Using the Bloch-Messiah reduction we show that squeezing is an “irreducible” resource which remains invariant under transformations by linear optical elements. In particular, this gives a decomposition of any optical circuit with linear input-output relations into a linear multiport interferometer followed by a unique set of single-mode squeezers and then another multiport interferometer. Using...

Let E be an elliptic curve over F=Fq(t) having conductor (p)·∞, where (p) is a prime ideal in Fq [t]. Let d ∈ Fq [t] be an irreducible polynomial of odd degree, and let K=F( √ d). Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗FK, 1). As a consequence, we obtain a formula for the order of the Tate–Shafarevich group I(E/K) when L(E⊗FK, 1) =...

We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal an ample groupoid algebra over field is intersection kernels induced representations from isotropy groups, with much shorter proof, by using the author's Disintegration Theorem for representations. also prove primitive kernel representation group; however, we are unable show, in general, it irreducible r...

We study in detail the irreducible representation of [Formula: see text] theory that corresponds to massless particles. This has little algebra and contains 128 physical states belong spinor text]. These are degrees freedom maximal supergravity eleven dimensions. smaller number freedom, compared what might be expected, is due an infinite duality relations which turn can traced existence a subag...

For a finite group, we present three algorithms to compute generating set of invariants simultaneously sets basic equivariants,i.e.,equivariants for the irreducible representations group. The main novelty resides in exploitation orthogonal complement ideal generated by invariants. Its symmetry adapted basis delivers fundamental equivariants. Fundamental equivariants allow assem...

An old question of T. Jech and K. Prikry asks if an existence of a precipitous ideal implies necessary existence of a normal precipitous ideal. The aim of the paper is to prove some results in the positive direction. Thus, it is shown that under some mild assumptions, an existence of a precipitous ideal over א1 implies an existence of a normal precipitous ideal over א1 once a Cohen subset is ad...

Abstract. In this paper, we classify all of the finite p-groups with at most three non linear irreducible character kernels.

In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference ideals, and group actions by difference tori. Connections between toric difference varieties and affine N[x]-semimodules are established by proving the corr...

Let {f1, f2, . . . , ft} ⊂ Q[z1, . . . , zN ] be a set of homogeneous polynomials. Let Z denote the complex, projective, algebraic set determined by the homogeneous ideal I = (f1, f2, . . . , ft) ⊂ C[z1, . . . , zN ]. Numerical continuation-based methods can be used to produce arbitrary precision numerical approximations of generic points on each irreducible component of Z. Consider the prime d...