Properties of canonical determinants and a test of fugacity expansion for finite density lattice QCD with Wilson fermions

We analyze canonical determinants, i.e., grand canonical determinants projected to a fixed net quark number. The canonical determinants are the coefficients in a fugacity expansion of the grand canonical determinant and we evaluate them as the Fourier moments of the grand canonical determinant with respect to imaginary chemical potential, using a dimensional reduction technique. The analysis is done for two mass-degenerate flavors of Wilson fermions at several temperatures below and above the confinement/deconfinement crossover. We discuss various properties of the canonical determinants and analyse the convergence of the fugacity series for different temperatures.Comment: Typo removed, paragraph added in the discussion. Version to appear in Phys. Rev.

Finite density phase transition of QCD with Nf=4N_f=4 and Nf=2N_f=2 using canonical ensemble method

In a progress toward searching for the QCD critical point, we study the finite density phase transition of $N_f = 4$ and 2 lattice QCD at finite temperature with the canonical ensemble approach. We develop a winding number expansion method to accurately project out the particle number from the fermion determinant which greatly extends the applicable range of baryon number sectors to make the study feasible. Our lattice simulation was carried out with the clover fermions and improved gauge action. For a given temperature, we calculate the baryon chemical potential from the canonical approach to look for the mixed phase as a signal for the first order phase transition. In the case of $N_f=4$, we observe an "S-shape" structure in the chemical potential-density plane due to the surface tension of the mixed phase in a finite volume which is a signal for the first order phase transition. We use the Maxwell construction to determine the phase boundaries for three temperatures below $T_c$. The intersecting point of the two extrapolated boundaries turns out to be at the expected first order transition point at $T_c$ with $\mu = 0$. This serves as a check for our method of identifying the critical point. We also studied the $N_f =2$ case, but do not see a signal of the mixed phase for temperature as low as 0.83 $T_c$.Comment: 28 pages, 11 figures,references added, final versio

The consequences of SU(3) colorsingletness, Polyakov Loop and Z(3) symmetry on a quark-gluon gas

Based on quantum statistical mechanics we show that the $SU(3)$ color singlet ensemble of a quark-gluon gas exhibits a $Z(3)$ symmetry through the normaized character in fundamental representation and also becomes equivalent, within a stationary point approximation, to the ensemble given by Polyakov Loop. Also Polyakov Loop gauge potential is obtained by considering spatial gluons along with the invariant Haar measure at each space point. The probability of the normalized character in $SU(3)$ vis-a-vis Polyakov Loop is found to be maximum at a particular value exhibiting a strong color correlation. This clearly indicates a transition from a color correlated to uncorrelated phase or vise-versa. When quarks are included to the gauge fields, a metastable state appears in the temperature range $145\le T({\rm{MeV}}) \le 170$ due to the explicit $Z(3)$ symmetry breaking in the quark-gluon system. Beyond $T\ge 170$ MeV the metastable state disappears and stable domains appear. At low temperature a dynamical recombination of ionized $Z(3)$ color charges to a color singlet $Z(3)$ confined phase is evident along with a confining background that originates due to circulation of two virtual spatial gluons but with conjugate $Z(3)$ phases in a closed loop. We also discuss other possible consequences of the center domains in the color deconfined phase at high temperature.Comment: Version published in J. Phys.

The strong thirteen spheres problem

The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen spheres problem (or the Tammes problem for 13 points) which asks to find an arrangement and the maximum radius of 13 equal size nonoverlapping spheres touching the unit sphere. In the paper we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on a enumeration of the so-called irreducible graphs.Comment: Modified lemma 2, 16 pages, 12 figures. Uploaded program packag

Classification of wines by means of multivariate data analysis using the SPME/CGC-chromatograms of volatile aroma compounds

The solid phase microextraction (SPME) is an effective solvent-free sample preparation technique for the capillary gas chromatographic (CGC) analysis of volatile aroma compounds of wines. Using discriminant analysis based upon only two terpene compounds, it was possible to analytically discern between the varieties Riesling, Muller-Thurgau and Silvaner grown in the same region. The discrimination of these varieties was unsuccessful for wines of different vintages (1988-1995). In order to obtain a highly significant classification, it was necessary to consider further aroma components described in wine literature. The differentiation between these wines by a similar high classification rate was obtained using a set of variables selected by mathematical methods. Wines prepared from known grape varieties were qualitatively recognized by factor- and cluster-analyses as well as the relative peak intensities of the terpene compounds in the SPME-CGC chromatograms. The composition of wine blends was quantitatively determined

The local atomic quasicrystal structure of the icosahedral Mg25Y11Zn64 alloy

A local and medium range atomic structure model for the face centred icosahedral (fci) Mg25Y11Zn64 alloy has been established in a sphere of r = 27 A. The model was refined by least squares techniques using the atomic pair distribution (PDF) function obtained from synchrotron powder diffraction. Three hierarchies of the atomic arrangement can be found: (i) five types of local coordination polyhedra for the single atoms, four of which are of Frank-Kasper type. In turn, they (ii) form a three-shell (Bergman) cluster containing 104 atoms, which is condensed sharing its outer shell with its neighbouring clusters and (iii) a cluster connecting scheme corresponding to a three-dimensional tiling leaving space for few glue atoms. Inside adjacent clusters, Y8-cubes are tilted with respect to each other and thus allow for overall icosahedral symmetry. It is shown that the title compound is essentially isomorphic to its holmium analogue. Therefore fci-Mg-Y-Zn can be seen as the representative structure type for the other rare earth analogues fci-Mg-Zn-RE (RE = Dy, Er, Ho, Tb) reported in the literature.Comment: 12 pages, 8 figures, 2 table

The Fermat-Torricelli problem in normed planes and spaces

We investigate the Fermat-Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat-Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat-Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach

A systematic approach for the accurate non-invasive estimation of blood glucose utilizing a novel light-tissue interaction adaptive modelling scheme

Diabetes is one of the biggest health challenges of the 21st century. The obesity epidemic, sedentary lifestyles and an ageing population mean prevalence of the condition is currently doubling every generation. Diabetes is associated with serious chronic ill health, disability and premature mortality. Long-term complications including heart disease, stroke, blindness, kidney disease and amputations, make the greatest contribution to the costs of diabetes care. Many of these long-term effects could be avoided with earlier, more effective monitoring and treatment. Currently, blood glucose can only be monitored through the use of invasive techniques. To date there is no widely accepted and readily available non-invasive monitoring technique to measure blood glucose despite the many attempts. This paper challenges one of the most difficult non-invasive monitoring techniques, that of blood glucose, and proposes a new novel approach that will enable the accurate, and calibration free estimation of glucose concentration in blood. This approach is based on spectroscopic techniques and a new adaptive modelling scheme. The theoretical implementation and the effectiveness of the adaptive modelling scheme for this application has been described and a detailed mathematical evaluation has been employed to prove that such a scheme has the capability of extracting accurately the concentration of glucose from a complex biological media

On affine maps on non-compact convex sets and some characterizations of finite-dimensional solid ellipsoids

Convex geometry has recently attracted great attention as a framework to formulate general probabilistic theories. In this framework, convex sets and affine maps represent the state spaces of physical systems and the possible dynamics, respectively. In the first part of this paper, we present a result on separation of simplices and balls (up to affine equivalence) among all compact convex sets in two- and three-dimensional Euclidean spaces, which focuses on the set of extreme points and the action of affine transformations on it. Regarding the above-mentioned axiomatization of quantum physics, our result corresponds to the case of simplest (2-level) quantum system. We also discuss a possible extension to higher dimensions. In the second part, towards generalizations of the framework of general probabilistic theories and several existing results including ones in the first part from the case of compact and finite-dimensional physical systems as in most of the literatures to more general cases, we study some fundamental properties of convex sets and affine maps that are relevant to the above subject.Comment: 25 pages, a part of this work is to be presented at QIP 2011, Singapore, January 10-14, 2011; (v2) References updated (v3) Introduction and references updated (v4) Re-organization of the paper (results not added