Finite Lorentz Transformations, Automorphisms, and Division Algebras

We give an explicit algebraic description of finite Lorentz transformations of vectors in 10-dimensional Minkowski space by means of a parameterization in terms of the octonions. The possible utility of these results for superstring theory is mentioned. Along the way we describe automorphisms of the two highest dimensional normed division algebras, namely the quaternions and the octonions, in terms of conjugation maps. We use similar techniques to define $SO(3)$ and $SO(7)$ via conjugation, $SO(4)$ via symmetric multiplication, and $SO(8)$ via both symmetric multiplication and one-sided multiplication. The non-commutativity and non-associativity of these division algebras plays a crucial role in our constructions.Comment: 24 pages, Plain TeX, 2 figures on 1 page submitted separately as uuencoded compressed tar fil

Rothberger gaps in fragmented ideals

The~\emph{Rothberger number} $\mathfrak{b} (\mathcal{I})$ of a definable ideal $\mathcal{I}$ on $\omega$ is the least cardinal $\kappa$ such that there exists a Rothberger gap of type $(\omega,\kappa)$ in the quotient algebra $\mathcal{P} (\omega) / \mathcal{I}$. We investigate $\mathfrak{b} (\mathcal{I})$ for a subclass of the $F_\sigma$ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is $\aleph_1$ while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consistent that there are infinitely many (even continuum many) different Rothberger numbers associated with fragmented ideals.Comment: 28 page

Relation between shear parameter and Reynolds number in statistically stationary turbulent shear flows

Studies of the relation between the shear parameter S^* and the Reynolds number Re are presented for a nearly homogeneous and statistically stationary turbulent shear flow. The parametric investigations are in line with a generalized perspective on the return to local isotropy in shear flows that was outlined recently [Schumacher, Sreenivasan and Yeung, Phys. Fluids, vol.15, 84 (2003)]. Therefore, two parameters, the constant shear rate S and the level of initial turbulent fluctuations as prescribed by an energy injection rate epsilon_{in}, are varied systematically. The investigations suggest that the shear parameter levels off for larger Reynolds numbers which is supported by dimensional arguments. It is found that the skewness of the transverse derivative shows a different decay behavior with respect to Reynolds number when the sequence of simulation runs follows different pathways across the two-parameter plane. The study can shed new light on different interpretations of the decay of odd order moments in high-Reynolds number experiments.Comment: 9 pages, 9 Postscript figure

Sequential Tunneling through Molecular Spin Rings

We consider electrical transport through molecules with Heisenberg-coupled spins arranged in a ring structure in the presence of an easy-axis anisotropy. The molecules are coupled to two metallic leads and a gate. In the charged state of the ring, a Zener double-exchange mechanism links transport properties to the underlying spin structure. This leads to a remarkable contact-site dependence of the current, which for an antiferromagnetic coupling of the spins can lead to a total suppression of the zero-bias conductance when the molecule is contacted at adjacent sites.Comment: 4 pages, 3 figure

Transfer of optical spectral weight in magnetically ordered superconductors

We show that, in antiferromagnetic superconductors, the optical spectral weight transferred to low frequencies below the superconducting transition temperature originates from energies that can be much larger than twice the superconducting gap $\Delta$. This contrasts to non-magnetic superconductors, where the optical spectrum is suppressed only for frequencies below $2\Delta$. In particular, we demonstrate that the superfluid condensate of the magnetically ordered superconductor is not only due to states of the magnetically reconstructed Fermi surface, but is enhanced by transfer of spectral weight from the mid infrared peak generated by the spin density wave gap. We apply our results to the iron arsenide superconductors, addressing the decrease of the zero-temperature superfluid density in the doping regime where magnetism coexists with unconventional superconductivity.Comment: 5 figures, 10 pages; revised versio

Strong Coordination over Noisy Channels: Is Separation Sufficient?

We study the problem of strong coordination of actions of two agents $X$ and $Y$ that communicate over a noisy communication channel such that the actions follow a given joint probability distribution. We propose two novel schemes for this noisy strong coordination problem, and derive inner bounds for the underlying strong coordination capacity region. The first scheme is a joint coordination-channel coding scheme that utilizes the randomness provided by the communication channel to reduce the local randomness required in generating the action sequence at agent $Y$. The second scheme exploits separate coordination and channel coding where local randomness is extracted from the channel after decoding. Finally, we present an example in which the joint scheme is able to outperform the separate scheme in terms of coordination rate.Comment: 9 pages, 4 figures. An extended version of a paper accepted for the IEEE International Symposium on Information Theory (ISIT), 201

Strong Coordination over Noisy Channels: Is Separation Sufficient?

We study the problem of strong coordination of actions of two agents $X$ and $Y$ that communicate over a noisy communication channel such that the actions follow a given joint probability distribution. We propose two novel schemes for this noisy strong coordination problem, and derive inner bounds for the underlying strong coordination capacity region. The first scheme is a joint coordination-channel coding scheme that utilizes the randomness provided by the communication channel to reduce the local randomness required in generating the action sequence at agent $Y$. The second scheme exploits separate coordination and channel coding where local randomness is extracted from the channel after decoding. Finally, we present an example in which the joint scheme is able to outperform the separate scheme in terms of coordination rate.Comment: 9 pages, 4 figures. An extended version of a paper accepted for the IEEE International Symposium on Information Theory (ISIT), 201