Dual Lukacs regressions for non-commutative variables

Dual Lukacs type characterizations of random variables in free probability are studied here. First, we develop a freeness property satisfied by Lukacs type transformations of free-Poisson and free-Binomial non-commutative variables which are free. Second, we give a characterization of non-commutative free-Poisson and free-Binomial variables by properties of first two conditional moments, which mimic Lukacs type assumptions known from classical probability. More precisely, our result is a non-commutative version of the following result known in classical probability: if $U$, $V$ are independent real random variables, such that $E(V(1-U)|UV)$ and $E(V^2(1-U)^2|UV)$ are non-random then $V$ has a gamma distribution and $U$ has a beta distribution

Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem

Fix an ordinary abelian variety defined over a finite field. The ideal class group of its endomorphism ring acts freely on the set of isogenous varieties with same endomorphism ring, by complex multiplication. Any subgroup of the class group, and generating set thereof, induces an isogeny graph on the orbit of the variety for this subgroup. We compute (under the Generalized Riemann Hypothesis) some bounds on the norms of prime ideals generating it, such that the associated graph has good expansion properties. We use these graphs, together with a recent algorithm of Dudeanu, Jetchev and Robert for computing explicit isogenies in genus 2, to prove random self-reducibility of the discrete logarithm problem within the subclasses of principally polarizable ordinary abelian surfaces with fixed endomorphism ring. In addition, we remove the heuristics in the complexity analysis of an algorithm of Galbraith for explicitly computing isogenies between two elliptic curves in the same isogeny class, and extend it to a more general setting including genus 2.Comment: 18 page

Multiple Chern-Simons Fields on a Torus

Intertwined multiple Chern-Simons gauge fields induce matrix statistics among particles. We analyse this theory on a torus, focusing on the vacuum structure and the Hilbert space. The theory can be mimicked, although not completely, by an effective theory with one Chern-Simons gauge field. The correspondence between the Wilson line integrals, vacuum degeneracy and wave functions for these two theories are discussed. Further, it is obtained in both of these cases that the two total momenta and Hamiltonian commute only in the physical Hilbert space.Comment: 20 pages, UMN-TH-1128/93, plain Te

Isogeny graphs of ordinary abelian varieties

Fix a prime number $\ell$. Graphs of isogenies of degree a power of $\ell$ are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called $\mathfrak l$-isogenies, resolving that they are (almost) volcanoes in any dimension. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as $(\ell, \ell)$-isogenies: those whose kernels are maximal isotropic subgroups of the $\ell$-torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.Comment: 36 pages, 4 figure

Politicians in times of transformation: "Transformational correctness" or genuine differences in perception?

The paper presents selected results of the 1996 study of top Polish politicians-members of the 1993-97 Parliament and leaders of those important political parties which failed to get into this Parliament. Presented results pertain to five aspects of politicians' attitudes: (1) opinions on qualities of persons who should be vs. actually are involved in politics, (2) normatively accepted definitions of politics, (3) visions of the good state, (4) visions of democracy, and (5) opinions on what defines political views as being either on the left or on the right. As a result, consistently found across all five domains, there is a strong attitudinal similarity among politicians of differing political parties and of divergent political orientations. This finding is interpreted as reflection of a fundamental track similarity in the way in which Polish politicians perceive the most important tasks confronting the whole political class in times of systemic transformation. Transformational correctness - believing that in such times politicians should have (or at least should display) certain views - might be a strong force behind this similarity. --