Leavitt RR-algebras over countable graphs embed into L2,RL_{2,R}

For a commutative ring $R$ with unit we show that the Leavitt path algebra $L_R(E)$ of a graph $E$ embeds into $L_{2,R}$ precisely when $E$ is countable. Before proving this result we prove a generalised Cuntz-Krieger Uniqueness Theorem for Leavitt path algebras over $R$.Comment: 17 pages. At the request of a referee the previous version of this paper has been split into two papers. This version is the first of these papers. The second will also be uploaded to the arXi

The Cuntz splice does not preserve ∗*-isomorphism of Leavitt path algebras over Z\mathbb{Z}

We show that the Leavitt path algebras $L_{2,\mathbb{Z}}$ and $L_{2-,\mathbb{Z}}$ are not isomorphic as $*$-algebras. There are two key ingredients in the proof. One is a partial algebraic translation of Matsumoto and Matui's result on diagonal preserving isomorphisms of Cuntz--Krieger algebras. The other is a complete description of the projections in $L_{\mathbb{Z}}(E)$ for $E$ a finite graph. This description is based on a generalization, due to Chris Smith, of the description of the unitaries in $L_{2,\mathbb{Z}}$ given by Brownlowe and the second named author. The techniques generalize to a slightly larger class of rings than just $\mathbb{Z}$.Comment: 17 pages. Since version 2 we extended the arguments from Z to more general ring

L2,Z⊗L2,ZL_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}} does not embed in L2,ZL_{2,\mathbb{Z}}

For a commutative ring $R$ with unit we investigate the embedding of tensor product algebras into the Leavitt algebra $L_{2,R}$. We show that the tensor product $L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ does not embed in $L_{2,\mathbb{Z}}$ (as a unital $*$-algebra). We also prove a partial non-embedding result for the more general $L_{2,R} \otimes L_{2,R}$. Our techniques rely on realising Thompson's group $V$ as a subgroup of the unitary group of $L_{2,R}$.Comment: 16 pages. At the request of a referee the paper arXiv:1503.08705v2 was split into two papers. This is the second of those paper

Almost Commuting Orthogonal Matrices

We show that almost commuting real orthogonal matrices are uniformly close to exactly commuting real orthogonal matrices. We prove the same for symplectic unitary matrices. This is in contrast to the general complex case, where not all pairs of almost commuting unitaries are close to commuting pairs. Our techniques also yield results about almost normal matrices over the reals and the quaternions.Comment: 13 pages, 3 figure

Genetic Improvement of Livestock for Organic Farming Systems

Organic farming which experienced a constant rise over the last two decades is a system based on sustainability and on a concept tending towards functional integrity. Legislation as well as the wish to produce separately from conventional farming raise the question whether organic farming should be conducted completely apart from conventional farming or not. This paper discusses the aspects that affect animal breeding under these circumstances, e.g., maintaining genetic diversity by using local breeds and possible G×E interactions which might occur when breeds adapted to conventional farming systems are used in organic farming. Ways of modelling G×E are presented, moreover examples of G×E in dairy cattle, swine, and poultry are given. Trends in selection index theory–designing multi-trait breeding goals including functional traits on one hand, and developing methods for using customised selection indices on the other hand–support breeding work for organic farming systems. It is concluded that before the technical issues can be addressed, all parties involved, farmers, consumers as well as legislators, have to agree on the socio-cultural conditions under which organic farming should be conducted

Opto-mechanical transducers for long-distance quantum communication

We describe a new scheme to interconvert stationary and photonic qubits which is based on indirect qubit-light interactions mediated by a mechanical resonator. This approach does not rely on the specific optical response of the qubit and thereby enables optical quantum interfaces for a wide range of solid state spin and charge based systems. We discuss the implementation of quantum state transfer protocols between distant nodes of a large scale network and evaluate the effect of the main noise sources on the resulting state transfer fidelities. For the specific examples of electronic spin qubits and superconducting charge qubits we show that high fidelity quantum communication protocols can be implemented under realistic experimental conditions.Comment: Version as accepted by PR

Invariance of the Cuntz splice

We show that the Cuntz splice induces stably isomorphic graph $C^*$-algebras.Comment: Our arguments to prove invariance of the Cuntz splice for unital graph C*-algebras in arXiv:1505.06773 applied with only minor changes in the general case. Since most of the results of that preprint have since been superseded by other forthcoming work, we do not intend to publish it, whereas this work is intended for publication. arXiv admin note: substantial text overlap with arXiv:1505.0677