Congruence and Metaplectic Covariance: Rational Biquadratic Reciprocity and Quantum Entanglement

Biquadratic reciprocity and a Lucasian primality test

Let {sk, k ≥ 0} be the sequence defined from a given initial value, the seed, s0, by the recurrence sk+1 = s 2 k − 2, k ≥ 0. Then, for a suitable seed s0, the number Mh,n = h · 2n − 1 (where h < 2n is odd) is prime iff sn−2 ≡ 0 mod Mh,n. In general s0 depends both on h and on n. We describe a slight modification of this test which determines primality of numbers h·2n±1 with a seed which depends...

Separability and entanglement of quantum states based on covariance matrices

Voltage-Controlled Entanglement between Quantum- Dot Molecule and its Spontaneous Emission Fields via Quantum Entropy

The time evolution of the quantum entropy in a coherently driven threelevel quantum dot (QD) molecule is investigated. The entanglement of quantum dot molecule and its spontaneous emission field is coherently controlled by the gat voltage and the intensity of applied field. It is shown that the degree of entanglement between a three-level quantum dot molecule and its spontaneous emission fields...

Modeling with rational biquadratic splines

We develop a rational biquadratic G analogue of the non-uniform C B-spline paradigm. These G splines can exactly reproduce parts of multiple basic shapes, such as cyclides and quadrics, and combine them into one smoothly-connected structure. This enables a design process that starts with basic shapes, re-pepresents them in spline form and uses the spline form to provide shape handles for locali...

Rational Quartic Reciprocity Ii

for every prime p ≡ 1 mod 4 such that (p/pj) = +1 for all 1 ≤ j ≤ r. This is ’the extension to composite values of m’ that was referred to in [3], to which this paper is an addition. Here I will fill in the details of a proof, on the one hand because I was requested to do so, and on the other hand because this general law can be used to derive general versions of Burde’s and Scholz’s reciprocit...