Questions and conjectures on extremal Hilbert series

Three conjectures in extremal spectral graph theory

We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of P2 and Pn−2 is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetković and Rowlinson from 1...

Some Hilbert Space Extremal Problems

is achieved. This is given by e¡ = (1 -\-'Kkj)~1hj, where X is the unique positive root of £"-0 kj(l +\k])-1\hj\2 = M2. Davis proved the result by first considering the extremal problem for ra dimensional Euclidean space Rn. In this case the problem has the following geometric interpretation : (a") Let E be the hyperellipse {(oi, • • • , an): ]£?„„ kjOJ^M2} in Rn and suppose that h = (hi, • ■ ■...

A New Series of Conjectures and Open Questions in Optimization and Matrix Analysis

We present below a new series of conjectures and open problems in the fields of (global) Optimization and Matrix analysis, in the same spirit as our recently published paper, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization, SIAM Review, Vol. 49, No 2, pp. 255-273 (published electronically on May 1, 2007). With each problem come a succinct presentation, a list ...

Tate Conjectures for Hilbert Modular Surfaces

the effects of cooperative versus individual directed reading thinking activities on students inferential and referential questions

two groups of students were assigned as experimental and control ones, and were given instruction on directed reading-thinking activities and after some treatment, they were post-tested. although the initial pre-test did not show any significant differences, the final post-test result revealed that the cooperative reading comprehension helped the experimental group. the cooperative students’...