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"counterexample" Definitions
  1. counterexample (to something) an example that provides evidence against an idea or theory

401 Sentences With "counterexample"

How to use counterexample in a sentence? Find typical usage patterns (collocations)/phrases/context for "counterexample" and check conjugation/comparative form for "counterexample". Mastering all the usages of "counterexample" from sentence examples published by news publications.

The counterexample everyone is going to cite isn't really a counterexample.
This latest one is "in between the 'technical' counterexample of the 1990s and a true counterexample," Horowitz said.
The counterexample is that North and South Korea don't get along.
But Michigan's intervention in Flint, an impoverished city, provides a striking counterexample.
It seems almost like a counterexample to the increased focus on results.
Canada thus represents a great counterexample for Glass-Steagall enthusiasts to ponder.
Can you think of a good counterexample to Trump on this front?
It's worth thinking about tax cuts for wealthy Americans as a counterexample here.
But in these same conversations, there's always someone else to think of a counterexample.
Ironically, he used music as a counterexample, a profes­sion where superstar effects were limited.
As a counterexample, he cited another European project to protect a freshwater mussel from pollution.
Blue states generally subsidize red states, so this a notable counterexample to the overall trend.
As a counterexample to Velcade, he points to Treanda, a leukemia drug manufactured by Teva Pharmaceuticals.
As Hitchens's counterexample demonstrates, it is possible to regret the opportunities missed while striving for top grades.
Both Chemaly and Traister see Thurman as a counterexample to the more unrestrainedly angry women they describe.
She described Medicaid expansion as a striking counterexample, where political preferences have sometimes outweighed the desire for resources.
Perhaps the sole counterexample is the sanctions that Congress overwhelmingly voted to impose on Russia for its election hack.
French Broad 13 Rebels ESB: Weirdly enough, this Extra Special Bitter is the counterexample to the DESTIHL six pack theory.
Knebel Mosel Riesling 218.99, $18.963 For anybody put off by German wine labels or terminology, here is a great counterexample.
The great counterexample, which comes up in just about every discussion of the subject, is the growth of Silicon Valley.
Another potential counterexample circulating is from a match between Saudi club Al Ahli Saudi and Barcelona last year in Doha.
Google's Pixelbook and the family of Chromebook devices from traditional PC makers give a great counterexample to the overbearing Windows experience.
Far from being a counterexample to the theory that moderation pays off, Trump's election is, if anything, a testament to its power.
Giving a counterexample, Koch denounces a 2014 French law intended to aid independent bookstores by banning online booksellers from offering free shipping.
And in an environment where there was no other counterexample to Apu on TV, the character made those stereotypes more popular and prominent.
As a counterexample, he offered President George H. W. Bush, whose campaign to expel the Iraqi Army from Kuwait ended successfully, in 1991.
My relative success is used as a counterexample for anyone else's perceived failures, the idea being: If Concepción did it, why can't they?
The Philadelphia 76ers are seen as the great counterexample, a team that embarked on an epic spasm of losing, year after grinding year.
Last year, the sociologists David Voas and Mark Chaves concluded that the United States is no longer a counterexample to the West's secularization.
In the past year, Cook has repeatedly cited Google (both implicitly and outright) as a counterexample to Apple on the issue of consumer privacy.
American Airlines – which entered Chapter 11 bankruptcy protection a few years back with little noticeable effect on its operations – provides an important counterexample here.
For the counterexample, just look to the self-denying Amelia, who mostly exists to be virtuous — and to show us that virtue is boring.
What other cities have passed The upcoming wage cut in St. Louis presented a stark counterexample to the increasing minimum wage seen across the country.
As a counterexample, he compared Asian enrollment at these schools to Caltech, where they don't have affirmative action: There are several flaws with this analysis.
If I could bring up a counterexample, there's a tendency among both parties to reject assessments that they disagree with, even when those assessments are nonpartisan.
For a counterexample of how a time of intense political bitterness can start to tear this country apart, look back exactly half a century to 1968.
Soundcore Spirit X Sports Headphones, available at Amazon, $35.99Many people think you can't buy good sports headphones under $50, but this pair is a perfect counterexample.
"Elon is the counterexample" to the argument that true innovation has stalled, Thiel said, comparing him to former Apple CEO Steve Jobs as a singular great innovator.
Given just the one counterexample of Judge Garland, the point is not that a Supreme Court nominee can't be confirmed with the opposing party in control of the Senate.
One obvious counterexample is the influx of African- Americans and Latinos into many parts of Brooklyn that started in the mid-20th century, both literally and figuratively changing its complexion.
A counterexample is Abe Fortas, whose nomination to be elevated from associate justice to chief justice in the summer of 1968 was killed by a filibuster by Republicans and Southern Democrats.
That makes Salesforce the rare counterexample to the dominance of the "Big Five" — Apple, Alphabet, Microsoft, Amazon, and Facebook — which have added a total of over $2100 trillion in value this year.
Consider Instagram as a counterexample: Users may not like that the app's main feed is no longer chronological, but it does not seem to have negatively affected the app's number of monthly users.
K comes from Charles Smith Wines, a leading Northwest producer that has often seemed to emphasize personality and clever marketing (K Syrah, get it?) over serious winemaking, but this is an excellent counterexample.
Even more than showing you she was a great dancer, Verdon convinced you that she was a good person, a counterexample to Fosse's cynicism, which, when he was wise, he knew was essential.
There's an obvious counterexample, one that Silver hesitates to acknowledge: the N.F.L. The controversy that resulted from Colin Kaepernick's kneeling during the national anthem in 23 has been a political flash point ever since.
When the only contact occurs under duress or stress, it is hard to imagine a counterexample to bias, never mind anyone ending up being a lifelong friend and honoring your civilian father 20 years later.
For a counterexample, you can try listening to the Jaybird X3s, which are very well tuned, but everything inside them sounds condensed like a closed accordion because of how close the sound driver is to the listener's ear.
But if the M.O. was reminiscent of Mr. Lichtenstein's — valorizing the avant-garde by presenting it in a movie palace — it also couldn't help demonstrating, by counterexample, what had happened to the avant-garde at the academy in the meantime.
But the recent restoration of the mansard-roofed A. & S. facade presents a welcome counterexample — thanks to Macy's, the building's owner, and Tishman Speyer, the big commercial real estate firm that partnered with Macy's to redevelop the Fulton Street property.
The strongest counterexample to this is "Climax," his collaboration with 6lack, which features a run where Thug pushes his voice to and past the point of breaking in a way that underscores that passage's emotional weight, but you'd be forgiven if the tape's opening two songs lull you out of rapt attention.
To take just one counterexample, the young former Goldman Sachs investment banker Fabrice Tourre, one of the few Wall Street figures actually found guilty of civil fraud for his role in the kind of complex mortgage deals that contributed to the crisis, is a graduate of France's École Centrale and Stanford University.
But there's a useful counterexample to consider, too: In the 1970s, young leftist radicals were so obsessed with the idea that there were no major differences between Nazi Germany and the postwar German Federal Republic that they made profound errors in judgment and, at times, ended up as terrorists and enemies of democracy.
White House press secretary Sarah Huckabee Sanders on Thursday defended President TrumpDonald John TrumpFacebook releases audit on conservative bias claims Harry Reid: 'Decriminalizing border crossings is not something that should be at the top of the list' Recessions happen when presidents overlook key problems MORE pulling out of denuclearization talks with North Korean leader Kim Jong Un, citing former President Obama's Iran deal as a counterexample.
Commerce Secretary Wilbur Ross criticized the European Union on Wednesday for not negotiating on trade thanks to the United States' steel and aluminum tariffs, citing China as a counterexample, CNBC reports: The big picture: Dutch Foreign Trade and Development Minister Sigrid Kaag said at an economic forum in Paris on Wednesday that EU lawmakers are open to trade talks, but won't "negotiate under threat," per CNBC.
So when I began to see signs of a different pattern emerging there in recent years — a stirring of economic life, in manufacturing, that communities like Twin Falls had never known — the idea gradually took root as I talked to people there and around the West, that a powerful counterexample to the national narrative of rural decline might be unfolding (as described in my front-page article on Tuesday).
If there is a counterexample, one has to enlarge the family of subgroups to a larger family which contains that counterexample.
Conjectures disproven through counterexample are sometimes referred to as false conjectures (cf. the Pólya conjecture and Euler's sum of powers conjecture). In the case of the latter, the first counterexample found for the n=4 case involved numbers in the millions, although it has been subsequently found that the minimal counterexample is actually smaller.
Many important logical ideas are explained in the book. For example, the difference between a counterexample to a lemma (a so-called 'local counterexample') and a counterexample to the specific conjecture under attack (a 'global counterexample' to the Euler characteristic, in this case) is discussed. Lakatos argues for a different kind of textbook, one that uses heuristic style. To the critics that say such a textbook would be too long, he replies: 'The answer to this pedestrian argument is: let us try.
In mathematics, a minimal counterexample is the smallest example which falsifies a claim, and a proof by minimal counterexample is a method of proof which combines the use of a minimal counterexample with the ideas of proof by induction and proof by contradiction.Chartrand, Gary, Albert D. Polimeni, and Ping Zhang. Mathematical Proofs: A Transition to Advanced Mathematics. Boston: Pearson Education, 2013. Print.
As of 2018, no counterexample to the new digraph reconstruction conjecture is known.
The Pólya conjecture was disproved by C. Brian Haselgrove in 1958. He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10361. An explicit counterexample, of n = 906,180,359 was given by R. Sherman Lehman in 1960; the smallest counterexample is n = 906,150,257, found by Minoru Tanaka in 1980. The conjecture fails to hold for most values of n in the region of 906,150,257 ≤ n ≤ 906,488,079.
A counterexample to the statement "all prime numbers are odd numbers" is the number 2, as it is a prime number but is not an odd number. Neither of the numbers 7 or 10 is a counterexample, as neither of them are enough to contradict the statement. In this example, 2 is in fact the only possible counterexample to the statement, even though that alone is enough to contradict the statement. In a similar manner, the statement "All natural numbers are either prime or composite" has the number 1 as a counterexample, as 1 is neither prime nor composite.
The textbook has an erroneous proof that localization preserves injectives, but a counterexample was given in .
Thus Socrates has proposed a counterexample to Callicles' claim, by looking in an area that Callicles perhaps did not expect — groups of people rather than individual persons. Callicles might challenge Socrates' counterexample, arguing perhaps that the common rabble really are better than the nobles, or that even in their large numbers, they still are not stronger. But if Callicles accepts the counterexample, then he must either withdraw his claim, or modify it so that the counterexample no longer applies. For example, he might modify his claim to refer only to individual persons, requiring him to think of the common people as a collection of individuals rather than as a mob.
In constructive mathematics, a statement may be disproved by giving a counterexample, as in classical mathematics. However, it is also possible to give a Brouwerian counterexample to show that the statement is non-constructive. This sort of counterexample shows that the statement implies some principle that is known to be non-constructive. If it can be proved constructively that a statement implies some principle that is not constructively provable, then the statement itself cannot be constructively provable.
However, not all Galois groups have generic polynomials, a counterexample being the cyclic group of order eight.
Richard Kirkham has proposed that it is best to start with a definition of knowledge so strong that giving a counterexample to it is logically impossible. Whether it can be weakened without becoming subject to a counterexample should then be checked. He concludes that there will always be a counterexample to any definition of knowledge in which the believer's evidence does not logically necessitate the belief. Since in most cases the believer's evidence does not necessitate a belief, Kirkham embraces skepticism about knowledge.
By way of counterexample, a password authenticator is not a cryptographic authenticator. See the #Examples section for details.
There are two alternative methods of disproving a conjecture that something is impossible: by counterexample (constructive proof) and by logical contradiction (non-constructive proof). The obvious way to disprove an impossibility conjecture by providing a single counterexample. For example, Euler proposed that at least n different nth powers were necessary to sum to yet another nth power. The conjecture was disproved in 1966, with a counterexample involving a count of only four different 5th powers summing to another fifth power: :275 \+ 845 \+ 1105 \+ 1335 = 1445.
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.
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.
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.
For the special purpose of searching for a counterexample to the Collatz conjecture, this precomputation leads to an even more important acceleration, used by Tomás Oliveira e Silva in his computational confirmations of the Collatz conjecture up to large values of . If, for some given and , the inequality : holds for all , then the first counterexample, if it exists, cannot be modulo . For instance, the first counterexample must be odd because , smaller than ; and it must be 3 mod 4 because , smaller than . For each starting value which is not a counterexample to the Collatz conjecture, there is a for which such an inequality holds, so checking the Collatz conjecture for one starting value is as good as checking an entire congruence class.
Also Bell's theorem: no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. While an impossibility assertion in science can never be absolutely proved, it could be refuted by the observation of a single counterexample. Such a counterexample would require that the assumptions underlying the theory that implied the impossibility be re-examined.
It is also known that a group of cohomological dimension 2 has a 3-dimensional Eilenberg−MacLane space. In 1997, Mladen Bestvina and Noel Brady constructed a group G so that either G is a counterexample to the Eilenberg–Ganea conjecture, or there must be a counterexample to the Whitehead conjecture; in other words, not both conjectures can be true.
In statistics, Hodges' estimator (or the Hodges–Le Cam estimator), named for Joseph Hodges, is a famous counterexample of an estimator which is "superefficient", i.e. it attains smaller asymptotic variance than regular efficient estimators. The existence of such a counterexample is the reason for the introduction of the notion of regular estimators. Hodges' estimator improves upon a regular estimator at a single point.
The result fails for p = 2 as the simple group PSL2(F7) of order 168 is a counterexample. gave a weaker version of this theorem.
As an example of this type of argument, consider the set of all binary trees. We will show that the number of leaves in a full binary tree is one more than the number of interior nodes. Suppose there is a counterexample; then there must exist one with the minimal possible number of interior nodes. This counterexample, C, has n interior nodes and leaves, where .
He introduced the Fourier–Mukai transform in 1981 in a paper on abelian varieties, which also made up his doctoral thesis. His research since has included work on vector bundles on K3 surfaces, three- dimensional Fano varieties, moduli theory, and non-commutative Brill-Noether theory. He also found a new counterexample to Hilbert's 14th problem (the first counterexample was found by Nagata in 1959).
The Whitehead conjecture is equivalent to the conjecture that every sub- presentation of an aspherical presentation is aspherical. In 1997, Mladen Bestvina and Noel Brady constructed a group G so that either G is a counterexample to the Eilenberg–Ganea conjecture, or there must be a counterexample to the Whitehead conjecture; in other words, it is not possible for both conjectures to be true.
After some results were obtained confirming Hilbert's conjecture in special cases and for certain classes of rings (in particular the conjecture was proved unconditionally for n = 1 and n = 2 by Zariski in 1954) then in 1959 Masayoshi Nagata found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants for the action of a linear algebraic group.
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.
If the form of the contradiction is that we can derive a further counterexample D, that is smaller than C in the sense of the working hypothesis of minimality, then this technique is traditionally called proof by infinite descent. In which case, there may be multiple and more complex ways to structure the argument of the proof. The assumption that if there is a counterexample, there is a minimal counterexample, is based on a well-ordering of some kind. The usual ordering on the natural numbers is clearly possible, by the most usual formulation of mathematical induction; but the scope of the method can include well-ordered induction of any kind.
A variation of the conjecture asserting that x, y, z (instead of A, B, C) must have a common prime factor is not true. A counterexample is 27^4 +162^3 = 9^7, in which 4, 3, and 7 have no common prime factor. (In fact, the maximum common prime factor of the exponents that is valid is 2; a common factor greater than 2 would be a counterexample to Fermat's Last Theorem.) The conjecture is not valid over the larger domain of Gaussian integers. After a prize of $50 was offered for a counterexample, Fred W. Helenius provided (-2+i)^3 + (-2-i)^3 = (1+i)^4.
In the 1899 German translation of the same textbook, he provided this surface as a counterexample to Serret's condition. At the point (0,0,0), Serret's conditions are met, but this point is a saddle point, not a local maximum. A related condition to Serret's was also criticized by , who used Peano's surface as a counterexample to it in an 1890 publication, credited to Peano. See in particular pp. 545–546.
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.
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.
Connelly has authored or co-authored several articles on mathematics, including Conjectures and open questions in rigidity; A flexible sphere; and A counterexample to the rigidity conjecture for polyhedra.
In logic (especially in its applications to mathematics and philosophy), a counterexample is an exception to a proposed general rule or law, and often appears as an example which disproves a universal statement. For example, the statement "all students are lazy" is a universal statement which makes the claim that a certain property (laziness) holds for all students. Thus, any student who is not lazy (e.g., hard-working) would constitute a counterexample to that statement.
A counterexample hence is a specific instance of the falsity of a universal quantification (a "for all" statement). In mathematics, the term "counterexample" is also used (by a slight abuse) to refer to examples which illustrate the necessity of the full hypothesis of a theorem. This is most often done by considering a case where a part of the hypothesis is not satisfied and the conclusion of the theorem does not hold.
Moreover, C must be nontrivial, because the trivial tree has and and is therefore not a counterexample. C therefore has at least one leaf whose parent node is an interior node. Delete this leaf and its parent from the tree, promoting the leaf's sibling node to the position formerly occupied by its parent. This reduces both n and by 1, so the new tree also has and is therefore a smaller counterexample.
Nevertheless, many disagree, claiming that (in this situation) telling the truth would result in needless death, would therefore be immoral, and that this scenario thus provides a counterexample contradicting SRU.
However, Joseph Horton provided a counterexample on 96 vertices, the Horton graph.Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 240, 1976.
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.
Higher-level mathematical papers may include variations on true/false, where the candidate is given a statement and asked to verify its validity by direct proof or stating a counterexample.
228 The computational technique underlying these lower bounds depends on some key results of Klee that make it possible to show that the smallest counterexample must be divisible by squares of the primes dividing its totient value. Klee's results imply that 8 and Fermat primes (primes of the form 2k + 1) excluding 3 do not divide the smallest counterexample. Consequently, proving the conjecture is equivalent to proving that the conjecture holds for all integers congruent to 4 (mod 8).
The octahedron is one of the most well-known examples of a spindle. Unfortunately, the Hirsch conjecture is not true in all cases, as shown by Francisco Santos in 2011. Santos' explicit construction of a counterexample comes both from the fact that the conjecture may be relaxed to only consider simple polytopes, and from equivalence between the Hirsch and d-step conjectures. In particular, Santos produces his counterexample by examining a particular class of polytopes called spindles.
The solution to that problem led to her well-known 1993 work in which she constructed a smooth counterexample to the Seifert conjecture. She has since continued to work in dynamical systems.
A simplified presentation of Kuranishi's proof is due to Sidney Webster. For n = 2 (i.e., real dimension 3), Nirenberg published a counterexample. The local embedding problem remains open in real dimension 5.
This publication marks the first known appearance of the Petersen graph in the mathematical literature, 12 years before Julius Petersen's use of the same graph as a counterexample to an edge coloring problem.
68, No.6, June 1980. Variations considered by Tamer Basar Basar, Tamer. "Variations on the theme of the Witsenhausen counterexample". 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9–11, 2008.
In mathematics, the Deuring–Heilbronn phenomenon, discovered by and , states that a counterexample to the generalized Riemann hypothesis for one Dirichlet L-function affects the location of the zeros of other Dirichlet L-functions.
This follows because flyping preserves writhe. This was proved earlier by Murasugi and Thistlethwaite. It also follows from Greene's work. For non- alternating knots this conjecture is not true; the Perko pair is a counterexample.
Thus, Hedetniemi's conjecture amounts to the assertion that tensor products cannot be colored with an unexpectedly small number of colors. A counterexample to the conjecture was discovered by (see ), thus disproving the conjecture in general.
Perron's counterexample shows that a negative largest Lyapunov exponent does not, in general, indicate stability, and that a positive largest Lyapunov exponent does not, in general, indicate chaos. Therefore, time-varying linearization requires additional justification.
Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem (i.e., if they did appear, one could make a smaller counter-example). Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps.
Formal mathematics is based on provable truth. In mathematics, any number of cases supporting a conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 1012 (over a trillion).
1043, pp. 238--266, Springer-Verlag, 1996. . ;Expressing important properties in formal verification :There are two main types of properties that can be expressed using linear temporal logic: safety properties usually state that something bad never happens (G eg\varphi), while liveness properties state that something good keeps happening (GF\psi or G(\varphi \rightarrowF\psi)). More generally, safety properties are those for which every counterexample has a finite prefix such that, however it is extended to an infinite path, it is still a counterexample.
The minimal counterexample method has been much used in the classification of finite simple groups. The Feit–Thompson theorem, that finite simple groups that are not cyclic groups have even order, was based on the hypothesis of some, and therefore some minimal, simple group G of odd order. Every proper subgroup of G can be assumed a solvable group, meaning that much theory of such subgroups could be applied. Euclid's proof of the fundamental theorem of arithmetic is a simple proof which uses a minimal counterexample.
It has since been extended to other small values of , and the Zarankiewicz conjecture is known to be true for the complete bipartite graphs with . The conjecture is also known to be true for , , and . If a counterexample exists, that is, a graph requiring fewer crossings than the Zarankiewicz bound, then in the smallest counterexample both and must be odd. For each fixed choice of , the truth of the conjecture for all can be verified by testing only a finite number of choices of .
If K1,2,2,2 could be shown not to have any planar covers, this would complete a proof of the conjecture. On the other hand, if the conjecture is false, K1,2,2,2 would necessarily be its smallest counterexample., pp.
In mathematics, more specifically in the field of analytic number theory, a Siegel zero, named after Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeroes of Dirichlet L-function.
Thus, the theorem is true!! > (punctuation in original) Care should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample.
The counterexample lies at the intersection of control theory and information theory. Due to its hardness, the problem of finding the optimal control law has also received attention from the theoretical computer science community. The importance of the problem was reflected upon in the 47th IEEE Conference on Decision and Control (CDC) 2008, Cancun, Mexico, where an entire session was dedicated to understanding the counterexample 40 years after it was first formulated. The problem is of conceptual significance in decentralized control because it shows that it is important for the controllers to communicateMitterrand and Sahai.
In 1997, the Oxford mathematician Peter M. Neumann proved the theorem that there is no ruler-and- compass construction for the general solution of Alhazen's problem (although in 1965 Elkin had already provided a counterexample to Euclidean construction).
In a certain sense, the 1/z counterexample is universal: For every analytic function that has no antiderivative on its domain, the reason for this is that 1/z itself does not have an antiderivative on ℂ − {0}.
And, similarly, why is it so very difficult for permissivists to devise a counterexample that can convince Uniqueness defenders to abandon it? Kopec, M. & Titlebaum, M. G.(2016). "The Uniqueness Thesis". Philosophy Compass, 11(4), pp. 189-200.
The answer is no. A counterexample occurs with R equal to the local ring of the curve at the origin. Thus the Quillen-Suslin theorem could never be proved by a simple induction on the number of variables.
This is essentially the same as the previous two examples. The quotient does exist as a scheme if every orbit is contained in an affine open subscheme; the counterexample above shows that this technical condition cannot be dropped.
Studying properties of this form, Bogomolov erroneously concluded that compact hyperkaehler manifolds do not exist, with the exception of K3 surfaces, tori, and their products. Almost four years passed since this publication before Akira Fujiki found a counterexample.
It is a necessary part of the conjecture that the cubes in the tiling all be congruent to each other, for if similar but not congruent cubes are allowed then the Pythagorean tiling would form a trivial counterexample in two dimensions.
The first of these two spindles has 48 facets and 322 vertices, while the spindle that actually disproves the conjecture has 86 facets and is 43-dimensional. This counterexample does not disprove the polynomial Hirsch conjecture, which remains an open problem.
His smallest counterexample was ::. A particular case of Elkies' solutions can be reduced to the identity :: where ::. This is an elliptic curve with a rational point at . From this initial rational point, one can compute an infinite collection of others.
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.
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 ,.
"You should rather play hide-and-seek against someone who cannot predict where you hide than against someone who can. Causal Decision Theory denies this. So Causal Decision Theory is false." Another recent counterexample is the "Psychopath Button":Greaves, Hilary.
Her results include the first known planar graph that requires five colors for list coloring, and a counterexample to a related conjecture that list coloring of planar graphs requires at most one more color than graph coloring for the same graphs.
The defiant and hopeful spring metaphor can be interpreted as a counterexample to the nature poetry of Peter Huchel, whose works take a drearier and elegiac tone.Jürgen Haupt: Natur und Lyrik. Naturbeziehungen im 20. Jahrhundert. Metzler, Stuttgart 1983, , pp. 190–192.
He invented a base 13 function as a counterexample to the converse of the intermediate value theorem: the function takes on every real value in each interval on the real line, so it has a Darboux property but is not continuous.
A second related conjecture, made by Furtwängler in 1936, instead relaxes the condition that the cubes form a tiling. Furtwängler asked whether a system of cubes centered on lattice points, forming a k-fold covering of space (that is, all but a measure-zero subset of the points in the space must be interior to exactly k cubes) must necessarily have two cubes meeting face to face. Furtwängler's conjecture is true for two- and three- dimensional space, but Hajós found a four-dimensional counterexample in 1938. characterized the combinations of k and the dimension n that permit a counterexample.
For , every -critical graph (that is, every odd cycle) can be generated as a -constructible graph such that all of the graphs formed in its construction are also -critical. For , this is not true: a graph found by as a counterexample to Hajós's conjecture that -chromatic graphs contain a subdivision of , also serves as a counterexample to this problem. Subsequently, -critical but not -constructible graphs solely through -critical graphs were found for all . For , one such example is the graph obtained from the dodecahedron graph by adding a new edge between each pair of antipodal vertices.
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.
Called in practical contexts "lazy evaluation". In implementations this "name" takes the form of a pointer, with the redex represented by a thunk. Applicative order is not a normalising strategy. The usual counterexample is as follows: define `Ω = ωω` where `ω = λx.xx`.
Then there is a -set which is a choice set for R , that is: # . # . A proof runs as follows: suppose for contradiction is a minimal counterexample, and fix , , and a good universal set for the -subsets of . Easily, must be a limit ordinal.
Beal is self-taught in number theory in mathematics. In 1993, he publicly stated a new mathematical hypothesis that implies Fermat's Last Theorem as a corollary. His hypothesis has become known as the Beal Conjecture. No counterexample has been found to the conjecture.
As a counterexample, a program in rural Virginia engages in culturally relevant teaching by explicitly avoiding high technology solutions and using locally relevant activities to guide learning. Specifically, fixing and using tools was used to teach engineering to rural middle school youth.
In the mathematical field of graph theory, the Errera graph is a graph with 17 vertices and 45 edges. Alfred Errera published it in 1921 as a counterexample to Kempe's erroneous proof of the four color theorem; it was named after Errera by .
For degree 6 surfaces in P3, showed that 65 is the maximum number of double points possible. The Barth sextic is a counterexample to an incorrect claim by Francesco Severi in 1946 that 52 is the maximum number of double points possible.
Various equivalent formulations of the problem had been given, such as the d-step conjecture, which states that the diameter of any 2d-facet polytope in d-dimensional Euclidean space is no more than d; Santos Leal's counterexample also disproves this conjecture., p. 84..
Two 7-Con quadrilaterals. Defining almost congruent triangles gives a binary relation on the set of triangles. This relation is clearly not reflexive, but it is symmetric. It is not transitive: As a counterexample, consider the three triangles with side lengths (8;12;18), (12;18;27), and (18;27;40.5).
A casual verifier of the counterexample may not think to change the colors of these regions, so that the counterexample will appear as though it is valid. Perhaps one effect underlying this common misconception is the fact that the color restriction is not transitive: a region only has to be colored differently from regions it touches directly, not regions touching regions that it touches. If this were the restriction, planar graphs would require arbitrarily large numbers of colors. Other false disproofs violate the assumptions of the theorem, such as using a region that consists of multiple disconnected parts, or disallowing regions of the same color from touching at a point.
Suppose that a mathematician is studying geometry and shapes, and she wishes to prove certain theorems about them. She conjectures that "All rectangles are squares", and she is interested in knowing whether this statement is true or false. In this case, she can either attempt to prove the truth of the statement using deductive reasoning, or she can attempt to find a counterexample of the statement if she suspects it to be false. In the latter case, a counterexample would be a rectangle that is not a square, such as a rectangle with two sides of length 5 and two sides of length 7.
Charles Akemann and Nik Weaver showed in 2003 that the statement "there exists a counterexample to Naimark's problem which is generated by ℵ1, elements" is independent of ZFC. Miroslav Bačák and Petr Hájek proved in 2008 that the statement "every Asplund space of density character ω1 has a renorming with the Mazur intersection property" is independent of ZFC. The result is shown using Martin's maximum axiom, while Mar Jiménez and José Pedro Moreno (1997) had presented a counterexample assuming CH. As shown by Ilijas Farah and N. Christopher Phillips and Nik Weaver, the existence of outer automorphisms of the Calkin algebra depends on set theoretic assumptions beyond ZFC.
The overall design is simple with comparatively little ornamentation, although a significant counterexample is the building's back wall, in which was placed a prominent petroglyph known as the Independence Slab.Owen, Lorrie K., ed. Dictionary of Ohio Historic Places. Vol. 1. St. Clair Shores: Somerset, 1999, 236.
Unprotected bore evacuators damaged by bullets have caused considerable problems in past conflicts, but up-armouring solved this problem. Bore evacuators are a common feature of most modern tanks. The Russian T-14 Armata is a counterexample, lacking a bore evacuator because its turret is uncrewed.
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.
In mathematics Antoine's necklace is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected. It also serves as a counterexample to the claim that all Cantor spaces are ambiently homeomorphic to each other. It was discovered by .
2, 289–319.Abbas Bahri. A counterexample to the second inequality of Corollary (19.10) in the monograph "Ricci Flow and the Poincare Conjecture" by J.Morgan and G.Tian. The error, originating in the incorrect calculation of a geometric evolution equation, was thereafter fixed by Morgan and Tian.
The Ornstein isomorphism theorem states that two Bernoulli schemes with the same entropy are isomorphic.Donald Ornstein, "Bernoulli shifts with the same entropy are isomorphic", Advances in Math. 4 (1970), pp.337–352 The result is sharp,Christopher Hoffman, "A K counterexample machine", Trans. Amer. Math. Soc.
Some of the finite structures considered in graph theory have names, sometimes inspired by the graph's topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many different contexts.
Example 1 One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example: :If Bill Gates owns Fort Knox, then Bill Gates is rich. :Bill Gates is rich. :Therefore, Bill Gates owns Fort Knox.
Substituting into the identity and removing common factors gives the numerical example cited above. In 1988, Roger Frye found the smallest possible counterexample :: for by a direct computer search using techniques suggested by Elkies. This solution is the only one with values of the variables below 1,000,000.
János Bolyai, Vol. 6, North-Holland, Amsterdam, 1973. This conjecture, if true, would have proven the Köthe conjecture through the equivalent statements above, however a counterexample was produced by Agata Smoktunowicz.Smoktunowicz, Agata. Polynomial rings over nil rings need not be nil J. Algebra 233 (2000), no.
A measure \mu is called a s-finite measure if it is the sum of at most countably many finite measures. Every σ-finite measure is s-finite, the converse is not true. For a proof and counterexample see s-finite measure#Relation to σ-finite measures.
One possible attack on the cycle double cover problem would be to show that there cannot exist a minimum counterexample, by proving that any graph contains a reducible configuration, a subgraph that can be replaced by a smaller subgraph in a way that would preserve the existence or nonexistence of a cycle double cover. For instance, if a cubic graph contains a triangle, a Δ-Y transform will replace the triangle by a single vertex; any cycle double cover of the smaller graph can be extended back to a cycle double cover of the original cubic graph. Therefore, a minimal counterexample to the cycle double cover conjecture must be a triangle-free graph, ruling out some snarks such as Tietze's graph which contain triangles. Through computer searches, it is known that every cycle of length 11 or less in a cubic graph forms a reducible configuration, and therefore that any minimal counterexample to the cycle double cover conjecture must have girth at least 12.. Unfortunately, it is not possible to prove the cycle double cover conjecture using a finite set of reducible configurations.
The philosopher and logician Vann McGee has argued that modus ponens can fail to be valid when the consequent is itself a conditional sentence.Vann McGee (1985). "A Counterexample to Modus Ponens", The Journal of Philosophy 82, 462–471. Here is an example: :Either Shakespeare or Hobbes wrote Hamlet.
The proof later of the Poincaré Conjecture simplified this to "always yes". Together, the two algorithms provided an algorithm that would find a counterexample to the Poincaré Conjecture, if one existed. In 2002, Martin Dunwoody posted a claimed proof of the Poincaré Conjecture. Rourke identified its fatal flaw.
However, it must have seemed at the time that they did. It only took one counterexample (Mercury's orbit) to prove that there was something wrong with his theory. This is typical of inductive logic. All of the observations that seem to validate the theory, do not prove its truth.
Any deviation from the above assumptions—a nonlinear state equation, a non-quadratic objective function, noise in the multiplicative parameters of the model, or decentralization of control—causes the certainty equivalence property not to hold. For example, its failure to hold for decentralized control was demonstrated in Witsenhausen's counterexample.
If the four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors. The proof showed that such a minimal counterexample cannot exist, through the use of two technical concepts:; ; # An unavoidable set is a set of configurations such that every map that satisfies some necessary conditions for being a minimal non-4-colorable triangulation (such as having minimum degree 5) must have at least one configuration from this set. # A reducible configuration is an arrangement of countries that cannot occur in a minimal counterexample. If a map contains a reducible configuration, the map can be reduced to a smaller map.
See for proofs showed that Barnette's conjecture is equivalent to a superficially stronger statement, that for every two edges e and f on the same face of a bipartite cubic polyhedron, there exists a Hamiltonian cycle that contains e but does not contain f. Clearly, if this statement is true, then every bipartite cubic polyhedron contains a Hamiltonian cycle: just choose e and f arbitrarily. In the other directions, Kelmans showed that a counterexample could be transformed into a counterexample to the original Barnette conjecture. Barnette's conjecture is also equivalent to the statement that the vertices of the dual of every cubic bipartite polyhedral graph can be partitioned into two subsets whose induced subgraphs are trees.
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.
The introduction usually is not included in the exposition repeat: the Pathétique is a possible counterexample. Much later, Chopin's Piano Sonata No. 2 (Op. 35) is a clear example where the introduction is also included. On occasion, the material of introduction reappears in its original tempo later in the movement.
The continued use of the Somali shilling as currency despite the lack of a functioning central government capable of raising taxes or a central bank to issue it has been cited as a counterargument to Chartalism. Bitcoin, a cryptocurrency not originated by a state, has also been cited as a counterexample.
Finally in 1867 Lindelöf systematically analysed all the earlier flawed proofs and was able to exhibit a specific counterexample where mixed derivatives failed to be equal. Six years after that, Schwarz succeeded in giving the first rigorous proof. Dini later contributed by finding more general conditions than those of Schwarz.
When a distinct order between two rules is required, a derivation must be shown. The derivation must consist of a correct application of rule ordering that proves the phonetic representation to be possible as well as a counterexample that proves, given the opposite ordering, an incorrect phonetic representation will be generated.
For liveness properties, on the other hand, every finite prefix of a counterexample can be extended to an infinite path that satisfies the formula. ;Specification language :One of the applications of linear temporal logic is the specification of preferences in the Planning Domain Definition Language for the purpose of preference-based planning.
As a counterexample take any neighbourhood of the particular point of an infinite particular point space. The neighbourhood itself may be compact but is not relatively compact because its closure is the whole non-compact space. Every compact subset of a (possibly non-Hausdorff) topological vector space is complete and relatively compact.
Being the 31st triangular number, 496 is the smallest counterexample to the hypothesis that one more than an even triangular number is a prime number. It is the largest happy number less than 500. There is no solution to the equation φ(x) = 496, making 496 a nontotient. E8 has real dimension 496.
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.
More specifically. in trying to prove a proposition P, one first assumes by contradiction that it is false, and that therefore there must be at least one counterexample. With respect to some idea of size (which may need to be chosen carefully), one then concludes that there is such a counterexample C that is minimal. In regard to the argument, C is generally something quite hypothetical (since the truth of P excludes the possibility of C), but it may be possible to argue that if C existed, then it would have some definite properties which, after applying some reasoning similar to that in an inductive proof, would lead to a contradiction, thereby showing that the proposition P is indeed true.
When F is the algebraic closure of a finite field, the result follows from Hilbert's Nullstellensatz. The Ax–Grothendieck theorem for complex numbers can therefore be proven by showing that a counterexample over C would translate into a counterexample in some algebraic extension of a finite field. This method of proof is noteworthy in that it is an example of the idea that finitistic algebraic relations in fields of characteristic 0 translate into algebraic relations over finite fields with large characteristic. Thus, one can use the arithmetic of finite fields to prove a statement about C even though there is no homomorphism from any finite field to C. The proof thus uses model- theoretic principles to prove an elementary statement about polynomials.
The interpretation of negation is different in intuitionist logic than in classical logic. In classical logic, the negation of a statement asserts that the statement is false; to an intuitionist, it means the statement is refutable (i.e., that there is a counterexample). There is thus an asymmetry between a positive and negative statement in intuitionism.
Applicative order is not a normalising strategy. The usual counterexample is as follows: define \Omega = \omega\omega where \omega = \lambda x . xx. This entire expression contains only one redex, namely the whole expression; its reduct is again \Omega. Since this is the only available reduction, \Omega has no normal form (under any evaluation strategy).
In the mathematical field of graph theory, the Fritsch graph is a planar graph with 9 vertices and 21 edges. It was obtained by FritschFritsch, R. and Fritsch, G. The Four-Color Theorem. New York: Springer-Verlag, 1998 as a minimal sized counterexample to the Alfred Kempe's attempt to prove the four- color theorem.
After Jason Cantarella suggested a possible counterexample, Hugh Nelson Howards weakened the conjecture to apply to any three planar curves that are not all circles. On the other hand, although there are infinitely many Brunnian links with three links, the Borromean rings are the only one that can be formed from three convex curves.
As a weak counterexample, suppose θ(x) is some decidable predicate of a natural number such that it is not known whether any x satisfies θ. For example, θ may say that x is a formal proof of some mathematical conjecture whose provability is not known. Let φ the formula . Then is trivially provable.
In TACAS, 2004 There exist fault localization methods to find the bug location as these model checking tools return a long counter example trace and it is hard to find the exact faulty location.Thomas Ball, Mayur Naik, and Sriram K. Rajamani. "From symptom to cause: localizing errors in counterexample traces". ACM SIGPLAN Notices, 2003.
It suffices to check that each prime gap starting at p is smaller than 2 \sqrt p. A table of maximal prime gaps shows that the conjecture holds to 4×1018.Jens Kruse Andersen, Maximal Prime Gaps. A counterexample near 1018 would require a prime gap fifty million times the size of the average gap.
As far as older regional styles, the best-known ones are from Jiangnan and Sichuan. Some have relocated several times, like the Zhucheng/Mei'an style. Major living Jiangnan lineages include Guangling, Zhe (Xumen) and Wumen. Others, like a "Jinling" style centered in Nanjing, don't really seem to exist anymore — though there's always an old master or two as counterexample.
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.
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.
Most unirational complex varieties of dimension 3 or larger are not rational. In characteristic p > 0 found examples of unirational surfaces (Zariski surfaces) that are not rational. At one time it was unclear whether a complex surface such that q and P1 both vanish is rational, but a counterexample (an Enriques surface) was found by Federigo Enriques.
He does research on the theory of functions of several complex variables with emphasis on their geometry and dynamics. With Nessim Sibony he constructed a Fatou-Julia theory in two complex variables. He is also known for constructing a counterexample in several complex variables in 1976. Fornæss is a fellow of the Norwegian Academy of Science and Letters.
Nagata arrived at the conjecture via work on the 14th problem of Hilbert, which asks whether the invariant ring of a linear group action on the polynomial ring over some field is finitely generated. Nagata published the conjecture in a 1959 paper in the American Journal of Mathematics, in which he presented a counterexample to Hilbert's 14th problem.
Again, a double cover of the resulting graph may be extended in a straightforward way to a double cover of the original graph: every cycle of the split off graph corresponds either to a cycle of the original graph, or to a pair of cycles meeting at v. Thus, every minimal counterexample must be cubic. But if a cubic graph can have its edges 3-colored (say with the colors red, blue, and green), then the subgraph of red and blue edges, the subgraph of blue and green edges, and the subgraph of red and green edges each form a collection of disjoint cycles that together cover all edges of the graph twice. Therefore, every minimal counterexample must be a non-3-edge-colorable bridgeless cubic graph, that is, a snark.
Schwarz devised his construction as a counterexample to the erroneous definition in J. A. Serret's book Cours de calcul differentiel et integral, second volume, page 296 of the first edition or page 298 of the second edition, in which it is said: In English Independently of Schwarz, Giuseppe Peano found the same counterexample while a student of his teacher Angelo Genocchi, who already knew about the difficulty on defining surface area from his communication with Schwarz. Genocchi informed Charles Hermite, who had been using Serret's erroneous definition in his course. After requesting details to Schwarz, Hermite revised his course and published the example in the second edition of his lecture notes (1883). The original note from Schwarz was not published until the second edition of his collected works in 1890.
Every apex graph has chromatic number at most five: the underlying planar graph requires at most four colors by the four color theorem, and the remaining vertex needs at most one additional color. used this fact in their proof of the case k = 6 of the Hadwiger conjecture, the statement that every 6-chromatic graph has the complete graph K6 as a minor: they showed that any minimal counterexample to the conjecture would have to be an apex graph, but since there are no 6-chromatic apex graphs such a counterexample cannot exist. conjectured that every 6-vertex-connected graph that does not have as a minor must be an apex graph. If this were proved, the Robertson–Seymour–Thomas result on the Hadwiger conjecture would be an immediate consequence.
Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. Joseph Liouville, for instance, proved the existence of transcendental numbers by constructing an explicit example. It can also be used to construct a counterexample to disprove a proposition that all elements have a certain property.
Because the earliest lower bound known for packings of tetrahedra was less than that of spheres, it was suggested that the regular tetrahedra might be a counterexample to Ulam's conjecture that the optimal density for packing congruent spheres is smaller than that for any other convex body. However, the more recent results have shown that this is not the case.
The Demazure character formula was introduced by . Victor Kac pointed out that Demazure's proof has a serious gap, as it depends on , which is false; see for Kac's counterexample. gave a proof of Demazure's character formula using the work on the geometry of Schubert varieties by and . gave a proof for sufficiently large dominant highest weight modules using Lie algebra techniques.
Richard T. Cox showed that Bayesian updating follows from several axioms, including two functional equations and a hypothesis of differentiability. The assumption of differentiability or even continuity is controversial; Halpern found a counterexample based on his observation that the Boolean algebra of statements may be finite. Other axiomatizations have been suggested by various authors with the purpose of making the theory more rigorous.
Idi Amin, who was Kakwa, recruited the Kawa and Nubians into his army, to kill the Acholi and Lango. Jonathan Owens argues that Nubi constitutes a major counterexample to Derek Bickerton's theories of creole language formation, showing "no more than a chance resemblance to Bickerton's universal creole features" despite fulfilling perfectly the historical conditions expected to lead to such features.
Francisco (Paco) Santos Leal (born May 28, 1968) is a Spanish mathematician at the University of Cantabria, known for finding a counterexample to the Hirsch conjecture in polyhedral combinatorics.. In 2015 he won the Fulkerson Prize for this research.2015 Fulkerson Prize citation, retrieved 2015-07-18. Santos francisco Santos was born in Valladolid, Spain.Curriculum vitae , retrieved 2015-07-18.
Graham's Hierarchy of Disagreement In reasoning and argument mapping, a counterargument is an objection to an objection. A counterargument can be used to rebut an objection to a premise, a main contention or a lemma. Synonyms of counterargument may include rebuttal, reply, counterstatement, counterreason, comeback and response. The attempt to rebut an argument may involve generating a counterargument or finding a counterexample.
This is the smallest T1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations. The real line can also be given the lower limit topology.
This grows extremely slowly, and suggests that the computer calculations do not provide much evidence for Vandiver's conjecture: for example, the probability argument (combined with the calculations for small primes) suggests that one should only expect about 1 counterexample in the first 10100 primes, suggesting that it is unlikely any counterexample will be found by further brute force searches even if there are an infinite number of exceptions. gave conjectural calculations of the class numbers of real cyclotomic fields for primes up to 10000, which strongly suggest that the class numbers are not randomly distributed mod p. They tend to be quite small and are often just 1. For example, assuming the generalized Riemann hypothesis, the class number of the real cyclotomic field for the prime p is 1 for p<163, and divisible by 4 for p=163.
For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number n for which the Mertens function M(n) equals or exceeds the square root of n) is known: all numbers less than 1014 have the Mertens property, and the smallest number that does not have this property is only known to be less than the exponential of 1.59 × 1040, which is approximately 10 to the power 4.3 × 1039. Since the number of particles in the universe is generally considered less than 10 to the power 100 (a googol), there is no hope to find an explicit counterexample by exhaustive search. The word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, group theory (see mathematical theory).
Their "distaste for bishops who had collaborated". with Rome came out of their broader view of the empire. Held out as a counterexample to the traditors was the venerated Saint Vincent of Saragossa who preferred to suffer martyrdom rather than agree to consign Scripture to the fire. He is depicted in religious paintings holding the book whose preservation he preferred to his own life.
Spinrad also pointed out that Moreau published several mathematical papers. In particular introduced Moreau's necklace-counting function, and described a variation of this that he credited to Moreau. pointed out a counterexample to a lemma used by Adrien-Marie Legendre in his attempt to prove Dirichlet's theorem on arithmetic progressions. describes Moreau's analysis of the mathematical game "red and black" invented by Arnous de Rivière.
Computer Aided Verification (CAV) is an annual academic conference on the theory and practice of computer aided formal analysis of software and hardware systems. It is one of the highest ranked conferences in computer science. For example, several important model checking techniques were published in CAV, such as Counterexample-Guided Abstraction Refinement and partial order reduction methods. The first CAV was held in 1989 in Grenoble, France.
The derivatives of the multipoles are not directly accessible to us and would require observations over time and space intervals on cosmological scales. In 1999 John Wainwright, M. J. Hancock and Claes Uggla show a counterexample in the non-tilted perfect fluid case. Thus an almost isotropic cosmic microwave temperature does not imply an almost isotropic universe. Using the methods of Wainwright et al.
When arguing against the unrestricted version of representationalism people will often bring up phenomenal mental states that appear to lack intentional content. The unrestricted version seeks to account for all phenomenal states. Thus, for it to be true, all states with phenomenal character must have intentional content to which that character is reduced. Phenomenal states without intentional content therefore serve as a counterexample to the unrestricted version.
In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the two ordinal spaces [0,\omega_1] and [0,\omega], where \omega is the first infinite ordinal and \omega_1 the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point \infty = (\omega_1,\omega).
In other words, in this example, transparency will encourage trust and good faith, that the effective business will not conceal what is in the interest of its audience. For the sake of counterexample, there may be a time when censorship is the more effective business practice: take the case of trade secrets, when a design method or management tactic is not openly revealed in the name of competitive advantage; or when terms of agreement/use that a business may have with a service provider forbids transparency. In the latter counterexample, a business may use social media to advertise, but the social media service provider may limit the conduct of its users. Here, if the business considers social media to be a valuable service to achieve its advertising, it may have to censor its product or service to preserve its agreement with the social media provider.
The Berkeley Lazy Abstraction Software verification Tool (BLAST) is a software model checking tool for C programs. The task addressed by BLAST is the need to check whether software satisfies the behavioral requirements of its associated interfaces. BLAST employs counterexample-driven automatic abstraction refinement to construct an abstract model that is then model-checked for safety properties. The abstraction is constructed on the fly, and only to the requested precision.
Painters arrange "conditions" in the paint/canvas medium, and dancers arrange the "conditions" of their bodily medium, for example. According to Beardsley’s first disjunct, art has an intended aesthetic function, but not all artworks succeed in producing aesthetic experiences. The second disjunct allows for artworks that were intended to have this capacity, but failed at it (bad art). Marcel Duchamp's Fountain is the paradigmatic counterexample to aesthetic definitions of art.
Walter Burley, a medieval scholastic philosopher, introduced donkey sentences in the context of the theory of suppositio, the medieval equivalent of reference theory. Peter Geach reintroduced donkey sentences as a counterexample to Richard Montague's proposal for a generalized formal representation of quantification in natural language (see Geach 1962). His example was reused by David Lewis (1975), Gareth Evans (1977) and many others, and is still quoted in recent publications.
In the mathematical field of graph theory, the Horton graph or Horton 96-graph is a 3-regular graph with 96 vertices and 144 edges discovered by Joseph Horton. Published by Bondy and Murty in 1976, it provides a counterexample to the Tutte conjecture that every cubic 3-connected bipartite graph is Hamiltonian.Tutte, W. T. "On the 2-Factors of Bicubic Graphs." Discrete Math. 1, 203-208, 1971/72.
In 1987, he proved that an elliptic curve over the rational numbers is supersingular at infinitely many primes. In 1988, he found a counterexample to Euler's sum of powers conjecture for fourth powers. His work on these and other problems won him recognition and a position as an associate professor at Harvard in 1990. In 1993, he was made a full, tenured professor at the age of 26.
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.
For a published proof or counterexample, banker Andrew Beal initially offered a prize of US $5,000 in 1997, raising it to $50,000 over ten years, but has since raised it to US $1,000,000. The American Mathematical Society (AMS) holds the $1 million prize in a trust until the Beal conjecture is solved. It is supervised by the Beal Prize Committee (BPC), which is appointed by the AMS president.
She attended the University of Texas, completing her B.A. in 1944 after just three years before moving into the graduate program in mathematics under Robert Lee Moore. Her graduate thesis presented a counterexample to one of "Moore's axioms". She completed her Ph.D. in 1949. During her time as an undergraduate, she was a member of the Phi Mu Women's Fraternity, and was elected to the Phi Beta Kappa society.
The vector space counterexample of the theorem above is G=F(Ω), for any set Ω with at least ℵ2 elements. This counterexample has been modified subsequently by Ploščica and Tůma to a direct semilattice construction. For a (∨,0)-semilattice, the larger semilattice R(S) is the (∨,0)-semilattice freely generated by new elements t(a,b,c), for a, b, c in S such that c ≤ a ∨ b, subjected to the only relations c=t(a,b,c) ∨ t(b,a,c) and t(a,b,c) ≤ a. Iterating this construction gives the free distributive extension D(S)=\bigcup(R^n(S)\mid n<\omega) of S. Now, for a set Ω, let L(Ω) be the (∨,0)-semilattice defined by generators 1 and ai,x, for i<2 and x in Ω, and relations a0,x ∨ a1,x=1, for any x in Ω. Finally, put G(Ω)=D(L(Ω)).
A related conjecture of Barnette states that every cubic polyhedral graph in which all faces have six or fewer edges is Hamiltonian. Computational experiments have shown that, if a counterexample exists, it would have to have more than 177 vertices.. The intersection of these two conjectures would be that every bipartite cubic polyhedral graph in which all faces have four or six edges is Hamiltonian. This was proved to be true by .
This theory of deductive reasoning – also known as term logic – was developed by Aristotle, but was superseded by propositional (sentential) logic and predicate logic. Deductive reasoning can be contrasted with inductive reasoning, in regards to validity and soundness. In cases of inductive reasoning, even though the premises are true and the argument is “valid”, it is possible for the conclusion to be false (determined to be false with a counterexample or other means).
At the symposium, he was not able to give a talk, but he did distribute a preprint containing a list of unsolved problems. One of these problems, regarding the Lusternik–Schnirelmann category, came to be known as Ganea's conjecture. A version of this conjecture for rational spaces was proved by Kathryn Hess in her 1989 MIT Ph.D. thesis. Many particular cases of Ganea's original conjecture were proved, until Norio Iwase provided a counterexample in 1998.
This can happen due to BCS teams turning them down in fear of an upset, or scheduling a traditionally strong school who turned out to be having a weak year. The 2009 TCU team is a counterexample, however. They defeated both Virginia and Clemson on the road, and won the rest of their games by an average of 31 points. They received a BCS bid to play against Boise State in the Fiesta Bowl.
Let n be the number of its facets, and let l be its length. Then there exists an (n-d)-spindle, P', with 2n-2d facets and a length bounded below by l+n-2d. In particular, if l>d, then P' violates the d-step conjecture. Santos then proceeds to construct a 5-dimensionsal spindle with length 6, hence proving that there exists another spindle that serves as a counterexample to the Hirsch conjecture.
A theory of art is intended to contrast with a definition of art. Traditionally, definitions are composed of necessary and sufficient conditions and a single counterexample overthrows such a definition. Theorizing about art, on the other hand, is analogous to a theory of a natural phenomenon like gravity. In fact, the intent behind a theory of art is to treat art as a natural phenomenon that should be investigated like any other.
In the mathematical field of graph theory, the Tutte graph is a 3-regular graph with 46 vertices and 69 edges named after W. T. Tutte. It has chromatic number 3, chromatic index 3, girth 4 and diameter 8. The Tutte graph is a cubic polyhedral graph, but is non-hamiltonian. Therefore, it is a counterexample to Tait's conjecture that every 3-regular polyhedron has a Hamiltonian cycle.. Reprinted in Scientific Papers, Vol.
For Banach spaces, the first example of an operator without an invariant subspace was constructed by Per Enflo. He proposed a counterexample to the invariant subspace problem in 1975, publishing an outline in 1976. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987; . Enflo's long "manuscript had a world-wide circulation among mathematicians" and some of its ideas were described in publications besides Enflo (1976).
In number theory, the Pólya conjecture stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an odd number of prime factors. The conjecture was posited by the Hungarian mathematician George Pólya in 1919, and proved false in 1958 by C. Brian Haselgrove. The size of the smallest counterexample is often used to show how a conjecture can be true for many cases, and still be false,.
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.
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.
This is true for Hall's (1966) notion of the pidgin-creole life cycle as well as Bickerton's language bioprogram theory. There are few undisputed examples of a creole arising from nativization of a pidgin by children. The Tok Pisin language reported by is one example where such a conclusion could be reached by scientific observation. A counterexample is the case where children of Gastarbeiter parents speaking pidgin German acquired German seamlessly without creolization.
"The Claims of Frederick Douglass Philosophically Considered." Pp. 155–56 in Frederick Douglass: A Critical Reader, edited by B. E. Lawson and F. M. Kirkland. Wiley-Blackwell. . "Moreover, though he does not make the point explicitly, again the very fact that Douglass is ably disputing this argument on this occasion celebrating a select few's intellect and will (or moral character)—this fact constitutes a living counterexample to the narrowness of the pro-slavery definition of humans."Hutchison, Michael. 2005.
Much of his work concerns the Geometry of Numbers, Hausdorff Measures, Analytic Sets, Geometry and Topology of Banach Spaces, Selection Theorems and Finite Dimensionsl Convex Geometry. In the theory of Banach spaces and summability, he proved the Dvoretzky-Rogers lemma and the Dvoretzky-Rogers theorem, both with Aryeh Dvoretzky. He constructed a counterexample to a conjecture related to the Busemann–Petty problem. In the geometry of numbers, the Rogers bound is a bound for dense packings of spheres.
Richard Jeffrey and later Judea Pearl showed that Savage's principle is only valid when the probability of the event considered (e.g., the winner of the election) is unaffected by the action (buying the property). Under such conditions, the sure-thing principle is a theorem in the do-calculus (see Bayes networks). Blyth constructed a counterexample to the sure-thing principle using sequential sampling in the context of Simpson's paradox, but this example violates the required action-independence provision.
"The Claims of Frederick Douglass Philosophically Considered." Pp. 155–56 in Frederick Douglass: A Critical Reader, edited by B. E. Lawson and F. M. Kirkland. Wiley-Blackwell. . "Moreover, though he does not make the point explicitly, again the very fact that Douglass is ably disputing this argument on this occasion celebrating a select few's intellect and will (or moral character)—this fact constitutes a living counterexample to the narrowness of the pro-slavery definition of humans."Hutchison, Michael. 2005.
Therefore, if you have > a certain day and time, the set of four pillars will repeat itself in 60 > years. However, since the same day may not appear in exactly the same month > - and even if it is in the same month, the day may not be found in the same > half month - it takes 240 years before the identical four pillars appear > again . . . Hee's proposal is incorrect and can be easily refuted by a counterexample.
These were difficult years for Japanese students and researchers because of World War II. The first paper published by Matsushima contained a proof that a conjecture of Hans Zassenhaus was false. Zassenhaus had conjectured that every semisimple Lie algebra L over a field of prime characteristic, with [L, L] = L, is the direct sum of simple ideals. Matsushima constructed a counterexample. He then developed a proof that Cartan subalgebras of a complex Lie algebra are conjugate.
Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles is false, it is sufficient that one provide a model in which there are two distinct (numerically nonidentical) things that have all the same properties. He claimed that in a symmetric universe wherein only two symmetrical spheres exist, the two spheres are two distinct objects even though they have all their properties in common.Metaphysics: An Anthology. eds.
In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below. One of the most important applications of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions.
The Errera graph, on the other hand, provides a counterexample to Kempe's entire method. When this method is run on the Errera graph, starting with no vertices colored, it can fail to find a valid coloring for the whole graph. Additionally, unlike the Poussin graph, all vertices in the Errera graph have degree five or more. Therefore, on this graph, it is impossible to avoid the problematic cases of Kempe's method by choosing lower-degree vertices.
The presence or absence of a magnetic field affects an upper atmosphere, and in turn the overall atmosphere. Impacts of solar wind particles create chemical reactions and ionic species, which may in turn affect magnetospheric phenomena. Earth serves as a counterexample to Venus and Mars, which have no planetary magnetospheres, and to Mercury, with a magnetosphere but negligible atmosphere. Jupiter's moon Io creates sulfur emissions, and a feature of sulfur and some sodium around that planet.
While editing the text, Chekhov extended rather than curtailed it, which was a rarity for the author. Chekhov relayed to Ivan Bunin that "The Student" was his favorite among his own short stories. According to L. M. O'Toole, it was also the story "that [Chekhov] considered to be structurally most perfect" among his oeuvre. Chekhov referenced it as a counterexample when challenged that his works were overtly pessimistic, describing it to his brother Alexander as his "manifesto for optimism".
Wilkins proposes a direct counterexample to Gleitman's example of /put/ and /look/, attested in the Central Australian Aboriginal language of Mparntwe Arrernte. In this case, the verbs /arrerne-/ 'put' and /are-/ 'look, see' share the same case array of {Ergative, Accusative, Dative} arguments (three nouns). Wilkins proposes that this is not a challenge against Gleitman's theory, rather data that may force reconsideration of Gleitman's claim that the theory manifests equally in all languages. ERG ACC VERB DAT a.
Evidently a space that is locally connected at x is weakly locally connected at x. The converse does not hold (a counterexample, the broom space, is given below). On the other hand, it is equally clear that a locally connected space is weakly locally connected, and here it turns out that the converse does hold: a space that is weakly locally connected at all of its points is necessarily locally connected at all of its points.Willard, Theorem 27.16, p.
Many are small and have heavily specialized economies, often relying on industries such as agriculture, mining, and tourism. An example of such a specialized town is La Rinconada, Peru, a gold-mining town and the highest elevation human habitation at . A counterexample is El Alto, Bolivia, at , which has a highly diverse service and manufacturing economy and a population of nearly 1 million. Traditional mountain societies rely on agriculture, with higher risk of crop failure than at lower elevations.
Intuitionistic logic can be understood as a weakening of classical logic, meaning that it is more conservative in what it allows a reasoner to infer, while not permitting any new inferences that could not be made under classical logic. Each theorem of intuitionistic logic is a theorem in classical logic, but not conversely. Many tautologies in classical logic are not theorems in intuitionistic logicin particular, as said above one of its chief points is to not affirm the law of the excluded middle so as to vitiate the use of non- constructive proof by contradiction which can be used to furnish existence claims without providing explicit examples of the objects that it proves exist. We say "not affirm" because while it is not necessarily true that the law is upheld in any context, no counterexample can be given: such a counterexample would be an inference (inferring the negation of the law for a certain proposition) disallowed under classical logic and thus is not allowed in a strict weakening like intuitionistic logic.
Paul Alexander Schweitzer, S. J., (born 21 July 1937, Yonkers, New York) is an American mathematician, specializing in differential topology, geometric topology, and algebraic topology. He has done research on foliations, knot theory, and 3-manifolds. In 1974 he found a counterexample to the Seifert conjecture that every non-vanishing vector field on the 3-sphere has a closed integral curve. In 1995 he demonstrated that Sergei Novikov's compact leaf theorem cannot be generalized to manifolds with dimension greater than 3\.
Given the partial correspondence between the 1-dimensional Hausdorff measure of a compact subset of C and its analytic capacity, it might be conjectured that γ(K) = 0 implies H1(K) = 0. However, this conjecture is false. A counterexample was first given by A. G. Vitushkin, and a much simpler one by J. Garnett in his 1970 paper. This latter example is the linear four corners Cantor set, constructed as follows: Let K0 := [0, 1] × [0, 1] be the unit square.
Petersen's interests in mathematics were manifold, including: geometry, complex analysis, number theory, mathematical physics, mathematical economics, cryptography and graph theory. His famous paper Die Theorie der regulären graphs was a fundamental contribution to modern graph theory as we know it today. In 1898, he presented a counterexample to Tait's claimed theorem about 1-factorability of 3-regular graphs, which is nowadays known as the "Petersen graph". In cryptography and mathematical economics he made contributions which today are seen as pioneering.
Cubes occasionally have the surjective property in other fields, such as in for such prime that ,The multiplicative group of is cyclic of order , and if it is not divisible by 3, then cubes define a group automorphism. but not necessarily: see the counterexample with rationals above. Also in only three elements 0, ±1 are perfect cubes, of seven total. −1, 0, and 1 are perfect cubes anywhere and the only elements of a field equal to the own cubes: .
Frobenius conjectured that if in addition the number of solutions to xn=1 is exactly n where n divides the order of G then these solutions form a normal subgroup. This has been proved as a consequence of the classification of finite simple groups. The symmetric group S3 has exactly 4 solutions to x4=1 but these do not form a normal subgroup; this is not a counterexample to the conjecture as 4 does not divide the order of S3.
This counterexample works because P can be assumed true in a world and false in another one. If however the same assumption is considered global, eg P is not allowed in any world of the model. These two problems can be combined, so that one can check whether B is a local consequence of A under the global assumption G. Tableaux calculi can deal with global assumption by a rule allowing its addition to every node, regardless of the world it refers to.
Based on this H. Hopf conjectured in 1956 that any immersed compact orientable constant mean curvature hypersurface in \R^nmust be a standard embedded n-1 sphere. This conjecture was disproven in 1982 by Wu-Yi Hsiang using a counterexample in \R^4. In 1984 Henry C. Wente constructed the Wente torus, an immersion into \R^3 of a torus with constant mean curvature. . Up until this point it had seemed that CMC surfaces were rare; new techniques produced a plethora of examples.
In the area of mathematics known as functional analysis, James' space is an important example in the theory of Banach spaces and commonly serves as useful counterexample to general statements concerning the structure of general Banach spaces. The space was first introduced in 1950 in a short paper by Robert C. James.James, Robert C. A Non-Reflexive Banach Space Isometric With Its Second Conjugate Space. Proceedings of the National Academy of Sciences of the United States of America 37, no.
Critics of evidentialism sometimes reject the claim that a conclusion is justified only if one's evidence supports that conclusion. A typical counterexample goes like this. Suppose, for example, that Babe Ruth approaches the batter's box believing that he will hit a home run despite his current drunkenness and overall decline in performance in recent games. He realizes that, however unlikely it is that his luck will change, it would increase his chances of hitting a home run if he maintains a confident attitude.
2009 This automates the reasoning about the program behavior with respect to the given correct specifications. Model checking and symbolic execution are used to verify the safety-critical properties of device drivers. The input to the model checker is the program and the temporal safety properties. The output is the proof that the program is correct or a demonstration that there exists a violation of the specification by means of a counterexample in the form of a specific execution path.
Several writers, including Philip Pullman,"Pullman attacks Narnia film plans" BBC News, 16 October 2005 Kyrie O'Connor,Kyrie O'Connor, "5th Narnia book may not see big screen" IndyStar.com, 1 December 2005 and Gregg Easterbrook,October 2001 of The Atlantic consider the use of Calormene characters as villains to be evidence of racism. Aravis is often presented as a counterexample to this (along with Emeth, who is accepted in Aslan's country for good deeds worthy of Aslan), since she is sympathetically portrayed as a largely virtuous Calormene heroine.
The proof of the strong perfect graph theorem by Chudnovsky et al. follows an outline conjectured in 2001 by Conforti, Cornuéjols, Robertson, Seymour, and Thomas, according to which every Berge graph either forms one of five types of basic building block (special classes of perfect graphs) or it has one of four different types of structural decomposition into simpler graphs. A minimally imperfect Berge graph cannot have any of these decompositions, from which it follows that no counterexample to the theorem can exist., Conjecture 5.1.
A skew partition is a partition of a graph's vertices into two subsets, one of which induces a disconnected subgraph and the other of which has a disconnected complement; had conjectured that no minimal counterexample to the strong perfect graph conjecture could have a skew partition. Chudnovsky et al. introduced some technical constraints on skew partitions, and were able to show that Chvátal's conjecture is true for the resulting "balanced skew partitions". The full conjecture is a corollary of the strong perfect graph theorem.
Certainly any law satisfied by all concrete Boolean algebras is satisfied by the prototypical one since it is concrete. Conversely any law that fails for some concrete Boolean algebra must have failed at a particular bit position, in which case that position by itself furnishes a one-bit counterexample to that law. Nondegeneracy ensures the existence of at least one bit position because there is only one empty bit vector. The final goal of the next section can be understood as eliminating "concrete" from the above observation.
Daebyeol-wang responds that they have solid trunks, but the younger brother wins by giving the counterexample of the bamboo, whose stems are hollow. The next riddle goes similarly. Sobyeol-wang asks whether grass grows thicker in the valleys below or the hills above (or why grass grows thicker in the former). The older brother explains why grass is thicker in the valleys, but Sobyeol-wang refutes him by asking why humans have more hair above, on the scalp, then below, on the feet.
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.
Model of the Peano surface in the Dresden collection In mathematics, the Peano surface is the graph of the two-variable function :f(x,y)=(2x^2-y)(y-x^2). It was proposed by Giuseppe Peano in 1899 as a counterexample to a conjectured criterion for the existence of maxima and minima of functions of two variables. The surface was named the Peano surface () by Georg Scheffers in his 1920 book Lehrbuch der darstellenden Geometrie. It has also been called the Peano saddle.
The Coxeter graph is also uniquely determined by its graph spectrum, the set of graph eigenvalues of its adjacency matrix.E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs. J. Algebraic Combin. 15, pages 189-202, 2003 As a finite connected vertex- transitive graph that contains no Hamiltonian cycle, the Coxeter graph is a counterexample to a variant of the Lovász conjecture, but the canonical formulation of the conjecture asks for an Hamiltonian path and is verified by the Coxeter graph.
There are very high lower bounds for Carmichael's conjecture that are relatively easy to determine. Carmichael himself proved that any counterexample to his conjecture (that is, a value n such that φ(n) is different from the totients of all other numbers) must be at least 1037, and Victor Klee extended this result to 10400. A lower bound of 10^{10^7}was given by Schlafly and Wagon, and a lower bound of 10^{10^{10}} was determined by Kevin Ford in 1998.Sándor & Crstici (2004) p.
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring. Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by .
Wall–Sun–Sun primes are named after Donald Dines Wall, Zhi Hong Sun and Zhi Wei Sun; Z. H. Sun and Z. W. Sun showed in 1992 that if the first case of Fermat's last theorem was false for a certain prime p, then p would have to be a Wall–Sun–Sun prime. As a result, prior to Andrew Wiles' proof of Fermat's last theorem, the search for Wall–Sun–Sun primes was also the search for a potential counterexample to this centuries-old conjecture.
Counterexample to a strengthening of the uniform convergence theorem, in which pointwise convergence, rather than uniform convergence, is assumed. The continuous green functions \sin^n(x) converge to the non- continuous red function. This can happen only if convergence is not uniform. If E and M are topological spaces, then it makes sense to talk about the continuity of the functions f_n,f:E\to M. If we further assume that M is a metric space, then (uniform) convergence of the f_n to f is also well defined.
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.
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.
The electromagnetism uniqueness theorem states that providing boundary conditions for Maxwell's equations uniquely fixes a solution for those equations. However, this theorem must not be misunderstood as that providing boundary conditions (or the field solution itself) uniquely fixes a source distribution. One counterexample is that the field outside a uniformly charged sphere may also be produced by a point charge placed at the center of the sphere instead, i.e. the source needed to produce such field at a boundary outside the sphere is not unique.
Ginzburg is best known for his work on the Conley conjecture, which asserts the existence of infinitely many periodic points for Hamiltonian diffeomorphisms in many cases, and for his counterexample (joint with Başak Gürel) to the Hamiltonian Seifert conjecture which constructs a Hamiltonian with an energy level with no periodic trajectories. Some of his other works concern coisotropic intersection theory, and Poisson–Lie groups.V. Ginzburg and A. Weinstein, Lie- Poisson structure on some Poisson Lie groups, J. Amer. Math. Soc. (2) 5, 445-453, 1992.
For a simplified proof of Läuchli's theorem by Mycielski, see . The De Bruijn–Erdős theorem for countable graphs can also be shown to be equivalent in axiomatic power, within a theory of second-order arithmetic, to Kőnig's infinity lemma. For a counterexample to the theorem in models of set theory without choice, let be an infinite graph in which the vertices represent all possible real numbers. In , connect each two real numbers and by an edge whenever one of the values is a rational number.
QCA can be performed probabilistically or deterministically with observations of categorical variables. For instance, the existence of a descriptive inference or implication is supported deterministically by the absence of any counter- example cases to the inference; i.e. if a researcher claims condition X implies condition Y, then, deterministically, there must not exist any counterexample cases having condition X, but not condition Y. However, if the researcher wants to claim that condition X is a probabilistic 'predictor' of condition Y, in another similar set of cases, then the proportion of counterexample cases to an inference to the proportion of cases having that same combination of conditions can be set at a threshold value of for example 80% or higher. For each prime implicant that QCA outputs via its logical inference reduction process, the "coverage" — percentage out of all observations that exhibit that implication or inference — and the "consistency" — the percentage of observations conforming to that combination of variables having that particular value of the dependent variable or outcome — are calculated and reported, and can be used as indicators of the strength of such an explorative probabilistic inference.
In July 2008, Slavic Village was one of six neighborhoods of the Neighborhood Strategic Investment Initiatives featured locally in an audio discussion titled "Conversations in Slavic Village" held by the Case Western Reserve Studies Symposium. Stakeholders discussed how neighborhoods are vital to the livable city. Speakers cited specific events occurring in direct counterexample to the increased crime reported in the media in 2007. Representatives included a savings and loan association official, steel mill corporate social responsibility manager, public art coordinator, community development corporation director, and local metropark, youth organization, and cultural center directors.
If an experiment is carefully conducted, the results usually either support or disprove the hypothesis. According to some philosophies of science, an experiment can never "prove" a hypothesis, it can only add support. On the other hand, an experiment that provides a counterexample can disprove a theory or hypothesis, but a theory can always be salvaged by appropriate ad hoc modifications at the expense of simplicity. An experiment must also control the possible confounding factors—any factors that would mar the accuracy or repeatability of the experiment or the ability to interpret the results.
" (Reid, p. 149)}} In his lecture in 1941 at Yale and the subsequent paper Gödel proposed a solution: "...that the negation of a universal proposition was to be understood as asserting the existence ... of a counterexample" (Dawson, p. 157)) Gödel's approach to the law of excluded middle was to assert that objections against "the use of 'impredicative definitions'" "carried more weight" than "the law of excluded middle and related theorems of the propositional calculus" (Dawson p. 156). He proposed his "system Σ ... and he concluded by mentioning several applications of his interpretation.
In graph theory, Vizing's conjecture concerns a relation between the domination number and the cartesian product of graphs. This conjecture was first stated by , and states that, if γ(G) denotes the minimum number of vertices in a dominating set for G, then : \gamma(G\,\Box\,H) \ge \gamma(G)\gamma(H). \, conjectured a similar bound for the domination number of the tensor product of graphs; however, a counterexample was found by . Since Vizing proposed his conjecture, many mathematicians have worked on it, with partial results described below.
Every reducible configuration contains a cycle, so for every finite set S of reducible configurations there is a number γ such that all configurations in the set contain a cycle of length at most γ. However, there exist snarks with arbitrarily high girth, that is, with arbitrarily high bounds on the length of their shortest cycle.. A snark G with girth greater than γ cannot contain any of the configurations in the set S, so the reductions in S are not strong enough to rule out the possibility that G might be a minimal counterexample.
This extremely controversial debate is centered around the efficacy of more radical and disruptive tactics (including targeted violence, riots, or general disorder) as opposed to more mainstream tactics (such as marches, rallies, and political lobbying). This issue is extremely complex because it seems to be extremely context-dependent. Gamson's original studyGamson, 1975 found that disruption did usually lead to movement success; however, it was with certain qualifications. First, the results have to face the striking counterexample of labor unions, which were greatly weakened by violent strikes most of the time.
The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order complete theory in a countable language is finite or ℵ0 or 2. Morley showed that the number of countable models is finite or ℵ0 or ℵ1 or 2, which solves the conjecture except for the case of ℵ1 models when the continuum hypothesis fails. For this remaining case, has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture.
The movement for net neutrality argues that the Internet should not be a resource that is dominated by one particular group, specifically those with more money to spend on Internet access. A counterexample to the tragedy of the commons is offered by Andrew Kahrl. Privatization can be a way to deal with the tragedy of the commons. However, Kahrl suggests that the privatization of beaches on Long Island, in an attempt to combat the overuse of Long Island beaches, made the residents of Long Island more susceptible to flood damage from Hurricane Sandy.
He was a co- editor of the forthcoming Handbook of Model Checking. In 2014, he was co-chair of the Vienna Summer of Logic 2014, the largest conference on logic and computer science in history. Veith is best known for his role in the development of Counterexample-guided Abstraction Refinement (CEGAR) which is a key ingredient in modern model checkers for software and hardware. His research applies formal and logical methods to problems in software technology and engineering, focusing on model checking, software verification and testing, embedded software and computer security.
To see the difference, first consider the class K (or simply the set) containing three models with linear orders, L1 of size one, L2 of size two, and L3 of size three. This class K has the joint embedding property because all three models can be embedded into L3. However, K does not have the amalgamation property. The counterexample for this starts with L1 containing a single element e and extends in two different ways to L3, one in which e is the smallest and the other in which e is the largest.
Then E. T. Parker found a counterexample of order 10 using a one-hour computer search on a UNIVAC 1206 Military Computer while working at the UNIVAC division of Remington Rand (this was one of the earliest combinatorics problems solved on a digital computer). In April 1959, Parker, Bose, and Shrikhande presented their paper showing Euler's conjecture to be false for all Thus, Graeco-Latin squares exist for all orders except In the November 1959 edition of Scientific American, Martin Gardner published this result. The front cover is the 10 × 10 refutation of Euler's conjecture.
In a military turbofan combat engine the bypass air is added into the exhaust, thereby increasing the core and afterburner efficiency. In turbojets the gain is limited to 50%, whereas in a turbofan it depends on the bypass ratio and can be as much as 70%."Basic Study of the Afterburner" Yoshiyuki Ohya, NASA TT F-13,657 However, as a counterexample, the SR-71 had reasonable efficiency at high altitude in afterburning ("wet") mode owing to its high speed (mach 3.2) and correspondingly high pressure due to ram intake.
1969–present. Poetry and translations have appeared in: (Print) Antaeus, Antenym, Bay Guardian, Beatitude, Caliban, City Lights Review, Compact Bone, Coracle, Gallery Works, Gas, Juxta, Mantis, Malthus, Melodeon, Mike & Dale's Younger Poets, The New College Review, Prosodia, Root & Branch, syllogism, Talisman, Terra, Velocities. (Web): The Alterran Poetry Assemblage #2, The Alterran Poetry Assemblage #3, Angel Poetry, Counterexample Poetics, black fire white fire, Deep Oakland, Duration Press Archive, Facture 1, Facture 2, Five Fingers Review, Issue 16, kayak, Montana Gothic, Orpheus Grid, ‘’The Pedestal Magazine’’, Processed World, ur- vox, MSNBC.com.
Shapiro generalized this into the "Contradiction Backtracing Algorithm" an algorithm for backtracking contradictions. This algorithm is applicable whenever a contradiction occurs between some conjectured theory and the facts. By testing a finite number of ground atoms for their truth in the model the algorithm can trace back a source for this contradiction, namely a false hypothesis, and can demonstrate its falsity by constructing a counterexample to it. The "Contradiction Backtracing Algorithm" is relevant both to the philosophical discussion on the refutability of scientific theories and in the aid for the debugging of logic programs. Prof.
At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology—what would later be called combinatorial topology and then algebraic topology. He was particularly interested in what topological properties characterized a sphere. Poincaré claimed in 1900 that homology, a tool he had devised based on prior work by Enrico Betti, was sufficient to tell if a 3-manifold was a 3-sphere. However, in a 1904 paper he described a counterexample to this claim, a space now called the Poincaré homology sphere.
Kempe observed that its vertices can represent the ten lines of the Desargues configuration, and its edges represent pairs of lines that do not meet at one of the ten points of the configuration. Donald Knuth states that the Petersen graph is "a remarkable configuration that serves as a counterexample to many optimistic predictions about what might be true for graphs in general." The Petersen graph also makes an appearance in tropical geometry. The cone over the Petersen graph is naturally identified with the moduli space of five-pointed rational tropical curves.
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.
3 Based on these two results, he conjectured that in fact every connected graph with a planar cover is projective.; , Conjecture 4, p. 4 As of 2013, this conjecture remains unsolved. It is also known as Negami's "1-2-∞ conjecture", because it can be reformulated as stating that the minimum ply of a planar cover, if it exists, must be either 1 or 2. K1,2,2,2, the only possible minimal counterexample to Negami's conjecture Like the graphs with planar covers, the graphs with projective plane embeddings can be characterized by forbidden minors.
169–200 Errera studied at the Université libre de Bruxelles, where he received his Ph.D. in 1921 with dissertation Du coloriage des cartes et de quelques questions d'analysis situs. In his dissertation he introduced what is now called the Errera graph,Errera Graph, Mathworld which is a counterexample to the validity of the alleged proof of the four color theorem by Alfred Kempe. From 1928 to 1956 he was a professor at the Université libre de Bruxelles. He did research on topology, especially the theory of polyhedra and the Jordan curve theorem.
The wheel W6 supplied a counterexample to a conjecture of Paul Erdős on Ramsey theory: he had conjectured that the complete graph has the smallest Ramsey number among all graphs with the same chromatic number, but Faudree and McKay (1993) showed W6 has Ramsey number 17 while the complete graph with the same chromatic number, K4, has Ramsey number 18. . That is, for every 17-vertex graph G, either G or its complement contains W6 as a subgraph, while neither the 17-vertex Paley graph nor its complement contains a copy of K4.
Their proof is similar to Suzuki's proof. It was about 17 pages long, which at the time was thought to be very long for a proof in group theory. The Feit–Thompson theorem can be thought of as the next step in this process: they show that there is no non-cyclic simple group of odd order such that every proper subgroup is solvable. This proves that every finite group of odd order is solvable, as a minimal counterexample must be a simple group such that every proper subgroup is solvable.
Raven can be a magician, a transformer, a potent creative force, ravenous debaucher but always a cultural hero. He is responsible for creating Haida Gwaii, releasing the sun from its tiny box and making the stars and the moon. In one story he released the first humans from a cockle shell on the beach; in another story he brought the first humans up out of the ground because he needed to fill up a party he was throwing. Raven stories on one level teach listeners how to live a good life, but usually by counterexample.
The fact that such sequences exist means that the collection of all computable real numbers does not satisfy the least upper bound principle of real analysis, even when considering only computable sequences. A common way to resolve this difficulty is to consider only sequences that are accompanied by a modulus of convergence; no Specker sequence has a computable modulus of convergence. More generally, a Specker sequence is called a recursive counterexample to the least upper bound principle, i.e. a construction that shows that this theorem is false when restricted to computable reals.
The Petkau effect is an early counterexample to linear-effect assumptions usually made about radiation exposure. It was found by Dr. Abram Petkau at the Atomic Energy of Canada Whiteshell Nuclear Research Establishment, Manitoba and published in Health Physics March 1972. The Petkau effect was coined by Swiss nuclear hazards commentator Ralph Graeub in 1985 in this book Der Petkau-Effekt und unsere strahlende Zukunft (The Petkau effect and our Radiating Future). Petkau had been measuring, in the usual way, the radiation dose that would rupture a simulated artificial cell membrane.
ABC is an imperative general-purpose programming language and programming environment developed at CWI, Netherlands by Leo Geurts, Lambert Meertens, and Steven Pemberton. It is interactive, structured, high-level, and intended to be used instead of BASIC, Pascal, or AWK. It is not meant to be a systems- programming language but is intended for teaching or prototyping. The language had a major influence on the design of the Python programming language (as a counterexample); Guido van Rossum, who developed Python, previously worked for several years on the ABC system in the early 1980s.
The conjecture attracted considerable interest when suggested that it implies Fermat's Last Theorem. He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply the existence of at least one non-modular elliptic curve. This argument was completed when identified a missing link (now known as the epsilon conjecture or Ribet's theorem) in Frey's original work, followed two years later by 's completion of a proof of the epsilon conjecture. Even after gaining serious attention, the Taniyama–Shimura–Weil conjecture was seen by contemporary mathematicians as extraordinarily difficult to prove or perhaps even inaccessible to proof .
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.
The groups T and V are (rare) examples of infinite but finitely-presented simple groups. The group F is not simple but its derived subgroup [F,F] is and the quotient of F by its derived subgroup is the free abelian group of rank 2. F is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2. It is conjectured that F is not amenable and hence a further counterexample to the long-standing but recently disproved von Neumann conjecture for finitely-presented groups: it is known that F is not elementary amenable.
A noteworthy counterexample is Columbia University, as the disputes between its President Nicholas Murray Butler and such faculty members as Charles A. Beard, Harry Thurston Peck, and Joel Spingarn testify. He similarly defended a student's anti-German poem with a statement of principle in defense of free speech within the academic community. In 1915, Kuno Meyer a professor at the University of Berlin who was considering a temporary Harvard appointment, protested the publication of an undergraduate's satirical poem in a college magazine. Lowell replied that freedom of speech played a different role in American universities than in their German counterparts.
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\.
ZetaGrid was at one time the largest distributed computing project, designed to explore the non-trivial roots of the Riemann zeta function, checking over one billion roots a day. Roots of the zeta function are of particular interest in mathematics, since the presence of even a single one that is out of line with the rest would disprove the Riemann hypothesis, with far-reaching consequences for all of mathematics. So far, every single one of them has failed to provide a counterexample to the Riemann hypothesis. The project ended in November 2005 due to instability of the hosting provider.
Sunada's work covers complex analytic geometry, spectral geometry, dynamical systems, probability, graph theory, discrete geometric analysis, and mathematical crystallography. Among his numerous contributions, the most famous one is a general construction of isospectral manifolds (1985), which is based on his geometric model of number theory, and is considered to be a breakthrough in the problem proposed by Mark Kac in "Can one hear the shape of a drum?" (see Hearing the shape of a drum). Sunada's idea was taken up by Carolyn S. Gordon, David Webb, and Scott A. Wolpert when they constructed a counterexample for Kac's problem.
Validity is defined in classical logic as follows: :An argument (consisting of premises and a conclusion) is valid if and only if there is no possible situation in which all the premises are true and the conclusion is false. For example a valid argument might run: :If it is raining, water exists (1st premise) :It is raining (2nd premise) :Water exists (Conclusion) In this example there is no possible situation in which the premises are true while the conclusion is false. Since there is no counterexample, the argument is valid. But one could construct an argument in which the premises are inconsistent.
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.
If the Erdős–Ulam problem has a positive solution, it would provide a counterexample to the Bombieri–Lang conjecture and to the abc conjecture. It would also solve Harborth's conjecture, on the existence of drawings of planar graphs in which all distances are integers. If a dense rational-distance set exists, any straight-line drawing of a planar graph could be perturbed by a small amount (without introducing crossings) to use points from this set as its vertices, and then scaled to make the distances integers. However, like the Erdős–Ulam problem, Harborth's conjecture remains unproven.
Part of the seventh of Hilbert's twenty three problems posed in 1900 was to prove, or find a counterexample to, the claim that ab is always transcendental for algebraic a ≠ 0, 1 and irrational algebraic b. In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 2. In 1919, he gave a lecture on number theory and spoke of three conjectures: the Riemann hypothesis, Fermat's Last Theorem, and the transcendence of 2. He mentioned to the audience that he didn't expect anyone in the hall to live long enough to see a proof of this final result.
Euler was aware of the equality involving sums of four fourth powers; this however is not a counterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in Plato's number or the taxicab number 1729. The general solution of the equation :x_1^3+x_2^3=x_3^3+x_4^3 is :x_1 = 1-(a-3b)(a^2+3b^2), \quad x_2 = (a+3b)(a^2+3b^2)-1 :x_3 = (a+3b)-(a^2+3b^2)^2, \quad x_4 = (a^2+3b^2)^2-(a-3b) where and are any integers.
András Ádám conjectured that these linear maps are the only ways of renumbering a circulant graph while preserving the circulant property: that is, if and are isomorphic circulant graphs, with different numberings, then there is a linear map that transforms the numbering for into the numbering for . However, Ádám's conjecture is now known to be false. A counterexample is given by graphs and with 16 vertices each; a vertex in is connected to the six neighbors , , and modulo 16, while in the six neighbors are , , and modulo 16. These two graphs are isomorphic, but their isomorphism cannot be realized by a linear map.
The Petersen graph The Petersen graph is an undirected graph with ten vertices and fifteen edges, commonly drawn as a pentagram within a pentagon, with corresponding vertices attached to each other. It has many unusual mathematical properties, and has frequently been used as a counterexample to conjectures in graph theory. The book uses these properties as an excuse to cover several advanced topics in graph theory where this graph plays an important role. It is heavily illustrated, and includes both open problems on the topics it discusses and detailed references to the literature on these problems.
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.
Jasmin Christian Blanchette, Lukas Bulwahn, Tobias Nipkow, "Automatic Proof and Disproof in Isabelle/HOL", in: Cesare Tinelli, Viorica Sofronie-Stokkermans (eds.), International Symposium on Frontiers of Combining Systems – FroCoS 2011, Springer, 2011. It also features two model finders (counterexample generators): NitpickJasmin Christian Blanchette, Mathias Fleury, Peter Lammich & Christoph Weidenbach, "A Verified SAT Solver Framework with Learn, Forget, Restart, and Incrementality", Journal of Automated Reasoning 61:333–365 (2018). and Nunchaku.Andrew Reynolds, Jasmin Christian Blanchette, Simon Cruanes, Cesare Tinelli, "Model Finding for Recursive Functions in SMT", in: Nicola Olivetti, Ashish Tiwari (eds.), 8th International Joint Conference on Automated Reasoning, Springer, 2016.
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}.
Delayed reaction to the attack contributed to what she later characterised as PTSD in June 1986."Tetris + Ecstasy = no PTSD", "Ask Nicola" Her writing and a women's self-defense course that she was teaching sustained her amid these difficulties, and Helena's counterexample helped persuade Griffith that the time to abandon all recreational drug use—including magic mushrooms, which she had relied on extensively—had come. By late 1987 Griffith had made her first professional fiction sale, of a short story, "Mirrors and Burnstone," to Interzone. She was also experiencing symptoms of multiple sclerosis, though her illness remained unrecognised.
Conversely, any solution to this equation could be factored to give a square congruum. (In particular, the squares (b^4-d^4-2b^2 d^2)^2, (b^4+d^4)^2, and (b^4-d^4+2b^2 d^2)^2 form an arithmetic progression with congruum 4b^2 d^2 (b^4-d^4) = (2bde)^2, which is a square itself.) Thus, the solvability of this equation is equivalent to the existence of a square congruum. But, if Fermat's last theorem were false for the exponent n=4, then squaring one of the three numbers in any counterexample would also give three numbers that solve this equation.
For example, as outlined in the Common Core State Standards Initiative, a standards-based education reform developed to increase high school graduation rates, educators are expected to present within the classroom "high level cognitive demands by asking students to demonstrate deep conceptual understanding through the application of content knowledge and skills to new situations." This guideline is the essence of cognitive flexibility, and a teaching style focused on promoting it has been seen to foster understanding especially in disciplines where information is complex and nonlinear. A counterexample is evident in cases where such material is presented in an oversimplified manner and learners fail to transfer their knowledge to a new domain.
Alveolar consonants are made with the tip or blade of the tongue at the alveolar ridge just behind the teeth and can similarly be apical or laminal. Crosslinguistically, dental consonants and alveolar consonants are frequently contrasted leading to a number of generalizations of crosslinguistic patterns. The different places of articulation tend to also be contrasted in the part of the tongue used to produce them: most languages with dental stops have laminal dentals, while languages with apical stops usually have apical stops. Languages rarely have two consonants in the same place with a contrast in laminality, though Taa (ǃXóõ) is a counterexample to this pattern.
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.
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.
A different kind of realist criticism (see for a discussion) stresses the role of nuclear weapons in maintaining peace. In realist terms, this means that, in the case of disputes between nuclear powers, respective evaluation of power might be irrelevant because of Mutual assured destruction preventing both sides from foreseeing what could be reasonably called a "victory". The 1999 Kargil War between India and Pakistan has been cited as a counterexample to this argument , though this was a small, regional conflict and the threat of WMDs being used contributed to its de-escalation . Some supporters of the democratic peace do not deny that realist factors are also important .
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.
Petrie also asked: What are necessary and sufficient conditions for the existence of a smooth G-map properly G-homotopic to F and transverse to the zero-section? DejterDejter I. J. "G-Transversality to CP^n", Springer-Verlag Lecture Notes in Mathematics, 652 (1976), 222–239 provided both types of conditions, which do not close to a necessary and sufficient condition due to a counterexample. The main tool involved in establishing the results above by reducing differential-topology problems into algebraic-topology solutions is equivariant algebraic K-theory, where equivariance is understood with respect to the group given by the circle, i.e. the unit circle of the complex plane.
Alveolar consonants are made with the tip or blade of the tongue at the alveolar ridge just behind the teeth and can similarly be apical or laminal. Crosslinguistically, dental consonants and alveolar consonants are frequently contrasted leading to a number of generalizations of crosslinguistic patterns. The different places of articulation tend to also be contrasted in the part of the tongue used to produce them: most languages with dental stops have laminal dentals, while languages with apical stops usually have apical stops. Languages rarely have two consonants in the same place with a contrast in laminality, though Taa (ǃXóõ) is a counterexample to this pattern.
This ice model provide an important 'counterexample' in statistical mechanics: the bulk free energy in the thermodynamic limit depends on boundary conditions. The model was analytically solved for periodic boundary conditions, anti-periodic, ferromagnetic and domain wall boundary conditions. The six vertex model with domain wall boundary conditions on a square lattice has specific significance in combinatorics, it helps to enumerate alternating sign matrices. In this case the partition function can be represented as a determinant of a matrix (whose dimension is equal to the size of the lattice), but in other cases the enumeration of W does not come out in such a simple closed form.
Searle claims that the picnickers, whose intentions are individually oriented and simply happen to coincide, do not display collective intentionality, while members of the dance troupe do, because they deliberately cooperate with one another. Searle's rebuttal to Tuomela and Miller's account begins with a counterexample involving a group of business school graduates who intend to pursue their own selfish interests, but believe that by doing so, they will indirectly serve humanity. These young businessmen believe that their fellow graduates will do likewise, but do not actively cooperate with one another in pursuing their goals. Searle holds that this example fulfills all of Tuomela and Miller's criteria for collective intentionality.
In the 1970s, the film critic Raymond Durgnat related the response to the left's own struggle to oppose the EEC without turning to patriotism. Durgnat wrote that it was "perhaps not too unkind to suggest that my left-wing colleagues were making their 'lost leader' a scapegoat for the very real difficulties of the left itself—or should one say the lefts themselves?" Gideon Bachmann wrote in Film Quarterly that many American critics gave Picnic on the Grass a "silly treatment", because they did not understand it through its director as a person. He used Hollis Alpert of the Saturday Review as a positive counterexample.
Pertti Mattila (born 28 March 1948) is a Finnish mathematician working in geometric measure theory, complex analysis and harmonic analysis. He is Professor of Mathematics in the Department of Mathematics and Statistics at the University of Helsinki, Finland. He is known for his work on geometric measure theory and in particular applications to complex analysis and harmonic analysis. His work include a counterexample to the general Vitushkin's conjecture and with Mark Melnikov and Joan Verdera he introduced new techniques to understand the geometric structure of removable sets for complex analytic functions which together with other works in the field eventually led to the solution of Painlevé's problem by Xavier Tolsa.
One of the first standard results using the class equation is that the center of a non-trivial finite p-group cannot be the trivial subgroup.proof This forms the basis for many inductive methods in p-groups. For instance, the normalizer N of a proper subgroup H of a finite p-group G properly contains H, because for any counterexample with H = N, the center Z is contained in N, and so also in H, but then there is a smaller example H/Z whose normalizer in G/Z is N/Z = H/Z, creating an infinite descent. As a corollary, every finite p-group is nilpotent.
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.
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.
241 and 244 Hilbert conjectured that this is not always possible. This was confirmed within the year by his student Max Dehn, who proved that the answer in general is "no" by producing a counterexample. The answer for the analogous question about polygons in 2 dimensions is "yes" and had been known for a long time; this is the Wallace–Bolyai–Gerwien theorem. Unknown to Hilbert and Dehn, Hilbert's third problem was also proposed independently by Władysław Kretkowski for a math contest of 1882 by the Academy of Arts and Sciences of Kraków, and was solved by Ludwik Antoni Birkenmajer with a different method than Dehn.
Reuleaux's original motivation for studying the Reuleaux triangle was as a counterexample, showing that three single-point contacts may not be enough to fix a planar object into a single position., p. 239. The existence of Reuleaux polygons shows that diameter measurements alone cannot verify that an object has a circular cross-section.. See in particular p. 200. Overlooking this fact may have played a role in the Space Shuttle Challenger disaster, as the roundness of sections of the rocket in that launch was tested only by measuring different diameters, and off-round shapes may cause unusually high stresses that could have been one of the factors causing the disaster.
Laman is often credited with proving, in 1970 , that a particular family of sparse graphs, since named Laman graphs, are precisely those that are minimally generically rigid in the plane. This result, however, had already been proven by Hilda Geiringer back in 1927.. Laman's original publication in 1970 went largely unnoticed at first. Only when Branko Grünbaum and G. C. Shephard wrote about Laman's paper in their Lectures on lost mathematics did this work receive more attention. Towards the end of his life, Laman worked to lift the original 'Laman graph' from its original two dimensions to three, inspired by a simple counterexample, the 'double banana graph'.
In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ2 compact elements using a construction based on Kuratowski's free set theorem.
The differences between the bacterial and eukaryotic ribosomes are exploited by pharmaceutical chemists to create antibiotics that can destroy a bacterial infection without harming the cells of the infected person. Due to the differences in their structures, the bacterial 70S ribosomes are vulnerable to these antibiotics while the eukaryotic 80S ribosomes are not. Even though mitochondria possess ribosomes similar to the bacterial ones, mitochondria are not affected by these antibiotics because they are surrounded by a double membrane that does not easily admit these antibiotics into the organelle. A noteworthy counterexample, however, includes the antineoplastic antibiotic chloramphenicol, which successfully inhibits bacterial 50S and mitochondrial 50S ribosomes.
With the axiom of choice we can well order the continuum; furthermore, we can do so in such a way that any proper initial portion does not have the cardinality of the continuum. We create a counterexample by transfinite induction on the set of strategies under this well ordering: We start with the set A undefined. Let T be the "time" whose axis has length continuum. We need to consider all strategies {s1(T)} of the first player and all strategies {s2(T)} of the second player to make sure that for every strategy there is a strategy of the other player that wins against it.
The conjecture of Chern can be considered a particular case of the following conjecture: > A closed aspherical manifold with nonzero Euler characteristic doesn't admit > a flat structure This conjecture was originally stated for general closed manifolds, not just for aspherical ones (but due to Smillie, there's a counterexample), and it itself can, in turn, also be considered a special case of even more general conjecture: > A closed aspherical manifold with nonzero simplicial volume doesn't admit a > flat structure While generalizing the Chern's conjecture on affine manifolds in these ways, it's known as the generalized Chern conjecture for manifolds that are locally a product of surfaces.
For instance, the terms rantans or brillig have no intension and hence no meaning. Such terms may be suggestive, but a term can be suggestive without being meaningful. For instance, ran tan is an archaic onomatopoeia for chaotic noise or din and may suggest to English speakers a din or meaningless noise, and brillig though made up by Lewis Carroll may be suggestive of 'brilliant' or 'frigid'. Such terms, it may be argued, are always intensional since they connote the property 'meaningless term', but this is only an apparent paradox and does not constitute a counterexample to the claim that without intension a word has no meaning.
Scientific realists argue that we have good reasons to believe that our presently successful scientific theories are true or approximately true. The pessimistic meta-induction undermines the realist's warrant for their epistemic optimism (the view that science tends to succeed in revealing what the world is like and that there are good reasons to take theories to be true or truthlike) via historical counterexample. Using meta-induction, Larry Laudan argues that if past scientific theories which were successful were found to be false, we have no reason to believe the realist's claim that our currently successful theories are approximately true. The pessimistic meta- induction argument was first fully postulated by Laudan in 1981.
A formal proof of a conjectured relation will then be sought – it is often easier to find a formal proof once the form of a conjectured relation is known. If a counterexample is being sought or a large-scale proof by exhaustion is being attempted, distributed computing techniques may be used to divide the calculations between multiple computers. Frequent use is made of general mathematical software such as Mathematica,A New Kind of Science although domain-specific software is also written for attacks on problems that require high efficiency. Experimental mathematics software usually includes error detection and correction mechanisms, integrity checks and redundant calculations designed to minimise the possibility of results being invalidated by a hardware or software error.
This smaller map has the condition that if it can be colored with four colors, this also applies to the original map. This implies that if the original map cannot be colored with four colors the smaller map cannot either and so the original map is not minimal. Using mathematical rules and procedures based on properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist. Their proof reduced the infinitude of possible maps to 1,834 reducible configurations (later reduced to 1,482) which had to be checked one by one by computer and took over a thousand hours.
This leaves only the case where G has a vertex of degree 5; but Kempe's argument was flawed for this case. Heawood noticed Kempe's mistake and also observed that if one was satisfied with proving only five colors are needed, one could run through the above argument (changing only that the minimal counterexample requires 6 colors) and use Kempe chains in the degree 5 situation to prove the five color theorem. In any case, to deal with this degree 5 vertex case requires a more complicated notion than removing a vertex. Rather the form of the argument is generalized to considering configurations, which are connected subgraphs of G with the degree of each vertex (in G) specified.
Although the truth of Barnette's conjecture remains unknown, computational experiments have shown that there is no counterexample with fewer than 86 vertices.; . If Barnette's conjecture turns out to be false, then it can be shown to be NP-complete to test whether a bipartite cubic polyhedron is Hamiltonian.. If a planar graph is bipartite and cubic but only of connectivity 2, then it may be non-Hamiltonian, and it is NP-complete to test Hamiltonicity for these graphs.. Another result was obtained by : if the dual graph can be vertex-colored with colors blue, red and green such that every red-green cycle contains a vertex of degree 4, then the primal graph is Hamiltonian.
Beckman and Quarles observe that the theorem is not true for the real line (one- dimensional Euclidean space). For, the function that returns if is an integer and returns otherwise obeys the preconditions of the theorem (it preserves unit distances) but is not an isometry. Beckman and Quarles also provide a counterexample for Hilbert space, the space of square-summable sequences of real numbers. This example involves the composition of two discontinuous functions: one that maps every point of the Hilbert space onto a nearby point in a countable dense subspace, and a second that maps this dense set into a countable unit simplex (an infinite set of points all at unit distance from each other).
Similarly, reduced the dimension in which a counterexample to the conjecture is known by finding a clique of size 28 in the Keller graph of dimension eight. Subsequently, showed that the Keller graph of dimension seven has a maximum clique of size 124 < 27. Because this is less than 27, the graph-theoretic version of Keller's conjecture is true in seven dimensions. However, the translation from cube tilings to graph theory can change the dimension of the problem, so this result doesn't settle the geometric version of the conjecture in seven dimensions. Finally, a 200-gigabyte computer- assisted proof in 2019 used Keller graphs to establish that the conjecture holds true in seven dimensions.
More generally, game semantics may be applied to predicate logic; the new rules allow a dominant quantifier to be removed by its "owner" (the Verifier for existential quantifiers and the Falsifier for universal quantifiers) and its bound variable replaced at all occurrences by an object of the owner's choosing, drawn from the domain of quantification. Note that a single counterexample falsifies a universally quantified statement, and a single example suffices to verify an existentially quantified one. Assuming the axiom of choice, the game-theoretical semantics for classical first-order logic agree with the usual model-based (Tarskian) semantics. For classical first- order logic the winning strategy for the Verifier essentially consists of finding adequate Skolem functions and witnesses.
They illustrate what happens over time as populations genetically diverge, specifically because they represent, in living populations, what normally happens over time between long-deceased ancestor populations and living populations, in which the intermediates have become extinct. The evolutionary biologist Richard Dawkins remarks that ring species "are only showing us in the spatial dimension something that must always happen in the time dimension". Formally, the issue is that interfertility (ability to interbreed) is not a transitive relation; if A breeds with B, and B breeds with C, it does not mean that A breeds with C, and therefore does not define an equivalence relation. A ring species is a species with a counterexample to the transitivity of interbreeding.
However, there cannot be an infinity of ever-smaller natural numbers, and therefore by mathematical induction, the original premise--that any solution exists-- is incorrect: its correctness produces a contradiction. An alternative way to express this is to assume one or more solutions or examples exists, from which a smallest solution or example--a minimal counterexample—can then be inferred. Once there, one would try to prove that if a smallest solution exists, then it must imply the existence of a smaller solution (in some sense), which again proves that the existence of any solution would lead to a contradiction. The earliest uses of the method of infinite descent appear in Euclid's Elements.
Thompson's normal p-complement theorem used conditions on two particular characteristic subgroups of a Sylow p-subgroup. Glauberman improved this further by showing that one only needs to use one characteristic subgroup: the center of the Thompson subgroup. used his ZJ theorem to prove a normal p-complement theorem, that if p is an odd prime and the normalizer of Z(J(P)) has a normal p-complement, for P a Sylow p-subgroup of G, then so does G. Here Z stands for the center of a group and J for the Thompson subgroup. The result fails for p = 2 as the simple group PSL2(F7) of order 168 is a counterexample.
The image above shows a counterexample, and many discontinuous functions could, in fact, be written as a Fourier series of continuous functions. The erroneous claim that the pointwise limit of a sequence of continuous functions is continuous (originally stated in terms of convergent series of continuous functions) is infamously known as "Cauchy's wrong theorem". The uniform limit theorem shows that a stronger form of convergence, uniform convergence, is needed to ensure the preservation of continuity in the limit function. More precisely, this theorem states that the uniform limit of uniformly continuous functions is uniformly continuous; for a locally compact space, continuity is equivalent to local uniform continuity, and thus the uniform limit of continuous functions is continuous.
The statement is still a conjecture since it has not yet been proven that if a number n is not prime (that is, n is composite), then the formula does not hold. It has been shown that a composite number n satisfies the formula if and only if it is both a Carmichael number and a Giuga number, and that if such a number exists, it has at least 13,800 digits (Borwein, Borwein, Borwein, Girgensohn 1996). Laerte Sorini, finally, in a work of 2001 showed that a possible counterexample should be a number n greater than 1036067 which represents the limit suggested by Bedocchi for the demonstration technique specified by Giuga to his own conjecture.
There have also been several results relating coloring to minimum degree in triangle-free graphs. proved that any n-vertex triangle-free graph in which each vertex has more than 2n/5 neighbors must be bipartite. This is the best possible result of this type, as the 5-cycle requires three colors but has exactly 2n/5 neighbors per vertex. Motivated by this result, conjectured that any n-vertex triangle-free graph in which each vertex has at least n/3 neighbors can be colored with only three colors; however, disproved this conjecture by finding a counterexample in which each vertex of the Grötzsch graph is replaced by an independent set of a carefully chosen size.
The configuration in which these three ordinary lines are replaced by a single line cannot be realized in the Euclidean plane, but forms a finite projective space known as the Fano plane. Because of this connection, the Kelly–Moser example has also been called the non-Fano configuration. The other counterexample, due to McKee,. consists of two regular pentagons joined edge-to-edge together with the midpoint of the shared edge and four points on the line at infinity in the projective plane; these 13 points have among them 6 ordinary lines. Modifications of Böröczky's construction lead to sets of odd numbers of points with 3\lfloor n/4\rfloor ordinary lines.
The Beal conjecture is the following conjecture in number theory: :If :: A^x +B^y = C^z, :where A, B, C, x, y, and z are non-zero integers with x, y, z ≥ 3, then A, B, and C have a common prime factor. Equivalently, :The equation A^x + B^y = C^z has no solutions in non-zero integers and pairwise coprime integers A, B, C if x, y, z ≥ 3. The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's last theorem. Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample.
That is, in the context of something like the institution of a market, a customer ordering potatoes, etc. would entail that they owe the grocer compensation equal to the service that was provided. While Anscombe does acknowledge that an institutional context is necessary for a particular description to make sense, it does not necessarily follow that a particular set of facts holding true in an institutional context entails the fact brute relative to it. To wit, if the example is indeed considered in the institutional context necessary for descriptions of 'owing', it could still be the case that the customer does not owe the grocer, per the counterexample of a film production.
Starting from XIII-2, in-house development tools, such as Moomle and Rosetta, have been developed to ensure all parts of the process were properly synchronized and centralized. In recent years, English language localization teams have tended to adopt two different approaches to translation and localization: either they remain quite faithful to the original Japanese, or they can make large changes as long as the story outline remains the same. The former method was adopted for Final Fantasy XIII and its sequels, although some alterations were made in order to make the English dialogue sound natural. In choosing voice actors, the company prefers to avoid well-known film and television actors, citing Ellen Page's casting in Beyond: Two Souls as a counterexample.
This is false for some noncommutative rings, and a counterexample can be constructed using the Eilenberg swindle as follows. Let X be an abelian group such that X ≅ X ⊕ X (for example the direct sum of an infinite number of copies of any nonzero abelian group), and let R be the ring of endomorphisms of X. Then the left R-module R is isomorphic to the left R-module R ⊕ R. Example: If A and B are any groups then the Eilenberg swindle can be used to construct a ring R such that the group rings R[A] and R[B] are isomorphic rings: take R to be the group ring of the restricted direct product of infinitely many copies of A ⨯ B.
Though long used informally, this term has found a formal definition in category theory. ; pathological:An object behaves pathologically (or, somewhat more broadly used, in a degenerated way) if it either fails to conform to the generic behavior of such objects, fails to satisfy certain context-dependent regularity properties, or simply disobeys mathematical intuition. In many occasions, these can be and often are contradictory requirements, while in other occasions, the term is more deliberately used to refer to an object artificially constructed as a counterexample to these properties. :Note for that latter quote that as the differentiable functions are meagre in the space of continuous functions, as Banach found out in 1931, differentiable functions are colloquially speaking a rare exception among the continuous ones.
One of the most notorious pathologies in topology is the Alexander horned sphere, a counterexample showing that topologically embedding the sphere S2 in R3 may fail to separate the space cleanly. As a counter-example, it motivated the extra condition of tameness, which suppresses the kind of wild behavior the horned sphere exhibits. Like many other pathologies, the horned sphere in a sense plays on infinitely fine, recursively generated structure, which in the limit violates ordinary intuition. In this case, the topology of an ever-descending chain of interlocking loops of continuous pieces of the sphere in the limit fully reflects that of the common sphere, and one would expect the outside of it, after an embedding, to work the same.
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.
His primary research area is Ramsey theory of infinite sets. He is known for solutions to the basis problem for uncountable linear orders and to the L space problem from general topologyJustin Tatch Moore: A SOLUTION TO THE L SPACE PROBLEM, JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 19, Number 3, Pages 717–736 and for his work in determining the consequences of relating the continuum to certain values of the aleph function. Moore, together with his PhD student Yash Lodha, produced the first torsion-free counterexample to the von Neumann-Day problem, originally described by mathematician John von Neumann in 1929. Lodha presented this solution at the London Mathematical Society's Geometric and Cohomological Group Theory symposium in August 2013.
Using the formula, Varchenko constructed a counterexample to V. I. Arnold's semicontinuity conjecture that the brightness of light at a point on a caustic is not less than the brightness at the neighboring points. Varchenko formulated a conjecture on the semicontinuity of the spectrum of a critical point under deformations of the critical point and proved it for deformations of low weight of quasi- homogeneous singularities. Using the semicontinuity, Varchenko gave an estimate from above for the number of singular points of a projective hypersurface of given degree and dimension. Varchenko introduced the asymptotic mixed Hodge structure on the cohomology, vanishing at a critical point of a function, by studying asymptotics of integrals of holomorphic differential forms over families of vanishing cycles.
One can generalize the definition of cellular automaton to those maps that are defined by rules for computing the new value of each position in a configuration based on the values of cells in a finite but variable neighborhood surrounding the position. In this case, as in the classical definition, the local rule is the same for all cells, but the neighborhood is also a function of the configuration around the position. The counterexample given above for a continuous and shift-equivariant map which is not a classical cellular automaton, is an example of a generalized cellular automaton. When the alphabet is finite, the definition of generalized cellular automata coincides with the classical definition of cellular automata due to the compactness of the shift space.
The disproof of Keller's conjecture, for sufficiently high dimensions, has progressed through a sequence of reductions that transform it from a problem in the geometry of tilings into a problem in group theory, and from there into a problem in graph theory. first reformulated Keller's conjecture in terms of factorizations of abelian groups. He shows that, if there is a counterexample to the conjecture, then it can be assumed to be a periodic tiling of cubes with an integer side length and integer vertex positions; thus, in studying the conjecture, it is sufficient to consider tilings of this special form. In this case, the group of integer translations, modulo the translations that preserve the tiling, forms an abelian group, and certain elements of this group correspond to the positions of the tiles.
Notoriously, Hume considers and dismisses the 'missing shade of blue' counterexample. Briefly examining impressions, Hume then distinguishes between impressions of sensation (found in sense experience) and impressions of reflection (found mainly in emotional experience), only to set aside any further discussion for Book 2's treatment of the passions. Returning to ideas, Hume finds two key differences between ideas of the memory and ideas of the imagination: the former are more forceful than the latter, and whereas the memory preserves the "order and position" of the original impressions, the imagination is free to separate and rearrange all simple ideas into new complex ideas. But despite this freedom, the imagination still tends to follow general psychological principles as it moves from one idea to another: this is the "association of ideas".
A trivial theorem may be a result that can be derived in an obvious and straightforward way from other known results, or which applies only to a specific set of particular objects such as the empty set. In some occasions, however, a statement of a theorem can be original enough to be considered deep — even though its proof is fairly obvious. In his A Mathematician's Apology, Hardy suggests that a beautiful proof or result possesses "inevitability", "unexpectedness", and "economy". Rota, however, disagrees with unexpectedness as a sufficient condition for beauty and proposes a counterexample: Perhaps ironically, Monastyrsky writes: This disagreement illustrates both the subjective nature of mathematical beauty and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them.
A possible weakness of Bohm's theory is that some (including Einstein, Pauli, and Heisenberg) feel that it looks contrived. (Indeed, Bohm thought this of his original formulation of the theory.) It was deliberately designed to give predictions that are in all details identical to conventional quantum mechanics. Bohm's original aim was not to make a serious counter proposal but simply to demonstrate that hidden-variable theories are indeed possible. (It thus provided a supposed counterexample to the famous proof by John von Neumann that was generally believed to demonstrate that no deterministic theory reproducing the statistical predictions of quantum mechanics is possible.) Bohm said he considered his theory to be unacceptable as a physical theory due to the guiding wave's existence in an abstract multi-dimensional configuration space, rather than three-dimensional space.
The conjecture of Thompson that F is not amenable was further popularized by R. Geoghegan --- see also the Cannon- Floyd-Parry article cited in the references below. Its current status is open: E. Shavgulidze published a paper in 2009 in which he claimed to prove that F is amenable, but an error was found, as is explained in the MR review. It is known that F is not elementary amenable, see Theorem 4.10 in Cannon-Floyd- Parry. If F is not amenable, then it would be another counterexample to the long-standing but recently disproved von Neumann conjecture for finitely- presented groups, which suggested that a finitely-presented group is amenable if and only if it does not contain a copy of the free group of rank 2.
Many authors (Lindley, 1973; De Groot, 1937; Kass and Wasserman, 1996) warn against the danger of over-interpreting those priors since they are not probability densities. The only relevance they have is found in the corresponding posterior, as long as it is well-defined for all observations. (The Haldane prior is a typical counterexample.) By contrast, likelihood functions do not need to be integrated, and a likelihood function that is uniformly 1 corresponds to the absence of data (all models are equally likely, given no data): Bayes' rule multiplies a prior by the likelihood, and an empty product is just the constant likelihood 1. However, without starting with a prior probability distribution, one does not end up getting a posterior probability distribution, and thus cannot integrate or compute expected values or loss.
The first X-4, AF serial number 46-676, was transferred to the United States Air Force Academy, Colorado Springs, Colorado, before being returned to Edwards Air Force Base. 46-676 has been restored as of August 2012, and is currently being held in storage pending placement in the Edwards Museum. The second X-4 went to the National Museum of the United States Air Force at Wright-Patterson Air Force Base near Dayton, Ohio, where it remains on display. The X-4's primary importance involved proving a negative, in that a swept-wing semi-tailless design was not suitable for speeds near Mach 1, although Vought's F7U Cutlass proved to be something of a counterexample—the developed version was the first aircraft to demonstrate stores separation above Mach 1\.
In May 2020 it was announced that he would be assuming the title chaire de combinatoire at the College de France beginning in October 2020, though he intends to continue to reside in Cambridge and maintain a part-time affiliation at the University, as well enjoy the privileges of his life Fellowship of Trinity College. Gowers initially worked on Banach spaces. He used combinatorial tools in proving several of Stefan Banach's conjectures in the subject, in particular constructing a Banach space with almost no symmetry, serving as a counterexample to several other conjectures.1998 Fields Medalist William Timothy Gowers from the American Mathematical Society With Bernard Maurey he resolved the "unconditional basic sequence problem" in 1992, showing that not every infinite-dimensional Banach space has an infinite- dimensional subspace that admits an unconditional Schauder basis.
Films featuring an empowered damsel date to the early days of movie making. One of the films most often associated with the stereotype damsel in distress, The Perils of Pauline (1914), also provides at least a partial counterexample, in that Pauline, played by Pearl White, is a strong character who decides against early marriage in favour of seeking adventure and becoming an author. Despite common belief, the film does not feature scenes with Pauline tied to a railroad track and threatened by a buzzsaw, although such scenes were incorporated into later re-creations and were also featured in other films made in the period around 1914. Academic Ben Singer has contested the idea that these "serial-queen melodramas" were male fantasies and has observed that they were marketed heavily at women.
The term "anywhere-continuous" means that there exists at least a single point at which is continuous. As shown below, if for all and , and is continuous at any single point , then is necessarily continuous everywhere. :: (As a counterexample, if one does not assume continuity or measurability, it is possible to prove the existence of an everywhere-discontinuous, non- measurable function with this property by using a Hamel basis for the real numbers over the rationals, as described in Hewitt and Stromberg.) :: Because is guaranteed for rational by the above properties (see below), one could also use monotonicity or other properties to enforce the choice of for irrational , but such alternatives appear to be uncommon. :: One could also replace the conditions that and that be Lebesgue-measurable or anywhere-continuous with the single condition that .
The problem of finding a single simple cycle that covers each vertex exactly once, rather than covering the edges, is much harder. Such a cycle is known as a Hamiltonian cycle, and determining whether it exists is NP-complete.. Much research has been published concerning classes of graphs that can be guaranteed to contain Hamiltonian cycles; one example is Ore's theorem that a Hamiltonian cycle can always be found in a graph for which every non-adjacent pair of vertices have degrees summing to at least the total number of vertices in the graph.. The cycle double cover conjecture states that, for every bridgeless graph, there exists a multiset of simple cycles that covers each edge of the graph exactly twice. Proving that this is true (or finding a counterexample) remains an open problem...
A speaker may very well be using the sentence as a joke, as a codephrase, or even simply as an example of a sentence with 15 letters. Which the sentence really means cannot be determined without the specific purpose a person might be using the statement for in a specific context. In an article titled "Pragmatism and Pseudo-pragmatism" Schiller defends his pragmatism against a particular counterexample in a way that sheds considerable light on his pragmatism: > The impossibility of answering truly the question whether the 100th (or > 10,000th) decimal in the evaluation of Pi is or is not a 9, splendidly > illustrates how impossible it is to predicate truth in abstraction from > actual knowing and actual purpose. For the question cannot be answered until > the decimal is calculated.
In the first chapter of his book Pyrronian Reflexions on Truth and Justification,Oxford, Oxford University Press , 1994 Robert Fogelin gives a diagnosis that leads to a dialogical solution to Gettier's problem. The problem always arises when the given justification has nothing to do with what really makes the proposition true. Now, he notes that in such cases there is always a mismatch between the information disponible to the person who makes the knowledge-claim of some proposition p and the information disponible to the evaluator of this knowledge-claim (even if the evaluator is the same person in a later time). A Gettierian counterexample arises when the justification given by the person who makes the knowledge-claim cannot be accepted by the knowledge evaluator because it does not fit with his wider informational setting.
Although it follows from ZFC that every measurable cardinal is inaccessible (and is ineffable, Ramsey, etc.), it is consistent with ZF that a measurable cardinal can be a successor cardinal. It follows from ZF + axiom of determinacy that ω1 is measurable, and that every subset of ω1 contains or is disjoint from a closed and unbounded subset. Ulam showed that the smallest cardinal κ that admits a non-trivial countably- additive two-valued measure must in fact admit a κ-additive measure. (If there were some collection of fewer than κ measure-0 subsets whose union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.) From there, one can prove (with the Axiom of Choice) that the least such cardinal must be inaccessible.
The birth and development of Galois theory was caused by the following question, which was one of the main open mathematical questions until the beginning of 19th century: The Abel–Ruffini theorem provides a counterexample proving that there are polynomial equations for which such a formula cannot exist. Galois' theory provides a much more complete answer to this question, by explaining why it is possible to solve some equations, including all those of degree four or lower, in the above manner, and why it is not possible for most equations of degree five or higher. Furthermore, it provides a means of determining whether a particular equation can be solved that is both conceptually clear and easily expressed as an algorithm. Galois' theory also gives a clear insight into questions concerning problems in compass and straightedge construction.
He showed that it was likely that the curve could link Fermat and Taniyama, since any counterexample to Fermat's Last Theorem would probably also imply that an elliptic curve existed that was not modular. Frey showed that there were good reasons to believe that any set of numbers (a, b, c, n) capable of disproving Fermat's Last Theorem could also probably be used to disprove the Taniyama–Shimura–Weil conjecture. Therefore, if the Taniyama–Shimura–Weil conjecture were true, no set of numbers capable of disproving Fermat could exist, so Fermat's Last Theorem would have to be true as well. Mathematically, the conjecture says that each elliptic curve with rational coefficients can be constructed in an entirely different way, not by giving its equation but by using modular functions to parametrise coordinates x and y of the points on it.
Tait conjectured that every cubic polyhedral graph (that is, a polyhedral graph in which each vertex is incident to exactly three edges) has a Hamiltonian cycle, but this conjecture was disproved by a counterexample of W. T. Tutte, the polyhedral but non-Hamiltonian Tutte graph. If one relaxes the requirement that the graph be cubic, there are much smaller non- Hamiltonian polyhedral graphs. The graph with the fewest vertices and edges is the 11-vertex and 18-edge Herschel graph,. and there also exists an 11-vertex non-Hamiltonian polyhedral graph in which all faces are triangles, the Goldner–Harary graph.. More strongly, there exists a constant α < 1 (the shortness exponent) and an infinite family of polyhedral graphs such that the length of the longest simple path of an n-vertex graph in the family is O(nα)...
It is hypohamiltonian, meaning that although it has no Hamiltonian cycle, deleting any vertex makes it Hamiltonian, and is the smallest hypohamiltonian graph. As a finite connected vertex-transitive graph that does not have a Hamiltonian cycle, the Petersen graph is a counterexample to a variant of the Lovász conjecture, but the canonical formulation of the conjecture asks for a Hamiltonian path and is verified by the Petersen graph. Only five connected vertex-transitive graphs with no Hamiltonian cycles are known: the complete graph K2, the Petersen graph, the Coxeter graph and two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with a triangle.Royle, G. "Cubic Symmetric Graphs (The Foster Census)." If G is a 2-connected, r-regular graph with at most 3r + 1 vertices, then G is Hamiltonian or G is the Petersen graph.
Goursat was the first to note that the generalized Stokes theorem can be written in the simple form :\int_S \omega = \int_T d \omega where \omega is a p-form in n-space and S is the p-dimensional boundary of the (p + 1)-dimensional region T. Goursat also used differential forms to state the Poincaré lemma and its converse, namely, that if \omega is a p-form, then d\omega=0 if and only if there is a (p − 1)-form \eta with d \eta=\omega. However Goursat did not notice that the "only if" part of the result depends on the domain of \omega and is not true in general. Élie Cartan himself in 1922 gave a counterexample, which provided one of the impulses in the next decade for the development of the De Rham cohomology of a differential manifold.
Hans S. Witsenhausen (6 May 1930 in Frankfurt/Main, Germany - 19 November 2016 in New York City, New York) is notable for his work in the fields of control and information theory, and their intersection. He has many foundational results including the intrinsic model in stochastic decentralized control, the Witsenhausen counterexample, his work on Turán graph, and the various notions of common information in information theory. He received the I.C.M.E. degree in electrical engineering in 1953 and the degree of Licence en Sciences in mathematical physics in 1956, both from the Universite Libre de Bruxelles, Brussels, Belgium. He received the S.M. and Ph.D. degrees in electrical engineering from the Massachusetts institute of technology, Cambridge, in 1964 and 1966, respectively From 1957 to 1959 he was engaged in problem analysis and programming at the European Computation Center, Brussels.
In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a perfect set. As nonempty perfect sets in a Polish space always have the cardinality of the continuum, and the reals form a Polish space, a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis, stated in the form that every uncountable set of reals has the cardinality of the continuum. The Cantor–Bendixson theorem states that closed sets of a Polish space X have the perfect set property in a particularly strong form: any closed subset of X may be written uniquely as the disjoint union of a perfect set and a countable set.
The cycle double cover conjecture posits that in every bridgeless graph one can find a collection of cycles covering each edge twice, or equivalently that the graph can be embedded onto a surface in such a way that all faces of the embedding are simple cycles. Snarks form the difficult case for this conjecture: if it is true for snarks, it is true for all graphs.. In this connection, Branko Grünbaum conjectured that it was not possible to embed any snark onto a surface in such a way that all faces are simple cycles and such that every two faces either are disjoint or share only a single edge; however, a counterexample to Grünbaum's conjecture was found by Martin Kochol.... Work by Peter Tait established that the 4-color theorem is true if and only every snark is non-planar. Thus all snarks are non-planar.
In 2000 Schick proved the Atiyah conjecture for a large class of special cases.. In 2007 he presented a method which proved the Baum-Connes conjecture for the full braid groups, and for other classes of groups which arise as (finite) extensions for which the Baum-Connes conjecture is known to be true. arXiv preprint In the 1990s there were proofs of many special cases of the Gromov-Lawson-Rosenberg conjecture concerning criteria for the existence of a metric with positive scalar curvature; in 1997 Schick published the first counterexample. arXiv preprint He is the coordinator of the Courant Research Center's Strukturen höherer Ordnung in der Mathematik (Structures of Higher Order in Mathematics) at the University of Göttingen. A major goal of the research center is the investigation of mathematical structures that could play a role in modern theoretical physics, especially string theory and quantum gravity.
She is noted for her work in algebraic geometry particularly as it pertains to variations of Hodge structures and mirror symmetry, and has written several books on Hodge theory. In 2002, Voisin proved that the generalization of the Hodge conjecture for compact Kähler varieties is false.A counterexample to the Hodge conjecture extended to Kähler varieties The Hodge conjecture is one of the seven Clay Mathematics Institute Millennium Prize Problems which were selected in 2000, each having a prize of one million US dollars. Voisin won the European Mathematical Society Prize in 1992 and the Servant Prize awarded by the Academy of Sciences in 1996.Prix Servant décerné par l’Académie des Sciences (1996) She received the Sophie Germain Prize in 2003Claire Voisin awarded the 2003 Sophie Germain Academy of Sciences and the Clay Research Award in 2008 for her disproof of the Kodaira conjecture on deformations of compact Kähler manifolds.
A nontrivial representation of 0 as a sum of three cubes would give a counterexample to Fermat's last theorem for the exponent three, as one of the three cubes would have the opposite sign as the other two and its negation would equal the sum of the other two. Therefore, by Leonhard Euler's proof of that case of Fermat's last theorem, there are only the trivial solutions :a^3 + (-a)^3 + 0^3 = 0. For representations of 1 and 2, there are infinite families of solutions :(9b^4)^3+(3b-9b^4)^3+(1-9b^3)^3=1 (discovered by K. Mahler in 1936) and :(1+6c^3)^3+(1-6c^3)^3+(-6c^2)^3=2. (discovered by A.S. Verebrusov in 1908, quoted by L.J. Mordell) These can be scaled to obtain representations for any cube or any number that is twice a cube.
A counterexample by Ernst S. Selmer shows that the Hasse–Minkowski theorem cannot be extended to forms of degree 3: The cubic equation 3x3 + 4y3 + 5z3 = 0 has a solution in real numbers, and in all p-adic fields, but it has no nontrivial solution in which x, y, and z are all rational numbers. Roger Heath-Brown showed that every cubic form over the integers in at least 14 variables represents 0, improving on earlier results of Davenport. Since every cubic form over the p-adic numbers with at least ten variables represents 0, the local–global principle holds trivially for cubic forms over the rationals in at least 14 variables. Restricting to non-singular forms, one can do better than this: Heath-Brown proved that every non-singular cubic form over the rational numbers in at least 10 variables represents 0, thus trivially establishing the Hasse principle for this class of forms.
A consequence of this rule is that technetium and promethium both have no stable isotopes, as each of the neighboring elements on the periodic table (molybdenum and ruthenium, and neodymium and samarium, respectively) have a beta-stable isotope for each mass number for the range in which the isotopes of the unstable elements usually would be stable to beta decay. (Note that although 147Sm is unstable, it is stable to beta decay; thus 147 is not a counterexample). These ranges can be calculated using the liquid drop model (for example the stability of technetium isotopes), in which the isobar with the lowest mass excess or greatest binding energy is shown to be stable to beta decay because energy conservation forbids a spontaneous transition to a less stable state. Thus no stable nuclides have proton number 43 or 61, and by the same reasoning no stable nuclides have neutron number 19, 21, 35, 39, 45, 61, 71, 89, 115, or 123.
Many of Petersen’s early contributions to mathematics were mainly focused on geometry. During the 1860s he wrote five textbooks along with some papers, all on geometry. One of his most remarkable works was a book, ‘Methods and Theories’. The first edition of this book appeared only in Danish, but the 1879 edition was translated into eight different languages including English, French, and Spanish, earning him an international reputation more than any of his other works. In graph theory, two of Petersen’s most famous contributions are: the Petersen graph, exhibited in 1898, served as a counterexample to Tait’s ‘theorem’ on the 4-colour problem: a bridgeless 3-regular graph is factorable into three 1-factors and the theorem: ‘a connected 3-regular graph with at most two leaves contains a 1-factor’. In 1891 Petersen published a paper in the Acta Mathematica (volume 15, pages 193–220) entitled ‘Die Theorie der regularen graphs’.
Another early line of research by Grinbergs at the Computer Center concerned the automated design of ship hulls, and the computations with spline curves and surfaces needed in this design. The goal of this research was to calculate patterns for cutting and then bending flat steel plates so that they could be welded together to form ship hulls without the need for additional machining after the bending step; the methods developed by Grinbergs were later used throughout the Soviet Union. In graph theory, Grinbergs is best known for Grinberg's theorem, a necessary condition for a planar graph to have a Hamiltonian cycle that has been frequently used to find non-Hamiltonian planar graphs with other special properties.. His researches in graph theory also concerned graph coloring, graph isomorphism, cycles in directed graphs, and a counterexample to a conjecture of András Ádám on the number of cycles in tournaments. Other topics in Grinbergs' research include Steiner triple systems, magnetohydrodynamics, operations research, and the mathematical modeling of hydrocarbon exploration.
According to the Pitman-Koopman-Darmois theorem, among families of probability distributions whose domain does not vary with the parameter being estimated, only in exponential families is there a sufficient statistic whose dimension remains bounded as sample size increases. Less tersely, suppose Xk, (where k = 1, 2, 3, ... n) are independent, identically distributed random variables. Only if their distribution is one of the exponential family of distributions is there a sufficient statistic T(X1, ..., Xn) whose number of scalar components does not increase as the sample size n increases; the statistic T may be a vector or a single scalar number, but whatever it is, its size will neither grow nor shrink when more data are obtained. As a counterexample if these conditions are relaxed, the family of uniform distributions (either discrete or continuous, with either or both bounds unknown) has a sufficient statistic, namely the sample maximum, sample minimum, and sample size, but does not form an exponential family, as the domain varies with the parameters.
Factor-critical graphs must always have an odd number of vertices, and must be 2-edge-connected (that is, they cannot have any bridges).. However, they are not necessarily 2-vertex- connected; the friendship graphs provide a counterexample. It is not possible for a factor-critical graph to be bipartite, because in a bipartite graph with a near-perfect matching, the only vertices that can be deleted to produce a perfectly matchable graph are the ones on the larger side of the bipartition. Every 2-vertex-connected factor-critical graph with edges has at least different near-perfect matchings, and more generally every factor-critical graph with edges and blocks (2-vertex-connected components) has at least different near-perfect matchings. The graphs for which these bounds are tight may be characterized by having odd ear decompositions of a specific form.. Any connected graph may be transformed into a factor-critical graph by contracting sufficiently many of its edges.
We need the first condition because if the leading coefficient is negative then f(x) < 0 for all large x, and thus f(n) is not a (positive) prime number for large positive integers n. (This merely satisfies the sign convention that primes are positive.) We need the second condition because if f(x) = g(x)h(x) where the polynomials g(x) and h(x) have integer coefficients, then we have f(n) = g(n)h(n) for all integers n; but g(x) and h(x) take the values 0 and \pm 1 only finitely many times, so f(n) is composite for all large n. The third condition, that the numbers f(n) have gcd 1, is obviously necessary, but is somewhat subtle, and is best understood by a counterexample. Consider f(x) = x^2 + x + 2, which has positive leading coefficient and is irreducible, and the coefficients are relatively prime; however f(n) is even for all integers n, and so is prime only finitely many times (namely when f(n)=2, in fact only at n =0,-1).
Throughout his career, he had made every effort to modify this situation which sometimes made it necessary to appoint wealthy men with no diplomatic experience as chiefs of mission in sensitive posts. After the outbreak of World War II, at the request of former President Hoover, Gibson remained in Great Britain to negotiate authorization for the organization of food relief for the civilian population in territories occupied by German forces. Winston Churchill, who had opposed relief in 1914, remained hostile to the idea in 1940 ("the idea does more credit to heart than to head," he is reported to have said)Hugh Gibson to his son Michael Francis Gibson, in 1954. The British feared that the Nazi authorities would take whatever food was sent in. Gibson responded to this argument in 1944 with a counterexample: In 1941, he said the Turkish Government addressed to the British Foreign Office a request for permission to send food to the suffering Greeks. In reply it received the same objection that had been made to Gibson in 1940–1941.
Hajós's theorem is named after Hajós, and concerns factorizations of Abelian groups into Cartesian products of subsets of their elements.. This result in group theory has consequences also in geometry: Hajós used it to prove a conjecture of Hermann Minkowski that, if a Euclidean space of any dimension is tiled by hypercubes whose positions form a lattice, then some pair of hypercubes must meet face-to-face. Hajós used similar group- theoretic methods to attack Keller's conjecture on whether cube tilings (without the lattice constraint) must have pairs of cubes that meet face to face; his work formed an important step in the eventual disproof of this conjecture.. Hajós's conjecture is a conjecture made by Hajós that every graph with chromatic number contains a subdivision of a complete graph . However, it is now known to be false: in 1979, Paul A. Catlin found a counterexample for ,. and Paul Erdős and Siemion Fajtlowicz later observed that it fails badly for random graphs.. The Hajós construction is a general method for constructing graphs with a given chromatic number, also due to Hajós.. As cited by .
The problem of epiphenomena is often a counterexample to theories of causation and is identified with situations in which an event E is caused by (or, is said to be caused by) an event C, which also causes (or, is said to cause) an event F. For example, take a simplified Lewisian counterfactual analysis of causation that the meaning of propositions about causal relationships between two events A and B can be explained in terms of counterfactual conditionals of the form “if A had not occurred then B would not have occurred”. Suppose that C causes E and that C has an epiphenomenon F. We then have that if E had not occurred, then F would not have occurred, either. But then according to the counterfactual analysis of causation, the proposition that there is a causal dependence of F on E is true; that is, on this view, E caused F. Since this is not in line with how we ordinarily speak about causation (we would not say that E caused F), a counterfactual analysis seems to be insufficient.
In 1956–1957, working as a student of Alfred Tarski, Kallin helped simplify Tarski's axioms for the first-order theory of Euclidean geometry, by showing that several of the axioms originally presented by Tarski did not need to be stated as axioms, but could instead be proved as theorems from the other axioms... Kallin earned her Ph.D. in 1963 from Berkeley under the supervision of John L. Kelley. Her thesis, only 14 pages long, concerned function algebras, and a summary of its results was published in the Proceedings of the National Academy of Sciences.; One of its results, that not every topological algebra is localizable, has become a "well-known counterexample".. See in particular p. 89. In the study of complex vector spaces, a set S is said to be polynomially convex if, for every point x outside of S, there exists a polynomial whose complex absolute value at x is greater than at any point of S. This condition generalizes the ordinary notion of a convex set, which can be separated from any point outside the set by a linear function.
Leonard Eugene Dickson studied generalizations of Waring's problem for seventh powers, showing that every non-negative integer can be represented as a sum of at most 258 non-negative seventh powers. All but finitely many positive integers can be expressed more simply as the sum of at most 46 seventh powers. If negative powers are allowed, only 12 powers are required. The smallest number that can be represented in two different ways as a sum of four positive seventh powers is 2056364173794800. The smallest seventh power that can be represented as a sum of eight distinct seventh powers is: :102^7=12^7+35^7+53^7+58^7+64^7+83^7+85^7+90^7. The two known examples of a seventh power expressible as the sum of seven seventh powers are :568^7 = 127^7+ 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7 (M. Dodrill, 1999); and : 626^7=625^7+309^7+258^7+255^7+158^7+148^7+91^7 (Maurice Blondot, 11/14/2000); any example with fewer terms in the sum would be a counterexample to Euler's sum of powers conjecture, which is currently only known to be false for the powers 4 and 5.
In an article titled "Pork No Longer Paves the Way to Reelection," the Amherst Times cited Deborah Pryce as a counterexample of that thesis: "[In] several races... the ability to bring home hundreds of federal projects might have made enough of a difference to withstand a Democratic tide. Representative Deborah Pryce of Ohio, the fourth- ranking Republican in the House, issued dozens of news releases over the last 18 months boasting of the projects she brought home to a district that is considered evenly divided between the two parties[:] $2.27 million to convert a mountain of garbage into a green energy center, $1.1 million to help keep residents of a fast-growing suburb from having to pay more in user fees for a new sewage system, and the latest installment in $2.7 million in federal disbursements to 'evaluate freeze-dried berries for their ability to inhibit cancer'.... [At one point] Ms. Pryce's district stood to get the largest single earmark in Ohio—$1.75 million for a health research institute. In total, the Columbus area lined up about $4.5 million in special money.... By comparison, Portland, Ore.—a similar-sized metropolitan area with no contested Congressional seats—was to receive $625,000 in earmarks." Two debates were held for the 2006 congressional race.
If a polynomial is SOS-convex, then it is also convex. Since establishing whether a polynomial is SOS-convex amounts to solving a semidefinite programming problem, SOS-convexity can be used as a proxy to establishing if a polynomial is convex. In contrast, deciding if a generic polynomial of degree large than four is convex is a NP-hard problem. The first counterexample of a polynomial which is convex but not SOS-convex was constructed by Amir Ali Ahmadi and Pablo Parrilo in 2009. The polynomial is a homogeneous polynomial that is sum-of-squares and given by: > p(x)= 32 x_{1}^{8}+118 x_{1}^{6} x_{2}^{2}+40 x_{1}^{6} x_{3}^{2}+25 > x_{1}^{4} x_{2}^{4} -43 x_{1}^{4} x_{2}^{2} x_{3}^{2}-35 x_{1}^{4} > x_{3}^{4}+3 x_{1}^{2} x_{2}^{4} x_{3}^{2} -16 x_{1}^{2} x_{2}^{2} > x_{3}^{4}+24 x_{1}^{2} x_{3}^{6}+16 x_{2}^{8} +44 x_{2}^{6} x_{3}^{2}+70 > x_{2}^{4} x_{3}^{4}+60 x_{2}^{2} x_{3}^{6}+30 x_{3}^{8} In the same year, Grigoriy Blekherman proved in a non-constructive manner that there exist convex forms that is not representable as sum of squares.

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