منابع مشابه
On the Number of Minimal Surfaces with a given Boundary
In [TT78], Tomi and Tromba used degree theory to solve a longstanding problem about the existence of minimal surfaces with a prescribed boundary: they proved that every smooth, embedded curve on the boundary of a convex subset of R must bound an embedded minimal disk. Indeed, they proved that a generic such curve must bound an odd number of minimal embedded disks. White [Whi89] generalized thei...
متن کاملThe number of permutations with a given number of sequences
P (n, s) denotes the number of permutations of 1, 2,. .. n that have exactly s sequences. Canfield and Wilf [math.CO/0609704] recently showed that P (n, s) can be written as a sum of s polynomials in n. We determine these polynomials explicitly and also obtain explicit expressions for P (n, s) and its fixed-s generating function us(x).
متن کاملOn the Number of Permutations Avoiding a Given Pattern
Let σ ∈ Sk and τ ∈ Sn be permutations. We say τ contains σ if there exist 1 ≤ x1 < x2 < . . . < xk ≤ n such that τ(xi) < τ(xj) if and only if σ(i) < σ(j). If τ does not contain σ we say τ avoids σ. Let F (n, σ) = |{τ ∈ Sn| τ avoids σ}|. Stanley and Wilf conjectured that for any σ ∈ Sk there exists a constant c = c(σ) such that F (n, σ) ≤ cn for all n. Here we prove the following weaker statemen...
متن کاملOn the Number of Markoff Numbers Below a Given Bound
According to a famous theorem of Markoff, the indefinite quadratic forms with exceptionally large minima (greater than f of the square root of the discriminant) are in 1 : 1 correspondence with the solutions of the Diophantine equation p2 + q2 + r1 = ~ipqr. By relating Markoffs algorithm for finding solutions of this equation to a problem of counting lattice points in triangles, it is shown tha...
متن کاملThe minimal number with a given number of divisors
A number n is said to be ordinary if the smallest number with exactly n divisors is p q1−1 1 · · · p qa−1 a where q1 · · · qa is the prime factorization of n and q1 ≥ . . . ≥ qa (and where pk denotes the k-th prime). We show here that all square–free numbers are ordinary and that the set of ordinary numbers has natural density one.
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ژورنال
عنوان ژورنال: Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics
سال: 1982
ISSN: 0263-6115
DOI: 10.1017/s1446788700017547