In this paper we study structures related to torsion of elliptic curves defined over number fields. The aim is to build families of elliptic curves more efficient to help factoring numbers of special form, including numbers from the Cunningham Project. We exhibit a family of curves with rational Z/4Z×Z/4Z torsion and positive rank over the field Q(ζ8) and a family of elliptic curves with ration...

We introduce a new concept of the bounded rank (with respect to a positive constant) for unital C∗-algebras as a modification of the usual real rank. We present a series of conditions insuring that bounded and real ranks coincide. These observations are then used to prove that for a given n and K > 0 there exists a separable unital C∗-algebra Z n such that every other separable unital C∗-algebr...

We consider the extension of the last-in-first-out graph searching game of Giannopoulou and Thilikos to digraphs. We show that all common variations of the game require the same number of searchers, and the minimal number of searchers required is one more than the cycle-rank of the digraph. We also obtain a tight duality theorem, giving a precise min-max characterization of obstructions for cyc...

Let G be a countable Abelian group with Zd as a subgroup so that G/Zd is a locally finite group. (An Abelian group is locally finite if every element has finite order.) We can construct a rank one action of G so that the Z-subaction is 2-simple, 2-mixing and only commutes with the other transformations in the action of G. Applications of this construction include a transformation with square ro...

With the E731 apparatus at Fermilab, we have simultaneously collected 6859 KL and Kz decays into n+z y. Using our sample of over 370000 z+z decays for normalization we have determined that the ratio I (K tt+tt y)/I (K tv+tv ) is (23.0+ 0.7)x IO for Kc and (7.10~0.22)x IO for Ks, for photon energies greater than 20 MeV in the kaon center of mass. After removing the innerbremsstrahlung contributi...

With nonnormal data, the typical confidence interval of the correlation (Fisher z') may be inaccurate. The literature has been unclear as to which of several alternative methods should be used instead, and how extreme a violation of normality is needed to justify an alternative. Through Monte Carlo simulation, 11 confidence interval methods were compared, including Fisher z', two Spearman rank-...

Let S = 〈s1, s2, s3, ..., sn〉 be a given vector of n real numbers. The rank of z ∈ R with respect to S is defined as the number of elements si ∈ S such that si ≤ z. We consider the following decision problem: determine whether the oddnumbered elements s1, s3, s5, . . . are precisely the elements of S whose rank with respect to S is odd. We prove a bound of Θ(n logn) on the number of operations ...

Let H be the class of complex-valued harmonic mappings f=h+g¯ defined in unit disk D:={z?C:|z|<1}, where h and g are analytic functions D with normalization h(0)=0=h?(0)?1 g(0)=0. H0={f=h+g¯?H:g?(0)=0}. PH0(M):={f=h+g¯?H0:Re(zh??(z))>?M+|zg??(z)|,z?DandM>0}. univalent [Ghosh N, Allu V. On some subclasses mappings. Bull Aust Math Soc. 2020;101:130–140.]. In this paper, we obtain sharp Bohr–Rogos...