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Amicable pairs and aliquot cycles for elliptic curves
An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good
reduction for E satisfying #E(F_p) = q and #E(F_q) = p. In this paper we study
elliptic amicable pairs and analogously defined longer elliptic aliquot cycles.
We show that there exist elliptic curves with arbitrarily long aliqout cycles,
but that CM elliptic curves (with j not 0) have no aliqout cycles of length
greater than two. We give conjectural formulas for the frequency of amicable
pairs. For CM curves, the derivation of precise conjectural formulas involves a
detailed analysis of the values of the Grossencharacter evaluated at a prime
ideal P in End(E) having the property that #E(F_P) is prime. This is especially
intricate for the family of curves with j = 0.Comment: 53 page
Constructions for orthogonal designs using signed group orthogonal designs
Craigen introduced and studied signed group Hadamard matrices extensively and
eventually provided an asymptotic existence result for Hadamard matrices.
Following his lead, Ghaderpour introduced signed group orthogonal designs and
showed an asymptotic existence result for orthogonal designs and consequently
Hadamard matrices. In this paper, we construct some interesting families of
orthogonal designs using signed group orthogonal designs to show the capability
of signed group orthogonal designs in generation of different types of
orthogonal designs.Comment: To appear in Discrete Mathematics (Elsevier). No figure
Free nilpotent and -type Lie algebras. Combinatorial and orthogonal designs
The aim of our paper is to construct pseudo -type algebras from the
covering free nilpotent two-step Lie algebra as the quotient algebra by an
ideal. We propose an explicit algorithm of construction of such an ideal by
making use of a non-degenerate scalar product. Moreover, as a bypass result, we
recover the existence of a rational structure on pseudo -type algebras,
which implies the existence of lattices on the corresponding pseudo -type
Lie groups. Our approach substantially uses combinatorics and reveals the
interplay of pseudo -type algebras with combinatorial and orthogonal
designs. One of the key tools is the family of Hurwitz-Radon orthogonal
matrices
Amicable pairs : a survey
In 1750, Euler [20, 21] published an extensive paper on amicable pairs, by which he added fifty-nine new amicable pairs to the three amicable pairs known thus far. In 1972, Lee and Madachy [45] published a historical survey of amicable pairs, with a list of the 1108 amicable pairs then known. In 1995, Pedersen [48] started to create and maintain an Internet site with lists of all the known amicable pairs. The current (February 2003) number of amicable pairs in these lists exceeds four million. The purpose of this paper is to update the 1972 paper of Lee and Madachy, in order to document the developments which have led to the explosion of known amicable pairs in the past thirty years. We hope that this may stimulate research in the direction of finding a proof that the number of amicable pairs is infinite
Non-existence of 6-dimensional pseudomanifolds with complementarity
In a previous paper the second author showed that if is a pseudomanifold
with complementarity other than the 6-vertex real projective plane and the
9-vertex complex projective plane, then must have dimension , and -
in case of equality - must have exactly 12 vertices. In this paper we prove
that such a 6-dimensional pseudomanifold does not exist. On the way to proving
our main result we also prove that all combinatorial triangulations of the
4-sphere with at most 10 vertices are combinatorial 4-spheres.Comment: 11 pages. To appear in Advances in Geometr
Beyond Geometry : Towards Fully Realistic Wireless Models
Signal-strength models of wireless communications capture the gradual fading
of signals and the additivity of interference. As such, they are closer to
reality than other models. However, nearly all theoretic work in the SINR model
depends on the assumption of smooth geometric decay, one that is true in free
space but is far off in actual environments. The challenge is to model
realistic environments, including walls, obstacles, reflections and anisotropic
antennas, without making the models algorithmically impractical or analytically
intractable.
We present a simple solution that allows the modeling of arbitrary static
situations by moving from geometry to arbitrary decay spaces. The complexity of
a setting is captured by a metricity parameter Z that indicates how far the
decay space is from satisfying the triangular inequality. All results that hold
in the SINR model in general metrics carry over to decay spaces, with the
resulting time complexity and approximation depending on Z in the same way that
the original results depends on the path loss term alpha. For distributed
algorithms, that to date have appeared to necessarily depend on the planarity,
we indicate how they can be adapted to arbitrary decay spaces.
Finally, we explore the dependence on Z in the approximability of core
problems. In particular, we observe that the capacity maximization problem has
exponential upper and lower bounds in terms of Z in general decay spaces. In
Euclidean metrics and related growth-bounded decay spaces, the performance
depends on the exact metricity definition, with a polynomial upper bound in
terms of Z, but an exponential lower bound in terms of a variant parameter phi.
On the plane, the upper bound result actually yields the first approximation of
a capacity-type SINR problem that is subexponential in alpha
Amicable Pairs
The ancient Greeks are often credited with making many new discoveries in the area of mathematics. Euclid, Aristotle, and Pythagoras are three such famous Greek mathematicians. One of their discoveries was the idea of an amicable pair. An Amicable pair is a pair of two whole numbers, each of which is the sum of the proper whole number divisors of the other
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