF), the first ionization energy of a molecular system is equal to the negative of
Koopmans'_theorem
Cosmological theory
Gödel's first incompleteness theorem. Tegmark replies that not only is the universe mathematical, but it is also computable. In 2014, Tegmark published
Mathematical universe hypothesis
Mathematical_universe_hypothesis
quickly as possible the computability of all partial recursive functions Péter's is perhaps the best; for proving their computability by Turing machines a
Counter-machine_model
Every natural number can be represented as the sum of four integer squares
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every nonnegative integer can be represented as a sum of four non-negative
Lagrange's four-square theorem
Lagrange's_four-square_theorem
Class of algorithms for pattern analysis
matrix from user-input according to the representer theorem. Kernel machines are slow to compute for datasets larger than a couple of thousand examples
Kernel_method
Theorem of quantum information processing
no-broadcasting theorem is a result of quantum information theory. In the case of pure quantum states, it is a corollary of the no-cloning theorem. The no-cloning
No-broadcasting_theorem
Claim that human mathematicians are not describable as formal proof systems
logical argument partially based on Kurt Gödel's first incompleteness theorem. In 1931, Gödel proved that every effectively generated theory capable
Penrose–Lucas_argument
Form of mathematical proof
1000 AD, who applied it to arithmetic sequences to prove the binomial theorem and properties of Pascal's triangle. Whilst the original work was lost
Mathematical_induction
Theorem of quantum information theory
The no-hiding theorem states that if information is lost from a system via decoherence, then it moves to the subspace of the environment and it cannot
No-hiding_theorem
Study of rates of change
derivatives and tangents (see The Method of Mechanical Theorems). The use of infinitesimals to compute rates of change was developed significantly by Bhāskara
Differential_calculus
Quantum states of two qubits
faster than the speed of light, a result known as the no-communication theorem. The Bell states are four specific maximally entangled quantum states of
Bell_state
Theorem that every subgroup of a free group is itself free
In group theory, a branch of mathematics, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free. It is named after Jakob
Nielsen–Schreier_theorem
Mathematical transform that expresses a function of time as a function of frequency
sufficient regularity and decay properties is given by the Fourier inversion theorem, i.e., Inverse transform The functions f {\displaystyle f} and f ^ {\displaystyle
Fourier_transform
Theorem in algebraic geometry
algebraic geometry, the main theorem of elimination theory states that every projective scheme is proper. A version of this theorem predates the existence of
Main theorem of elimination theory
Main_theorem_of_elimination_theory
Logical principle
(see Nouveaux Essais, IV,2)" (ibid p 421) The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica
Law_of_excluded_middle
Algorithm for public-key cryptography
λ(pq)). This is part of the Chinese remainder theorem, although it is not the significant part of that theorem. Although the original paper of Rivest, Shamir
RSA_cryptosystem
Programming language and theorem prover
language, an extensible theory in a first-order logic, and an automated theorem prover. ACL2 is designed to support automated reasoning in inductive logical
ACL2
Polish mathematician (1905–1981)
Approximation property Banach–Mazur theorem Banach–Mazur game Compact operator Gelfand–Mazur theorem Mazur–Ulam theorem Schauder basis Stanisław Mazur at the Mathematics
Stanisław_Mazur
BASIS THEOREM-COMPUTABILITY
BASIS THEOREM-COMPUTABILITY
Female
Hebrew
 Variant spelling of Hebrew Basya, BASIA means "daughter of God."
Girl/Female
Arabic
Happines
Boy/Male
Greek American English
Royal. Kingly. St Basil the Great was Bishop of Caesarea in the latter half of the 4th century....
Boy/Male
Greek
Royal. Kingly. St Basil the Great was Bishop of Caesarea in the latter half of the 4th century....
Boy/Male
Muslim
Vast, Spacious, One who stretches, Enlarges
Surname or Lastname
English
English : from Old French bas(se) ‘low’, ‘short’ (Latin bassus ‘thickset’; see Basso), either a descriptive nickname for a short person or a status name meaning ‘of humble origin’, not necessarily with derogatory connotations.English : in some instances, from Middle English bace ‘bass’ (the fish), hence a nickname for a person supposedly resembling this fish, or a metonymic occupational name for a fish seller or fisherman.Scottish : habitational name from a place in Aberdeenshire, of uncertain origin.Jewish (Ashkenazic) : metonymic occupational name for a maker or player of bass viols, from Polish, Ukrainian, and Yiddish bas ‘bass viol’.German : see Basse.
Boy/Male
Muslim
King, Basil the herb (1)
Girl/Female
Greek
Watcher.
Boy/Male
Muslim
Vision, Propitious, Auspicious, Prudent, Bringer of glad tidings
Boy/Male
Indian
Smiling, Happy
Boy/Male
Indian
Vast, Spacious, One who stretches, Enlarges
Boy/Male
Muslim
Clear
Surname or Lastname
English and French
English and French : from a medieval personal name, ultimately from Greek Basileios ‘royal’. The name was borne by a 4th-century bishop of Caesarea in Cappadocia, regarded as one of the four Fathers of the Eastern Church; he wrote important theological works and established a rule for religious orders of monks. Various other saints are also known under these and cognate names. The popularity of Vasili as a Russian personal name is largely due to the fact that this was the ecclesiastical name of St. Vladimir (956–1015), Prince of Kiev, who was chiefly responsible for the introduction of Christianity to Russia. As an American surname, this has also absorbed some Greek, Russian, and other derivatives of Greek Vasili.
Boy/Male
Indian
Vision, Propitious, Auspicious, Prudent, Bringer of glad tidings
Girl/Female
Egyptian
Great.
Boy/Male
Tamil
King, Basil the herb
Boy/Male
Hindu
King, Basil the herb
Surname or Lastname
English
English : probably a variant spelling of Bevis.
Boy/Male
Muslim
Smiling, Happy
Male
English
 English form of French Basile, BASIL means "king." Also sometimes given as an herb name.
BASIS THEOREM-COMPUTABILITY
BASIS THEOREM-COMPUTABILITY
Girl/Female
Hindu, Indian, Marathi
Chand Ki Raah
Boy/Male
Muslim
The maker of order
Girl/Female
Tamil
Emotional
Boy/Male
Arabic, Muslim
Irritable; Impatient
Girl/Female
Indian, Telugu
Small Utensil
Girl/Female
Indian
Surname or Lastname
English
English : patronymic from Bicker.
Boy/Male
Hindu
Bestowed of boons
Surname or Lastname
English
English : variant of Partridge.
Boy/Male
Arabic
Father of a Toothless Old Woman
BASIS THEOREM-COMPUTABILITY
BASIS THEOREM-COMPUTABILITY
BASIS THEOREM-COMPUTABILITY
BASIS THEOREM-COMPUTABILITY
BASIS THEOREM-COMPUTABILITY
n.
The science, as distinguished from the art; as, the theory and practice of medicine.
pl.
of Bass
a.
Relating to, or skilled in, theory; theoretically skilled.
n.
The philosophical explanation of phenomena, either physical or moral; as, Lavoisier's theory of combustion; Adam Smith's theory of moral sentiments.
a.
A bass, or deep, sound or tone.
n.
The name given to several aromatic herbs of the Mint family, but chiefly to the common or sweet basil (Ocymum basilicum), and the bush basil, or lesser basil (O. minimum), the leaves of which are used in cookery. The name is also given to several kinds of mountain mint (Pycnanthemum).
n.
An isolated or circumscribed formation, particularly where the strata dip inward, on all sides, toward a center; -- especially applied to the coal formations, called coal basins or coal fields.
n.
The southern, red, or channel bass (Sciaena ocellata). See Redfish.
n.
The quantity contained in a basin.
a.
Of or pertaining to a theorem or theorems; comprised in a theorem; consisting of theorems.
v. t.
To formulate into a theorem.
n.
Speculation; theory.
n.
Species of Serranus, the sea bass and rock bass. See Sea bass.
a.
One who sings, or the instrument which plays, bass.
n.
The two American fresh-water species of black bass (genus Micropterus). See Black bass.
pl.
of Basis
a.
Hence, basic; metallic; not acid; -- opposed to negative, and said of metals, bases, and basic radicals.
pl.
of Theory