We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Zp , whenever p is a fixed prime. For the induction step, we introduce the ...

We consider the isomorphism problem for groups specified by their multiplication tables. Until recently, the best published bound for the worst-case was achieved by the np n+O(1) generator-enumeration algorithm. In previous work with Fabian Wagner, we showed an n logp n+O(logn/ log logn) time algorithm for testing isomorphism of p-groups by building graphs with degree bounded by p + O(1) that r...

The authors use a series of lattice dualities to build exactly solvable models for supercohomology symmetry-protected topological phases in (3+1)-dimensions. They also construct gapped topologically ordered boundaries the fermionic by extending global symmetry.

We introduce the notion of an orbit series in a quandle. Using this we define four families quandles based on finiteness conditions their series. Intuitively, classes t OS and n correspond to finitary compositions trivial while connected quandles. study properties these explore relationships with several previously studied quandles: reductive, -reductive, locally -locally solvable

Applications of hypergroups have mainly appeared in special subclasses. One of the important subclasses is the class of polygroups. In this paper, we study the notions of nilpotent and solvable polygroups by using the notion of heart of polygroups. In particular, we give a necessary and sufficient condition between nilpotent (solvable) polygroups and fundamental groups.

For the N−plets of bound states in a quasi-exactly solvable (QES) toy model (sextic oscillator), the spectrum is known to be given as eigenvalues of an N by N matrix. Its determination becomes purely numerical for all the larger N > N0 = 9. We propose a new perturbative alternative to this construction. It is based on the fact that at any N , the problem turns solvable in the limit of very larg...

Let k be an algebraic number field. Let α be a root of a polynomial f ∈ k[x] which is solvable by radicals. Let L be the splitting field of α over k. Let n be a natural number divisible by the discriminant of the maximal abelian subextension of L, as well as the exponent of G(L/k), the Galois group of L over k. We show that an optimal nested radical with roots of unity for α can be effectively ...

In the 1990s, a series of dramatic results led to the completion of the Gromov program for lattices in semisimple Lie groups. The next natural class of examples to consider are lattices in solvable Lie groups, and even results for the simplest examples were elusive for a considerable time. Fisher’s joint work with Eskin and Whyte in which they proved the first results on quasi-isometric classif...

In this paper we study the representation theory associated to dilute lattice models. Given a solvable lattice model then often there are dilute versions of the model which are also solvable. These dilute models are studied in [Roc92], [GP93], [Gri94] and [WN93]. In this paper we discuss the algebras generated by the single bond transfer matrices and their representation theory. Our motivation ...