Number theory is considered, by proposing quantum mechanical models and string-like models at zero and finite temperatures, where the factorization of number into prime numbers is viewed as the decay of particle into elementary particles conserving energy. In these models, energy of a particle labeled by an integer n is assumed or derived to being proportional to lnn. The one-loop vacuum amplit...

We present a commitment scheme allowing commitment to arbitrary size integers, based on any Abelian group with certain properties, most importantly that it is hard for the committer to compute its order. Potential examples include RSA and class groups. We also give efficient zero-knowledge protocols for proving knowledge of the contents of a commitment and for verifying multiplicative relations...

where / is an odd prime and x, y, z are rational integers prime to each other and none zero, we shall refer to the case where xyz is prime to / as case I ; if xyz = 0 (mod I) then we call this case II . I now give a sketch of a proof of a theorem which appears to be the principal result I have so far found concerning the first case of the last theorem. THEOREM 1. If (1) is possible in case I, t...

We give a description of the minimal primes of the ideal generated by the 2×2 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent m×m minors of an m×n generic matrix when the characteristic of the ground field is zero. A key intermediate result is the proof that the ideals which appear as minimal primes are, in fact, prime ideals. This ...

If ~q1, . . . , ~qm : Z → Z are polynomials with zero constant terms and E ⊂ Z has positive upper Banach density, then we show that the set E ∩ (E − ~q1(p− 1))∩ . . .∩(E−~qm(p−1)) is nonempty for some prime p. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results ...

let $mathcal{a}$ and $mathcal{b}$ be two $c^{*}$-algebras such that $mathcal{b}$ is prime. in this paper, we investigate the additivity of maps $phi$ from $mathcal{a}$ onto $mathcal{b}$ that are bijective, unital and satisfy $phi(ap+eta pa^{*})=phi(a)phi(p)+eta phi(p)phi(a)^{*},$ for all $ainmathcal{a}$ and $pin{p_{1},i_{mathcal{a}}-p_{1}}$ where $p_{1}$ is a nontrivial projection in $mathcal{a...

Let n > 1 be a positive integer and p be the smallest prime divisor of n. Let S be a sequence of elements from Zn = Z/nZ of length n + k where k ≥ np − 1. If every element of Zn appears in S at most k times, we prove that there must be a subsequence of S of length n whose sum is zero when n has only two distinct prime divisors.

We describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra. Our methods come from the theory of Hopf algebra crossed products. For primitive ideals we obtain an analogue of Duflo’s Theorem, which says that every primitive ideal is the annihilator of a simple highest weight...