There are counterexamples, of course, to this line of thinking.
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But we have a lot of counterexamples to suggest otherwise.
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It may seem, however, like history has abounded with counterexamples.
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Counterexamples — self-alienating writers such as Beckett and Conrad — are exceptional.
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But the acute counterexamples of pain and suffering are getting worse.
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But there were many counterexamples where seemingly massive results had unravelled immediately on publication.
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But Bloomberg didn't roll over in this debate, answering the criticism sufficiently with counterexamples.
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We also live in an era in which the counterexamples are few and far between.
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But the counterexamples that can show another way and another path are out there too.
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Counterexamples, such as an account of the value of bootlegging in Malaysia, are notably absent from his argument.
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His list of counterexamples in the paragraph above is just one example of not considering the counterfactual after another.
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The rejected faces then become training counterexamples, and increase the detector's ability to avoid similar false positives in the future.
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Given how hostile mainstream economists can be to counterexamples, the biggest surprise might be that this study was published at all.
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But despite some prominent liberal counterexamples, rich Americans tend to support the economic policies from which they have so greatly benefited.
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With anti-Muslim sentiment reaching frightening levels, it's more important than ever to see counterexamples of the Islamophobic stereotypes we've been taught.
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But he pushed back on that perception, citing the Trump administration's reforms on criminal justice and his NCAA bill as two counterexamples.
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Still, successes like "My Fair Lady" and "Carousel," to name two currently prominent counterexamples, keep tempting big talents to believe they can work the same magic.
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Since just about all the winners have been vigorously engaged in research at the time of the award, they provide a string of counterexamples to Hardy's declaration.
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Photo: AP.Prior to Plasco, it was difficult to produce counterexamples that would immediately disprove these truther axioms because out-of-control high-rise fires just don't happen very often.
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The variables that go into producing good wine can be so many that even laudable efforts to characterize the general traits of a particular place inevitably give way to counterexamples.
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The reason you don't see counterexamples of the many healthy celibate gay priests is that they're afraid to come out—now, more than ever, in this environment of blaming and stereotyping.
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While characters like "Cletus the Slack-Jawed Yokel" and Groundskeeper Willie are satirical stereotypes, they're specific stereotypes of narrow subgroups, and there are plenty of counterexamples of Southern people or Scottish people on television.
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Graham wrote an essay, "Mean People Fail," in which—ignoring such possible counterexamples as Jeff Bezos and Larry Ellison—he declared that "being mean makes you stupid" and discourages good people from working for you.
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The history of players coming back from wrist surgeries shows an inconsistent pattern with respect to short-term shooting, and while J.R. Smith and Iman Shumpert stand out as recent worrisome examples, there are counterexamples, too.
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If you're using one to recognize faces, you don't write lines of code related to things like hair color and nose length; instead, you "train" the neural network, by repeatedly giving it large numbers of labelled examples and counterexamples—"cow"; "not cow"—which it compares, beginning at the pixel level.
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Later, when asked why Facebook does not just remove Infowars completely, he gave some counterexamples of places where the platform was used to encourage violence, such as Myanmar and Sri Lanka, and offered: The principles that we have on what we remove from the service are: If it's going to result in real harm, real physical harm, or if you're attacking individuals, then that content shouldn't be on the platform.
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In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable theorems. It is sometimes said that mathematical development consists primarily in finding (and proving) theorems and counterexamples.
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Jørgensen's conjecture remains unproven.. However, if false, it has only finitely many counterexamples.
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It is an example of pathological function which provides counterexamples to many situations.
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Furthermore, the person takes the relevant information, in the form of counterexamples, and ignores it.
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These counterexamples suggest that the gender paradox could be limited to western cultures or languages.
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The abc conjecture would imply that there are at most finitely many counterexamples to Beal's conjecture.
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Frankfurt cases (also known as Frankfurt counterexamples or Frankfurt-style cases) were presented by philosopher Harry Frankfurt in 1969 as counterexamples to the principle of alternate possibilities (PAP), which holds that an agent is morally responsible for an action only if that person could have done otherwise.
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D. E. Over (1987). "Assumption and the Supposed Counterexamples to Modus Ponens", Analysis 47, 142–146.Bledin (2015).
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This phenomenon, along with the lack of counterexamples of fatigue occurring in the opposite direction, supports Marcan priority.
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Gettier's paper used counterexamples (see also thought experiment) to argue that there are cases of beliefs that are both true and justified—therefore satisfying all three conditions for knowledge on the JTB account—but that do not appear to be genuine cases of knowledge. Therefore, Gettier argued, his counterexamples show that the JTB account of knowledge is false, and thus that a different conceptual analysis is needed to correctly track what we mean by "knowledge". Gettier's case is based on two counterexamples to the JTB analysis. Each relies on two claims.
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Though the book is written as a narrative, it aims to develop an actual method of investigation based upon "proofs and refutations". In Appendix I, Lakatos summarizes this method by the following list of stages: # Primitive conjecture. # Proof (a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures). # "Global" counterexamples (counterexamples to the primitive conjecture) emerge.
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Kempe's proof can be translated into an algorithm to color planar graphs, which is also erroneous. Counterexamples to his proof were found in 1890 and 1896 (the Poussin graph), and later, the Fritsch graph and Soifer graph provided two smaller counterexamples. However, until the work of Errera, these counterexamples did not show that the whole coloring algorithm fails. Rather, they assumed that all but one vertex of the graph had already been colored, and showed that Kempe's method (which purportedly would modify the coloring to extend it to the whole graphs) failed in those precolored instances.
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So in this case one can really use the family of finite subgroups. On the other hand this shows that the isomorphism conjecture for algebraic K-Theory and the family of finite subgroups is not true. One has to extend the conjecture to a larger family of subgroups which contains all the counterexamples. Currently no counterexamples for the Farrell–Jones conjecture are known.
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He further sought to defend the Jews against bigoted charges of parasitic greed and cowardice with anecdotal counterexamples of Jewish industriousness and martial courage.
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A proof by counterexample is a constructive proof, in that an object disproving the claim is exhibited. In contrast, a non-constructive proof of an impossibility claim would proceed by showing it is logically contradictory for all possible counterexamples to be invalid: At least one of the items on a list of possible counterexamples must actually be a valid counterexample to the impossibility conjecture. For example, a conjecture that it is impossible for an irrational power raised to an irrational power to be rational was disproved, by showing that one of two possible counterexamples must be a valid counterexample, without showing which one it is.
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The result is always homeomorphic to S4. Many cases over the years were ruled out as possible counterexamples to the smooth 4 dimensional Poincaré conjecture. For example, , , , , , , , .
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Later, Mark Ellingham constructed two more counterexamples: the Ellingham–Horton graphs.Ellingham, M. N. "Non-Hamiltonian 3-Connected Cubic Partite Graphs."Research Report No. 28, Dept. of Math.
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With John Erik Fornæss, Stensønes is an author of the book Lectures on Counterexamples in Several Complex Variables (AMS Chelsea Publishing, American Mathematical Society, 1987; reprinted 2007).
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2, Elsevier, 1996, 1447-1540. but both conjectures remain widely open. It is not even known if a single counterexample would necessarily lead to a series of counterexamples.
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For directed Cayley graphs (digraphs) the Lovász conjecture is false. Various counterexamples were obtained by Robert Alexander Rankin. Still, many of the below results hold in this restrictive setting.
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In 1992, with Fulvio Ricci, Damek published a family of counterexamples to a form of the Lichnerowicz conjecture according to which harmonic Riemannian manifolds must be locally symmetric. The asymmetric spaces they found as counterexamples are at least seven-dimensional; they are called Damek–Ricci spaces. Damek is the coauthor, with D. Buraczewski and T. Mikosch, of the book Stochastic Models with Power Law Tails: The Equation X=AX+B (Springer, 2016).
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Testability, a property applying to an empirical hypothesis, involves two components: #Falsifiability or defeasibility, which means that counterexamples to the hypothesis are logically possible. #The practical feasibility of observing a reproducible series of such counterexamples if they do exist. In short, a hypothesis is testable if there is a possibility of deciding whether it is true or false based on experimentation. This allows to decide whether a theory can be supported or refuted by data.
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II, pp. 85–98. Published by Tutte in 1946, it is the first counterexample constructed for this conjecture.. Other counterexamples were found later, in many cases based on Grinberg's theorem.
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The conjecture was proved in 1994 by Henning Haahr Andersen, Jens Carsten Jantzen and Wolfgang Soergel for sufficiently large group-specific characteristics (without explicit bound) and later by Peter Fiebig for a very high explicitly stated bound. Williamson found several infinite families of counterexamples to the generally suspected validity limits of Lusztig's conjecture. He also found counterexamples to a 1990 conjecture of Gordon James on symmetric groups. His work also provided new perspectives on the respective conjectures.
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Byrne, R.M.J., Espino, O. & Santamaria, C. (1999). Counterexamples and the suppression of inferences." Journal of Memory & Language", 40, 347-373. Other investigations of propositional inference examine how people think about disjunctive alternatives, e.g.
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However, despite having found rectangles that were not squares, all the rectangles she did find had four sides. She then makes the new conjecture "All rectangles have four sides". This is logically weaker than her original conjecture, since every square has four sides, but not every four-sided shape is a square. The above example explained — in a simplified way — how a mathematician might weaken her conjecture in the face of counterexamples, but counterexamples can also be used to demonstrate the necessity of certain assumptions and hypothesis.
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Euler's sum of powers conjecture was disproved by counterexample. It asserted that at least n nth powers were necessary to sum to another nth power. This conjecture was disproved in 1966, with a counterexample involving n = 5; other n = 5 counterexamples are now known, as well as some n = 4 counterexamples. Witsenhausen's counterexample shows that it is not always true (for control problems) that a quadratic loss function and a linear equation of evolution of the state variable imply optimal control laws that are linear.
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The affirmative answer to Hilbert's nineteenth problem given by Ennio De Giorgi and John Forbes Nash raised the question if the same conclusion holds also for Euler-lagrange equations of more general functionals: at the end of the 1960s, ,See , , , , , and . and constructed independently several counterexamples,See , . showing that in general there is no hope to prove such kind of regularity results without adding further hypotheses. Precisely, gave several counterexamples involving a single elliptic equation of order greater than two with analytic coefficients:See , and .
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Euler's conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when, through a direct computer search on a CDC 6600, they found a counterexample for . This was published in a paper comprising just two sentences. A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known: :: (Lander & Parkin, 1966), :: (Scher & Seidl, 1996), and :: (Frye, 2004). In 1986, Noam Elkies found a method to construct an infinite series of counterexamples for the case.
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In mathematics, the Andrews–Curtis conjecture states that every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relators, named after James J. Andrews and Morton L. Curtis who proposed it in 1965. It is difficult to verify whether the conjecture holds for a given balanced presentation or not. It is widely believed that the Andrews–Curtis conjecture is false. While there are no counterexamples known, there are numerous potential counterexamples.
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Over an algebraically closed field of characteristic 0, any unipotent abelian connected algebraic group is isomorphic to a product of copies of the additive group G_a. The analogue of this for fields of characteristic p is false: the truncated Witt schemes are counterexamples. (We make them into algebraic groups by forgetting the multiplication and just using the additive structure.) However, these are essentially the only counterexamples: over an algebraically closed field of characteristic p, any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.
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However, it is entirely possible that Goldbach's conjecture may have a constructive proof (as we do not know at present whether it does), in which case the quoted statement would have a constructive proof as well, albeit one that is unknown at present. The main practical use of weak counterexamples is to identify the "hardness" of a problem. For example, the counterexample just shown shows that the quoted statement is "at least as hard to prove" as Goldbach's conjecture. Weak counterexamples of this sort are often related to the limited principle of omniscience.
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Conversely, other bodies (including extrasolar ones) may provide additional examples, edge cases, and counterexamples to earthbound processes; without a greater context, studying these phenomena in relation to Earth alone may result in low sample sizes and observational biases.
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This computer evidence is not a proof that the conjecture is true. As shown in the cases of the Pólya conjecture, the Mertens conjecture, and Skewes' number, sometimes a conjecture's only counterexamples are found when using very large numbers.
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Every Hilbert space has this property. There are, however, Banach spaces which do not; Per Enflo published the first counterexample in a 1973 article. However, much work in this area was done by Grothendieck (1955). Later many other counterexamples were found.
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The first counterexample to the Tutte conjecture was the Horton graph, published by . After the Horton graph, a number of smaller counterexamples to the Tutte conjecture were found. Among them are a 92-vertex graph by ,. a 78-vertex graph by ,.
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He served on the editorial board of the Pacific Journal of Mathematics for 14 years 1965-1979\. There are three topological spaces named for Arens in the book Counterexamples in Topology, including Arens–Fort space. Arens died in Los Angeles, California.
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In organic chemistry, benzenoids are a class of chemical compounds with at least one benzene ring. These compounds have increased stability from resonance in the benzene rings. Most aromatic hydrocarbons are benzenoid. Notable counterexamples are cyclooctadecanonaene, azulene and trans- bicalicene.
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It is known that the conjecture is true for q = 3 . Informal probability arguments suggest that the "expected" number of counterexamples to the Feit–Thompson conjecture is very close to 0, suggesting that the Feit–Thompson conjecture is likely to be true.
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Mazya's early achievements include: his work on Sobolev spaces, in particular the discovery of the equivalence between Sobolev and isoperimetric/isocapacitary inequalities (1960),. his counterexamples related to Hilbert's 19th and Hilbert's 20th problem (1968),, , (p. 6, footnote 7, and p. 343 of the English translation).
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But by hypothesis, C was already the smallest counterexample; therefore, the supposition that there were any counterexamples to begin with must have been false. The partial ordering implied by 'smaller' here is the one that says that S < T whenever S has fewer nodes than T.
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The answer to the decision problem for the existential theory of the reals, given this sentence as input, is the Boolean value false: there are no counterexamples. Therefore, this sentence does not belong to the existential theory of the reals, despite being of the correct grammatical form.
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Artin conjectured that p-adic fields were C2, but Guy Terjanian found p-adic counterexamples for all p.Lang (1997) p.247 The Ax–Kochen theorem applied methods from model theory to show that Artin's conjecture was true for Qp with p large enough (depending on d).
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Counterexamples of cards which intentionally ignore ISO standards include hotel key cards, most subway and bus cards, and some national prepaid calling cards (such as for the country of Cyprus) in which the balance is stored and maintained directly on the stripe and not retrieved from a remote database.
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The converse is not necessarily true. The counterexamples are equirectangular and equal-area cylindrical projections (of normal aspects). These projections expand meridian-wise and parallel-wise by different ratios respectively. Thus, parallels and meridians cross rectangularly on the map, but these projections do not preserve other angles; i.e.
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Not all specialists in Balto-Slavic historical linguistics accept Winter's law. A study of counterexamples led Patri (2006) to conclude that there is no law at all. According to him, exceptions to the law create a too heterogeneous and voluminous set of data to allow any phonological generalization.
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Although their paper has been very widely cited, Julius Ross and David Witt Nyström found counterexamples to the regularity results of Chen and Tian in 2015.Ross, Julius; Nyström, David Witt. Harmonic discs of solutions to the complex homogeneous Monge-Ampère equation. Publ. Math. Inst. Hautes Études Sci.
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However, the failure to find a counterexample after extensive search does not constitute a proof that no counterexample exists, nor that the conjecture is true—because the conjecture might be false but with a very large minimal counterexample. Instead, a conjecture is considered proven only when it has been shown that it is logically impossible for it to be false. There are various methods of doing so; see methods of mathematical proof for more details. One method of proof, applicable when there are only a finite number of cases that could lead to counterexamples, is known as "brute force": in this approach, all possible cases are considered and shown not to give counterexamples.
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Kempe's (incorrect) proof is based on alternating chains, and as those chains prove useful in graph theory mathematicians remain interested in such counterexamples. More were found later: first, the Errera graph in 1921,Errera, A. "Du coloriage des cartes et de quelques questions d'analysis situs." Ph.D. thesis. 1921.Peter Heinig.
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Sun Zhihong (, born October 16, 1965) is a Chinese mathematician, working primarily on number theory, combinatorics, and graph theory. Sun and his twin brother Sun Zhiwei proved a theorem about what are now known as the Wall–Sun–Sun primes that guided the search for counterexamples to Fermat's last theorem.
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For example, two half-moon shaped clusters intertwined in space do not separate well when projected onto PCA subspace. k-means should not be expected to do well on this data. It is straightforward to produce counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.
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In topology, an irrational winding of a torus is a continuous injection of a line into a two-dimensional torus that is used to set up several counterexamples. A related notion is the Kronecker foliation of a torus, a foliation formed by the set of all translates of a given irrational winding.
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Weak rule utilitarianism (WRU) attempts to handle SRU counterexamples as legitimate exceptions. One such response is two-level utilitarianism; more systematic WRUs attempt to create sub-rules to handle the exceptions. But as David LyonsForms and Limits of Utilitarianism, 1965. and others have argued, this will necessarily tend to collapse into act utilitarianism.
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Counterexamples to this conjecture were later discovered, but the possibility that a finite bound on toughness might imply Hamiltonicity remains an important open problem in graph theory. A simpler proof both of Fleischner's theorem, and of its extensions by , was given by ,; . and another simplified proof of the theorem was given by .; .
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Journal de Mathématiques Pures et Appliquées. Vol. 1, 139–216. The mathematician Mikhail Yakovlevich Suslin found that error about ten years later, and his following research has led to descriptive set theory. The fundamental mistake of Lebesgue was to think that projection commutes with decreasing intersection, while there are simple counterexamples to that.
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This states that there is no number with the property that for all other numbers , , . See Ford's theorem above. As stated in the main article, if there is a single counterexample to this conjecture, there must be infinitely many counterexamples, and the smallest one has at least ten billion digits in base 10.
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In mathematics, particularly topology, a comb space is a particular subspace of \R^2 that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb space. The deleted comb space is a variation on the comb space.
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This theorem stumped mathematicians for more than a hundred years, until a proof was developed which ruled out large classes of possible counterexamples, yet still left open enough possibilities that a computer program was needed to finish the proof. That proof was controversial for some time, but, later, it was verified using Coq.
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Looking Into the Eyes of the Sun was shot in just 31 working days. Although Bulajić's name was usually associated with high-budget productions, he gave Looking Into the Eyes of the Sun and The Man to Destroy as counterexamples, describing them as "the two cheapest films in the history of Jadran Film".
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Just as standard mathematical induction is equivalent to the well-ordering principle, structural induction is also equivalent to a well-ordering principle. If the set of all structures of a certain kind admits a well-founded partial order, then every nonempty subset must have a minimal element. (This is the definition of "well- founded".) The significance of the lemma in this context is that it allows us to deduce that if there are any counterexamples to the theorem we want to prove, then there must be a minimal counterexample. If we can show the existence of the minimal counterexample implies an even smaller counterexample, we have a contradiction (since the minimal counterexample isn't minimal) and so the set of counterexamples must be empty.
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The students are attempting to prove the formula for the Euler characteristic in algebraic topology, which is a theorem about the properties of polyhedra, namely that for all polyhedra the number of their Vertices minus the number of their Edges plus the number of their Faces is 2: (V – E + F = 2). The dialogue is meant to represent the actual series of attempted proofs which mathematicians historically offered for the conjecture, only to be repeatedly refuted by counterexamples. Often the students paraphrase famous mathematicians such as Cauchy, as noted in Lakatos's extensive footnotes. Lakatos termed the polyhedral counterexamples to Euler's formula monsters and distinguished three ways of handling these objects: Firstly, monster-barring, by which means the theorem in question could not be applied to such objects.
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If none of the interior vertices of a Halin graph has degree three, then it is necessarily pancyclic.. observed that many classical conditions for the existence of a Hamiltonian cycle were also sufficient conditions for a graph to be pancyclic, and on this basis conjectured that every 4-connected planar graph is pancyclic. However, found a family of counterexamples.
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Thus intuitionists absolutely disallow the blanket assertion: "For all propositions P concerning infinite sets D: P or ~P" (Kleene 1952:48). :For more about the conflict between the intuitionists (e.g. Brouwer) and the formalists (Hilbert) see Foundations of mathematics and Intuitionism. Putative counterexamples to the law of excluded middle include the liar paradox or Quine's paradox.
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In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither Lindelöf nor separable). Therefore, it serves as one of the basic counterexamples of topology.
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Note that the conclusion is false if the sectional curvatures are allowed to take values in the closed interval [1,4]. The standard counterexample is complex projective space with the Fubini–Study metric; sectional curvatures of this metric take on values between 1 and 4, with endpoints included. Other counterexamples may be found among the rank one symmetric spaces.
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The theorem does not apply if one of the bodies is not convex. If one of A or B is not convex, then there are many possible counterexamples. For example, A and B could be concentric circles. A more subtle counterexample is one in which A and B are both closed but neither one is compact.
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There exist numerous topologies on any given finite set. Such spaces are called finite topological spaces. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general. Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite.
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During the 20th century, compatibilists presented novel arguments that differed from the classical arguments of Hume, Hobbes, and John Stuart Mill. Importantly, Harry Frankfurt popularized what are now known as Frankfurt counterexamples to argue against incompatibilism,Kane 2005, p. 83 and developed a positive account of compatibilist free will based on higher- order volitions.Kane 2005, p.
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After the Horton graph, a number of smaller counterexamples to the Tutte conjecture were found. Among them are a 92 vertex graph by Horton published in 1982, a 78 vertex graph by Owens published in 1983, and the two Ellingham-Horton graphs (54 and 78 vertices).Horton, J. D. "On Two-Factors of Bipartite Regular Graphs." Discrete Math. 41, 35-41, 1982.
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Almost all modern essays are written in prose, but works in verse have been dubbed essays (e.g., Alexander Pope's An Essay on Criticism and An Essay on Man). While brevity usually defines an essay, voluminous works like John Locke's An Essay Concerning Human Understanding and Thomas Malthus's An Essay on the Principle of Population are counterexamples. In some countries (e.g.
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In constructive mathematics, the limited principle of omniscience (LPO) and the lesser limited principle of omniscience (LLPO) are axioms that are nonconstructive but are weaker than the full law of the excluded middle . The LPO and LLPO axioms are used to gauge the amount of nonconstructivity required for an argument, as in constructive reverse mathematics. They are also related to weak counterexamples in the sense of Brouwer.
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Lo argues that much of what behaviorists cite as counterexamples to economic rationality—loss aversion, overconfidence, overreaction, and other behavioral biases—are, in fact, consistent with an evolutionary model of individuals adapting to a changing environment using simple heuristics. Even fear and greed, which are viewed as the usual culprits in the failure of rational thinking by the behaviorists, are driven by evolutionary forces.
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Prover9 is the successor of the Otter theorem prover also developed by William McCune. Prover9 is noted for producing relatively readable proofs and having a powerful hints strategy. Prover9 is intentionally paired with Mace4, which searches for finite models and counterexamples. Both can be run simultaneously from the same input, with Prover9 attempting to find a proof, while Mace4 attempts to find a (disproving) counter-example.
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This basic research process iterates until one runs out of new ideas for refuting the explanation. Stopping to write up results is the next step. These developments are merged into a wider stream of his past research work and where feasible, the relevant literatures. Because of the importance he places on counterexamples, he eschews the use of mathematical statistics in favor of strong inference.
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However, scrutiny to detail will often produce counterexamples. In the case given above, holding a PhD may lower lifetime income, because of the years of lost earnings it implies and because many PhD holders enter academia instead of higher-paid fields. Nonetheless, broadly speaking, people with more education tend to earn more, so the above example is true in the sense of a stylized fact.
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He also proved that any element in an at most countable conical refinement monoid is measured by a unique (up to isomorphism) V-measure on a unique at most countable Boolean algebra. He raised there the problem whether any conical refinement monoid is measurable. This was answered in the negative by Friedrich Wehrung in 1998. The counterexamples can have any cardinality greater than or equal to ℵ2.
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In the ninth chapter titled "Syntax and Semantics", Chomsky reminds that his analysis so far has been "completely formal and non- semantic." He then offers many counterexamples to refute some common linguistic assertions about grammar's reliance on meaning. He concludes that the correspondence between meaning and grammatical form is "imperfect", "inexact" and "vague." Consequently, it is "relatively useless" to use meaning "as a basis for grammatical description".
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In mathematics, the Markus–Yamabe conjecture is a conjecture on global asymptotic stability. The conjecture states that if a continuously differentiable map on an n-dimensional real vector space has a fixed point, and its Jacobian matrix is everywhere Hurwitz, then the fixed point is globally stable. The conjecture is true for the two-dimensional case. However, counterexamples have been constructed in higher dimensions.
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All of the New Math projects emphasized some form of discovery learning. Students worked in groups to invent theories about problems posed in the textbooks. Materials for teachers described the classroom as "noisy." Part of the job of the teacher was to move from table to table assessing the theory that each group of students had developed and "torpedoing" wrong theories by providing counterexamples.
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They summarized their work in Counterexamples in Topology (1978). In 1975 Seebach and Steen became co-editors of Mathematics Magazine. Steen wrote: :Arthur’s sense of whimsy, his love of puns, and his proclivity for obscure connections totally transformed the image of Mathematics Magazine. Cover art, viewed as radical at the time, has since been emulated... Seebach welcomed the rise of computers when he assembled a Heathkit H8.
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The Bousso bound evades all known counterexamples to the spacelike bound. It was proven to hold when the entropy is approximately a local current, under weak assumptions. In weakly gravitating settings, the Bousso bound implies the Bekenstein bound and admits a formulation that can be proven to hold in any relativistic quantum field theory. The lightsheet construction can be inverted to construct holographic screens for arbitrary spacetimes.
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Counterexamples by Fujiwara and Sudo show that the Hasse–Minkowski theorem is not extensible to forms of degree 10n + 5, where n is a non-negative integer. On the other hand, Birch's theorem shows that if d is any odd natural number, then there is a number N(d) such that any form of degree d in more than N(d) variables represents 0: the Hasse principle holds trivially.
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None of these counterexamples are finitely presented, and for some years it was considered possible that the conjecture held for finitely presented groups. However, in 2003, Alexander Ol'shanskii and Mark Sapir exhibited a collection of finitely- presented groups which do not satisfy the conjecture. In 2013, Nicolas Monod found an easy counterexample to the conjecture. Given by piecewise projective homeomorphisms of the line, the group is remarkably simple to understand.
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Intuition why the above theorem should be true, is only partially true and sometimes completely wrong (explicit counterexamples). This is why this result has attracted much attention. The mathematical proof does not use new mathematical methods but is subtle. Apart from a classical result on so-called complete convergence, it is mainly based on theorems for stopping times on sums of independent and identically distributed order statistics (ref.
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Thalberg published two books of philosophical studies through the Muirhead Library of Philosophy: Enigmas of Agency (Allen & Unwin, London, 1972), and Perception, Emotion & Action (Blackwells, Oxford, 1977). Unlike most epistemologists, Thalberg published articles that defended the Platonic tripartite analysis of knowledge (justified true belief, a.k.a. "JTB") against the more popular view that Gettier counterexamples refuted the JTB account. Specifically, Thalberg argued that justification is not transmissible through valid deduction.
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Kollár is known for his contributions to the minimal model program for threefolds and hence the compactification of moduli of algebraic surfaces, for pioneering the notion of rational connectedness (i.e. extending the theory of rationally connected varieties for varieties over the complex field to varieties over local fields), and finding counterexamples to a conjecture of John Nash. (In 1952 Nash conjectured a converse to a famous theorem he proved,, . See and Kollár was able to provide many 3-dimensional counterexamples from an important new structure theory for a class of 3-dimensional algebraic varieties.) Kollár also gave the first algebraic proof of effective Nullstellensatz: let f_1,\ldots,f_m be polynomials of degree at most d \ge 3 in n\ge 2 variables; if they have no common zero, then the equation g_1 f_1+\cdots +g_m f_m=1 has a solution such that each polynomial g_j has degree at most d^n - d.
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Since, under traditional descriptivism, these descriptions are what define the name Jonah, these descriptivists must say that Jonah did not exist. But this does not follow. But under Katz's version of descriptivism, the sense of Jonah contains no information derived from the Biblical accounts but contains only the term "Jonah" itself in the phrase "the thing that is a bearer of 'Jonah'." Hence, it is not vulnerable to these kinds of counterexamples.
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It has been asserted that the relaxed solution of -means clustering, specified by the cluster indicators, is given by the principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace. However, that PCA is a useful relaxation of -means clustering was not a new result, and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.
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A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa. Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpinski space is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.
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François Fages [1994] found a syntactic condition on logic programs that eliminates such counterexamples and guarantees the stability of every model of the program's completion. The programs that satisfy his condition are called tight. Fangzhen Lin and Yuting Zhao [2004] showed how to make the completion of a nontight program stronger so that all its nonstable models will be eliminated. The additional formulas that they add to the completion are called loop formulas.
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The first four stages of the construction of the bucket handle as the limit of a series of nested intersections In point-set topology, an indecomposable continuum is a continuum that is indecomposable, i.e. that cannot be expressed as the union of any two of its proper subcontinua. In 1910, L. E. J. Brouwer was the first to describe an indecomposable continuum. Indecomposable continua have been used by topologists as a source of counterexamples.
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In short, Kirk sees Eliade's theory of eternal return as a universalization of the Australian Dreamtime concept. As two counterexamples to the eternal return, Kirk cites Native American mythology and Greek mythology. The eternal return is nostalgic: by retelling and reenacting mythical events, Australian Aborigines aim to evoke and relive the Dreamtime. However, Kirk believes that Native American myths "are not evocative or nostalgic in tone, but tend to be detailed and severely practical".
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In the mathematical field of graph theory, the Kittell graph is a planar graph with 23 vertices and 63 edges. Its unique planar embedding has 42 triangular faces. The Kittell graph is named after Irving 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 Poussin graph (both published earlier than Kittell) and the Fritsch graph and Soifer graph.
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Nagata's compactification theorem shows that varieties can be embedded in complete varieties. The Chevalley–Iwahori–Nagata theorem 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. His 1962 book on local rings contains several other counterexamples he found, such as a commutative Noetherian ring that is not catenary, and a commutative Noetherian ring of infinite dimension.
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Hence Chalmers's argument needn't go through. Moreover, while Chalmers defuses criticisms of the view that conceivability can tell us about possibility, he provides no positive defense of the principle. As an analogy, the generalized continuum hypothesis has no known counterexamples, but this doesn't mean we must accept it. And indeed, the fact that Chalmers concludes we have epiphenomenal mental states that don't cause our physical behavior seems one reason to reject his principle.
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Those solutions that are not exact are called non-exact solutions. Such solutions mainly arise due to the difficulty of solving the EFE in closed form and often take the form of approximations to ideal systems. Many non-exact solutions may be devoid of physical content, but serve as useful counterexamples to theoretical conjectures. Al Momin argues that Kurt Gödel's solution to these equations do not describe our universe and are therefore approximations.
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Small bodies such as comets, some asteroid types, and dust grains, on the other hand, serve as counterexamples. Assumed to have experienced little or no heating, these materials may contain (or be) samples representing the early Solar System, which have since been erased from Earth or any other large body. Some extrasolar planets are covered entirely in lava oceans, and some are tidally locked planets, whose star-facing hemisphere is entirely lava.
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List of Fellows of the American Mathematical Society, retrieved 2013-07-07. Rudin is best known in topology for her constructions of counterexamples to well-known conjectures. In 1958, she found an unshellable triangulation of the tetrahedron. Most famously, Rudin was the first to construct a Dowker space, which she did in 1971, thus disproving a conjecture of Clifford Hugh Dowker that had stood, and helped drive topological research, for more than twenty years.
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The strong real Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. That is equivalent to asking whether such a map is topologically a proper map, in which case it is a covering map of a simply connected manifold, hence invertible. constructed two variable counterexamples of total degree 25 and higher. It is well-known that the Dixmier conjecture implies the Jacobian conjecture (see Bass et al. 1982).
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According to Kvanvig, an adequate account of knowledge should resist counterexamples and allow an explanation of the value of knowledge over mere true belief. Should a theory of knowledge fail to do so, it would prove inadequate. One of the more influential responses to the problem is that knowledge is not particularly valuable and is not what ought to be the main focus of epistemology. Instead, epistemologists ought to focus on other mental states, such as understanding.
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A graph that can be proven non-Hamiltonian using Grinberg's theorem In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengths of its face cycles. The result has been widely used to construct non-Hamiltonian planar graphs with further properties, such as to give new counterexamples to Tait's conjecture (originally disproved by W.T. Tutte in 1946). This theorem was proved by Latvian mathematician Emanuel Grinberg in 1968.
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In mathematics, the Kurosh problem is one general problem, and several more special questions, in ring theory. The general problem is known to have a negative solution, since one of the special cases has been shown to have counterexamples. These matters were brought up by Aleksandr Gennadievich Kurosh as analogues of the Burnside problem in group theory. Kurosh asked whether there can be a finitely-generated infinite-dimensional algebraic algebra (the problem being to show this cannot happen).
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Although the macrostructure of the two hemispheres appears to be almost identical, different composition of neuronal networks allows for specialized function that is different in each hemisphere. Lateralization of brain structures is based on general trends expressed in healthy patients; however, there are numerous counterexamples to each generalization. Each human's brain develops differently, leading to unique lateralization in individuals. This is different from specialization, as lateralization refers only to the function of one structure divided between two hemispheres.
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He works on operator algebras, K-theory of operator algebras, groupoids, locally compact quantum groups and singular foliations. In 2002 with Nigel Higson and Vincent Lafforgue, Skandalis published counterexamples to a generalization of the Baum–Connes conjecture (i.e. Baum-Connes conjecture with coefficients) in various special cases, based on work by Mikhail Gromov.. In 1990 Skandalis was an invited speaker at the International Congress of Mathematicians in Kyoto (Operator Algebras and Duality). He was a member of Bourbaki.
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Simon Brendle has solved major open problems regarding the Yamabe equation in conformal geometry. This includes his counterexamples to the compactness conjecture for the Yamabe problem, and the proof of the convergence of the Yamabe flow in all dimensions (conjectured by Richard Hamilton). In 2007, he proved the differentiable sphere theorem (in collaboration with Richard Schoen), a fundamental problem in global differential geometry. In 2012, he proved the Hsiang–Lawson's conjecture, a longstanding problem in minimal surface theory.
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Angluin joined the faculty at Yale in 1979. Dana Angluin has authored many papers and been a pioneer in many fields specifically learning regular sets from queries and counterexamples, robot navigation with distance queries, self-stabilizing universal algorithms and query learning of regular tree languages. A lot of Angluin's work involves queries, a field in which she has made many great contributions. Angluin also has worked in the field of robotics as well dealing with navigation with distance queries.
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Several counterexamples have been offered by philosophers claiming to show that there are cases when an "ought" logically follows from an "is." First of all, Hilary Putnam, by tracing back the quarrel to Hume's dictum, claims fact/value entanglement as an objection, since the distinction between them entails a value. A. N. Prior points out, from the statement "He is a sea captain," it logically follows, "He ought to do what a sea captain ought to do."Alasdair MacIntyre, After Virtue (2007), p.
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He goes on and gives further stages that might sometimes take place: 5. Proofs of other theorems are examined to see if the newly found lemma or the new proof-generated concept occurs in them: this concept may be found lying at cross-roads of different proofs, and thus emerge as of basic importance. 6. The hitherto accepted consequences of the original and now refuted conjecture are checked. 7. Counterexamples are turned into new examples - new fields of inquiry open up.
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Originally from Bridgeport, Connecticut, he majored in Mathematics with a B.A. degree from Carnegie Mellon University in 1970. From 1972 to 1973, he was a research and teaching assistant at Ohio State University, where he earned the Master of Science degree in Mathematics. In 1976, he went to work at Wayne State University, where he concentrated the research on chromatic numbers and Brooks' theorem. As a result, Paul A. Catlin published one of the most cited papers in that series: Hajós graph coloring conjecture: variations and counterexamples.
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In at least one version, it is Daebyeol-wang who asks the riddles and refutes Sobyeol-wang's responses. In one account, it is Sobyeol-wang who asks the questions, but Daebyeol-wang wins by successfully answering his counterexamples. The bamboo keeps its leaves in winter because, although its internodes are hollow, bamboo leaves actually grow from the nodes, which are solid. Humans have more hair on the scalp than on the feet because newborns come out head-first during childbirth, and so the head is originally below.
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The five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proven in the late 19th century; however, proving that four colors suffice turned out to be significantly harder. A number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. The four color theorem was ultimately proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer.
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Showing this with hundreds of pages of hand analysis, Appel and Haken concluded that no smallest counterexample exists because any must contain, yet do not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by mathematicians at all because the computer-assisted proof was infeasible for a human to check by hand. However, the proof has since then gained wider acceptance, although doubts still remain.
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On the other hand, Sosa also found problems in the foundationalist approach to epistemology. Foundationalism arguably encounters a problem when attempting to describe how foundational beliefs relate to the sensory experiences that support them. Coherentism and foundationalism developed as a response to the problems with the "traditional" account of knowledge (as justified true belief) developed by Edmund Gettier in 1963. As a result of Gettier's counterexamples, competing theories had been developed by a variety of philosophers, but the dispute between coherentists and foundationalists proved to be intractable.
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Ford also proved that if there exists a counterexample to the Conjecture, then a positive proportion (in the sense of asymptotic density) of the integers are likewise counterexamples. Although the conjecture is widely believed, Carl Pomerance gave a sufficient condition for an integer n to be a counterexample to the conjecture . According to this condition, n is a counterexample if for every prime p such that p − 1 divides φ(n), p2 divides n. However Pomerance showed that the existence of such an integer is highly improbable.
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The construction of Árpád Line was 5-10 times cheaper per kilometer than the German and French counterexamples and it was able to hold off the enemy for an incomparably longer period. Despite the almost tenfold numerical superiority of Soviet forces they were unable to occupy the line and serious damage occurred in only a few "völgyzárs". In several cases (e.g. Gyimesbük) the company-scaled defensive groups (250-400 border guards) successfully faced greater-than-division-sized Soviet forces (10000-15000 soldiers supported by heavy artillery).
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The prefix quasi- came to denote methods that are "almost" or "socially approximate" an ideal of truly empirical methods. It is unnecessary to find all counterexamples to a theory; all that is required to disprove a theory logically is one counterexample. The converse does not prove a theory; Bayesian inference simply makes a theory more likely, by weight of evidence. One can argue that no science is capable of finding all counter-examples to a theory, therefore, no science is strictly empirical, it's all quasi-empirical.
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After the counterexamples were found, suggested that a reformulation of the conjecture should still hold. The current formulation of the generalized Ramanujan conjecture is for a globally generic cuspidal automorphic representation of a connected reductive group, where the generic assumption means that the representation admits a Whittaker model. It states that each local component of such a representation should be tempered. It is an observation due to Langlands that establishing functoriality of symmetric powers of automorphic representations of will give a proof of the Ramanujan–Petersson conjecture.
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In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space for which there are only finitely many points. While topology has mainly been developed for infinite spaces, finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can lend good insight to a variety of questions".
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Secondly, monster-adjustment whereby by making a re-appraisal of the monster it could be made to obey the proposed theorem. Thirdly, exception handling, a further distinct process. These distinct strategies have been taken up in qualitative physics, where the terminology of monsters has been applied to apparent counterexamples, and the techniques of monster-barring and monster- adjustment recognized as approaches to the refinement of the analysis of a physical issue. What Lakatos tried to establish was that no theorem of informal mathematics is final or perfect.
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Let M and N be closed and aspherical topological manifolds, and let :f \colon M \to N be a homotopy equivalence. The Borel conjecture states that the map f is homotopic to a homeomorphism. Since aspherical manifolds with isomorphic fundamental groups are homotopy equivalent, the Borel conjecture implies that aspherical closed manifolds are determined, up to homeomorphism, by their fundamental groups. This conjecture is false if topological manifolds and homeomorphisms are replaced by smooth manifolds and diffeomorphisms; counterexamples can be constructed by taking a connected sum with an exotic sphere.
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Modernisation refers to a model of an evolutionary transition from a 'pre-modern' or 'traditional' to a 'modern' society. The teleology of modernization is described in social evolutionism theories, existing as a template that has been generally followed by societies that have achieved modernity. While it may theoretically be possible for some societies to make the transition in entirely different ways, there have been no counterexamples provided by reliable sources. Historians link modernization to the processes of urbanization and industrialisation, as well as to the spread of education.
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For Weyl groups (special Coxeter groups, which are connected to Lie groups), David Kazhdan and George Lusztig succeeded in doing so by identifying the polynomials with certain invariants (local intersection cohomology) of Schubert varieties. Elias and Williamson were able to follow this path of proof also for more general groups of reflection (Coxeter groups), although there is no geometrical interpretation in contrast to the case of the Weyl groups. He is also known for several counterexamples. In 1980, Lusztig suggested a character formula for simple modules of reductive groups over fields of finite characteristic p.
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Those that are in the center manifold are susceptible to small perturbations that generally push them about randomly, and often push them out of the center manifold. That is, small perturbations tend to destabilize points in the center manifold: the center manifold behaves like a saddle point, or rather, an extended collection of saddle points. There are dramatic counterexamples to this idea of instability at the center manifold; see Lagrangian coherent structure for detailed examples. A much more sophisticated example is the Anosov flow on tangent bundles of Riemann surfaces.
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After the Battle of Kursk, only the Árpád Line was able to detain the Russian army for more than three weeks. With regard to effectiveness per cost rate, it was the most potent fortification system during World War 2. The construction of Árpád Line was 5-10 times cheaper per kilometer than the German and French counterexamples, and it was able to hold off the enemy for an incomparably longer period. The losses were extremely low, despite the enemy's numerical superiority and the poor equipment of Hungarian Armed Forces.
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Several counterexamples have been offered by philosophers claiming to show that there are cases when an evaluative statement does indeed logically follow from a factual statement. A. N. Prior points out, from the statement "He is a sea captain," it logically follows, "He ought to do what a sea captain ought to do."Alasdair MacIntyre, After Virtue (1984), p. 57 Alasdair MacIntyre points out, from the statement "This watch is grossly inaccurate and irregular in time-keeping and too heavy to carry about comfortably," the evaluative conclusion validly follows, "This is a bad watch."ibid.
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This condition is controversial, but Grice argues that apparent counterexamples—cases in which a speaker apparently says something without meaning it—are actually examples of what he calls "making as if to say", which can be thought of as a kind of "mock saying" or "play saying".Neale 1992, p.554. Another point of controversy surrounding Grice's notion of saying is the relationship between what a speaker says with an expression and the expression's timeless meaning. Although he attempts to spell out the connection in detail several times,Grice 1989, pp.87–88.
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In the case of borderline cases of art and prima facie counterexamples, open concepts "call for some sort of decision on our part to extend the use of the concept to cover this, or to close the concept and invent a new one to deal with the new case and its new property" (p. 31 ital. in original). The question of whether a new artifact is art "is not factual, but rather a decision problem, where the verdict turns on whether or not we enlarge our set of conditions for applying the concept" (p. 32).
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On the other hand, spaces that are regular but not completely regular, or preregular but not regular, are usually constructed only to provide counterexamples to conjectures, showing the boundaries of possible theorems. Of course, one can easily find regular spaces that are not T0, and thus not Hausdorff, such as an indiscrete space, but these examples provide more insight on the T0 axiom than on regularity. An example of a regular space that is not completely regular is the Tychonoff corkscrew. Most interesting spaces in mathematics that are regular also satisfy some stronger condition.
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Navy's doctoral thesis, "Nonparacompactness in Para- Lindelöf Spaces", was important in the development of metrizability theory. The paper examines the properties of para-Lindelöf topological spaces, which are a generalization of both Lindelöf spaces and paracompact spaces. In a para-Lindelöf space, every open cover has a locally countable open refinement, that is, one such that each point of the space has a neighborhood that intersects only countably many elements of the refinement. The spaces constructed by Navy are counterexamples to the conjecture that all para- Lindelöf spaces are paracompact.
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Chomsky also criticized Kirkpatrick's assertion that authoritarian governments "create no refugees," citing counterexamples such as Haiti and Somoza-era Nicaragua. Ted Galen Carpenter of the Cato Institute has also disputed the doctrine, noting that while Communist movements tend to depose rival authoritarians, the traditional authoritarian regimes supported by the US came to power by overthrowing democracies. He thus concludes that while Communist regimes are more difficult to eradicate, traditional autocratic regimes "pose the more lethal threat to functioning democracies.""The United States and Third World Dictatorships: A Case for Benign Detachment" Ted Galen Carpenter.
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Donald Dines Wall (August 13, 1921 – November 28, 2000) was a mathematician working primarily on number theory. He obtained his Ph.D. on normal numbers from University of California, Berkeley in 1949, where his adviser was Derrick Henry Lehmer. His better known papers include the first modern analysis of Fibonacci sequence modulo a positive integer. Drawing on Wall's work, Zhi-Hong Sun and his twin brother Zhi-Wei Sun proved a theorem about what are now known as the Wall–Sun–Sun primes that guided the search for counterexamples to Fermat's last theorem.
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In just two and a half pages, Gettier argued that there are situations in which one's belief may be justified and true, yet fail to count as knowledge. That is, Gettier contended that while justified belief in a true proposition is necessary for that proposition to be known, it is not sufficient. According to Gettier, there are certain circumstances in which one does not have knowledge, even when all of the above conditions are met. Gettier proposed two thought experiments, which have become known as Gettier cases, as counterexamples to the classical account of knowledge.
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One less common response to the Gettier problem is defended by Richard Kirkham, who has argued that the only definition of knowledge that could ever be immune to all counterexamples is the infallibilist definition. To qualify as an item of knowledge, goes the theory, a belief must not only be true and justified, the justification of the belief must necessitate its truth. In other words, the justification for the belief must be infallible. While infallibilism is indeed an internally coherent response to the Gettier problem, it is incompatible with our everyday knowledge ascriptions.
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Another problem stemming from Wilkie's result, which remains open, is that which asks what the smallest algebra is for which W(x, y) is not true but the eleven axioms above are. In 1985 an algebra with 59 elements was found that satisfied the axioms but for which W(x, y) was false. Smaller such algebras have since been found, and it is now known that the smallest such one must have either 11 or 12 elements.Jian Zhang, Computer search for counterexamples to Wilkie's identity, Automated Deduction – CADE-20, Springer (2005), pp.
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Generally, the simplest, though invalid, counterexamples attempt to create one region which touches all other regions. This forces the remaining regions to be colored with only three colors. Because the four color theorem is true, this is always possible; however, because the person drawing the map is focused on the one large region, they fail to notice that the remaining regions can in fact be colored with three colors. This trick can be generalized: there are many maps where if the colors of some regions are selected beforehand, it becomes impossible to color the remaining regions without exceeding four colors.
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In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by Alexander Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski p-group for every prime p > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.
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Plemelj's claim that the system can be made Fuchsian at the last point as well is wrong. (Il'yashenko has shown that if one of the monodromy operators is diagonalizable, then Plemelj's claim is true.) Indeed found a counterexample to Plemelj's statement. This is commonly viewed as providing a counterexample to the precise question Hilbert had in mind; Bolibrukh showed that for a given pole configuration certain monodromy groups can be realised by regular, but not by Fuchsian systems. (In 1990 he published the thorough study of the case of regular systems of size 3 exhibiting all situations when such counterexamples exists.
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Mathematician Bertram Kostant discussed the background of Lisi's work in a colloquium presentation at UC Riverside. Some follow-ups to Lisi's original preprint have been published in peer-reviewed journals. Lee Smolin's "The Plebanski action extended to a unification of gravity and Yang–Mills theory", December 6, 2007, proposes a symmetry-breaking mechanism to go from an E8 symmetric action to Lisi's action for the Standard Model and gravity. Roberto Percacci's "Mixing internal and spacetime transformations: some examples and counterexamples" addresses a general loophole in the Coleman–Mandula theorem also thought to work in E8 Theory.
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The Lovász–Plummer conjecture remained open until 2015, when a construction for infinitely many counterexamples was published.. The Halin graphs are sometimes also called skirted trees or roofless polyhedra.. However, these named are ambiguous. Some authors use the name "skirted trees" to refer to planar graphs formed from trees by connecting the leaves into a cycle, but without requiring that the internal vertices of the tree have degree three or more. And like "based polyhedra", the "roofless polyhedra" name may also refer to the cubic Halin graphs.. The convex polyhedra whose graphs are Halin graphs have also been called domes..
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George Mark Bergman, born on 22 July 1943 in Brooklyn, New York,CV Berkeley is an American mathematician. He attended Stuyvesant High School in New York City and received his Ph.D. from Harvard University in 1968, under the direction of John Tate. The year before he had been appointed Assistant Professor of mathematics at the University of California, Berkeley, where he has taught ever since, being promoted to Associate Professor in 1974 and to Professor in 1978. His primary research area is algebra, in particular associative rings, universal algebra, category theory and the construction of counterexamples.
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In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted F \subseteq T \subseteq V, which were introduced by Richard 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, and is sometimes referred to as the Thompson group or Thompson's group. The Thompson groups, and F in particular, have a collection of unusual properties which have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented.
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Essentially, one can show that if the first k primes p congruent to 1 (mod q) (where q is a prime) are all less than qk+1, then such an integer will be divisible by every prime and thus cannot exist. In any case, proving that Pomerance's counterexample does not exist is far from proving Carmichael's Conjecture. However if it exists then infinitely many counterexamples exist as asserted by Ford. Another way of stating Carmichael's conjecture is that, if A(f) denotes the number of positive integers n for which φ(n) = f, then A(f) can never equal 1\.
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One had the remarkable extensions of Clifford theory by Green to the indecomposable modules of group algebras. During this era, the field of computational group theory became a recognized field of study, due in part to its tremendous success during the first generation classification. In discrete groups, the geometric methods of Jacques Tits and the availability the surjectivity of Serge Lang's map allowed a revolution in algebraic groups. The Burnside problem had tremendous progress, with better counterexamples constructed in the 1960s and early 1980s, but the finishing touches "for all but finitely many" were not completed until the 1990s.
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A closely related question is the so-called "non-compact Yamabe problem", which asks: Is it true that on every smooth complete Riemannian manifold which is not compact, there exists a metric that is conformal to g, has constant scalar curvature and is also complete? The answer is no, due to counterexamples given by . Various additional criteria under which a solution to the Yamabe problem for a non- compact manifold can be shown to exist are known (for example ); however, obtaining a full understanding of when the problem can be solved in the non- compact case remains a topic of research.
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For smooth, but not necessarily affine varieties, there is an isomorphism relating the hypercohomology of algebraic the de Rham complex to the singular cohomology. A proof of this comparison result using the concept of a Weil cohomology was given by . Counter-examples in the singular case can be found with non-Du Bois singularities such as the graded ring k[x,y]/(y^2-x^3) with y where \deg(y)=3 and \deg(x)=2. Other counterexamples can be found in algebraic plane curves with isolated singularities whose Milnor and Tjurina numbers are non-equal.
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Such works are said to be counterexamples because they are artworks that don't possess an intended aesthetic function. Beardsley replies that either such works are not art or they are "comments on art" (1983): "To classify them [Fountain and the like] as artworks just because they make comments on art would be to classify a lot of dull and sometimes unintelligible magazine articles and newspaper reviews as artworks" (p. 25). This response has been widely considered inadequate (REF). It is either question-begging or it relies on an arbitrary distinction between artworks and commentaries on artworks.
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For example, a particular statement may be shown to imply the law of the excluded middle. An example of a Brouwerian counterexample of this type is Diaconescu's theorem, which shows that the full axiom of choice is non-constructive in systems of constructive set theory, since the axiom of choice implies the law of excluded middle in such systems. The field of constructive reverse mathematics develops this idea further by classifying various principles in terms of "how nonconstructive" they are, by showing they are equivalent to various fragments of the law of the excluded middle. Brouwer also provided "weak" counterexamples.
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There exist planar non- Hamiltonian graphs in which all faces have five or eight sides. For these graphs, Grinberg's formula taken modulo three is always satisfied by any partition of the faces into two subsets, preventing the application of his theorem to proving non-Hamiltonicity in this case . It is not possible to use Grinberg's theorem to find counterexamples to Barnette's conjecture, that every cubic bipartite polyhedral graph is Hamiltonian. Every cubic bipartite polyhedral graph has a partition of the faces into two subsets satisfying Grinberg's theorem, regardless of whether it also has a Hamiltonian cycle .
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Hrushovski is well known for several fundamental contributions to model theory, in particular in the branch that has become known as geometric model theory, and its applications. His PhD thesis revolutionized stable model theory (a part of model theory arising from the stability theory introduced by Saharon Shelah). Shortly afterwards he found counterexamples to the Trichotomy Conjecture of Boris Zilber and his method of proof has become well known as Hrushovski constructions and found many other applications since. One of his most famous results is his proof of the geometric Mordell–Lang conjecture in all characteristics using model theory in 1996.
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In the mathematical field of graph theory, the Ellingham–Horton graphs are two 3-regular graphs on 54 and 78 vertices: the Ellingham–Horton 54-graph and the Ellingham–Horton 78-graph. They are named after Joseph D. Horton and Mark N. Ellingham, their discoverers. These two graphs provide counterexamples to the conjecture of W. T. Tutte that every cubic 3-connected bipartite graph is Hamiltonian.. The book thickness of the Ellingham-Horton 54-graph and the Ellingham-Horton 78-graph is 3 and the queue numbers 2Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, Universität Tübingen, 2018.
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In order to have a valid claim of knowledge for any proposition, one must be justified in believing "p" and "p" must be true. Since Gettier proposed his counterexamples the traditional analysis has included the further claim that knowledge must be more than justified true belief. Reliabilist theories of knowledge are sometimes presented as an alternative to that theory: rather than justification, all that is required is that the belief be the product of a reliable process. But reliabilism need not be regarded as an alternative, but instead as a further explication of the traditional analysis.
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Wachsmann argues that the concentration camps were only peripheral to the Final Solution, because most Jewish victims of the Holocaust died in shootings, gas vans, or dedicated extermination camps rather than in the concentration camp system. Although Jews made up a majority of deaths in concentration camps, they ranged from 10–30% of the population depending on the time period. Throughout the book, Wachsmann presents a generalization and then complicates the picture with counterexamples. The book is a work of synthetic history drawing mainly on published German sources, although it also incorporates the author's archival research.
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There are innumerable "counterexamples" where, it is argued, a straightforward application of CDT fails to produce a defensibly "sane" decision. Philosopher Andy Egan argues this is due to a fundamental disconnect between the intuitive rational rule, "do what you expect will bring about the best results", and CDT's algorithm of "do whatever has the best expected outcome, holding fixed our initial views about the likely causal structure of the world." In this view, it is CDT's requirement to "hold fixed the agent’s unconditional credences in dependency hypotheses" that leads to irrational decisions. An early alleged counterexample is Newcomb's problem.
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The counterexamples 7^3 + 13^2 = 2^9 and 1^m + 2^3 = 3^2 show that the conjecture would be false if one of the exponents were allowed to be 2. The Fermat–Catalan conjecture is an open conjecture dealing with such cases. If we allow at most one of the exponents to be 2, then there may be only finitely many solutions (except the case 1^m + 2^3 = 3^2). If A, B, C can have a common prime factor then the conjecture is not true; a classic counterexample is 2^{10} + 2^{10} = 2^{11}.
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Bolibrukh was born on 30 January 1950 in Moscow and studied at the 45th Physics-Mathematics School in Saint Petersburg. After receiving his mathematical education at the Lomonosov Moscow State University, with Mikhail Mikhailovich Postnikov and Alexey Chernavskii as thesis advisers, he started working on the proof of the existence of linear differential equations having a prescribed monodromic group. He applied modern methods of complex analytic geometry to classical problems about ordinary differential equations and was an expert on Hilbert's twenty-first problem. In 1989, Bolibrukh produced his famous counterexamples which invalidated the Josip Plemelj's 1908 solution of Hilbert's twenty-first problem.
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In contract bridge, the Law of total tricks (abbreviated here as LoTT) is a guideline used to help determine how high to bid in a competitive auction. It is not really a law (because counterexamples are easy to find) but a method of hand evaluation which describes a relationship that seems to exist somewhat regularly. Written by Jean-René Vernes for French players in the 1950s as a rule of thumb, it was first described in English in 1966 International Bridge Academy Annals. It received more notice from appearing in The Bridge World in June 1969.
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These observers are also concerned about competition among nations for footloose industries. This interesting book systematically examines the original European Community, the Single European Act, the General Agreement on Tariffs and Trade and its 1979 standards code, the U.S.-Canada Free Trade Area, and the more recent North American Free Trade Agreement, as well as the states in the free trade area of the United States. Vogel finds that, while counterexamples do exist, trade liberalization on balance has strongly reinforced environment-improving regulations. A good example is auto emissions requirements, which have gradually stiffened and leveled up in the trading system over time.
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" Ryan claims that this dictum has generally been understood to generate four 'humility principles' about the necessary and sufficient conditions for epistemic humility, depending on one's interpretation of the text. She formulates the first two humility principles as follows: > (HP1) S is wise iff S believes s/he is not wise > (HP2) S is wise iff S believes that S does not know anything. Ryan rejects (HP1) and (HP2) as plausible interpretations of the parable because she believes that they do not offer sufficient conditions of wisdom. She writes: "I take myself, and most people I've known, to be clear counterexamples to (HP1).
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It says that a successful scientific explanation must deduce the occurrence of the phenomena in question from a scientific law. This view has been subjected to substantial criticism, resulting in several widely acknowledged counterexamples to the theory. It is especially challenging to characterize what is meant by an explanation when the thing to be explained cannot be deduced from any law because it is a matter of chance, or otherwise cannot be perfectly predicted from what is known. Wesley Salmon developed a model in which a good scientific explanation must be statistically relevant to the outcome to be explained.
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Among philosophers, he was for a time best known for his interpretation of Descartes's rationalism. His most influential work, however, has been on freedom of the will (on which he has written numerous important papersFeinberg; Shafer-Landau: Reason & Responsibility, p. 486.) based on his concept of higher-order volitions and for developing what are known as "Frankfurt cases" or "Frankfurt counterexamples" (i.e., thought experiments designed to show the possibility of situations in which a person could not have done other than he/she did, but in which our intuition is to say nonetheless that this feature of the situation does not prevent that person from being morally responsible).
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No true Scotsman, or appeal to purity, is an informal fallacy in which one attempts to protect a universal generalization from counterexamples by changing the definition in an ad hoc fashion to exclude the counterexample.No True Scotsman, Internet Encyclopedia of Philosophy Rather than denying the counterexample or rejecting the original claim, this fallacy modifies the subject of the assertion to exclude the specific case or others like it by rhetoric, without reference to any specific objective rule: "no Scotsman would do such a thing"; i.e., those who perform that action are not part of our group and thus criticism of that action is not criticism of the group.
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This particular example is true, because any natural number could be substituted for n and the statement "2·n = n + n" would be true. In contrast, > For all natural numbers n, 2·n > 2 + n is false, because if n is substituted with, for instance, 1, the statement "2·1 > 2 + 1" is false. It is immaterial that "2·n > 2 + n" is true for most natural numbers n: even the existence of a single counterexample is enough to prove the universal quantification false. On the other hand, for all composite numbers n, 2·n > 2 + n is true, because none of the counterexamples are composite numbers.
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Many results supporting the digraph reconstruction conjecture appeared between 1964 and 1976. However, this conjecture was proved to be false when P. K. Stockmeyer discovered several infinite families of counterexample pairs of digraphs (including tournaments) of arbitrarily large order.. Erratum, J. Graph Th. 62 (2): 199–200, 2009, , ... The falsity of the digraph reconstruction conjecture caused doubt about the reconstruction conjecture itself. Stockmeyer even observed that “perhaps the considerable effort being spent in attempts to prove the (reconstruction) conjecture should be balanced by more serious attempts to construct counterexamples.” In 1979, Ramachandran revived the digraph reconstruction conjecture in a slightly weaker form called the new digraph reconstruction conjecture.
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In other cases, an increase in size may in fact represent a transition to an optimal body size, and not imply that populations always develop to a larger size. However, many palaeobiologists are skeptical of the validity of Cope's rule, which may merely represent a statistical artefact. Purported examples of Cope's rule often assume that the stratigraphic age of fossils is proportional to their "clade rank", a measure of how derived they are from an ancestral state; this relationship is in fact quite weak. Counterexamples to Cope's rule are common throughout geological time; although size increase does occur more often than not, it is by no means universal.
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Popper was not able to find any counterexamples of human behavior in which the behavior could not be explained in the terms of Adler's or Freud's theory. Popper argued it was that the observation always fitted or confirmed the theory which, rather than being its strength, was actually its weakness. In contrast, Popper gave the example of Einstein's gravitational theory, which predicted "light must be attracted by heavy bodies (such as the Sun), precisely as material bodies were attracted." Following from this, stars closer to the Sun would appear to have moved a small distance away from the Sun, and away from each other.
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He will > not know how to apply what he has learned in a new situation. - Pierre van > Hiele, 1959 The best known part of the van Hiele model are the five levels which the van Hieles postulated to describe how children learn to reason in geometry. Students cannot be expected to prove geometric theorems until they have built up an extensive understanding of the systems of relationships between geometric ideas. These systems cannot be learned by rote, but must be developed through familiarity by experiencing numerous examples and counterexamples, the various properties of geometric figures, the relationships between the properties, and how these properties are ordered.
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Existing serial or parallel processing models of PRP have successfully accounted for a variety of PRP phenomena; however, each also encounters at least 1 experimental counterexample to its predictions or modeling mechanisms. This article describes a queuing network-based mathematical model of PRP that is able to model various experimental findings in PRP with closed-form equations including all of the major counterexamples encountered by the existing models with fewer or equal numbers of free parameters. This modeling work also offers an alternative theoretical account for PRP and demonstrates the importance of the theoretical concepts of “queuing” and “hybrid cognitive networks” in understanding cognitive architecture and multitask performance.
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Although von Neumann's name is popularly attached to the conjecture, its first written appearance seems to be due to Mahlon Marsh Day in 1957. The Tits alternative is a fundamental theorem which, in particular, establishes the conjecture within the class of linear groups. The historically first potential counterexample is Thompson group F. While its amenability is a wide open problem, the general conjecture was shown to be false in 1980 by Alexander Ol'shanskii; he demonstrated that Tarski monster groups, constructed by him, which are easily seen not to have free subgroups of rank 2, are not amenable. Two years later, Sergei Adian showed that certain Burnside groups are also counterexamples.
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There is also a connection to the Hadwiger conjecture, an important open problem in combinatorics concerning the relationship between chromatic number and the existence of large cliques as minors in a graph.. A variant of the Hadwiger conjecture, stated by György Hajós, is that every n-chromatic graph contains a subdivision of K_n; if this were true, the Albertson conjecture would follow, because the crossing number of the whole graph is at least as large as the crossing number of any of its subdivisions. However, counterexamples to the Hajós conjecture are now known,; . so this connection does not provide an avenue for proof of the Albertson conjecture.
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In practice, if some "naturally occurring" set of primes has a Dirichlet density, then it also has a natural density, but it is possible to find artificial counterexamples: for example, the set of primes whose first decimal digit is 1 has no natural density, but has Dirichlet density log(2)/log(10).This is attributed by J.-P. Serre to a private communication from Bombieri in A course in arithmetic; an elementary proof based on the prime number theorem is given in: A. Fuchs, G. Letta, Le problème du premier chiffre décimal pour les nombres premiers [The first digit problem for primes] (French) The Foata Festschrift. Electron. J. Combin.
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The essay starts with a definition of Gettier's theory, followed by multiple reiterations of the idea of causal connections, figures to explain knowledge through a visual perspective, and references to perception and memory through causal chains. The essay tends to focus on examples in which knowledge or other sensations do not exist, rather than proving a certain fact to be known. Goldman also states on multiple occasions that he does not wish to explain the causal process in detail, instead pointing out counterexamples. At numerous times in the essay, he also points out that he does not intend to give definitive answers to each of the propositions mentioned.
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In mathematics, a Raynaud surface is a particular kind of algebraic surface that was introduced in and named for . To be precise, a Raynaud surface is a quasi-elliptic surface over an algebraic curve of genus g greater than 1, such that all fibers are irreducible and the fibration has a section. The Kodaira vanishing theorem fails for such surfaces; in other words the Kodaira theorem, valid in algebraic geometry over the complex numbers, has such surfaces as counterexamples, and these can only exist in characteristic p. Generalized Raynaud surfaces were introduced in , and give examples of surfaces of general type with global vector fields.
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After rediscovering Mendel's laws of heredity, which can be explained with chromosomal inheritance, he undertook experiments with the four o'clock plant Mirabilis jalapa to investigate apparent counterexamples to Mendel's laws in the heredity of variegated (green and white mottled) leaf color. Correns found that, while Mendelian traits behave independently of the sex of the source parent, leaf color depended greatly on which parent had which trait. For instance, pollinating an ovule from a white branch with pollen from another white area resulted in white progeny, the predicted result for a recessive gene. Green pollen used on a green stigma resulted in all green progeny, the expected result for a dominant gene.
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It has been an elusive goal to prove the normality of numbers that are not artificially constructed. While , π, ln(2), and e are strongly conjectured to be normal, it is still not known whether they are normal or not. It has not even been proven that all digits actually occur infinitely many times in the decimal expansions of those constants (for example, in the case of π, the popular claim "every string of numbers eventually occurs in π" is not known to be true). It has also been conjectured that every irrational algebraic number is absolutely normal (which would imply that is normal), and no counterexamples are known in any base.
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In philosophy, counterexamples are usually used to argue that a certain philosophical position is wrong by showing that it does not apply in certain cases. Alternatively, the first philosopher can modify their claim so that the counterexample no longer applies; this is analogous to when a mathematician modifies a conjecture because of a counterexample. For example, in Plato's Gorgias, Callicles, trying to define what it means to say that some people are "better" than others, claims that those who are stronger are better. But Socrates replies that, because of their strength of numbers, the class of common rabble is stronger than the propertied class of nobles, even though the masses are prima facie of worse character.
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But the conclusion, considered by itself and with the possible authors not limited to just Shakespeare and Hobbes, is dubious, because if Shakespeare is ruled out as _Hamlet_ 's author, there are many more plausible alternatives than Hobbes. The general form of McGee-type counterexamples to modus ponens is simply P, P \rightarrow (Q \rightarrow R), therefore Q \rightarrow R; it is not essential that P be a disjunction, as in the example given. That these kinds of cases constitute failures of modus ponens remains a minority view among logicians, but opinions vary on how the cases should be disposed of.Sinnott-Armstrong, Moor, and Fogelin (1986). "A Defense of Modus Ponens", The Journal of Philosophy 83, 296–300.
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Katz's theory, to take this example, is based on the fundamental idea that sense should not have to be defined in terms of, nor determine, referential or extensional properties but that it should be defined in terms of, and determined by, all and only the intensional properties of names. He illustrates the way a metalinguistic description theory can be successful against Kripkean counterexamples by citing, as one example, the case of "Jonah." Kripke’s Jonah case is very powerful because in this case the only information that we have about the Biblical character Jonah is just what the Bible tells us. Unless we are fundamentalist literalists, it is not controversial that all of this is false.
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Euler was unable to solve the problem, but in this work he demonstrated methods for constructing Graeco-Latin squares where is odd or a multiple of 4. Observing that no order two square exists and being unable to construct an order six square, he conjectured that none exist for any oddly even number The non-existence of order six squares was confirmed in 1901 by Gaston Tarry through a proof by exhaustion. However, Euler's conjecture resisted solution until the late 1950s, but the problem has led to important work in combinatorics. In 1959, R.C. Bose and S. S. Shrikhande constructed some counterexamples (dubbed the Euler spoilers) of order 22 using mathematical insights.
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An undirected graph G is Hamiltonian if it contains a cycle that touches each of its vertices exactly once. It is 2-vertex-connected if it does not have an articulation vertex, a vertex whose deletion would leave the remaining graph disconnected. Not every 2-vertex- connected graph is Hamiltonian; counterexamples include the Petersen graph and the complete bipartite graph K2,3. The square of G is a graph G2 that has the same vertex set as G, and in which two vertices are adjacent if and only if they have distance at most two in G. Fleischner's theorem states that the square of a finite 2-vertex-connected graph with at least three vertices must always be Hamiltonian.
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Scholars such as Peter Boghossian suggest that although the method improves creative and critical thinking, there is a flip side to the method. He states that the teachers who use this method wait for the students to make mistakes, thus creating a kind of negative feelings in the class, exposing the student to possible ridicule and humiliation. Some have countered this thought by stating that the humiliation and ridicule is not caused by the method, rather it is due to the lack of knowledge of the student. Boghossian mentions that even though the questions may be perplexing, they are not originally meant for it, in fact such questions provoke the students and can be countered by employing counterexamples.
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The question of whether one can speak of a theory of art without employing a concept of art is also discussed below. The motivation behind seeking a theory, rather than a definition, is that our best minds have not been able to find definitions without counterexamples. The term 'definition' assumes there are concepts, in something along Platonic lines, and a definition is an attempt to reach in and pluck out the essence of the concept and also assumes that at least some of us humans have intellectual access to these concepts. In contrast, a 'conception' is an individual attempt to grasp at the putative essence behind this common term while nobody has "access" to the concept.
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The absence of TATA boxes in bidirectional promoters suggests that TATA boxes play a role in determining the directionality of promoters, but counterexamples of bidirectional promoters do possess TATA boxes and unidirectional promoters without them indicates that they cannot be the only factor. Although the term "bidirectional promoter" refers specifically to promoter regions of mRNA-encoding genes, luciferase assays have shown that over half of human genes do not have a strong directional bias. Research suggests that non-coding RNAs are frequently associated with the promoter regions of mRNA-encoding genes. It has been hypothesized that the recruitment and initiation of RNA polymerase II usually begins bidirectionally, but divergent transcription is halted at a checkpoint later during elongation.
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The Theory of Communicative Action, Volume 1 sets out "to develop a concept of rationality that is no longer tied to, and limited by, the subjectivistic and individualistic premises of modern philosophy and social theory." With this failure of the search for ultimate foundations by "first philosophy" or "the philosophy of consciousness", an empirically tested theory of rationality must be a pragmatic theory based on science and social science. This implies that any universalist claims can only be validated by testing against counterexamples in historical (and geographical) contexts – not by using transcendental ontological assumptions. This leads him to look for the basis of a new theory of communicative action in the tradition of sociology.
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In 1821 Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is 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 be incorrect. Completely standard notions of convergence did not exist at the time, and Cauchy handled convergence using infinitesimal methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. The failure of a merely pointwise-convergent limit of continuous functions to converge to a continuous function illustrates the importance of distinguishing between different types of convergence when handling sequences of functions.
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However, in 2002 Sapir and Olshanskii found finitely presented counterexamples: non-amenable finitely presented groups that have a periodic normal subgroup with quotient the integers. For finitely generated linear groups, however, the von Neumann conjecture is true by the Tits alternative: every subgroup of GL(n,k) with k a field either has a normal solvable subgroup of finite index (and therefore is amenable) or contains the free group on two generators. Although Tits' proof used algebraic geometry, Guivarc'h later found an analytic proof based on V. Oseledets' multiplicative ergodic theorem. Analogues of the Tits alternative have been proved for many other classes of groups, such as fundamental groups of 2-dimensional simplicial complexes of non-positive curvature.
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Significantly, this work further formulated and first appreciated the all-important star-product, the cornerstone of this formulation of the theory, ironically often also associated with Moyal's name, even though it is not featured in Moyal's papers and was not fully understood by Moyal. Moreover, Groenewold first understood and demonstrated that the Moyal bracket is isomorphic to the quantum commutator, and thus that the latter cannot be made to faithfully correspond to the Poisson bracket, as had been envisioned by Paul Dirac. This observation and his counterexamples contrasting Poisson brackets to commutators have been generalized and codified to what is now known as the Groenewold – Van Hove theorem.See Theorem 13.13 for one version of this result See Groenewold's theorem for one version.
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Additionally, combining both Furtwängler's and Keller's conjectures, Robinson showed that k-fold square coverings of the Euclidean plane must include two squares that meet edge to edge. However, for every k > 1 and every n > 2 there is a k-fold tiling of n-dimensional space by cubes with no shared faces . Once counterexamples to Keller's conjecture became known, it became of interest to ask for the maximum dimension of a shared face that can be guaranteed to exist in a cube tiling. When the dimension n is at most six, this maximum dimension is just n − 1, by Perron's proof of Keller's conjecture for small dimensions, and when n is at least eight, then this maximum dimension is at most n − 2\.
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Feyerabend's position was seen as radical in the philosophy of science, because it implies that philosophy can neither succeed in providing a general description of science, nor in devising a method for differentiating products of science from non-scientific entities like myths. (Feyerabend's position also implies that philosophical guidelines should be ignored by scientists, if they are to aim for progress.) To support his position that methodological rules generally do not contribute to scientific success, Feyerabend provides counterexamples to the claim that (good) science operates according to a certain fixed method. He took some examples of episodes in science that are generally regarded as indisputable instances of progress (e.g. the Copernican revolution), and argued that these episodes violated all common prescriptive rules of science.
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The basis sets in the product topology have almost the same definition as the above, except with the qualification that all but finitely many Ui are equal to the component space Xi. The product topology satisfies a very desirable property for maps fi : Y → Xi into the component spaces: the product map f: Y → X defined by the component functions fi is continuous if and only if all the fi are continuous. As shown above, this does not always hold in the box topology. This actually makes the box topology very useful for providing counterexamples--many qualities such as compactness, connectedness, metrizability, etc., if possessed by the factor spaces, are not in general preserved in the product with this topology.
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Spherical geometry obeys two of Euclid's postulates: the second postulate ("to produce [extend] a finite straight line continuously in a straight line") and the fourth postulate ("that all right angles are equal to one another"). However, it violates the other three: contrary to the first postulate, there is not a unique shortest route between any two points (antipodal points such as the north and south poles on a spherical globe are counterexamples); contrary to the third postulate, a sphere does not contain circles of arbitrarily great radius; and contrary to the fifth (parallel) postulate, there is no point through which a line can be drawn that never intersects a given line.Gowers, Timothy, Mathematics: A Very Short Introduction, Oxford University Press, 2002: pp. 94 and 98.
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After her return to the United States, she taught in Baltimore for three years until a second scholarship, by the Baltimore Association for the Promotion of University Education of Women, permitted her to return to Bryn Mawr college to complete her Ph.D. under the direction of Charlotte Scott. Her dissertation, "On the Arrangement of the Real Branches of Plane Algebraic Curves," was published in 1906 by the American Journal of Mathematics. Her dissertation addressed the 16th of Hilbert's problems, for which Ragsdale formulated a conjecture that provided an upper bound on the number of topological circles of a certain type. Her result is called the Ragsdale conjecture; it was an open problem for 90 years until counterexamples were derived by Oleg Viro (1979) and Ilya Itenberg (1994).
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In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed by and disproved by , who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38 vertices, were later proved minimal by . The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian.
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The first detailed analysis of circumnutation was Charles Darwin's The Power of Movement in Plants; he concluded that most plant movements were modifications of circumnutation, but many counterexamples are now known. Circumnutation is not a direct response to gravity or the direction of illumination, but these factors and many physiological processes can influence its direction, timing and amplitude. Although the function of circumnutation in most plants is not known, many twining plants have adapted these movements to help them find and twine around vertical objects such as tree trunks, and to help tendrils find and wind around smaller supports. The growing tips of the vine or tendril initially swings in wide circles that maximize its chance of bumping into an obstacle (a potential support).
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Classifying categories of mimicry in 1890, Edward Bagnell Poulton placed this into the category of aggressive mimicry, a deceptive mechanism in which one species resembles another in order to approach it without arousing suspicion to carry out a detrimental end. However, Gregory Bateson criticized this view in 1892 by pointing out that the Volucella example fits much better as an instance of protective mimicry, now commonly known as Batesian mimicry. By appearing as bees, palatable flies gain protection from predators that recognize bees as noxious and therefore unappetizing. Bateson argued with the counterexamples that Volucella females entered bumblebee nests belonging to species that they did not mimic and that a European species with similar habits actually benefited the host because the fly larvae, once hatched, acted as scavengers inside the nest.
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Boundaries may exist between objects before the entity lays claim to them, (that is, if it intends to lay claim to them.) Breach of these boundaries constitute theft. A hypothetical entity empowered to lay claim upon any object can also approach a grey area between legitimate possession and theft if its possession of an object is not clearly defined, that is, if this entity itself has doubts as to whether it possesses the object in question. Boundaries exist in empirical reality because people and things obviously do not melt together upon touching. While some systems of thought would contest this even on a limited level, and there are counterexamples, especially when dealing with ideas, in general, it is accepted that boundaries exist at least in some areas of consensus reality.
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In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings.
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Generalized to directed graphs, the conjecture has simple counterexamples, as observed by . Here, the chromatic number of a directed graph is just the chromatic number of the underlying graph, but the tensor product has exactly half the number of edges (for directed edges g→g' in G and h→h' in H, the tensor product G × H has only one edge, from (g,h) to (g',h'), while the product of the underlying undirected graphs would have an edge between (g,h') and (g',h) as well). However, the Weak Hedetniemi Conjecture turns out to be equivalent in the directed and undirected settings . The problem cannot be generalized to infinite graphs: gave an example of two infinite graphs, each requiring an uncountable number of colors, such that their product can be colored with only countably many colors.
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Rabbi Levi, or some say Rabbi Jonathan, said that a tradition handed down from the Men of the Great Assembly taught that wherever the Bible employs the term "and it was" or "and it came to pass" (, wa-yehi), as it does in , it indicates misfortune, as one can read wa-yehi as wai, hi, "woe, sorrow." Thus the words, "And it came to pass when man began to multiply," in , are followed by the words, "God Saw that the wickedness of man was great," in . And the Gemara also cited the instances of followed by ; followed by ; followed by the rest of ; followed by ; followed by ; followed by ; close after ; followed by ; followed by the rest of ; and followed by Haman. But the Gemara also cited as counterexamples the words, "And there was evening and there was morning one day," in , as well as , and .
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This conjecture states that every algebra homomorphism from the Banach algebra C(X) (continuous complex-valued functions on X, where X is a compact Hausdorff space) into any other Banach algebra, is necessarily continuous. The conjecture is equivalent to the statement that every algebra norm on C(X) is equivalent to the usual uniform norm. (Kaplansky himself had earlier shown that every complete algebra norm on C(X) is equivalent to the uniform norm.) In the mid-1970s, H. Garth Dales and J. Esterle independently proved that, if one furthermore assumes the validity of the continuum hypothesis, there exist compact Hausdorff spaces X and discontinuous homomorphisms from C(X) to some Banach algebra, giving counterexamples to the conjecture. In 1976, R. M. Solovay (building on work of H. Woodin) exhibited a model of ZFC (Zermelo-Fraenkel set theory + axiom of choice) in which Kaplansky's conjecture is true.
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What makes this tutoring system stand out is the additional levels of abstraction involved in its results. The system presents exercises, including the facts of a problem and a set of on-line cases and instructions to make, or respond to, a legal argument about the problem. The student/user will have a set of tools to analyze the problem and fashion an answer comparing it to other cases. Instead of simply generating precedent cases, the system actually functions in such a way as to interpret student responses, comparing them against a list of possibilities and responding to student entries, for example, by citing counterexamples, and providing feedback on a student's problem solving activities with explanations of correctness or giving further hints as to what may be wrong with evaluating a student's ability to perform legal reasoning and argumentation, examples and follow-up assignments by employing HYPO's model of case-based structure.
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But if one did not make that assumption, then the simplest definition might not be the right one for the most useful concept; in any case, it would destroy the (transitive) entailment of Ti by Tj, allowing (for example) non-Hausdorff regular spaces. Topologists working on the metrisation problem generally did assume T1; after all, all metric spaces are T1. Thus, they used the simplest definitions for the Ti. Then, for those occasions when they did not assume T1, they used words ("regular" and "normal") for the more complicated definitions, in order to contrast them with the simpler ones. This approach was used as late as 1970 with the publication of Counterexamples in Topology by Lynn A. Steen and J. Arthur Seebach, Jr. In contrast, general topologists, led by John L. Kelley in 1955, usually did not assume T1, so they studied the separation axioms in the greatest generality from the beginning.
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The Tower of Babel (1594 painting by Lucas van Valckenborch at the Louvre) Rabbi Levi, or some say Rabbi Jonathan, said that a tradition handed down from the Men of the Great Assembly taught that wherever the Bible employs the term "and it was" or "and it came to pass" (, wa-yehi), as it does in , it indicates misfortune, as one can read wa-yehi as wai, hi, "woe, sorrow." Thus the words, "And it came to pass," in are followed by the words, "Come, let us build us a city," in . And the Gemara also cited the instances of followed by ; followed by ; followed by the rest of ; followed by ; followed by ; followed by ; close after ; followed by ; followed by the rest of ; and followed by Haman. But the Gemara also cited as counterexamples the words, "And there was evening and there was morning one day," in , as well as , and .
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As an historical note, Cass never really liked this terminology, preferring instead to think of these models as ones of General Financial Equilibrium (GFE) to emphasize the presence of financial assets and the frictions these introduced. The earliest work on market incompleteness goes back to Arrow in the 1950s, Diamond in the mid-‘60’s and a number of related papers in the finance literature between the late 1950s and early ‘70’s (Geanakoplos provides an excellent survey of this literature). The canonical GEI model was formulated by Radner in the early 1970s in a paper which also pointed up one of the fundamental puzzles about models with incomplete markets: the possible loss of dimensionality in the span of the asset payoffs as prices vary. This potential for non-existence of equilibrium (which was formally developed in Hart’s counterexamples to existence of equilibrium) left the literature in limbo for almost a decade, until Cass’s work on existence in economies with purely financial assets pointed the way out.
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In 1966, Chính luận published a translation of the letter from James R. Kipp, a US Navy serviceman stationed in Vietnam, who criticised Vietnamese for their bad habits based on his observations of daily life in Vietnam and refuted the claim of Vietnam's "four thousand years of civilization". Kipp's letter made many Vietnamese readers angry and produced an avalanche of responses from them, which provided counterexamples to refute Kipp's negative stereotypes, highlighted historical events to justify their claim to "four thousand years of civilization," and defended the reputation of Vietnamese women. The Kipp affair represents, in miniature, some of the ways that the American intervention in the Republic of Vietnam shaped South Vietnamese ideas about their cultural identity and political community. Overall, from 1965 to 1969, many articles published in the inner pages of Chính luận show that Vietnamese responded the impact of the American presence on Vietnamese society by categorizing Vietnamese and Americans as separate groups, by expressing an emotional sense of group membership, and by attempting to discursively reify Vietnameseness through definitions of collective identity.
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Rabbi Levi, or some say Rabbi Jonathan, said that a tradition handed down from the Men of the Great Assembly taught that wherever the Bible employs the term "and it was" or "and it came to pass" (, wa-yehi), as it does in , it indicates misfortune, as one can read wa-yehi as wai, hi, "woe, sorrow." Thus the words, "And it came to pass in those days of Amraphel, Arioch, Kenderlaomer, Tidal, Shemeber, Shinab, Backbrai, and Lama the kings of Shinar, Ellasar, Elam, Goiim, Zeboiim, Admah, Bela, and Lasha" in , are followed by the words, "they made war with Bera, Birsta, Nianhazel, and Melchizedek the kings of Sodom, Gomorrah, Zoar, and Salem" in . And the Gemara also cited the instances of followed by ; followed by ; followed by the rest of ; followed by ; 1 Samuel followed by ; followed by ; close after ; followed by ; followed by the rest of ; and followed by Haman. But the Gemara also cited as counterexamples the words, "And there was evening and there was morning one day," in , as well as , and .
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C. Lloyd Morgan's (1852-1936) observations suggested to him that prima facie intelligent behavior in animals is often the result of either instincts or trial and error. For instance, most visitors watching Morgan's dog smoothly lifting a latch with the back of its head (and thereby opening a garden gate and escaping) were convinced that the dog's actions involved thinking. Morgan, however, carefully observed the dog's prior, random, purposeless actions and argued that they involved “continued trial and failure, until a happy effect is reached,” rather than “methodical planning.” E. L. Thorndike (1874 –1949) placed hungry cats and dogs in enclosures “from which they could escape by some simple act, such as pulling at a loop of cord.” Their behavior suggested to him that they did not “possess the power of rationality.” Most books about animal behavior, Thorndike wrote, “do not give us a psychology, but rather a eulogy of animals.” Although Wolfgang Köhler's experiments are often cited as providing support for the animal cognition hypothesis, his book is replete with counterexamples.
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His distinctive historical analysis of scientific methodology based on research programmes suggests: "scientists regard the successful theoretical prediction of stunning novel facts – such as the return of Halley's comet or the gravitational bending of light rays – as what demarcates good scientific theories from pseudo- scientific and degenerate theories, and in spite of all scientific theories being forever confronted by 'an ocean of counterexamples'". Lakatos offers a "novel fallibilist analysis of the development of Newton's celestial dynamics, [his] favourite historical example of his methodology" and argues in light of this historical turn, that his account answers for certain inadequacies in those of Karl Popper and Thomas Kuhn. "Nonetheless, Lakatos did recognize the force of Kuhn's historical criticism of Popper – all important theories have been surrounded by an 'ocean of anomalies', which on a falsificationist view would require the rejection of the theory outright...Lakatos sought to reconcile the rationalism of Popperian falsificationism with what seemed to be its own refutation by history". The boundary between science and pseudoscience is disputed and difficult to determine analytically, even after more than a century of study by philosophers of science and scientists, and despite some basic agreements on the fundamentals of the scientific method.
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