If C is a compact subset of R and H is a halfspace bounded by a hyperplane π, the set C̃ obtained by shaking C on π is defined as the set contained in H, such that for every line ` orthogonal to π, C̃ ∩ ` is a segment of the same length as C ∩ `, and one of its endpoints is on π. It is shown that there exist d + 1 hyperplanes such that every compact set can be reduced to a simplex, via repeated s...

We study an interesting class of Banach function algebras of innitely dierentiable functions onperfect, compact plane sets. These algebras were introduced by Honary and Mahyar in 1999, calledLipschitz algebras of innitely dierentiable functions and denoted by Lip(X;M; ), where X is aperfect, compact plane set, M = fMng1n=0 is a sequence of positive numbers such that M0 = 1 and(m+n)!Mm+n ( m!Mm)...

We study the general problem of behaviour continuum function in presence non-supercompact strongly compact cardinals. begin by showing that it is possible to force violations GCH at an arbitrary cardinal using only strong compactness as our initial assumption. This result due third author. then investigate realising Easton functions and above least measurable limit supercompact cardinals starti...

It is well-known that the Toffoli gate and the negation gate together yield a universal gate set, in the sense that every permutation of {0, 1}n can be implemented as a composition of these gates. Since every bit operation that does not use all of the bits performs an even permutation, we need to use at least one auxiliary bit to perform every permutation, and it is known that one bit is indeed...

We construct a strongly minimal set which is not a finite cover of one with DMP. We also prove that for a strongly minimal theory T generic automorphisms exist iff T has DMP, thus proving a conjecture of Kikyo and