‎we investigate graham higman's paper enumerating $p$-groups‎, ‎ii‎, ‎in which he formulated his famous porc conjecture‎. ‎we are able to simplify some of the theory‎. ‎in particular‎, ‎higman's paper contains five pages of homological algebra which he uses in‎ ‎his proof that the number of solutions in a finite field to a finite set of‎ ‎monomial equations is porc‎. ‎it turns out tha...

We prove a strong form of the Brumer–Stark Conjecture and, as a consequence, a strong form of Rubin’s integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum kp∞ := kp · k∞ of the maximal pro-p abelian extension kp/k and the maximal...

We give a polynomial counterexample to a discrete version of the Markus-Yamabe Conjecture and a conjecture of Deng, Meisters and Zampieri, asserting that if F : C → C is a polynomial map with det(JF ) ∈ C∗, then for all λ ∈ R large enough λF is global analytic linearizable. These counterexamples hold in any dimension ≥ 4.

We give a polynomial counterexample to both the Markus-Yamabe Conjecture and the discrete Markus-Yamabe problem for all dimensions ≥ 3.

Håstad established that any predicate P ⊆ {0,1}m containing Parity of width at least three is approximation resistant for almost-satisfiable instances. In comparison to for example the approximation hardness of 3SAT, this general result however left open the hardness of perfectly-satisfiable instances. This limitation was addressed by O’Donnell and Wu, and subsequently generalized by Huang, to ...

All continuous endomorphisms f∞ of the shift dynamical system S on the 2-adic integers Z2 are induced by some f : Bn → {0, 1}, where n is a positive integer, Bn is the set of n-blocks over {0, 1}, and f∞ (x) = y0y1y2 . . . where for all i ∈ N, yi = f (xixi+1 . . . xi+n−1). Define D : Z2 → Z2 to be the endomorphism of S induced by the map {(00, 0) , (01, 1) , (10, 1) , (11, 0)} and V : Z2 → Z2 b...

Currently, it has been claimed that certain Hermitian Hamiltonians have parity (P) and they are PT-invariant. We propose generalized definitions of time-reversal operator (T) and orthonormality such that all Hermitian Hamil-tonians are P,T,PT,CPT invariant. The PT-norm and CPT-norm are indefinite and definite respectively. The energy-eigenstates are either E-type (e.g., even) or O-type (e.g., o...

We prove the following theorem: Let F be an algebraic closure of a finite field of characteristic p > 3. Let ρ be a continuous homomorphism from the absolute Galois group of Q to GL(3,F) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. We assume that the image of ρ is contained in the intersection of the stabilizers of the line spanned by ...

Classical low-density parity-check (LDPC) codes were first introduced by Robert Gallager in the 1960's and have reemerged as one of the most influential coding schemes. We present new families of quantum low-density parity-check error-correcting codes derived from regular tessellations of Platonic 2-manifolds and from embeddings of the Lubotzky-Phillips-Sarnak Ramanujan graphs. These families o...