Logic and Sets
... been done, with the following results: (1) 28 is the only 2-digit perfect number; (2) 496 is the only 3-digit perfect number; (3) 8128 is the only 4-digit perfect number; (4) there are no k-digit perfect numbers for k = 5, 6, 7; in fact, the next perfect number after 8128 is 33550336. So perfect num ...
... been done, with the following results: (1) 28 is the only 2-digit perfect number; (2) 496 is the only 3-digit perfect number; (3) 8128 is the only 4-digit perfect number; (4) there are no k-digit perfect numbers for k = 5, 6, 7; in fact, the next perfect number after 8128 is 33550336. So perfect num ...
39(1)
... A natural number n is harmonic if its positive divisors have integral harmonic mean H{ri). In our paper [1], we gave an algorithm for determining all harmonic squarefree multiples of a given harmonic number, based on our concept of a harmonic seed, but our program to implement the algorithm was faul ...
... A natural number n is harmonic if its positive divisors have integral harmonic mean H{ri). In our paper [1], we gave an algorithm for determining all harmonic squarefree multiples of a given harmonic number, based on our concept of a harmonic seed, but our program to implement the algorithm was faul ...
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... Functional admits the geometric interpretation that A is more symmetric than B i (A) (B ). For A 2 D we have (A ) = (A). De nition of norm and metric, as well as many topological properties of (D; +; ; ) are given in [6]. Some other properties of the extended interval arithmetic can be fo ...
... Functional admits the geometric interpretation that A is more symmetric than B i (A) (B ). For A 2 D we have (A ) = (A). De nition of norm and metric, as well as many topological properties of (D; +; ; ) are given in [6]. Some other properties of the extended interval arithmetic can be fo ...
Compositions of n with parts in a set
... Let A be any set of positive integers and n ∈ N. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. A palindromic composition of n with parts in A is a composition of n with parts in A in which the summands are the same in the given or in reverse ...
... Let A be any set of positive integers and n ∈ N. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. A palindromic composition of n with parts in A is a composition of n with parts in A in which the summands are the same in the given or in reverse ...
Modal logic and the approximation induction principle
... A context C[] denotes a formula containing one occurrence of []. The formula C[φ ] is obtained by replacing this occurrence of [] by the formula φ . It is well-known, and easy to see, that φ ⇒ ψ yields C[φ ] ⇒ C[ψ] for all contexts C[] over HML+ (here ϕ ⇒ ψ denotes that for any state s, s |= ϕ ⇒ s | ...
... A context C[] denotes a formula containing one occurrence of []. The formula C[φ ] is obtained by replacing this occurrence of [] by the formula φ . It is well-known, and easy to see, that φ ⇒ ψ yields C[φ ] ⇒ C[ψ] for all contexts C[] over HML+ (here ϕ ⇒ ψ denotes that for any state s, s |= ϕ ⇒ s | ...
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS
... REMARKS. The natural number N of the preceding definition surely depends on the positive number . If 0 is a smaller positive number than , then the corresponding N 0 very likely will need to be larger than N. Sometimes we will indicate this dependence by writing N () instead of simply N. It is a ...
... REMARKS. The natural number N of the preceding definition surely depends on the positive number . If 0 is a smaller positive number than , then the corresponding N 0 very likely will need to be larger than N. Sometimes we will indicate this dependence by writing N () instead of simply N. It is a ...