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formal languages, the interchange lemma states a necessary condition for a language to be context-free, just like the pumping lemma for context-free languages
Interchange_lemma
Type of pumping lemma
lemma does not suffice to guarantee that a language is context-free; there are other necessary conditions, such as Ogden's lemma, or the Interchange lemma
Pumping lemma for context-free languages
Pumping_lemma_for_context-free_languages
lemma Isolation lemma Switching lemma Forking lemma Leftover hash lemma Piling-up lemma (linear cryptanalysis) Yao's XOR lemma Interchange lemma Newman's
List_of_lemmas
Lemma in measure theory
In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior
Fatou's_lemma
Mathematical theorem
In mathematics, Grönwall's inequality (also called Grönwall's lemma or the Grönwall–Bellman inequality) allows one to bound a function that is known to
Grönwall's_inequality
Theorems on the convergence of bounded monotonic sequences
\infty }f_{k}} almost everywhere. The interchange of limits and integrals is then an easy consequence of Fatou's lemma. One has ∫ X f d μ = ∫ X lim inf k
Monotone_convergence_theorem
Commutativity of certain mathematical operations
Cafiero convergence theorem Fatou's lemma Monotone convergence theorem for integrals (Beppo Levi's lemma) Interchange of derivative and integral: Leibniz
Interchange of limiting operations
Interchange_of_limiting_operations
Petersen in 1891 and Kőnig in 1931). To prove Berge's theorem, we first need a lemma. Take a graph G and let M and M′ be two matchings in G. Let G′ be the resultant
Berge's_theorem
1970 American film
Zorns Lemma is a 1970 American structural experimental film by Hollis Frampton. Originally starting as a series of photographs, the non-narrative film
Zorns_Lemma
Election result probability theorem
Bertrand's ballot theorem is related to the cycle lemma. They give similar formulas, but the cycle lemma considers circular shifts of a given ballot counting
Bertrand's_ballot_theorem
of Γ[a,b]. The generalisation of Morse's lemma to CAT(-1) spaces is often referred to as the Morse–Mostow lemma and can be proved by a straightforward generalisation
Busemann_function
Drawings of dense graphs have many crossings
mathematics of graph drawing, the crossing number inequality or crossing lemma gives a lower bound on the minimum number of edge crossings in a plane drawing
Crossing_number_inequality
Volume of a sound or note
via YouTube. Ferneyhough, Brian (1982). Lemma-Icon-Epigram. London: Edition Peters 7233. Brian Ferneyhough - Lemma-Icon-Epigram (w/ score) (for piano solo)
Dynamics_(music)
Sums of sets of vectors are nearly convex
The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. The lemma may be intuitively
Shapley–Folkman_lemma
Central object of study in category theory
completely known and easy to describe; this is the content of the Yoneda lemma. Saunders Mac Lane, one of the founders of category theory, is said to have
Natural_transformation
{\displaystyle |T|<2(1-\varepsilon )\gamma n\,} has a unique neighbour. Lemma 1 proves the case | T | ≤ γ n {\displaystyle |T|\leq \gamma n\,} , so suppose
Expander_code
Theorem in measure theory
condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded
Dominated_convergence_theorem
Extension of Laplace's method for approximating integrals
. Proof of complex Morse lemma The following proof is a straightforward generalization of the proof of the real Morse Lemma, which can be found in . We
Method_of_steepest_descent
Mathematical object in abstract algebra
homomorphism from a pure submodule can be extended to the whole module. "Lemma 47.7.5 (08Z6)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-02-25
Injective_module
Visual representation of the structure of an argument
show co-premises, objections, counterarguments, rebuttals, inferences, and lemmas. There are different styles of argument map but they are often functionally
Argument_map
Theorem in classical electromagnetism
electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time-harmonic electric current densities (sources) and the resulting
Reciprocity (electromagnetism)
Reciprocity_(electromagnetism)
Axiom of set theory
are the same as that of AC). The most important among them are Zorn's lemma and the well-ordering theorem. In fact, Zermelo initially introduced the
Axiom_of_choice
Graph representing edges of another graph
representative graph, and the θ-obrazom, as well as the edge graph, the interchange graph, the adjoint graph, and the derived graph. Hassler Whitney (1932)
Line_graph
Recursive integer sequence
{2n+1}{n}}\,,} which can be directly interpreted in terms of the cycle lemma; see below. The Catalan numbers satisfy the recurrence relations C 0 = 1
Catalan_number
Generalization of category
n-category Doctrine (mathematics) Pseudofunctor String diagram 2-Yoneda lemma Pasting theorem Ehresmann 1965 Bénabou 1967 Kelly & Street 1974, § 1.2.
2-category
Exterior algebraic map taking tensors from p forms to n-p forms
{\frac {\partial }{\partial x_{n}}}\right\}} . In analogy to the Poincaré lemma for exterior derivative, one can define its version for codifferential,
Hodge_star_operator
Average value of a random variable
Fatou's lemma: Let { X n ≥ 0 : n ≥ 0 } {\displaystyle \{X_{n}\geq 0:n\geq 0\}} be a sequence of non-negative random variables. Fatou's lemma states that
Expected_value
Obsolete theories in natural history and natural philosophy
be exchanged in chemical reactions just as chemical elements can be interchanged in inorganic compounds. Vitalism – See section on Biology. Nascent state
List of superseded scientific theories
List_of_superseded_scientific_theories
logical systems that deviate from classical logic. diagonalization lemma A lemma used in the proof of Gödel's incompleteness theorems, stating that for
Glossary_of_logic
Mathematical theorem
_{y}\partial _{x}f} . (In modern language this is one version of the Poincaré lemma.) These can also be rephrased in terms of the action of operators on Schwartz
Symmetry of second derivatives
Symmetry_of_second_derivatives
Tool in linear algebra and matrix analysis
A and D interchanged. By equating the expressions for M−1 obtained in these two different ways, one can establish the matrix inversion lemma, which relates
Schur_complement
Statement about integration on manifolds
basis vector), the result follows. Mathematics portal Chandrasekhar–Wentzel lemma For mathematicians this fact is known, therefore the circle is redundant
Generalized_Stokes_theorem
Mathematical ring with well-behaved ideals
ring is defined in terms of ascending chains of ideals. The Artin–Rees lemma, on the other hand, gives some information about a descending chain of ideals
Noetherian_ring
Theorem
_{C}{(z-a)^{n} \over (w-a)^{n+1}}f(w)\,\mathrm {d} w.\end{aligned}}} Interchange of the integral and infinite sum is justified by observing that f ( w
Analyticity of holomorphic functions
Analyticity_of_holomorphic_functions
Branch of topology
closed sets can be separated by a continuous function; this is Urysohn's lemma.) X is completely normal, or T5 or completely T4, if it is T1 and if any
General_topology
Since M ( x ) ⊂ C 2 ( x ) {\displaystyle M(x)\subset C^{2}(x)} , this lemma establishes the equivalence of the M-regions M ( x ) {\displaystyle M(x)}
Convex_cap
Mathematical use of "for all" and "there exists"
but not uniform continuity (its slope is unbounded). In contrast, interchanging the two initial universal quantifiers in the definition of pointwise
Quantifier_(logic)
Awareness of facts, or competency
Barn Cases Ichikawa & Steup 2018, § 3. The Gettier Problem, § 4. No False Lemmas, § 5. Modal Conditions, § 6. Doing Without Justification? Steup & Neta 2020
Knowledge
Random process independent of past history
again time-homogeneous. If X {\displaystyle X} is a CTMC, then by Kelly's lemma X ← {\displaystyle {\overleftarrow {X}}} has the same stationary distribution
Markov_chain
Method of mathematical integration
} The value of any of the integrals is allowed to be infinite. Fatou's lemma: If {fk}k∈N is a sequence of non-negative measurable functions, then ∫ lim inf
Lebesgue_integral
Differentiation under the integral sign formula
does not consider the possibility of the surface deforming as it moves. Lemma. One has: ∂ ∂ b ( ∫ a b f ( x ) d x ) = f ( b ) , ∂ ∂ a ( ∫ a b f ( x )
Leibniz_integral_rule
Mathematical concept
Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels
Limit_(category_theory)
Mathematical version of an order change
2017. Retrieved 31 March 2018. Heap, B. R. (1963). "Permutations by Interchanges". The Computer Journal. 6 (3): 293–298. doi:10.1093/comjnl/6.3.293. Mütze
Permutation
Type of cipher
is required to securely interchange symmetric keys or PINs with other actors in the banking industry. This secure interchange is performed using the AKB
Block_cipher
Type of vector space in math
every x, then {ek} is an orthonormal basis. As a consequence of Zorn's lemma, every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal
Hilbert_space
In mathematics, invariant of square matrices
\left(W\right)\det \left(W^{-1}+BZ^{-1}A\right)} (see Matrix determinant lemma), where Z is an m × m invertible matrix and W is an n × n invertible matrix
Determinant
Mathematical operation on matrices
unique solution, if and only if A and B are invertible (Horn & Johnson 1991, Lemma 4.3.1). If X and C are row-ordered into the column vectors u and v, respectively
Kronecker_product
Non-commutative group with 6 elements
A result closely related to the orbit-stabilizer theorem is Burnside's lemma: | X / G | = 1 | G | ∑ g ∈ G | X g | {\displaystyle \left|X/G\right|={\frac
Dihedral_group_of_order_6
Correspondence between properties of a category and its opposite
statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms
Dual_(category_theory)
Conformal structure admits a Hodge dual of 1-forms without even specifying a metric
{\partial }}+{\bar {\partial }}\partial =0.} On a Riemann surface the Poincaré lemma states that every closed 1-form or 2-form is locally exact. Thus if ω is
Differential forms on a Riemann surface
Differential_forms_on_a_Riemann_surface
Mathematical table used in logic
op Q = ¬(P op2 Q). The Dual row shows the dual operation obtained by interchanging T with F, and AND with OR. The L id row shows the operator's left identities
Truth_table
Algebraic manipulation of "true" and "false"
algebra is unchanged when all dual pairs are interchanged. One change not needed to make as part of this interchange was to complement. Complement is a self-dual
Boolean_algebra
Mathematical description of quantum state
mechanics, only separable Hilbert spaces are considered, which using Zorn's Lemma, implies it admits a countably infinite Schauder basis rather than an orthonormal
Wave_function
Heliostatic model of solar system by Nicolaus Copernicus
publication of his book. Copernicus used what is now known as the Urdi lemma and the Tusi couple in the same planetary models as found in Arabic sources
Copernican_heliocentrism
Organized notations & citations regarding the history & content of a primary source work
parts: The location of the variant in the text (act, scene, line number) The lemma, which is the portion of the text to which the note applies A right bracket
Critical_apparatus
Earliest civilian block ciphers
on a 64-bit half-block of data, together with a 64-bit subkey and 8 "interchange control bits" (ICBs). The ICBs control a swapping operation. The 64-bit
Lucifer_(cipher)
In group theory, equivalence class under the relation of conjugation
product of two cyclic groups of order p . {\displaystyle p.} By Burnside's lemma, the number of conjugacy classes of a finite group G {\displaystyle G} is
Conjugacy_class
Creation of knowledge from structured and unstructured sources
knowledge extraction include: part-of-speech (POS) tagging lemmatization (LEMMA) or stemming (STEM) word sense disambiguation (WSD, related to semantic
Knowledge_extraction
f^{*}} and is denoted by f ∗ {\displaystyle f_{*}} . Ken Brown's lemma Ken Brown's lemma gives a sufficient condition for a functor to preserve weak equivalences
Glossary_of_category_theory
Computer approximation for real numbers
difference is computed exactly because the numbers are close—the Sterbenz lemma guarantees this, even in case of underflow when gradual underflow is supported
Floating-point_arithmetic
Power series with negative powers
constricted closed annulus, so the integration and summation can be interchanged. Substituting the identity ∮ γ ( z − c ) n − k − 1 d z = 2 π i δ n k
Laurent_series
Mathematical category formed by reversing morphisms
category C {\displaystyle C} is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields
Opposite_category
Function that is its own inverse
called Desargues's Involution Theorem. Its origins can be seen in Lemma IV of the lemmas to the Porisms of Euclid in Volume VII of the Collection of Pappus
Involution_(mathematics)
Characterizes the diagonal of a Hermitian matrix with given eigenvalues
{x}}.} The following lemma characterizes the permutation polytope of a vector in R n . {\displaystyle \mathbb {R} ^{n}.} Lemma—If x 1 ≥ ⋯ ≥ x n , {\displaystyle
Schur–Horn_theorem
Mathematical concept for comparing objects
the orbits of the action of H on G—are the right cosets of H in G. Interchanging a and b yields the left cosets. Related thinking can be found in Rosen
Equivalence_relation
correspondence with codewords. Hence, this step outputs the list of codewords. Lemma: Interpolation step implies ( r + 1 2 ) {\displaystyle
Guruswami–Sudan list decoding algorithm
Guruswami–Sudan_list_decoding_algorithm
Integral criterion for holomorphy
Similarly, in the case of the zeta function, the M-test justifies interchanging the integral along the closed curve and the sum. The hypotheses of Morera's
Morera's_theorem
Root-finding algorithm
{\displaystyle p^{*}} , and T p {\displaystyle Tp} are related as follows. Lemma Let p {\displaystyle p} be a complex polynomial and δ = ( T p ) ( 0 ) {\displaystyle
Lehmer–Schur_algorithm
Theorem used in quantum mechanics for angular momentum calculations
each irreducible representation occurs only once. This allows Schur's lemma to be used. Starting with the definition of a spherical tensor operator
Wigner–Eckart_theorem
Extinct Indo-European language in Asia
623–638. doi:10.1017/S0035869X00168807. JSTOR 25220813. S2CID 162418763. A dictionary of Tocharian B by Douglas Q. Adams Tocharian B lemmas in Wiktionary
Tocharian_B
Value indicating the relation of a proposition to truth
Formal proof Formal semantics Foundations of mathematics Information theory Lemma Logical consequence Model Theorem Theory Type theory Theorems (list), paradoxes
Truth_value
On chains and antichains in partial orders
Spinrad, J. (1999), Recognition Algorithms for Orders of Small Width and Graphs of Small Dilworth Number Weisstein, Eric W. "Dilworth's Lemma". MathWorld.
Dilworth's_theorem
Formal language that can be expressed using a regular expression
is not regular, one often uses the Myhill–Nerode theorem and the pumping lemma. Other approaches include using the closure properties of regular languages
Regular_language
Decimal representation of real numbers in computing
difference is computed exactly because the numbers are close—the Sterbenz lemma guarantees this, even in case of underflow when gradual underflow is supported
Decimal_floating_point
Graphical representation of a morphism
the same equivalence class of the congruence relation generated by the interchanger: d ⊗ dom ( d ′ ) ∘ cod ( d ) ⊗ d ′ = dom ( d ) ⊗ d ′ ∘ d ⊗ cod
String_diagram
Traditional maritime greeting
earlier uses of the term are lacking, because ahoi did not get its own lemma in the Woordenboek der Nederlandsche Taal (WNT), even though this comprehensive
Ahoy
Construction in group theory
g(l)) ∈ I. For PSL (except PSL(2, 2) and PSL(2, 3)) this follows by Grün's lemma because SL is a perfect group (hence center equals hypercenter), but for
Projective_linear_group
Mathematical logic concept
)\to (\neg \phi \to \neg \psi )} , is proven below, using the following lemmas proven here: (DN1) ¬ ¬ p → p {\displaystyle \neg \neg p\to p} - Double negation
Contraposition
Generating polynomial of the number of ways to place non-attacking rooks on a chessboard
stationary. The polynomial also remains the same if rows are interchanged or columns are interchanged. The term "rook polynomial" was coined by John Riordan
Rook_polynomial
Quadrilateral whose vertices lie on a circle
and q2. Then (the first equality is Proposition 11 in Archimedes' Book of Lemmas) D 2 = p 1 2 + p 2 2 + q 1 2 + q 2 2 = a 2 + c 2 = b 2 + d 2 {\displaystyle
Cyclic_quadrilateral
Concept in calculus of variations
{\displaystyle \phi } is also an arbitrary function, applying the fundamental lemma of calculus of variations to the last line, the functional derivative is
Functional_derivative
Identities and relationships involving sets
transformed into the other by interchanging ∪ {\displaystyle \cup } and ∩ {\displaystyle \cap } , while also interchanging ∅ {\displaystyle \varnothing
Algebra_of_sets
Axioms in topology defining notions of "separation"
closed sets can be separated by a continuous function; this is Urysohn's lemma.) X is normal regular if it is both R0 and normal. Every normal regular
Separation_axiom
Special mathematical function
Retrieved 2024-10-14. Bateman & Erdélyi 1953, p. 27 Guillera & Sondow 2008, Lemma 2.1 and 2.2 Bateman & Erdélyi 1953, p. 28 "The Analytic Continuation of
Lerch_transcendent
Orthogonal group of an indefinite quadratic form
The signature of the form determines the group up to isomorphism; interchanging p {\displaystyle p} with q {\displaystyle q} amounts to replacing the
Indefinite_orthogonal_group
Mathematical methods used in Bayesian inference and machine learning
{\displaystyle q^{*}} over all variables not in the partition: refer to Lemma 4.1 of for a derivation of the distribution q j ∗ ( Z j ∣ X ) {\displaystyle
Variational_Bayesian_methods
Vector satisfying some of the criteria of an eigenvector
x {\displaystyle \mathbf {x} } is then obtained using the relation (8). Lemma: Given the following chain of generalized eigenvectors of length r , {\displaystyle
Generalized_eigenvector
incorrectly list the ISBNs of these volumes, for example, sometimes interchanging those of volumes II and III(1/2). Review PDF by Arthur W. Burks in the
Charles Sanders Peirce bibliography
Charles_Sanders_Peirce_bibliography
Grammar of the Cebuano language
root is the simplest version of a verb that conveys its overall meaning or lemma and cannot be broken down any further (excluding morphological processes
Cebuano_grammar
Expression in commutative algebra
_{j=0}^{k}{\binom {n+k-1}{k-j}}h_{j}(X_{1},\ldots ,X_{n}).} The proof, as found in Lemma 3.5 of, relies on the combinatorial properties of increasing k {\displaystyle
Complete homogeneous symmetric polynomial
Complete_homogeneous_symmetric_polynomial
Game class in game theory
symmetric. All players have the same options so if players interchange their moves, they also interchange their payoffs. By design, symmetric games are fair in
Simultaneous_game
Economic model
is stable, but Cournot remarks that if the red and blue curves were interchanged then this would cease to be true. He adds that it is easy to see that
Cournot_competition
\scriptstyle d\geq 2} proof By Chebyshev's inequality and the Borel–Cantelli lemma, there is the equation below: P ( ρ r ≤ lim inf t → ∞ T t t ≤ lim sup t
Voter_model
Representation theory of the symplectic group
g ( z ) ) . {\displaystyle \pi (g)W(z)\pi (g)^{*}=W(g(z)).} By Schur's lemma the unitary π(g) is unique up to multiplication by a scalar ζ with |ζ| =
Oscillator_representation
(1959), Theorem 7.2 Liao (1954) Dold and Thom (1958), 4.7 McCord (1969), Lemma 6.2 McCord (1969), Corollary 6.9 McCord (1969), Theorem 11.5 McCord (1969)
Symmetric_product_(topology)
structure. Note that this process is essentially an application of the Yoneda lemma to the definition of the groupoid schemes in the category Aff {\displaystyle
Hopf_algebroid
INTERCHANGE LEMMA
INTERCHANGE LEMMA
Girl/Female
Hindu
The name lemma means a creeper, A deer, A lady
Girl/Female
Muslim
The name lemma means a creeper, A deer, A lady
Girl/Female
Tamil
The name lemma means a creeper, A deer, A lady
Surname or Lastname
English (mainly Lancashire and Cheshire)
English (mainly Lancashire and Cheshire) : unexplained.Probably an altered form of German Dornig, which is probably a nickname for someone with a sharp tongue, from an adjectival derivative of Middle High German, Middle Low German dorn ‘thorn’. The suffixes -ig and -ing were often interchanged in Pennsylvania German and elsewhere. The name may also refer to a sloe bush.
Surname or Lastname
English
English : probably a habitational name from Burnworthy in Devon, which is named with the Old English personal name Beorna + Old English worð or worðig ‘enclosure’; the interchange between worth and worthy is common in Middle English names in the southwest. The surname has died out in the British Isles.
INTERCHANGE LEMMA
INTERCHANGE LEMMA
Girl/Female
Arabic, Muslim
Present; Manifest
Boy/Male
Christian, German, Hebrew
The Eagle Rules; Strong as an Eagle; Exalted; On High; Diminutive of Arnold; Eagle Power
Surname or Lastname
English (mainly Yorkshire)
English (mainly Yorkshire) : habitational name from any of various places, for example in Durham and Yorkshire, so called from Old English scīr ‘bright’ + burna ‘stream’. (In southern English dialects, burna became modern bourne, and Sherborne in Dorset is one of several places so called.)Americanized form of French Charbon (see Jarboe) or Charbonneau.
Boy/Male
American, Anglo, British, Christian, Danish, Dutch, English, French, Gaelic, German, Scottish, Swedish
Friend of the Sea; Form of Irving; Handsome and Fair; Place Name; Beautiful; Honored Friend
Boy/Male
Muslim
Who has the secret, Confidant
Girl/Female
Spanish
Noble. Of the nobility.
Boy/Male
Hindu, Indian, Telugu
White Page
Girl/Female
Muslim
Beautiful
Surname or Lastname
English
English : nickname for a respectable or decent person, or else a good-looking one, both these senses belonging to Middle English tall (Old English getæl ‘swift’, ‘prompt’). The modern sense ‘of high stature’ did not develop until the end of the 16th century; the usual Middle English equivalents were Long and High.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Lord Vishnu
INTERCHANGE LEMMA
INTERCHANGE LEMMA
INTERCHANGE LEMMA
INTERCHANGE LEMMA
INTERCHANGE LEMMA
n.
The quality of correlation; reciprocation; interchange; interaction; interdependence.
n.
A meeting for consultation, discussion, or an interchange of opinions.
n.
The act of mutually changing; the act of mutually giving and receiving; exchange; as, the interchange of civilities between two persons.
a.
Capable of being commuted or interchanged.
v. t.
To interchange; to transfer reciprocally.
n.
Interchange of speech; dialogue; conversation; conference.
n.
The mutual exchange of commodities between two persons or countries; barter; commerce.
v. t.
To put each in the place of the other; to give and take mutually; to exchange; to reciprocate; as, to interchange places; they interchanged friendly offices and services.
imp. & p. p.
of Interchange
n.
Familiar discourse; free interchange of thoughts or views; conversation; chat.
v. i.
To make an interchange; to alternate.
n.
Interchange; mutual or reciprocal change.
n.
Alternate succession; alternation; a mingling.
a.
Capable of being exchanged or interchanged; reciprocal; interchangeable.
v. t.
To cause to follow alternately; to intermingle; to vary; as, to interchange cares with pleasures.
n.
Reciprocal migration; interchange of dwelling place by migration.
n.
Colloquial discourse; oral interchange of sentiments and observations; informal dialogue.
v. t.
To communicate mutually; to interchange.
n.
Interchange of opinions; mutual advising; consultation.
p. pr. & vb. n.
of Interchange