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Longer titles found: Counterexample-guided abstraction refinement (view), Counterexamples in Probability (view), Counterexamples in Probability and Statistics (view), Counterexamples in Topology (view), Minimal counterexample (view), Witsenhausen's counterexample (view)

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Hilbert's fourteenth problem (828 words) [view diff] exact match in snippet view article find links to article

1954). Then in 1959 Masayoshi Nagata found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants

appeared. He said he had a counterexample to a lemma which had been used in the proof. An hour or two later, he produced a counterexample to the theorem itself

the first counterexample in a 1973 article. However, much work in this area was done by Grothendieck (1955). Later many other counterexamples were found

Tutte (1946), who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38

conjecture, or $50 for a counterexample; it is one of many conjectures of Erdős. If the conjecture is false, a counterexample would take the form of a

non-compact homogeneous spaces (specifically spaces of lattices). As a counterexample take any finite neighbourhood of the particular point of an infinite

In chemistry, stacking refers to superposition of molecules or atomic sheets owing to attractive interactions between these molecules or sheets. Metal

Norio Iwase gave a counterexample to the general case in 1998. In a follow-up paper from 2002, Iwase gave an even stronger counterexample, with X a closed

of a holomorphic function along a closed curve is zero. The standard counterexample is the function f(z) = 1/z, which is holomorphic on C − {0}. On any

unique, i.e. any other weak solution must have the same law. Another counterexample of this type is Tsirelson's stochastic differential equation. Øksendal

the two iterated integrals have different values; see the section on counterexamples below for an example of this. Tonelli's theorem and the Fubini–Tonelli

43-dimensional polytope of 86 facets with a diameter of more than 43. The counterexample has no direct consequences for the analysis of the simplex method, as

In mathematics, specifically in operator theory, each linear operator A {\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint)

Shm-reduplication is a form of reduplication originating in Yiddish in which the original word or its first syllable (the base) is repeated with the copy

scheme if every orbit is contained in an affine open subscheme; the counterexample above shows that this technical condition cannot be dropped. For quasi-projective

these proteins in a similar way, acquiring GAG side chains in vivo. One counterexample is the protein chymase, which directly binds to heparin. Dermatan sulfate

In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | {\displaystyle |H|} is a

generators that serves as a counterexample to Naimark's problem. More precisely, they showed that the existence of a counterexample generated by ℵ 1 {\displaystyle

(A)<\infty } is necessary. To see this, it is simple to construct a counterexample when μ is the Lebesgue measure: consider the sequence of real-valued

subsequently found its way into Rolfsen's table. The Perko pair gives a counterexample to a "theorem" claimed by Little in 1900 that the writhe of a reduced

a group G so that either G is a counterexample to the Eilenberg–Ganea conjecture, or there must be a counterexample to the Whitehead conjecture; in other

Published by Tutte in 1946, it is the first counterexample constructed for this conjecture. Other counterexamples were found later, in many cases based on

In economics, the Lucas paradox or the Lucas puzzle is the observation that capital does not flow from developed countries to developing countries despite

Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, F is the most widely studied

if either space is assumed to be only a normed vector space; see § Counterexample. The proof is based on the following lemmas, which are also somewhat

spacetime. It contains a non-curvature singularity and is an important counterexample to various hypotheses in general relativity. Michio Kaku develops the

existence of the minimal counterexample implies an even smaller counterexample, we have a contradiction (since the minimal counterexample isn't minimal) and

The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian

The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian

2007) has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture. As of 2021, the counterexample has not been verified

that the boundaries should always be the tonic and dominant notes. One counterexample comes from the Minuet of Wolfgang Amadeus Mozart's String Quartet in

Dehn, who proved that the answer in general is "no" by producing a counterexample. The answer for the analogous question about polygons in 2 dimensions

Source-specific multicast (SSM) is a method of delivering multicast packets in which the only packets that are delivered to a receiver are those originating

Kolmogorov's counterexample in L1 was unbounded in any interval, but it was thought to be only a matter of time before a continuous counterexample was found

who was Kakwa. Jonathan Owens argues that Nubi constitutes a major counterexample to Derek Bickerton's theories of creole language formation, showing

semidirect product, though not in general a direct product. To form a counterexample, take the smallest non-abelian group B ≅ S3, the symmetric group on

and Mark N. Ellingham, their discoverers. These two graphs provide counterexamples to the conjecture of W. T. Tutte that every cubic 3-connected bipartite

Joseph Horton. Published by Bondy and Murty in 1976, it provides a counterexample to the Tutte conjecture that every cubic 3-connected bipartite graph

derivation, one can indeed verify one of the constructed point sets as a counterexample for n = 1325 (as well as all higher dimensions up to 1560). Their result

logically, it is unnecessary to find all counterexamples to a theory; all that is required is one counterexample. The converse does not prove a theory;

originally published in CAV are techniques in model checking, such as Counterexample-Guided Abstraction Refinement and partial order reduction. It is often

In commutative algebra, a complete intersection ring is a commutative ring similar to the coordinate rings of varieties that are complete intersections

computer-aided proof. This came after many false proofs and mistaken counterexamples in the preceding decades. The Appel–Haken proof proceeds by analyzing

In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives

products cannot be colored with an unexpectedly small number of colors. A counterexample to the conjecture was discovered by Yaroslav Shitov (2019) (see Kalai

space, whose complement is not simply connected. It also serves as a counterexample to the claim that all Cantor spaces are ambiently homeomorphic to each

general if the extension is abelian but not cyclic. Hasse gave the counterexample that 3 is a local norm everywhere for the extension Q ( − 3 , 13 ) /

exists a counterexample to the conjecture, then a positive proportion (in the sense of asymptotic density) of the integers are likewise counterexamples. Although

definitions are composed of necessary and sufficient conditions, and a single counterexample overthrows such a definition. Theorizing about art, on the other hand

false when P. K. Stockmeyer discovered several infinite families of counterexample pairs of digraphs (including tournaments) of arbitrarily large order

theory need not have an infinite number of axioms by coming up with a counterexample: Robinson arithmetic Q. Q is finitely axiomatizable because it lacks

phenomenon, discovered by Deuring (1933) and Heilbronn (1934), states that a counterexample to the generalized Riemann hypothesis for one Dirichlet L-function affects

neighborhood is also a function of the configuration around the position. The counterexample given above for a continuous and shift-equivariant map which is not

in 2003. The counterexample provided in that reference uses advanced measure-theoretical ideas. Subsequently, many other counterexamples have been provided

instance in which four fifth powers sum to a fifth power. This is a counterexample to a conjecture by Euler that at least n nth powers are required to

the maximum number of double points possible. The Barth sextic is a counterexample to an incorrect claim by Francesco Severi in 1946 that 52 is the maximum

not true that f {\displaystyle f} will necessarily be continuous. A counterexample in two dimensions is given by f ( x , y ) = x y x 2 + y 2 . {\displaystyle

define an equivalence relation. A ring species is a species with a counterexample to the transitivity of interbreeding. However, it is unclear whether

convergence. More generally, a Specker sequence is called a recursive counterexample to the least upper bound principle, i.e. a construction that shows that

the closed interval [ 1 , 4 ] {\displaystyle [1,4]} . The standard counterexample is complex projective space with the Fubini–Study metric; sectional

constant mean curvature, discovered by Henry C. Wente (1986). It is a counterexample to the conjecture of Heinz Hopf that every closed, compact, constant-mean-curvature

several extensions over this earlier program, including Generation of counterexamples for conflicts Location tracking (e.g., file, line, column) Rich and

Cayley graph. A counterexample was proposed by Diestel and Leader in 2001. In 2005, Eskin, Fisher, and Whyte confirmed the counterexample. Edge-transitive

non-linear elasticity. In 1992, he won an EMS Prize for producing a counterexample to a problem first posed by Charles B. Morrey, Jr. in 1950, whether

of their broader view of the empire.[citation needed] Held out as a counterexample to the traditors was the venerated Saint Vincent of Saragossa who preferred

p) for p ≡ ±1 (mod 5) is false and has e.g. a 6-digit counterexample. The smallest counterexample for +1 (mod 5) is 6601 = 7 × 23 × 41 and the smallest

this inequality has a larger counterexample, proving the Riemann hypothesis to be false, the smallest such counterexample must be a superabundant number

concrete graph on 10 vertices that appears as a minimal example or counterexample in many different contexts. Balaban 10-cage Balaban 11-cage Bidiakis

for the domination number of the tensor product of graphs; however, a counterexample was found by Klavžar & Zmazek (1996). Since Vizing proposed his conjecture

statistically independent. However, this is untrue, as can be demonstrated by counterexample. Likewise, it is sometimes mistakenly thought that a linear combination

spaces on large class of pseudoconvex domains. Boas also provided a counterexample to the Lu Qi-Keng Conjecture which was raised by Lu Qi-Keng in 1966

symmetrically differentiable, but the converse is not true. A well-known counterexample is the absolute value function f(x) = |x|, which is not differentiable

Analytical-numerical localization of hidden oscillation in counterexamples to Aizerman's and Kalman's conjectures Discrete-time counterexample in Maplecloud

numbers; so if it is false, then some natural number is a counterexample. If this counterexample existed within the standard natural numbers, its existence

unequal sizes are allowed, then the Pythagorean tiling would form a counterexample in two dimensions. The conjecture as stated does not require all of

later that he found the first counterexample, [2184, 2185, …, 2200], with k = 16. The existence of this counterexample shows that 16 is an Erdős–Woods

example in the theory of Banach spaces and commonly serves as useful counterexample to general statements concerning the structure of general Banach spaces

Ph.D. under Claude Chevalley in 1970, and at that time published a counterexample to the original form of a conjecture of Emil Artin, which suitably modified

Kittell, who used it as a counterexample to Alfred Kempe's flawed proof of the four-color theorem. Simpler counterexamples include the Errera graph and

is a function created by British mathematician John H. Conway as a counterexample to the converse of the intermediate value theorem. In other words, it

book thickness 3 and queue number 2. Published in 1973, it provides a counterexample to the Crispin Nash-Williams conjecture that every 4-regular 4-vertex-connected

The Petkau effect is an early counterexample to linear-effect assumptions usually made about radiation exposure. It was found by Dr. Abram Petkau at the

with 17 vertices and 45 edges. Alfred Errera published it in 1921 as a counterexample to Kempe's erroneous proof of the four color theorem; it was named after

In geometry, the Reeve tetrahedra are a family of polyhedra with vertices at ( 0 , 0 , 0 ) , ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 1 , 1 , r ) , {\displaystyle

true that f {\displaystyle f} will necessarily be differentiable. A counterexample in two dimensions is given by f ( x , y ) = 2 x 2 y + y 3 x 2 + y 2

In algebraic geometry, a divisorial scheme is a scheme admitting an ample family of line bundles, as opposed to an ample line bundle. In particular, a

excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable now, but provable in the future to show that

in the hypothesis cannot be relaxed; see an example in the section Counterexamples and uniqueness. This version of the separation theorem does generalize

Dieter Puppe. A proof due to Kathleen Lewis that Puppe's counterexample is indeed a counterexample can be found in the book of J. Peter May and Johann Sigurdsson

closed in localizations of the ring. Brenner & Monsky (2010) found a counterexample to the localization property of tight closure. However, there is still

modern sense, then it would be a counterexample to the modern version of the third conjecture (without being a counterexample to the original version). The

theory in two complex variables. He is also known for constructing a counterexample in several complex variables in 1976. Fornæss is a fellow of the Norwegian

to each other, but the converse is not necessarily true. One obvious counterexample is R n {\displaystyle \mathbb {R} ^{n}} and T n {\displaystyle \mathbb

describes the geometry around a rotating object), Taub–NUT vacuum (a famous counterexample describing the exterior gravitational field of an isolated object with

Reuleaux's original motivation for studying the Reuleaux triangle was as a counterexample, showing that three single-point contacts may not be enough to fix a

knowledge. Attributed to American philosopher Edmund Gettier, Gettier-type counterexamples (called "Gettier-cases") challenge the long-held justified true belief

Peano surface, a counterexample to some criteria of local maxima of the 19th century

through a direct computer search on a CDC 6600, they have found the counterexample 275 + 845 + 1105 + 1335 = 1445. Their paper announcing the result is

research on foliations, knot theory, and 3-manifolds. In 1974 he found a counterexample to the Seifert conjecture that every non-vanishing vector field on the

K\subset \mathbb {S} ^{3}} has Property P, then one cannot construct a counterexample to the Poincaré conjecture by surgery along K {\displaystyle K} . A

applications to complex analysis and harmonic analysis. His work include a counterexample to the general Vitushkin's conjecture and with Mark Melnikov and Joan

dissertation he introduced what is now called the Errera graph, which is a counterexample to the validity of the alleged proof of the four color theorem by Alfred

K\subset \mathbb {S} ^{3}} has Property P, then one cannot construct a counterexample to the Poincaré conjecture by surgery along K {\displaystyle K} . A

that group. However, not all Galois groups have generic polynomials, a counterexample being the cyclic group of order eight. The symmetric group Sn. This

paper in the American Journal of Mathematics, in which he presented a counterexample to Hilbert's 14th problem. Nagata Conjecture. Suppose p1, ..., pr are

known planar graph that requires five colors for list coloring, and a counterexample to a related conjecture that list coloring of planar graphs requires

The bystander effect, or bystander apathy, is a social psychological theory that states that individuals are less likely to offer help to a victim in the

Pomerance, Selfridge, and Wagstaff offered $30 for the discovery of a counterexample, that is, a composite number that passed this test. Richard Guy incorrectly

scales. In 1999 John Wainwright, M. J. Hancock and Claes Uggla show a counterexample in the non-tilted perfect fluid case. Thus an almost isotropic cosmic

If there exists such a counterexample, then there also exists a minimal (p/q)-counterexample, which is a (p/q)-counterexample with a smallest number of

combining three Walther graphs to produce the Tutte graph, the first known counterexample to Tait's conjecture that every 3-regular polyhedron has a Hamiltonian

embedded into L3. However, K does not have the amalgamation property. The counterexample for this starts with L1 containing a single element e and extends in

Akemann and Nik Weaver showed in 2003 that the statement "there exists a counterexample to Naimark's problem which is generated by ℵ1, elements" is independent

conjecture were proved, until Norio Iwase provided a counterexample in 1998. A minimum dimensional counterexample to Ganea’s conjecture was constructed by Don

that problem led to her 1993 work in which she constructed a smooth counterexample to the Seifert conjecture. She has since continued to work in dynamical

who take after their parents. The story Kate Crackernuts serves as a counterexample where the daughter of the evil stepparent is a loving stepsister. Many

gymnasium. In 1914 he finally moved to Stockholm. He is best known for his counterexample of a continuous function for which the Newton potential is not twice

be able to announce its correctness; and the teacher might present a counterexample to any representation arbitrarily long after. Gold defined two types

the sum of a positive semidefinite matrix and a nonnegative matrix. A counterexample for order 5 is given by a copositive matrix known as Horn-matrix: (

Brill–Noether theory. He also found a new counterexample to Hilbert's 14th problem (the first counterexample was found by Nagata in 1959). Mukai, Shigeru;

intrinsic model in stochastic decentralized control, the Witsenhausen counterexample, his work on Turán graph, and the various notions of common information

named for Richard Friederich Arens. Its role is mainly to serve as a counterexample. The Arens square is the topological space ( X , τ ) , {\displaystyle

is the set of real numbers, the proposition is false, with x = √2 as counterexample; if the domain is the set of natural numbers, the proposition is true

2\leq x<3\end{cases}}} for which every value has period 3, would be a counterexample. Similarly essential is the assumption of f {\displaystyle f} being

(Barrouillet, et al., 2000). They reject a conclusion if they find a counterexample, i.e., a possibility in which the premises hold, but the conclusion

ISBN 9781139503143, Dougherty, Freiling, and Zeger (2005) showed via an ingenious counterexample that unlike the multicast case, linear network coding fails to achieve

1165R. doi:10.1119/1.1973819. ISSN 0002-9505. Rindler, W. (1968). "Counterexample to the Lenz-Schiff Argument". American Journal of Physics. 36 (6). American

differentiability or even continuity is controversial; Halpern found a counterexample based on his observation that the Boolean algebra of statements may

1090/s0002-9947-1976-0431253-2. ISSN 0002-9947. MR 0431253. Haydon, Richard (1990). "A counterexample to several questions about scattered compact spaces". Bulletin of the

therefore be proven by showing that a counterexample over C {\displaystyle \mathbb {C} } would translate into a counterexample in some algebraic extension of

and particle trajectories are everywhere dense. The last result is a counterexample to some statements in traditional textbooks on fluid mechanics that

A. Bolibrukh (1990) found a counterexample to Plemelj's statement. This is commonly viewed as providing a counterexample to the precise question Hilbert

where such a conclusion could be reached by scientific observation. A counterexample is the case where children of Gastarbeiter parents speaking pidgin German

disproved in characteristic 2 by Giles Gardam by exhibiting an explicit counterexample in a crystallographic group, namely the fundamental group of the Hantzsche–Wendt

functions that will converge to the counterexample to the common fixed point problem. Although the discovery of counterexamples by Boyce and Huneke meant that

and Józef H. Przytycki Conway base 13 function – a function used as a counterexample to the converse of the intermediate value theorem Conway chained arrow

lattices with at most ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ2 compact elements using a construction

general. z w = z w {\displaystyle {\sqrt {zw}}={\sqrt {z}}{\sqrt {w}}} Counterexample for the principal square root: z = −1 and w = −1 This equality is valid

cardinals. In particular, then, if κ {\displaystyle \kappa } is the least counterexample to the singular cardinals hypothesis, then c f ( κ ) = ω {\displaystyle

weakened. In 1977, Steven Zucker showed that it is possible to construct a counterexample to the Hodge conjecture as complex tori with analytic rational cohomology

every topological algebra is localizable, has become a "well-known counterexample". In the study of complex vector spaces, a set S is said to be polynomially

simple with comparatively little ornamentation, although a significant counterexample is the building's back wall, in which was placed a prominent petroglyph

vertices and 21 edges. It was used by Fritsch & Fritsch (1998) as a small counterexample to Alfred Kempe's false proof of the four color theorem using Kempe

following 171 and preceding 173. 172 is a part of a near-miss for being a counterexample to Fermat's last theorem, as 1353 + 1383 = 1723 − 1. This is only the

A proof runs as follows: suppose for contradiction θ is a minimal counterexample, and fix ≺, R, and a good universal set U ⊆ (ωω)3 for the Γ-subsets

because it lacks the antecedents of a "bona fide nonstandard word." A counterexample is provided in ain't, which has an "ancient genealogy" at which scholars

non-alternating knots this conjecture is not true; the Perko pair is a counterexample. This result also implies the following conjecture: Alternating amphicheiral

can be expressed as the sum of k − 1 other k-th powers, providing counterexamples to Euler's sum of powers conjecture. Specifically, 275 + 845 + 1105

If a counterexample exists, that is, a graph Km,n requiring fewer crossings than the Zarankiewicz bound, then in the smallest counterexample both m

Hilbert's proof did not exhibit any explicit counterexample: only in 1967 the first explicit counterexample was constructed by Motzkin. Furthermore, if

assumption of the axiom of choice, we present two separate constructions of counterexamples to the axiom of determinacy. It follows that the axiom of determinacy

to modern graph theory as we know it today. In 1898, he presented a counterexample to Tait's claimed theorem about 1-factorability of 3-regular graphs

stable. The conjecture is true for the two-dimensional case. However, counterexamples have been constructed in higher dimensions. Hence, in the two-dimensional

describes the quotient of a variety by a group. In 1959, he introduced a counterexample to the general case of Hilbert's fourteenth problem on invariant theory

London A counterexample to the first Zassenhaus conjecture Leo Margolis Winner (ex aequo)  Germany/ Russia Vrije Universiteit Brussel A counterexample to the

term', but this is only an apparent paradox and does not constitute a counterexample to the claim that without intension a word has no meaning. Part of its

Dvoretzky–Rogers theorem, both with Aryeh Dvoretzky. He constructed a counterexample to a conjecture related to the Busemann–Petty problem. In the geometry

Have Symmetric Solutions?" and the second was in 1995 for his paper "A Counterexample for Germain". The latter has been characterized as "a historical and

are positive integers, then his conjecture was that n ≥ k. In 1966, a counterexample to Euler's sum of powers conjecture was found by Leon J. Lander and

Conjectures and open questions in rigidity; A flexible sphere; and A counterexample to the rigidity conjecture for polyhedra. "4816 Connelly (1981 PK)"

on one level teach listeners how to live a good life, but usually by counterexample. Raven has been described as the greediest, most lecherous and mischievous

periodic points for Hamiltonian diffeomorphisms in many cases, and for his counterexample (joint with Başak Gürel) to the Hamiltonian Seifert conjecture which

all of its spanning trees, to be independent. The wheel W6 supplied a counterexample to a conjecture of Paul Erdős on Ramsey theory: he had conjectured that

for early contributions to dynamical systems and topology, including counterexamples to the Denjoy conjecture and the Seifert conjecture. Harrison grew

his fieldwork on the language A 1994 message from Jacques Guy, citing Tolomako as a counterexample to the thesis that all languages are equally complex

not resolved until Mary Ellen Rudin constructed one in 1971. Rudin's counterexample is a very large space (of cardinality ℵ ω ℵ 0 {\displaystyle \aleph

of Alhazen's problem (although in 1965 Elkin had already provided a counterexample to Euclidean construction). Researchers have extended Alhazen's problem

the same distance sets. The claim is incorrect, but there is only one counterexample, a pair of six-point Golomb rulers with a shared distance set. The equilateral

proved, it could be refuted by the observation of a single counterexample. Such a counterexample would require that the assumptions underlying the theory

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is

properties. Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles is false, it

Mathematica. 39 (2–3): 287–296. doi:10.1007/BF01165793. —— (1985). "A counterexample in 3-space to a conjecture of H. Hopf". Arbeitstagung Bonn 1984. Lecture

about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number n for which the Mertens function M(n) equals

Dries proved that if the Jacobian conjecture is false, then it has a counterexample with integer coefficients and Jacobian determinant 1. In consequence

Mathematica. 39 (2–3): 287–296. doi:10.1007/BF01165793. —— (1985). "A counterexample in 3-space to a conjecture of H. Hopf". Arbeitstagung Bonn 1984. Lecture

not necessarily reliable, because del does not commute in general. A counterexample that demonstrates the divergence ( ∇ ⋅ v {\displaystyle \nabla \cdot

σ-finite measure is s-finite, the converse is not true. For a proof and counterexample see relation to σ-finite measures. Finite measure Sigma additivity Klenke

The necessity of the first condition can be seen by considering the counterexample where the functions are f ( x ) = x + 1 {\displaystyle f(x)=x+1} and

Tarski, and others worked on the problem, but failed to find a proof or counterexample. William McCune proved the conjecture in 1996, using the automated theorem

Girgensohn 1996). Laerte Sorini, in a work of 2001 showed that a possible counterexample should be a number n greater than  1036067 which represents the limit

One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example: If

probability less than 1, leading to regress. There are innumerable "counterexamples" where, it is argued, a straightforward application of CDT fails to

original on 2012-02-16, retrieved 2008-07-27 Connelly, Robert (1977), "A counterexample to the rigidity conjecture for polyhedra", Publications Mathématiques

The proof is by contradiction. We consider a minimal counterexample, that is, a counterexample with a smallest m and fewest input numbers. Denote the

conjecture might not be true, and provided a family of knots that could be counterexamples to it. The conjecture was further strengthened when a famous potential

original on 2008-03-16. Retrieved 2008-03-11. Drury, S.W. (2011). "A counterexample to a conjecture of Matsaev". Linear Algebra and Its Applications. 435

for a patient's clinical signs and symptoms, and his triad provides a counterexample to the commonly used diagnostic principle that "the explanation of any

1936 proof in a lecture in 1938 in what came to be known as the no-counterexample interpretation. Both the original proof and the reformulation can be

in history. Veith is best known for his role in the development of Counterexample-guided Abstraction Refinement (CEGAR), which is a key ingredient in

term satisfies the condition since all derivatives are zero. Another counterexample (which is separable) is Xp+1 − Xp. The conjecture is known to hold in

{\displaystyle x} since all of these numbers are prime. This isn't a counterexample to Bunyakovsky conjecture since the condition (2) fails. The third condition

generalized tests are called semiquantum nonlocal games. While, as the counterexample of Werner states shows, Bell nonlocality is known not to be equivalent

Theory, Serie A, Vol.29, 1980, p. 1-19 Kahn, Jeff; Kalai, Gil (1993), "A counterexample to Borsuk's conjecture", Bulletin of the American Mathematical Society

then F is an epiphenomenon. The problem of epiphenomena is often a counterexample to theories of causation. For example, take a simplified Lewisian counterfactual

published a counterexample to Alfred Kempe's false proof of the four color theorem. The Poussin graph, the graph he used for this counterexample, is named

Dirichlet. It is an example of a pathological function which provides counterexamples to many situations. The Dirichlet function is nowhere continuous. Proof

mathematics and has offered a US$1 million prize for a solution or a counterexample. The solution of the equations is a flow velocity. It is a vector field—to

methods in the study of algebraic cycles Voisin, Claire (2002). "A counterexample to the Hodge conjecture extended to Kähler varieties". International

(1): 43–58, doi:10.1016/S0166-218X(00)00386-3. Lubiw, Anna (1990), "Counterexample to a conjecture of Szymanski on hypercube routing", Information Processing

a complex surface such that q and P1 both vanish is rational, but a counterexample (an Enriques surface) was found by Federigo Enriques. Bordiga surfaces:

of the graph exactly twice. Proving that this is true (or finding a counterexample) remains an open problem. Several important classes of graphs can be

Hilbert's twenty-three problems posed in 1900 was to prove, or find a counterexample to, the claim that ab is always transcendental for algebraic a ≠ 0, 1

l p space nor a c0 space can be embedded. The Tsirelson's drift, a counterexample in the theory of stochastic differential equations, is an SDE which

is unbounded. Moreover, it is not even necessarily a closed set: a counterexample is a sphere passing through the origin, with the conical hull being

numbers is supersingular at infinitely many primes. In 1988, he found a counterexample to Euler's sum of powers conjecture for fourth powers. His work on these

305. PMC 534126. PMID 16589478. Herstein, I. N. (October 1965). "A counterexample in Noetherian rings". Proc Natl Acad Sci U S A. 54 (4): 1036–1037. Bibcode:1965PNAS

Accordingly, he was described by abolitionists in his time as a living counterexample to claims by supporters of slavery that enslaved people lacked the intellectual

is false while its premises are true. This means that there are no counterexamples: the conclusion is true in all such cases, not just in most cases.

rejected. The arithmetical example above provides what is called a weak counterexample. The existence claim ∃x (φ → θ) cannot be provable by intuitionistic

modern tanks and some self-propelled guns. The French Leclerc tank is a counterexample; it instead uses overpressure from an internal compartment to force

was not resolved until M.E. Rudin constructed one in 1971. Rudin's counterexample is a very large space (of cardinality ℵ ω ℵ 0 {\displaystyle \aleph

algebraic variety over an uncountable field is not first-countable. Another counterexample is the ordinal space ω 1 + 1 = [ 0 , ω 1 ] {\displaystyle \omega _{1}+1=\left[0

complex plane. The function f(x) defined in the paragraph above is a counterexample, as it is not defined for x = ±i. This explains why the Taylor series

Joseph Horton provided a counterexample on 96 vertices, the Horton graph. Later, Mark Ellingham constructed two more counterexamples: the Ellingham–Horton

doi:10.1090/S0002-9939-96-03732-X. Kahn, Jeff; Kalai, Gil (1993), "A counterexample to Borsuk's conjecture", Bulletin of the American Mathematical Society

complex plane. The function f(x) defined in the paragraph above is a counterexample, as it is not defined for x = ±i. This explains why the Taylor series

doi:10.1090/S0002-9939-96-03732-X. Kahn, Jeff; Kalai, Gil (1993), "A counterexample to Borsuk's conjecture", Bulletin of the American Mathematical Society

vice versa. A planar snark would therefore necessarily be dual to a counterexample to the four-color theorem. Thus, the subsequent proof of the four-color

12-crossing and many 14-crossing amphicheiral knots in the late 1910s. But a counterexample to Tait's conjecture, a 15-crossing amphicheiral knot, was found by

to determine whether a compact Hausdorff space must be Blumberg. A counterexample was given in 1974 by Ronnie Levy, conditional on Luzin's hypothesis

with positive scalar curvature; in 1997 Schick published the first counterexample. He is the coordinator of the Courant Research Center's Strukturen höherer

suggests there are infinitely many counterexamples. In particular, the heuristic shows that such counterexamples have asymptotic density greater than

minimal counterexample to the conjecture would have to be an apex graph, but since there are no 6-chromatic apex graphs such a counterexample cannot exist

L] = L, is the direct sum of simple ideals. Matsushima constructed a counterexample. He then developed a proof that Cartan subalgebras of a complex Lie

paradox is not a semantic paradox: The Pinocchio paradox is, in a way, a counterexample to solutions to the Liar that would exclude semantic predicates from

normal magic square and n-Queens Problem for n = 25. Also the first counterexample in the Boolean Pythagorean triples problem. 7832 - pronic number 7841

2 ϕ a ϕ b {\displaystyle R_{ab}=2\phi _{a}\phi _{b}} which forms a counterexample to many formulations of the cosmic censorship hypothesis was found by

be applied. The necessity of the first condition can be seen by the counterexample where the function f ( x ) = | x | {\displaystyle f(x)=|x|} on [-1,1]

These principles appeared across all cultures studied, and only one counterexample was found: an instance of the “respecting property” value clashing with

sphere. This conjecture was disproven in 1982 by Wu-Yi Hsiang using a counterexample in R 4 {\displaystyle \mathbb {R} ^{4}} . In 1984 Henry C. Wente constructed

judgments. Critics of welfarism frequently concentrate on specific counterexamples in which these general intuitions seem to fail, including cases of

(2007) "Metric Entropy of Dynamical System" Christopher Hoffman, "A K counterexample machine", Trans. Amer. Math. Soc. 351 (1999), pp 4263–4280 M. Keane

sources. Proprietary software is an example. Copyleft software is a counterexample where copyleft advocates use copyright licenses to guarantee the right

ergodicity theorems. In particular, scarred states provide a striking visual counterexample to the assumption that the eigenstates of a classically chaotic system

stored counterexamples. Thinking and reasoning, 11, 349–381. De Neys, W., Schaeken, W., d'Ydewalle, G. (2005). Working memory and counterexample retrieval

noticed a counterexample, namely 959.) This corresponds to the case of a = 2 {\displaystyle a=2} in the original statement. The counterexample of 959 was

specifically the law that a universal statement is true just in case no counterexample exists: ∀ x P ( x ) ≡ ¬ ∃ x ¬ P ( x ) {\displaystyle \forall x\,P(x)\equiv

degree, then the theorem does not hold, and P(x) = x + ixi + 1 = 0 is a counterexample with no solutions. Eilenberg–Niven theorem can also be generalized to

while being irreducible over the rational numbers. They are a standard counterexample in number theory. Given a finite set P {\displaystyle P} of prime numbers

functional on E {\displaystyle E} . Already in two dimensions one obtains a counterexample. Let E = R 2 ,   K = { ( x , y ) : y > 0 } ∪ { ( x , 0 ) : x > 0 }

object position, and is the one getting hit. Wilkins proposes a direct counterexample to Gleitman's example of /put/ and /look/, attested in the Central Australian

depends on the bypass ratio and can be as much as 70%. However, as a counterexample, the SR-71 had reasonable efficiency at high altitude in afterburning

graph' from its original two dimensions to three, inspired by a simple counterexample, the 'double banana graph'. Laman 1959. Laman 1970. Pollaczek‐Geiringer

Mathematical Society 7(6):266–71. Levinson, Norman (1970). "Coding Theory: A Counterexample to G. H. Hardy's Conception of Applied Mathematics". The American Mathematical

of spheres, it was suggested that the regular tetrahedra might be a counterexample to Ulam's conjecture that the optimal density for packing congruent

5749 – super-prime 5768 – tribonacci number 5776 = 762 5777 – smallest counterexample to the conjecture that all odd numbers are of the form p + 2a2 5778

Joseph; Henk, Martin; Martin, Alexander; Weismantel, Robert (1999), "A counterexample to an integer analogue of Carathéodory's theorem", Journal für die reine

593–605. doi:10.2307/3647744. JSTOR 3647744. Wente, Henry C. (1986). "Counterexample to a conjecture of H. Hopf". Pacific Journal of Mathematics. 121 (1):

of being a relative norm for a cyclic extension of number fields. A counterexample by Ernst S. Selmer shows that the Hasse–Minkowski theorem cannot be

weaving. The weaving is, however, usually not done under water (see counterexample below). An issue of The American Philatelist from 1956 refers to an

much-publicized 1994–1996 feud with Warner Bros. Records provides a strong counterexample, as does Roger McGuinn's claim, made in July 2000 before a US Senate

However, not every ruled surface is developable; the helicoid provides a counterexample. The tangent developable of a curve containing a point of zero torsion

{E} \mathrm {e} ^{|M_{t}|}<\infty } is still not sufficient; for a counterexample see Example 3 above. A special case: M t = f ( t , W t ) , {\displaystyle

in which each element has order a power of p. The group G provides a counterexample to the generalised Burnside conjecture: it is a finitely generated infinite

associativity. In the absence of associativity, the sedenions provide a counterexample. The converse does not hold: an element which is not a zero divisor

unipotent group is a nilpotent group, though the converse is not true (counterexample: the diagonal matrices of GLn(k)). For example, the standard representation

minor third down (e.g. C6 has the same notes as an Am7), although a counterexample is The 5th Dimension's recorded version of "Stoned Soul Picnic" (on

vertex-transitive graph that contains no Hamiltonian cycle, the Coxeter graph is a counterexample to a variant of the Lovász conjecture, but the canonical formulation

+ y {\displaystyle x+y} is also of that form. Gauss replied with a counterexample: 15 11 + 8 11 {\displaystyle 15^{11}+8^{11}} can be written as h 2 +

(M⟨y:=P⟩)⟨x:=(N⟨y:=P⟩)⟩ (where x≠y and x not free in P) A surprising counterexample, due to Melliès, shows that the way this rule is encoded in the original

1090/s0002-9904-1961-10514-4. MR 0123174. Lomonosov, Victor (2000). "A counterexample to the Bishop-Phelps theorem in complex spaces". Israel Journal of Mathematics

sponsored by Ohio State University. Page 586 in Halmos 1990. Per Enflo: A counterexample to the approximation problem in Banach spaces. Acta Mathematica vol

paradoxes of material implication). The general form of McGee-type counterexamples to modus ponens is simply P , P → ( Q → R ) {\displaystyle P,P\rightarrow

LGBTQ people cannot succeed in maths and sciences, and presenting a counterexample to the widespread belief that people can be analytical or creative but

p-adic fields are C2), but Guy Terjanian found the following 2-adic counterexample for d = 4. Define G ( x ) = G ( x 1 , x 2 , x 3 ) = ∑ x i 4 − ∑ i <

lattice-theoretic approach is provided by Korkee, Mattila & Haukkanen (2019). The counterexample presented in Haukkanen, Wang & Sillanpää (1997) is S = { 1 , 2 , 3

as villains to be evidence of racism. Aravis is often presented as a counterexample to this (along with Emeth, who is accepted in Aslan's country for good

always continuous, to which Niels Henrik Abel in 1826 found purported counterexamples in the context of Fourier series, arguing that Cauchy's proof had to

However, it is now known to be false: in 1979, Paul A. Catlin found a counterexample for k = 8, and Paul Erdős and Siemion Fajtlowicz later observed that

give a counterexample for singular varieties with non-log canonical singularities, and also,Lauritzen & Rao (1997) gave elementary counterexamples inspired

might not be isomorphic to [ X , S 1 ] {\displaystyle [X,S^{1}]} . A counterexample is given by the Warsaw circle, whose first cohomology group vanishes

but the converse is not generally true, as the Kerr metric provides a counterexample. In a stationary spacetime satisfying the vacuum Einstein equations

conjecture, leading to many interesting results discussed below. However, a counterexample was eventually found. There are also some generic rigidity results with

Var(X)-a} for all a ∈ R {\displaystyle a\in \mathbb {R} } , and a simple counterexample for monotonicity can be found. The standard deviation is a deviation

contains all the counterexamples. Currently no counterexamples for the Farrell–Jones conjecture are known. If there is a counterexample, one has to enlarge

Counterexamples in Probability And Statistics. CRC Press. pp. 194–196. ISBN 978-0-412-98901-8. Hardy, Michael (2003). "An Illuminating Counterexample"

Z-module), Z(p∞) is Artinian but not Noetherian. It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every

the highest elevation human habitation at 5,100 metres (16,700 ft). A counterexample is El Alto, Bolivia, at 4,150 metres (13,620 ft), which has a highly

case if the Euler characteristic is positive (a topological sphere). A counterexample is provided by a cube where one face is replaced by a square pyramid:

\operatorname {sgn}(a_{n})\to 1\neq \operatorname {sgn}(a)} . This counterexample confirms more formally the discontinuity of sgn ⁡ x {\displaystyle \operatorname

partionable. However, not every epigroup has this property. A simple counterexample is the Brandt semigroup with five elements B2 because the unipotency

S. Gordon, David Webb, and Scott A. Wolpert when they constructed a counterexample for Kac's problem. For this work, Sunada was awarded the Iyanaga Prize

is not bounded (see unbounded operator) exists and thus serves as a counterexample. The Hausdorff–Young inequality says that the Fourier transformation

graph KL−1 with no blue Kr subgraph and no red Ks subgraph. Such a counterexample is called a Ramsey graph. Brendan McKay maintains a list of known Ramsey

doi:10.1137/060661569. S2CID 5676548. Shitov, Yaroslav (2018). "A Counterexample to Comon's Conjecture". SIAM Journal on Applied Algebra and Geometry

always fail in non-classical intuitionistic logics. A simple explicit counterexample is that of Gödel many valued logics, which are a fuzzy logic where truth

X\oplus X\oplus X} but not X ⊕ X {\displaystyle X\oplus X} serves as a counterexample for weakening the premises of the Schroeder–Bernstein theorem List of

Springer. pp. 91–93. ISBN 9783540404606. Connelly, Robert (1977). "A counterexample to the rigidity conjecture for polyhedra" (PDF). Publications Mathématiques

doi:10.1016/0001-8708(72)90002-3, MR 0294454 Neeman, Amnon (2002), "A counterexample to a 1961 "theorem" in homological algebra (with appendix by Pierre

Annals of Mathematics (2) 95 1972 66–82. D.B.A. Epstein and E. Vogt, A counterexample to the periodic orbit conjecture in codimension 3. Annals of Mathematics

intervals). Robinson called this example "[p]ossibly the best known counterexample for Neyman's version of confidence interval theory." To Welch, it showed

Amer. Math. Soc. Transl., Series 2, 39 (1964), 1-36. Christopher Hoffman, "A K counterexample machine", Trans. Amer. Math. Soc. 351 (1999), pp 4263–4280.

PMID 9905557 Naito, Hiroyoshi; Okuda, Masahiro; Zhong-Can, Ou-Yang (1993), "Counterexample to some shape equations for axisymmetric vesicles", Physical Review

Yash Lodha, produced the first torsion-free and finitely presented counterexample to the von Neumann-Day problem, originally described by mathematician

\omega } and is not true in general. Élie Cartan himself in 1922 gave a counterexample, which provided one of the impulses in the next decade for the development

metric as it does not satisfy the triangle inequality. The simplest counterexample of this is given by the three sets X = { a } {\displaystyle X=\{a\}}

would consider acceptable to do. This is pointed out as a falsifying counterexample to the claim that "no descriptive statement can in itself become normative"

\omega } and is not true in general. Élie Cartan himself in 1922 gave a counterexample, which provided one of the impulses in the next decade for the development

\ldots ,k\}} . The converse is not necessarily true, as the following counterexample shows: [ 1 2 0 3 ] [ 1 1 0 1 ] = [ 1 3 0 3 ] ≠ [ 1 5 0 3 ] = [ 1 1 0

direction is necessary for the theorem to work. Without that, we have this counterexample: there are only two outcomes A , B {\displaystyle A,B} , and the agent

The Scientific Image, Oxford University Press 1980. Derivation and Counterexample: An Introduction to Philosophical Logic (with Karel Lambert), Dickenson

four years passed since this publication before Akira Fujiki found a counterexample. Bogomolov's paper on "Holomorphic tensors and vector bundles on projective

form need not be a sum of simple and abelian Lie algebras. A typical counterexample is G = L[t]/tnL[t] where n>1, L is a simple complex Lie algebra with

is the set of real numbers, the proposition is false, with x = √2 as counterexample; if the domain is the set of naturals, the proposition is true, since

arXiv:math/0408173. doi:10.1016/j.jctb.2006.05.002. Santos, Francisco (2011). "A counterexample to the Hirsch conjecture". Annals of Mathematics. 176 (1): 383–412.

Counterexample to a strengthening of the uniform limit theorem, in which pointwise convergence, rather than uniform convergence, is assumed. The continuous

union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.) From there, one can prove (with the Axiom of

particular constructing a Banach space with almost no symmetry, serving as a counterexample to several other conjectures. With Bernard Maurey he resolved the "unconditional

"theorems"[definition needed] are still being clarified. For example in 2020 a counterexample to such a "theorem" was found, where it turns out that an elementary

for Syracuse University as a professor. The use of Syracuse as the counterexample is done to show how de facto educational segregation is prevalent in

contain no such subgraph if, for example, the bottom edge in the present subgraph were replaced by a blue edge – thus proving by counterexample that N* > 3.

2307/1969503, JSTOR 1969503. Dilworth, Robert P.; Greene, Curtis (1971), "A counterexample to the generalization of Sperner's theorem", Journal of Combinatorial

2307/1994094, JSTOR 1994094, MR 0165344. James, Robert C. (1971), "A counterexample for a sup theorem in normed space", Israel Journal of Mathematics, 9

N {\displaystyle {\mathcal {N}}} . But the diagonal lemma yields a counterexample to this equivalence, by giving a "liar" formula S {\displaystyle S}

near Mach 1, although Vought's F7U Cutlass proved to be something of a counterexample—the developed version was the first aircraft to demonstrate stores separation

Planar graph used as counterexample

non-Hermitian matrices (or more accurately, for non-normal matrices). For a counterexample, let t > 0 {\displaystyle t>0} be arbitrarily small, and consider M

As S(T) jumps by at least 2 at any counterexample to the Riemann hypothesis, one might expect any counterexamples to the Riemann hypothesis to start appearing

finitely generated; the rank 1 group Q {\displaystyle \mathbb {Q} } is one counterexample, and the rank-0 group given by a direct sum of countably infinitely

. The graph of the octahedral pyramid is the only possible minimal counterexample to Negami's conjecture, that the connected graphs with planar covers

implies condition Y, then, deterministically, there must not exist any counterexample cases having condition X, but not condition Y. However, if the researcher

although 147Sm is unstable, it is stable to beta decay; thus 147 is not a counterexample). These ranges can be calculated using the liquid drop model (for example

calculating the second derivative—but does not provide a definition. A counterexample is the sign function sgn ⁡ ( x ) {\displaystyle \operatorname {sgn}(x)}

should be credited with the first correct proof. Dickson's search for a counterexample to Wedderburn's theorem led him to investigate nonassociative algebras

Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. Enflo, P. (1973). "A counterexample to the approximation problem in Banach spaces". Acta Mathematica. 130

both epic and monic is not necessarily an isomorphism. An elementary counterexample: in the category consisting of two objects A and B, the identity morphisms

until a proof was developed that ruled out large classes of possible counterexamples, yet still left open enough possibilities that a computer program was

Sloane, he invented the icosians. He invented a base 13 function as a counterexample to the converse of the intermediate value theorem: the function takes

A005421, OEIS, retrieved 2015-12-13. Wang, Venecia (October 2012), "A counterexample to the prime conjecture of expressing numbers using just ones", Journal

seem to exist anymore—though there's always an old master or two as counterexample. For several generations the major qin lineage in Zhejiang has been

Palamodov, Appl. Categ. Structures 20 (5) (2012) 531–541. Wolfgang Rump, A counterexample to Raikov's conjecture, Bull. London Math. Soc. 40, 985–994 (2008).

by means of shared quantum states. In 2008 Matthew Hastings proved a counterexample where the minimum output entropy is not additive for all quantum channels

variation of this that he credited to Moreau. Moreau (1873) pointed out a counterexample to a lemma used by Adrien-Marie Legendre in his attempt to prove Dirichlet's

109–110, doi:10.1007/BF01589099, MR 1017214. Santos, Francisco (2012), "A counterexample to the Hirsch conjecture", Annals of Mathematics, 176 (1), Princeton

representations θ10 over local and global fields in their construction of counterexamples to the generalized Ramanujan conjecture for the symplectic group. Adams

Phenomenal states without intentional content therefore serve as a counterexample to the unrestricted version. If the state has no intentional content

the language lacks grammatical recursion, then it is proposed as a counterexample to the theory proposed by Chomsky, Hauser and Fitch (2002) that recursion

This paper introduces the k-creative sets, which form a potential counterexample to the Berman–Hartmanis conjecture. Hopcroft, John; Joseph, Deborah;

section 2), which is false; see (Joseph 1985, section 4) for Kac's counterexample. Andersen (1985) gave a proof of Demazure's character formula using

asymmetric distribution with zero skewness. This figure serves as a counterexample that zero skewness does not imply symmetric distribution necessarily

have a taxicab length 2, but the hypotenuses are not congruent. This counterexample eliminates AAS, ASA, and SAS. It also eliminates AASS, AAAS, and even

≥ n + r + k {\displaystyle p+q\geq n+r+k} . Demailly (1988) gave a counterexample, which is as follows: Conjecture of Sommese (1978): Let X be a n-dimensional

≥ n + r + k {\displaystyle p+q\geq n+r+k} . Demailly (1988) gave a counterexample, which is as follows: Conjecture of Sommese (1978): Let X be a n-dimensional

same dimension. This is false for some noncommutative rings, and a counterexample can be constructed using the Eilenberg swindle as follows. Let X be

([ŋ]; cp. Svantesson et al. 2005: 18) Rinchen 1979: 222-223, for a counterexample see 225 Sanžaa and Tujaa 2001 From the material cited in this article

reformulation of Gentzen's first consistency proof of arithmetic: the no-counterexample interpretation". Bulletin of Symbolic Logic. 11 (2): 225–238. JSTOR 1556751

number 893,871,739 = 197 895,745,041 = 1734 906,150,257 = smallest counterexample to the Polya conjecture 916,132,832 = 625 923,187,456 = 303842, the

ISSN 0895-7177. Chen, Kani; Ying, Zhiliang (1996-04-01). "A counterexample to a conjecture concerning the Hall-Wellner band". The Annals of Statistics

checker detects witnesses and counterexamples. A witness is a path where the property is satisfied, whereas a counterexample is a path in the execution of

makes a generalization true by changing the generalization to exclude a counterexample. Cherry picking (suppressed evidence, incomplete evidence, argument

29–31, doi:10.1090/s0002-9904-1943-07825-1. Bloom, Gary S. (1977), "A counterexample to a theorem of S. Piccard", Journal of Combinatorial Theory, Series

The rational numbers Q {\displaystyle \mathbb {Q} } seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the

unexpectedness as a sufficient condition for beauty and proposed a counterexample: A great many theorems of mathematics, when first published, appear

disagrees, Venia then takes the extreme position of making his case by counterexample and he strips naked in the classroom to “prove” the error of her ways

Fp for such prime p that p ≠ 1 (mod 3), but not necessarily: see the counterexample with rationals above. Also in F7 only three elements 0, ±1 are perfect

there are any exotic 4-spheres; such an exotic 4-sphere would be a counterexample to the smooth generalized Poincaré conjecture in dimension 4. Some plausible

not view the security dilemma as being the root cause of all war. A counterexample frequently given is the Second World War, where there was no dilemma

y_{i}=q\left(s_{i}\right)} for every i ∈ I {\displaystyle i\in I} ). This counterexample can be generalized to nets by letting ( A , ≤ ) {\displaystyle (A,\leq

codices in this group, dated the late 12th century. As bird-like A counterexample is also given where the illustrated sirens (group of three) are bird-like

same initial data was subsequently called the Perron effect. Perron's counterexample shows that a negative largest Lyapunov exponent does not, in general

that γ(K) = 0 implies H1(K) = 0. However, this conjecture is false. A counterexample was first given by A. G. Vitushkin, and a much simpler one by John B

a non-Noetherian local ring whose maximal ideal is principal (see a counterexample to Krull's intersection theorem at Local ring#Commutative case.) If

colors; however, Häggkvist (1981) disproved this conjecture by finding a counterexample in which each vertex of the Grötzsch graph is replaced by an independent

adding covariates to the adjustment set Z can introduce bias. A typical counterexample occurs when Z is a common effect of X and Y, a case in which Z is not

Indeed, counterexamples exist satisfying the canonical commutation relations but not the Weyl relations. (These same operators give a counterexample to the

Together, the two algorithms provided an algorithm that would find a counterexample to the Poincaré Conjecture, if one existed. In 2002, Martin Dunwoody

example has also been called the non-Fano configuration. The other counterexample, due to McKee, consists of two regular pentagons joined edge-to-edge

any ring R with unity such that R/J(R) is not a semisimple ring is a counterexample. Another formulation is this: a finitely generated module M is one for

the converse of the second statement is not true (see graphic for a counterexample). The first two statements remain true if the two rays get replaced

Heinzer, William J.; Lantz, David C. (1994), "ACCP in polynomial rings: a counterexample", Proceedings of the American Mathematical Society, 121 (3): 975–977

space H R {\displaystyle H_{\mathbb {R} }} may not be symmetric. As a counterexample, define A : R 2 → R 2 {\displaystyle A:\mathbb {R} ^{2}\to \mathbb {R}

is equivalent to Diff. Diff: False generally, with the first known counterexample in dimension 7. True in some dimensions including 1, 2, 3, 5, 6, 12

no such elliptic curve could have a modular form, so no odd prime counterexample to Fermat's equation could exist). Wiles also proved that the conjecture

individual was Adam. Thus, Paul points out Adam as "precedent" (in form of "counterexample") for "the universality of Christ's atonement". Therefore as sin came

electrostatic potential (Galvani potential), but this is not exactly true. As a counterexample, multi-material devices such as p–n junctions contain internal electrostatic

"[7][permanent dead link] the Amherst Times cited Deborah Pryce as a counterexample of that thesis: "[In] several races... the ability to bring home hundreds

high dose could be survived, although he does not rule out whether a counterexample might exist (for updated information on the question of how to interpret

translation length), and it is the same group that was proven to be a counterexample to the unit conjecture. More generally the topic of orderability of

Mathematics Genealogy Project Brittenham M, Hermiller S (2021). "A Counterexample to the Bernhard–Jablan Unknotting Conjecture". Experimental Mathematics:

_{n}\}\models \psi } , can then be stated as its absence of a counterexample, where a counterexample is defined as a case I {\displaystyle {\mathcal {I}}} in

is one of the smallest non-representable matroids, and served as a counterexample to a conjecture of Ingleton that the matroids on eight or fewer elements

graph coloring, graph isomorphism, cycles in directed graphs, and a counterexample to a conjecture of András Ádám on the number of cycles in tournaments

still be the case that the customer does not owe the grocer, per the counterexample of a film production. This fundamental ambiguity is essentially what

must be capable of being parsed into a sequence of discrete events. [counterexample: Ligeti, computer music] Constraint 2: The musical surface must be available

because although the facts supposed in the paradox would present a counterexample to the claim that all contingent truths are determined by sufficient

still be the case that the customer does not owe the grocer, per the counterexample of a film production. This fundamental ambiguity is essentially what

( A n ) {\displaystyle \mu (A_{n})} is also zero, which proves the counterexample. One may define additive functions with values in any additive monoid

closely supervised by a benevolent deity. The Lisbon disaster provided a counterexample for Voltaire. Theodor Adorno wrote, "the earthquake of Lisbon sufficed

literature, 12 years before Julius Petersen's use of the same graph as a counterexample to an edge coloring problem. As a projective configuration, the Desargues

Robert; Hahn, Jeremy; Levy, Ishan; Schlank, Tomer (2023). "K-theoretic counterexamples to Ravenel's telescope conjecture". arXiv:2310.17459 [math.AT]. Dimitrov

always give the minimum possible number of multiplications. The smallest counterexample is for a power of 15, when the binary method needs six multiplications

Maurizio Cornalba: Cornalba, Maurizio; Shiffman, Bernard (1972). "A counterexample to the "Transcendental Bezout problem"". Annals of Mathematics. Second

community. The defiant and hopeful spring metaphor can be interpreted as a counterexample to the nature poetry of Peter Huchel, whose works take a drearier and

implies Fermat's Last Theorem. He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply the existence of at least one non-modular

embedded into L3. However, K does not have the amalgamation property. The counterexample for this starts with L1 containing a single element e and extends in

S. Shrikhande and E. T. Parker) a Graeco-Latin square of size 10, a counterexample to Euler's conjecture that no Graeco-Latin square of size 4k + 2 exists

research, Daniel Everett has claimed that the Pirahã language is a counterexample to the basic tenets of universal grammar because it lacks clausal embedding

PMID 9902666. Reinhard F. Werner and Alexander S. Holevo (2002). "Counterexample to an additivity conjecture for output purity of quantum channels".

1007/BF01080698. MR 0420305. S2CID 121421267. Lomonosov, Victor (2000). "A counterexample to the Bishop-Phelps theorem in complex spaces". Israel Journal of Mathematics

(square roots, cube roots, etc)? The Abel–Ruffini theorem provides a counterexample proving that there are polynomial equations for which such a formula

function. While the conjecture is known to hold for certain conditions, counterexamples have also been found. Khabibullin's conjecture (version 1, 1992). Let

not easily admit these antibiotics into the organelle. A noteworthy counterexample is the antineoplastic antibiotic chloramphenicol, which inhibits bacterial

ISSN 0045-7825. Caffarelli, Luis A.; Stefanelli, Ulisse (2008-07-03). "A Counterexample to C 2,1 Regularity for Parabolic Fully Nonlinear Equations". Communications

distress, The Perils of Pauline (1914), also provides at least a partial counterexample, in that Pauline, played by Pearl White, is a strong character who decides

Cauchy sequence is convergent in a general metric space. A classic counterexample is the rational numbers, Q {\displaystyle \mathbb {Q} } , with the usual

several other authors. Ball (1988) pointed out a particularly simple counterexample: all sections of the unit volume cube have measure at most √2, while

Dynamical System" Hoffman, Christopher (1999). "A K {\displaystyle K} Counterexample Machine". Transactions of the American Mathematical Society. 351 (10):

{\displaystyle n=2} (i.e., real dimension 3), Nirenberg published a counterexample. The local embedding problem remains open in real dimension 5. Heisuke

orthogonality, even with orthogonal columns, as shown by the following counterexample. [ 2 0 0 1 2 ] {\displaystyle {\begin{bmatrix}2&0\\0&{\frac {1}{2}}\end{bmatrix}}}

However, it must have seemed at the time that they did. It only took one counterexample (Mercury's orbit) to prove that there was something wrong with his theory

choosing, drawn from the domain of quantification. Note that a single counterexample falsifies a universally quantified statement, and a single example suffices

branches of mathematics, as well as contemporary theoretical physics. As a counterexample, the general linear group is neither simple, nor semisimple. This is

mathematics under Robert Lee Moore. Her graduate thesis presented a counterexample to one of "Moore's axioms". She completed her Ph.D. in 1949. During

must be Hamiltonian; this was disproved when W. T. Tutte provided a counterexample, the Tutte graph, which is much larger than the Herschel graph. A refinement

justified only if one's evidence supports that conclusion. A typical counterexample goes like this. Suppose, for example, that Babe Ruth approaches the

and the satisfaction of desires as distinct sources. An influential counterexample to hedonism, proposed by philosopher Robert Nozick (1938–2002), imagines

counterexamples to several longstanding conjectures in group theory, including the von Neumann conjecture. In 1980, he constructed a counterexample to

states "At this writing, in twenty years I have encountered not a single counterexample" (McWhorter 2018). Nevertheless, the existence of a creole prototype

E + tu|| > /kiyetu/ 'they said to him'. (However, Einaudi cites one counterexample, ||ki + į|| > /kiį/ 'they were drinking it for him', perhaps with a

characteristic n > 0 is positive, the identity in n dimensions is a counterexample, as tr ⁡ ( I n k ) = tr ⁡ ( I n ) = n ≡ 0 {\displaystyle \operatorname

the Zapotec (a group of indigenous Mexican peoples) acts as a useful counterexample to the trend. She claims the Zapotec used harmony ideology in a "counter-hegemonic"

Du Bois, and the poet Paul Laurence Dunbar stood as high-profile counterexamples to widespread stereotypes of black intellectual inferiority. In Britain

https://frontiersinai.com/ecai/ecai2000/pdf/p0571.pdf Joseph Y. Halpern, "A counterexample to theorems of Cox and Fine," Journal of AI research, 10, 67–85 (1999)

that proves the phonetic representation to be possible as well as a counterexample that proves, given the opposite ordering, an incorrect phonetic representation

constant scalar curvature and is also complete? The answer is no, due to counterexamples given by Jin (1988). Various additional criteria under which a solution

again ... Hee's proposal is incorrect and can be easily refuted by a counterexample. For example, the four-pillar quadruplets for 1984-03-18 and 2044-03-03