The Mathematical Tripos

MATHEMATICAL TRIPOS : PART II Michaelmas Term 2013 OPTIMIZATION AND CONTROL

1. Suppose that the matrix Mk is of dimension nk × nk+1, k ∈ {1, . . . , h}. We wish to compute the product M1M2 · · ·Mh. Notice that the order of multiplication makes a difference. For example, if (n1, n2, n3, n4) = (1, 10, 1, 10), the calculation (M1M2)M3 requires 20 scalar multiplications, but the calculation M1(M2M3) requires 200 scalar multiplications. Indeed, multiplying a m × n matrix by...

Tripos theory

Introduction. One of the most important constructions in topos theory ia that of the category Shv (̂ 4) of sheaves on a locale (= complete Heyting algebra) A. Normally, the objects of this category are described as 'presheaves on A satisfying a gluing condition'; but, as Higgs(7) and Fourman and Scott(5) have observed, they may also be regarded as 'sets structured with an A -valued equality pred...

Permutation Models for Set Theory An essay submitted for Part III of the Mathematical Tripos

Tripos Theory in Retrospect

The notion of tripos (Hyland, Johnstone, and Pitts 1980; Pitts 1981) was motivated by the desire to explain in what sense Higg’s description of sheaf toposes as H-valued sets and Hyland’s realizability toposes are instances of the same construction. The construction itself can be seen as the universal solution to the problem of realizing the predicates of a first order hyperdoctrine as subobjec...

Computational Statistics and Statistical Modelling: Mathematical Tripos, Part IIA, questions from 1997 onwards

where H is a matrix that you should define. If now Xβ is written as Xβ = X1β1 +X2β2, where X = (X1 : X2), β T = (β 1 : β T 2 ), and β2 is of dimension p2, state without proof the form of the F−test for testing H0 : β2 = 0. ii) The data in the GLIM analysis shown were obtained as part of a health survey in the US investigating cholesterol levels in women from two states, Iowa and Nebraska (denot...