Ultra Low Expansion (ULE) material of Corning Glass Work was chosen for the Gemini primary mirrors. The ULE mirror blank becomes monolithic by a fusion process which seals together 55 piece parts from a total of 44 hexagonal segments (hexes). As a consequence of this fusion process, an optical surface distortion due to inhomogeniety in the coefficient of thermal expansion (CTE) is induced. The ...

Let n = pq, p, q distinct odd primes. We can divide the numbers in Zn into four classes depending on their membership in QRp and QRq. Let Q11 n be those numbers that are quadratic residues mod both p and q; let Q10 n be those numbers that are quadratic residues mod p but not mod q; let Q 01 n be those numbers that are quadratic residues mod q but not mod p; and let Q00 n be those numbers that a...

Fermat’s little theorem says for prime p that ap−1 ≡ 1 mod p for all a 6≡ 0 mod p. A naive extension of this to a composite modulus n ≥ 2 would be: for a 6≡ 0 mod n, an−1 ≡ 1 mod n. Let’s call this “Fermat’s little congruence.” It may or may not be true. When n is prime, it is true for all a 6≡ 0 mod n. But when n is composite it usually has many counterexamples. Example 1.1. When n = 15, the t...

On primitive roots of 1 mod p k , divisors of p 2 − 1 , Wieferich primes and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k > 2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by...

On primitive roots of 1 mod p k , divisors of p ± 1, Wieferich primes, and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k >2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by p+...

The structural and thermodynamic features of three polymorphic forms of the local anesthetic drug oxybuprocaine hydrochloride (OBPHC) were characterized by hot-stage microscopy, differential scanning calorimetry (DSC), pycnometry, Fourier transform infrared spectroscopy (FTIR), FT-Raman and solid-state NMR (SSNMR) spectroscopy as well as X-ray powder and single-crystal diffractometry. Mod II° c...

Recall the collection of functions {BlumN : QRN → QRN} were BlumN = x mod N for so-called Blum integers N that are products of two distinct primes congruent to 3 mod 4 are candidate one-way functions. As we mentioned earlier, inverting for algorithms for BlumN yield algorithms for factoring N . We now show that for N = pq, p 6= q prime, p, q ≡ 3 (mod 4), the pair (p, q) of factors of N forms tr...

Modafinil (MOD) exhibits therapeutic efficacy for treating sleep and psychiatric disorders; however, its mechanism is not completely understood. Compared with other psychostimulants inhibiting dopamine (DA) uptake, MOD weakly interacts with the dopamine transporter (DAT) and modestly elevates striatal dialysate DA, suggesting additional targets besides DAT. However, the ability of MOD to induce...