Exercise 25.2. Recall that for A ∈Mn(R) with A = A , the Rayleigh quotient is defined as RA(x) = xTAx xT x . For symmetric φ : V → V , the Rayleigh quotient is defined as Rφ(x) = 〈x,φx〉 ||x||2 . Use the above theorem to show that argmaxxRφ(x) is achieved on V \ {0}. Proof. Note that Rφ(αx) = Rφ(x). So any value achieved by Rφ on V \ {0} is achieved on the unit sphere {x ∈ V | ||x|| = 1}. Restri...

a system has a solution. Recall that if x =  x1 .. x` , and the matrix A is written as A = [ a1 . . . a` ] , where ai is the ith column of the matrix, then Ax = ∑i=` i=1 xiai is simply a linear combination of the columns. Therefore if Ax = b holds, then b ∈ Span{a1, . . . ,a`}. Recall that the span of the ai is called the column space of the matrix A. It follows that rk(A) = rk ([A|b]), whe...

This work presents a new, simple O(log |G|) algorithm, the Fibonacci cube algorithm, for producing random group elements in black box groups. After the initial O(log |G|) group operations, ε-uniform random elements are produced using O((log 1/ε) log |G|) operations each. This is the first major advance over the ten year old result of Babai [Bab91], which had required O(log |G|) group operations...

The genus Hippasa is a group of web-weaving lycosids, whose web is funnel-like, quite similar to those of the Agelenidae. In this paper, based on published papers and specimens from China, we diagnose and discuss the composition of this genus. Two species groups are recognised, the Hippasa greenalliae-group (12 species and a subspecies) mainly from Asia and the Hippasa partita-group (17 species...

The elaterid beetle fauna has been thoroughly investigated on Yaku Island, one of the Ôsumi Islands, Ryukyu Japan, and many species have added to or excluded from fauna. However, is difficult assess because there discordance in information distribution records. In this study, we examined specimens collected by us as well reviewed previous records Island understand island. During our f...