How many knots have the same group?

How Many Pairs of Products of Consecutive Integers Have the Same Prime Factors?

e _ 2'/z 2 4 4 6 6 8 1 / 8 2 () (33) 1/4 ( 5577) which is proved as follows . For v > 2, the Pth factor is [2' '. . 2P/(2p'+ 1) . (2 1)]'áz [(2"-'-1)!!z2"!!z/2 .2v-'!!z(2v-1)!!z]'/z where n!!=n(n-2) . . .4.2 if n is even, n(n-2) . . .3 .1 if n is odd. Since 2°!!=2 z " '2° '1 and (2°-1)!!=2°! /2°!!=2°!/2z" '2°'!, this expression becomes [2z"2°-'16/2 .2 zí42'!z]I/z By induction on v, the product ...

How many Adjunctions give Rise to the same Monad?

Given an adjoint pair of functors F,G, the composite GF naturally gets the structure of a monad. The same monad may arise from many such adjoint pairs of functors, however. Can one describe all of the adjunctions giving rise to a given monad? In this paper we single out a class of adjunctions with especially good properties, and we develop methods for computing all such adjunctions, up to natur...

How many species have mass M?

Within large taxonomic assemblages, the number of species with adult body mass M is characterized by a broad but asymmetric distribution, with the largest mass being orders of magnitude larger than the typical mass. This canonical shape can be explained by cladogenetic diffusion that is bounded below by a hard limit on viable species mass and above by extinction risks that increase weakly with ...

Perspective. How many women have osteoporosis?

Osteoporosis is widely viewed as a major public health concern, but the exact magnitude of the problem is uncertain and likely to depend on how the condition is defined. Noninvasive bone mineral measurements can be used to define a state of heightened fracture risk (osteopenia), or the ultimate clinical manifestation of fracture can be assessed (established osteoporosis). If bone mineral measur...

How many doctors should Sri Lanka have?