Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?

We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension $q$, containing a square, into its factors. Illustrative low dimensional examples are the quartit ($q=4$) and two-qubit ($q=2^2$) systems, the octit ($q=8$), qubit/quartit ($q=2\times 4$) and three-qubit ($q=2^3$) systems, and so on. In the single qudit case, e.g. $q=4,8,12,...$, one defines a bijection between the $\sigma (q)$ maximal commuting sets [with $\sigma[q)$ the sum of divisors of $q$] of Pauli observables and the maximal submodules of the modular ring $\mathbb{Z}_q^2$, that arrange into the projective line $P_1(\mathbb{Z}_q)$ and a independent set of size $\sigma (q)-\psi(q)$ [with $\psi(q)$ the Dedekind psi function]. In the multiple qudit case, e.g. $q=2^2, 2^3, 3^2,...$, the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if $q=2^2$) and GQ(3,3) (if $q=3^2$). More precisely, in dimension $p^n$ ($p$ a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the $2n$-dimensional vector space over the field $\mathbb{F}_p$. In this space, one makes use of the commutator to define a symplectic polar space $W_{2n-1}(p)$ of cardinality $\sigma(p^{2n-1})$, that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of $W_{2n-1}(p)$ are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function $\psi(p^{2n-1})$. For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, ponctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.Comment: 18 pages, version submiited to J. Phys. A: Math. Theo

Spatial Analyses of the Flow of Slaughter Livestock in 1955 and 1960

In this study attention is focused on the spatial aspects of slaughter livestock movements from production to slaughtering. Given the regional levels of production, slaughtering and the costs of moving one unit of various types of slaughter livestock from any one region to another region, this study is concerned with ascertaining the regional price differentials, and the volume and direction of regional imports and exports that are consistent with minimizing the total cost of moving the livestock from production to slaughter. In addition, questions about the consequences of changes in the existing structure of the livestock economy may be evaluated with regard to their impact on regional prices and slaughter livestock flows

Joint Spatial Analysis of Regional Slaughter and the Flows and Pricing of Livestock and Meat

The purpose of this study is to (1) develop a model to handle the simultaneous solution for the processing and flow problem, (2) develop estimates of slaughtering capacity for cattle and hogs in each region, and (3) apply the model using estimates of regional levels of production, regional levels of consumption, regional slaughtering capacities, and transportation costs of live slaughter animals and meats. Attention is focused at determining what regional levels of slaughter and directions and levels of interregional livestock and meat flows satisfy the regional production consumption, and capacity constraints and make the total cost of transportation of live slaughter animals and meat a minimum. The analysis is broadened to also obtain the impacts of alternative regional slaughter capacity restrictions and .regional differences in the labor cost of slaughtering livestock

Spatial Analyses of the Meat Marketing Sector in 1955 and 1960

The livestock products sector is a complex composed of the activities of production, farm marketing, slaughtering, distribution and consumption. The level of each of these activities varies spatially and thus regional imbalances are generated which make necessary product flows between the geographical areas. Within this setting this study is concerned with an interregional analysis of the livestock meat sector of the U. S. economy. Thus, spatial slaughter-consumption relations will be basic observations for this analysis. In this study regional demands are reflected by price dependent demand relations or specific estimates of consumption. Regional supplies are dressed carcass weights of livestock slaughter within the regions. In particular for the beef, pork, veal, and lamb and mutton sectors for the years 1955 and 1960 answers will be sought to the following questions: 1. What are the levels of regional demand for each of these meat products? 2. What are the levels of regional supply for each of these products? 3. What is the aggregate interregional trade for each meat product for each year? 4. For each commodity and for each year, what regions import, export or do neither? 5. What are the levels of regional exports and imports for each region, commodity and year? 6. What is the optimum volume and direction of trade between all possible pairs of regions for each commodity and each year? 7. What are the optimum price differentials between regions for each commodity and year? 8. What is the total transport cost for the aggregate trade of each commodity and year? 9. What is the impact of alternative ways of estimating regional meat consumption on the interregional flows and price differentials? In the following pages the results that are generated by these questions will be given and the implications and uses of the results will be discussed

Scedosporium and Pseudallescheria low molecular weight metabolites revealed by database search

The potential of mMass software search tool with new compound libraries was demonstrated on metabolomics of Scedosporium prolificans, S. apiospermum and Pseudallescheria boydii sensu stricto. Cyclic peptides pseudacyclins, small molecular weight tyroscherin analogues and various lipids were annotated by open source mass spectometry tool utilising accurate matrix-assisted laser desorption/ionisation mass spectral data of intact fungal spores. Electrospray ionisation combined with tandem mass spectrometry was used for monohexosylceramide characterisation in fungal extracts

Pauli graphs, Riemann hypothesis, Goldbach pairs

Let consider the Pauli group $\mathcal{P}_q=$ with unitary quantum generators $X$ (shift) and $Z$ (clock) acting on the vectors of the $q$-dimensional Hilbert space via $X|s> =|s+1>$ and $Z|s> =\omega^s |s>$, with $\omega=\exp(2i\pi/q)$. It has been found that the number of maximal mutually commuting sets within $\mathcal{P}_q$ is controlled by the Dedekind psi function $\psi(q)=q \prod_{p|q}(1+\frac{1}{p})$ (with $p$ a prime) \cite{Planat2011} and that there exists a specific inequality $\frac{\psi (q)}{q}>e^{\gamma}\log \log q$, involving the Euler constant $\gamma \sim 0.577$, that is only satisfied at specific low dimensions $q \in \mathcal {A}=\{2,3,4,5,6,8,10,12,18,30\}$. The set $\mathcal{A}$ is closely related to the set $\mathcal{A} \cup \{1,24\}$ of integers that are totally Goldbach, i.e. that consist of all primes $p2$) is equivalent to Riemann hypothesis. Introducing the Hardy-Littlewood function $R(q)=2 C_2 \prod_{p|n}\frac{p-1}{p-2}$ (with $C_2 \sim 0.660$ the twin prime constant), that is used for estimating the number $g(q) \sim R(q) \frac{q}{\ln^2 q}$ of Goldbach pairs, one shows that the new inequality $\frac{R(N_r)}{\log \log N_r} \gtrapprox e^{\gamma}$ is also equivalent to Riemann hypothesis. In this paper, these number theoretical properties are discusssed in the context of the qudit commutation structure.Comment: 11 page

Validation of equilibrium tools on the COMPASS tokamak

SOFT 2014 conference, submitted to Fusion Engineering and DesignInternational audienceVarious MHD (magnetohydrodynamic) equilibrium tools, some of which being recently developed or considerably updated, are used on the COMPASS tokamak at IPP Prague. MHD equilibrium is a fundamental property of the tokamak plasma, whose knowledge is required for many diagnostics and modelling tools. Proper benchmarking and validation of equilibrium tools is thus key for interpreting and planning tokamak experiments. We present here benchmarks and comparisons to experimental data of the EFIT++ reconstruction code [L.C. Appel et al., EPS 2006, P2.184], the free-boundary equilibrium code FREEBIE [J.-F. Artaud, S.H. Kim, EPS 2012, P4.023], and a rapid plasma boundary reconstruction code VacTH [B. Faugeras et al., PPCF 56, 114010 (2014)]. We demonstrate that FREEBIE can calculate the equilibrium and corresponding poloidal field (PF) coils currents consistently with EFIT++ reconstructions from experimental data. Both EFIT++ and VacTH can reconstruct equilibria generated by FREEBIE from synthetic, optionally noisy diagnostic data. Hence, VacTH is suitable for real-time control. Optimum reconstruction parameters are estimated