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proof and general
For the frequent case of propositional logic, the problem is decidable but Co-NP-complete, and hence only exponential-time algorithms are believed to exist for general proof tasks.
A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.
Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.
The extension ψ is in general not uniquely specified by φ, and the proof gives no explicit method as to how to find ψ: in the case of an infinite dimensional space V, it depends on Zorn's lemma, one formulation of the axiom of choice.
In these papers he sketched a proof of the Poincaré conjecture and a more general conjecture, Thurston's geometrization conjecture, completing the Ricci flow program outlined earlier by Richard Hamilton.
The estrangement between the secular and the spiritual authority which Ganganelli strove to avert is now irreparable, and his pontificate remains an exceptional episode in the general history of the Papacy, and a proof how little the logical sequence of events can be modified by the virtues and abilities of an individual.
The proof works just as well if we have an algorithm for deciding any other nontrivial property of programs, and will be given in general below.
She outlined a strategy for a general proof of Fermat's Last Theorem, including a proof for a special case ( see Best Work on Fermat's Last Theorem ).
* NATO's secretary general, Lord Robertson announces that the United States provided " clear and compelling proof " in oral briefings to NATO al-Qaeda's responsibility who affirm the invocation of the mutual defense clause of the organization's charter.
In like manner, I apprehend, the sole evidence it is possible to produce that anything is desirable, is that people do actually desire it … No reason can be given why the general happiness is desirable, except that each person, so far as he believes it to be attainable, desires his own happiness … we have not only all the proof which the case admits of, but all which it is possible to require, that happiness is a good: that each person's happiness is a good to that person, and the general happiness, therefore, a good to the aggregate of all persons .”
Another proof of the general case for n can be done by technique used to prove Inequality of arithmetic and geometric means.
Proof: We only give the proof in the simplified case ; the general case is similar.
The technique of pi – theorem ’ s usage (`` the method of dimensions ’’) became widely known due to the works of Rayleigh ( the first application of the pi – theorem in the general case to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892, a heuristic proof with the use of series expansion, to 1894 ).
However, it demonstrates a powerful and general technique that has since been used in a wide range of proofs, also known as diagonal arguments by analogy with the argument used in this proof.
His other contributions include a simplified proof of the positive energy theorem involving spinors in general relativity, his work relating supersymmetry and Morse theory, his introduction of topological quantum field theory and related work on mirror symmetry, knot theory, twistor theory and D-branes and their intersections.
While the proof that is a Noetherian ring uses the order structure of, typical proofs in ring theory in general do not assume such additional structure on the ring.
Lack of proper content is not an affirmative defense ; the plaintiff must prove the allegations even if uncontroverted ; proof is made according to the general rules of evidence.
The cut-elimination ( or equivalently the normalization of the underlying calculus if there is one ) implies the consistency of the calculus: since there is obviously no cut-free proof of falsity, there is no contradiction in general.
When Emil Post in his 1921 Introduction to a general theory of elementary propositions extended his proof of the consistency of the propositional calculus ( i. e. the logic ) beyond that of Principia Mathematica ( PM ) he observed that with respect to a generalized set of postulates ( i. e. axioms ) he would no longer be able to automatically invoke the notion of " contradiction " – such a notion might not be contained in the postulates:
Actually, the correctness of the Bieberbach conjecture was only the most important consequence of de Branges's proof, which covers a more general problem, the Milin conjecture.
In the other one ( 57 pages ), he claims to modify his earlier approach on the subject by means of spectral theory and harmonic analysis to obtain a proof of RH for Hecke L-functions, a group even more general than Dirichlet L-functions ( which would imply an even more powerful result if his claim were proved correct ).
The group concluded a general acceptance of the need to decommission, though there was no conclusive proof of moves towards this end.
As an ecclesiastical statesman, he showed the same fiery zeal and versatility of which he had given proof in his academic career ; but the general tendency of modern writers has been to exaggerate his political and ecclesiastical services, and to neglect his performance as a scientist and scholar.

proof and case
The reader will find it helpful to think of the special case when the primes are of degree 1, and even more particularly, to think of the proof of Theorem 10, a special case of this theorem.
In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.
In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis ( in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i. e. no parallel postulate.
More recently, both Elms and librarian Lee Weinstein have gathered circumstantial evidence to support the case for Linebarger's being " Allen ," but both concede there is no direct proof that Linebarger was ever a patient of Lindner's or that he suffered from a disorder similar to that of " Kirk Allen.
The completed proof of the classification was announced by after Aschbacher and Smith published a 1221 page proof for the missing quasithin case.
In propositional logic, disjunction elimination ( sometimes named proof by cases or case analysis ), is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof.
Gödel's original proof of the theorem proceeded by reducing the problem to a special case for formulas in a certain syntactic form, and then handling this form with an ad hoc argument.
proved the theorem ( for the special case of polynomial rings over a field ) in the course of his proof of finite generation of rings of invariants.
However, there is a body of case law governing the civil commitment of individuals under the Fourteenth Amendment through U. S. Supreme Court rulings beginning with Addington v. Texas in 1979 which set the bar for involuntary commitment for treatment by raising the burden of proof required to commit persons from the usual civil burden of proof of " preponderance of the evidence " to the higher standard of " clear and convincing " evidence.
' In the second case, he cites an example that demonstrates ignorance of statistical principles in the lay press: ' Since no such proof is possible genetically modified food is harmless, the article in The New York Times was what is called a " bad rap " against the U. S. Department of Agriculture-a bad rap based on a junk-science belief that it's possible to prove a null hypothesis.
This might come in the form of a proof that the number in question is in fact irrational ( or rational, as the case may be ); or a finite algorithm that could determine whether the number is rational or not.
In the case of seL4, complete formal verification of the implementation has been achieved, i. e. a mathematical proof that the kernel's implementation is consistent with its formal specification.
It is widely believed that ( 1 ) is the case, although no proof as to the truth of either statement has yet been discovered.
In most legal proceedings, one party has a burden of proof, which requires it to present prima facie evidence for all of the essential facts in its case.
Rules of evidence govern whether, when, how, and for what purpose, proof of a legal case may be placed before a trier of fact for consideration.
Boehm claimed to have cured gay people in the fourth part of his series on homosexuality, but presented as proof a case in which “ the homosexuality never became conscious for the patient and had never expressed itself in manifest activity .” This patient does not appear to have been homosexual.

proof and requires
The ownership and purchase of firearms requires a specific reason: if one wants to have a gun for hunting or target shooting, a person must obtain a proof of their membership in a shooting sports organization.
The latter requires a proof of rain, whereas the former merely requires a proof that rain would not be contradictory.
This crime requires an extra level of satisfactory proof, as prosecutors must show not only that perjury occurred, but also that the defendant positively induced said perjury.
One of the Millennium Prize Problems announced by the Clay Mathematics Institute requires a claimant to produce such a proof.
The more severe penalties available in criminal law also means that it requires a higher burden of proof to be discharged than the related tort.
Under a no-fault divorce system, divorce requires no allegation or proof of fault of either party.
The fallacy of petitio principii, or " begging the question ", is committed " when a proposition which requires proof is assumed without proof ", or more generally denotes when an assumption is used, " in some form of the very proposition to be proved, as a premise from which to deduce it ".
In the United Kingdom and countries such as Malaysia and Singapore, informed consent in medical procedures requires proof as to the standard of care to be expected as a recognised standard of acceptable professional practice ( the Bolam Test ), that is, what risks would a medical professional usually disclose in the circumstances ( see Loss of right in English law ).
For many natural P-complete graph problems, where the graph is expressed in a natural representation such as an adjacency matrix, solving the same problem on a succinct circuit representation is EXPTIME-complete, because the input is exponentially smaller ; but this requires nontrivial proof, since succinct circuits can only describe a subclass of graphs.
According to Quebec law, Montreuil could not change her record of sex because this requires proof of a completed sex reassignment surgery, which she has not had.
Gödel's original statement and proof of the incompleteness theorem requires the assumption that the theory is not just consistent but ω-consistent.
J. Barkley Rosser ( 1936 ) strengthened the incompleteness theorem by finding a variation of the proof ( Rosser's trick ) that only requires the theory to be consistent, rather than ω-consistent.
Another proposed scheme is DHAES, whose proof requires an assumption that is weaker than the DDH assumption.
The section requires proof that the announcement is both provably false, and has actual or likely market impact.
In this standard, a greater degree of believability must be met than the common standard of proof in civil actions, " Preponderance of the Evidence ", which requires that the facts as a threshold be more likely than not to prove the issue for which they are asserted.
This falls out of the fact that is an equivalence relation and also requires a proof that ( 1 ) and ( 2 ) are independent of the choice of class representatives.
A Consistency proof requires ( i ) an axiomatic system ( ii ) a demonstration that it is not the case that both the formula p and its negation ~ p can derived in the system.
This proof requires the most basic elements of group theory.
Theophrastus introduced his Physics with the proof that all natural existence, being corporeal and composite, requires principles, and first and foremost, motion, as the basis of all change.
It is submitted to the state by an auto insurance company to serve as proof that a driver has the minimum liability insurance that the states requires. SR-22s are typically filed with the respective State's DMV, and in some States must be carried by the licensed driver, or in the registered vehicle ( particularly if the licensee has been cited for coverage lapses, DUI or other administrative infractions ).

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