What Can Mathematical Chemistry Contribute to the Development of
... and other mathematical databases. The number of citations of the above paper has increased up to more than 900 now and become the second highest among all published papers in BCSJ. With these facts as the background I will convey my thoughts on how mathematical chemistry could and will be able to co ...
... and other mathematical databases. The number of citations of the above paper has increased up to more than 900 now and become the second highest among all published papers in BCSJ. With these facts as the background I will convey my thoughts on how mathematical chemistry could and will be able to co ...
Using linear logic to reason about sequent systems
... • The Forum specifications do not deal with context explicitly (side formulas): they only mention the formulas that are directly involved in the inference rule. • The distinction between additive and multiplicative inference rules is achieved using the appropriate linear logic connective. • Object-l ...
... • The Forum specifications do not deal with context explicitly (side formulas): they only mention the formulas that are directly involved in the inference rule. • The distinction between additive and multiplicative inference rules is achieved using the appropriate linear logic connective. • Object-l ...
Round 1 Solutions
... where “division by a” is well-defined modulo c. (Indeed, with a = 689, c = 1489 we can check that a1 ≡ 389 (mod c), since (389)(689) = 268021 = (180)(1489) + 1.) So there is a unique v with 0 < v < c satisfying (1). To verify that this also satisfies (2), note that since a2 + b2 = c2 , we have b2 ≡ ...
... where “division by a” is well-defined modulo c. (Indeed, with a = 689, c = 1489 we can check that a1 ≡ 389 (mod c), since (389)(689) = 268021 = (180)(1489) + 1.) So there is a unique v with 0 < v < c satisfying (1). To verify that this also satisfies (2), note that since a2 + b2 = c2 , we have b2 ≡ ...
The greatest common divisor: a case study for program extraction
... Yiannis Moschovakis suggested the following example of a classical existence proof with a quantifier–free kernel which does not obviously contain an algorithm: the gcd of two natural numbers a1 and a2 is a linear combination of the two. Here we treat that example as a case study for program extracti ...
... Yiannis Moschovakis suggested the following example of a classical existence proof with a quantifier–free kernel which does not obviously contain an algorithm: the gcd of two natural numbers a1 and a2 is a linear combination of the two. Here we treat that example as a case study for program extracti ...
full text (.pdf)
... local; doing so can lead to anomalies such as the Dead Variable Paradox [Kozen and Patron 2000]. One major distinguishing factor between tests and correctness assertions that may not be immediately apparent is that the former are classical in nature, whereas the latter are intuitionistic. For exampl ...
... local; doing so can lead to anomalies such as the Dead Variable Paradox [Kozen and Patron 2000]. One major distinguishing factor between tests and correctness assertions that may not be immediately apparent is that the former are classical in nature, whereas the latter are intuitionistic. For exampl ...
cmsc203-sep9
... Theorem. x=2p-1(2p-1) is a perfect number when 2p-1 is prime. Proof. For x to be a perfect number, it must be equal to the sum of its proper divisors. The divisors of 2p-1 are 1, 2, 22, ..., 2p-1. Since 2p-1 is prime, its only divisors are 1 and itself. Therefore, the proper divisors of x are 1, ...
... Theorem. x=2p-1(2p-1) is a perfect number when 2p-1 is prime. Proof. For x to be a perfect number, it must be equal to the sum of its proper divisors. The divisors of 2p-1 are 1, 2, 22, ..., 2p-1. Since 2p-1 is prime, its only divisors are 1 and itself. Therefore, the proper divisors of x are 1, ...
![arXiv:math/0511682v1 [math.NT] 28 Nov 2005](http://s1.studyres.com/store/data/014696627_1-8def914a5ac3ed74bde3727e1309931c-300x300.png)
Russell`s logicism
... talking about the number 3, and number in general, as properties or characteristics. Now he is moving from this to talking about the number 3, and number in general, as sets. The next question is, what sets are they? Russell says: “Reurning now to the definition of number, it is clear that number i ...
... talking about the number 3, and number in general, as properties or characteristics. Now he is moving from this to talking about the number 3, and number in general, as sets. The next question is, what sets are they? Russell says: “Reurning now to the definition of number, it is clear that number i ...
DOC - Rose
... life. A less common way to represent a number is the so called Cantor expansion. Often presented as exercises in discrete math and computer science courses [8.2, 8.5], this system uses factorials rather than exponentials as the basis for the representation. It can be shown that the expansion is uniq ...
... life. A less common way to represent a number is the so called Cantor expansion. Often presented as exercises in discrete math and computer science courses [8.2, 8.5], this system uses factorials rather than exponentials as the basis for the representation. It can be shown that the expansion is uniq ...
ON DIOPHANTINE APPROXIMATIONS^)
... of the three classes (i) p, q both odd, (ii) p odd, q even, or (iii) p even, q odd, then there are infinitely many such p/q satisfying (1). Other proofs of this result have been given by Robinson [22], Oppenheim [20] and Kuipers and Meulenbeld [ll]. Robinson also showed that if any pair of these cla ...
... of the three classes (i) p, q both odd, (ii) p odd, q even, or (iii) p even, q odd, then there are infinitely many such p/q satisfying (1). Other proofs of this result have been given by Robinson [22], Oppenheim [20] and Kuipers and Meulenbeld [ll]. Robinson also showed that if any pair of these cla ...
vmcai - of Philipp Ruemmer
... In program verification, an interpolating theorem prover often interacts tightly with various decision procedures. It is therefore advantageous for the interpolants computed by the prover to be expressible in simple logic fragments. Unfortunately, interpolation procedures for expressive first-order ...
... In program verification, an interpolating theorem prover often interacts tightly with various decision procedures. It is therefore advantageous for the interpolants computed by the prover to be expressible in simple logic fragments. Unfortunately, interpolation procedures for expressive first-order ...
Lebesgue`s Dominated Convergence Theorem in Bishop`s Style
... resulting proof is thus very different, both in spirit and technically, from Fremlin’s one. Spitters’ proof cannot be adapted in the most general setting of uniform spaces since it is given for an archimedean lattice vector space that is also an algebra with a multiplicative unit. In fairness to him ...
... resulting proof is thus very different, both in spirit and technically, from Fremlin’s one. Spitters’ proof cannot be adapted in the most general setting of uniform spaces since it is given for an archimedean lattice vector space that is also an algebra with a multiplicative unit. In fairness to him ...
ON A VARIATION OF PERFECT NUMBERS Douglas E. Iannucci
... Φ97 (3) has any prime divisors q > 103 . By Lemma 1, q ≡ 1 (mod 194), so it suffices merely to test all such primes q < 103 to see if 397 ≡ 1 (mod q). Thus we obtain the product, say R, of all primes less than 103 which divide Φ97 (3). If ln R < 96 ln 3, then Φ97 (3) must contain a prime divisor gre ...
... Φ97 (3) has any prime divisors q > 103 . By Lemma 1, q ≡ 1 (mod 194), so it suffices merely to test all such primes q < 103 to see if 397 ≡ 1 (mod q). Thus we obtain the product, say R, of all primes less than 103 which divide Φ97 (3). If ln R < 96 ln 3, then Φ97 (3) must contain a prime divisor gre ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.