Fibonacci numbers
... Theorem 3. The Fibanacci number un is the nearest whole number to the nth term αn of the geometric progression whose first term is √α5 and whose common ratio is α. αn That is, un is the nearest whole number to αn = √ ...
... Theorem 3. The Fibanacci number un is the nearest whole number to the nth term αn of the geometric progression whose first term is √α5 and whose common ratio is α. αn That is, un is the nearest whole number to αn = √ ...
Uniform distribution of zeros of Dirichlet series,
... Consequently under the analogue of the Riemann Hypothesis for this class, the imaginary parts of zeros of elements of this class are uniformly distributed mod 1. However, proving these facts unconditionally is a new and a difficult problem in analytic number theory. To establish some unconditional r ...
... Consequently under the analogue of the Riemann Hypothesis for this class, the imaginary parts of zeros of elements of this class are uniformly distributed mod 1. However, proving these facts unconditionally is a new and a difficult problem in analytic number theory. To establish some unconditional r ...
Removing Independently Even Crossings
... of adjacent edges are trivial, and easily got rid of.” While this is true for the standard crossing number, it is not at all obvious for other variants. Székely [13] later commented “We interpret this sentence as a philosophical view and not a mathematical claim.” In [5], Pach and Tóth suggest a s ...
... of adjacent edges are trivial, and easily got rid of.” While this is true for the standard crossing number, it is not at all obvious for other variants. Székely [13] later commented “We interpret this sentence as a philosophical view and not a mathematical claim.” In [5], Pach and Tóth suggest a s ...
Mathematical Olympiad in China : Problems and Solutions
... score for a team is 252 marks. About half of the participants will be awarded a medal, where 1/12 will be awarded a gold medal. The numbers of gold, silver and bronze medals awarded are in the ratio of 1:2:3 approximately. In the case when a participant provides a better solution than the official a ...
... score for a team is 252 marks. About half of the participants will be awarded a medal, where 1/12 will be awarded a gold medal. The numbers of gold, silver and bronze medals awarded are in the ratio of 1:2:3 approximately. In the case when a participant provides a better solution than the official a ...
CS1231 - Lecture 09
... Which means (by definition of ‘;’), we need to show that there exists a bijection from 2Z to Z+ (or vice versa). Define f : Z2Z such that f(n) = 2n Now, f is a bijection from Z to 2Z. Which means that |2Z|;|Z| But we have also shown that |Z|;|Z+|. So |2Z|;|Z+| (Since ; is transitive) ...
... Which means (by definition of ‘;’), we need to show that there exists a bijection from 2Z to Z+ (or vice versa). Define f : Z2Z such that f(n) = 2n Now, f is a bijection from Z to 2Z. Which means that |2Z|;|Z| But we have also shown that |Z|;|Z+|. So |2Z|;|Z+| (Since ; is transitive) ...
A Judgmental Reconstruction of Modal Logic
... An alternative way to understand local completeness is to reconsider our meaning explanation of conjunction. We have said that a verification of A ∧ B consists of a verification of A and a verification of B. Local completeness entails that it is always possible to bring the verification of A ∧ B int ...
... An alternative way to understand local completeness is to reconsider our meaning explanation of conjunction. We have said that a verification of A ∧ B consists of a verification of A and a verification of B. Local completeness entails that it is always possible to bring the verification of A ∧ B int ...
Supplementary Notes
... there is a randomized polynomial-time algorithm A(x, r) such that for every input x, A(x, r) outputs the correct answer with probability at least 1 − 2−1000 . So when we discover an efficient randomized algorithm for a problem, it is reasonable to consider that problem to be solved for all practical ...
... there is a randomized polynomial-time algorithm A(x, r) such that for every input x, A(x, r) outputs the correct answer with probability at least 1 − 2−1000 . So when we discover an efficient randomized algorithm for a problem, it is reasonable to consider that problem to be solved for all practical ...
and let A,B be finitely generated graded S-modules. If T is a
... d, then I t = mdt for all t 0. This theorem relies on our specialization results in Section 5. The following theorem proves Conjecture 1.1 in the case n = 3, and gives more precise information than Theorem 1.2. It is perhaps the most surprising result of this paper. THEOREM 1.3. Suppose I and J ar ...
... d, then I t = mdt for all t 0. This theorem relies on our specialization results in Section 5. The following theorem proves Conjecture 1.1 in the case n = 3, and gives more precise information than Theorem 1.2. It is perhaps the most surprising result of this paper. THEOREM 1.3. Suppose I and J ar ...
Uniform satisfiability in PSPACE for local temporal logics over
... next and (universal) until has the same expressive power as first order logic over traces [4]. Moreover, local temporal logics have usually a low complexity, i.e., satisfiability can be solved in PSPACE. We cannot expect a lower complexity since already the classical temporal logic LTL over sequence ...
... next and (universal) until has the same expressive power as first order logic over traces [4]. Moreover, local temporal logics have usually a low complexity, i.e., satisfiability can be solved in PSPACE. We cannot expect a lower complexity since already the classical temporal logic LTL over sequence ...
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.