Fast arithmetic with general Gauß periods

Density Estimates Related to Gauß Periods

Given two integers q and k, for any prime r not dividing q with r ≡ 1 mod k, we denote by indr(q) the index of q mod r. In [2] the question was raised of calculating the density of the primes r for which indr(q) and (r− 1)/k are coprime; this is the condition that the Gauß period in Fq(r−1)/k defined by these data be normal over Fq. We assume the Generalized Riemann Hypothesis and calculate a f...

On periods modulo p in arithmetic dynamics ∗ †

We prove the following mod p version of a case of the dynamical André-Oort conjecture obtained in [GKN ]. Theorem. There are constants c1, c2 depending on d and h such that the following holds. For almost all P, there is a finite subset T ⊂ F̄P , |T | ≤ c1 such that if t ∈ F̄P \ T at least one of the sets { f (`) t (0) : ` = 1, 2, · · · , [c2 logN ] } , { g (`) t (0) : ` = 1, 2, · · · , [c2 logN ...

Fast Arithmetic Computing with Neural Networks

Neural networks can be viewed as circuits of highly interconnected parallel processing units called ‘neurons’. The most commonly used models of neurons are linear threshold gates or, when continuity or differentiability is required, elements with a sigmoid input-output function. Because of the recent advance in VLSI technology, neural network has also emerged as a new technology and has found w...

Twofold fast arithmetic

Can we assure math computations by automatic verifying floating-point accuracy? We define fast arithmetic (based on Dekker [1]) over twofold approximations z ≈ z0 + z1, such that z0 is standard result and z1 assesses inaccuracy ∆z0 = z − z0. We propose on-fly tracking z1, detecting if ∆z0 appears too high. We believe permanent tracking is worth its cost. C++ test code for Intel AVX available vi...

Fast and Parallel Interval Arithmetic