Congruences concerning Jacobi polynomials and Ap\'ery-like formulae

Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case $t=(-1)^{d}/16$ of the former sum, where the congruences hold modulo $p^{5-d}$.Comment: to appear in Int. J. Number Theor

A tour on Hermitian symmetric manifolds

Hermitian symmetric manifolds are Hermitian manifolds which are homogeneous and such that every point has a symmetry preserving the Hermitian structure. The aim of these notes is to present an introduction to this important class of manifolds, trying to survey the several different perspectives from which Hermitian symmetric manifolds can be studied.Comment: 56 pages, expanded version. Written for the Proceedings of the CIME-CIRM summer course "Combinatorial Algebraic Geometry". Comments are still welcome

Spherical designs and lattices

In this article we prove that integral lattices with minimum <= 7 (or <= 9) whose set of minimal vectors form spherical 9-designs (or 11-designs respectively) are extremal, even and unimodular. We furthermore show that there does not exist an integral lattice with minimum <=11 which yields a 13-design.Comment: The final publication is available at http://link.springer.com/article/10.1007%2Fs13366-013-0155-

Multiple CSLs for the body centered cubic lattice

Ordinary Coincidence Site Lattices (CSLs) are defined as the intersection of a lattice $\Gamma$ with a rotated copy $R\Gamma$ of itself. They are useful for classifying grain boundaries and have been studied extensively since the mid sixties. Recently the interests turned to so-called multiple CSLs, i.e. intersections of $n$ rotated copies of a given lattice $\Gamma$, in particular in connection with lattice quantizers. Here we consider multiple CSLs for the 3-dimensional body centered cubic lattice. We discuss the spectrum of coincidence indices and their multiplicity, in particular we show that the latter is a multiplicative function and give an explicit expression of it for some special cases.Comment: 4 pages, SSPCM (31 August - 7 September 2005, Myczkowce, Poland

Unified N=2 Maxwell-Einstein and Yang-Mills-Einstein Supergravity Theories in Four Dimensions

We study unified N=2 Maxwell-Einstein supergravity theories (MESGTs) and unified Yang-Mills Einstein supergravity theories (YMESGTs) in four dimensions. As their defining property, these theories admit the action of a global or local symmetry group that is (i) simple, and (ii) acts irreducibly on all the vector fields of the theory, including the ``graviphoton''. Restricting ourselves to the theories that originate from five dimensions via dimensional reduction, we find that the generic Jordan family of MESGTs with the scalar manifolds [SU(1,1)/U(1)] X [SO(2,n)/SO(2)X SO(n)] are all unified in four dimensions with the unifying global symmetry group SO(2,n). Of these theories only one can be gauged so as to obtain a unified YMESGT with the gauge group SO(2,1). Three of the four magical supergravity theories defined by simple Euclidean Jordan algebras of degree 3 are unified MESGTs in four dimensions. Two of these can furthermore be gauged so as to obtain 4D unified YMESGTs with gauge groups SO(3,2) and SO(6,2), respectively. The generic non-Jordan family and the theories whose scalar manifolds are homogeneous but not symmetric do not lead to unified MESGTs in four dimensions. The three infinite families of unified five-dimensional MESGTs defined by simple Lorentzian Jordan algebras, whose scalar manifolds are non-homogeneous, do not lead directly to unified MESGTs in four dimensions under dimensional reduction. However, since their manifolds are non-homogeneous we are not able to completely rule out the existence of symplectic sections in which these theories become unified in four dimensions.Comment: 47 pages; latex fil

The flavor symmetry in the standard model and the triality symmetry

A Dirac fermion is expressed by a 4 component spinor which is a combination of two quaternions and which can be treated as an octonion. The octonion possesses the triality symmetry, which defines symmetry of fermion spinors and bosonic vector fields. The triality symmetry relates three sets of spinors and two sets of vectors, which are transformed among themselves via transformations $G_{23}, G_{12}, G_{13}$, $G_{123}$ and $G_{132}$. If the electromagnetic (EM) interaction is sensitive to the triality symmetry, i.e. EM probe selects one triality sector, EM signals from the 5 transformed world would not be detected, and be treated as the dark matter. According to an astrophysical measurement, the ratio of the dark to ordinary matter in the universe as a whole is almost exactly 5. We expect quarks are insensitive to the triality, and triality will appear as three times larger flavor degrees of freedom in the lattice simulation.Comment: 16 pages 8 figures, To be published in International Journal of Modern Physics

Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups

We study the symmetries of generalized spacetimes and corresponding phase spaces defined by Jordan algebras of degree three. The generic Jordan family of formally real Jordan algebras of degree three describe extensions of the Minkowskian spacetimes by an extra "dilatonic" coordinate, whose rotation, Lorentz and conformal groups are SO(d-1), SO(d-1,1) XSO(1,1) and SO(d,2)XSO(2,1), respectively. The generalized spacetimes described by simple Jordan algebras of degree three correspond to extensions of Minkowskian spacetimes in the critical dimensions (d=3,4,6,10) by a dilatonic and extra (2,4,8,16) commuting spinorial coordinates, respectively. The Freudenthal triple systems defined over these Jordan algebras describe conformally covariant phase spaces. Following hep-th/0008063, we give a unified geometric realization of the quasiconformal groups that act on their conformal phase spaces extended by an extra "cocycle" coordinate. For the generic Jordan family the quasiconformal groups are SO(d+2,4), whose minimal unitary realizations are given. The minimal unitary representations of the quasiconformal groups F_4(4), E_6(2), E_7(-5) and E_8(-24) of the simple Jordan family were given in our earlier work hep-th/0409272.Comment: A typo in equation (37) corrected and missing titles of some references added. Version to be published in JHEP. 38 pages, latex fil

Lectures on Spectrum Generating Symmetries and U-duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace

We review the underlying algebraic structures of supergravity theories with symmetric scalar manifolds in five and four dimensions, orbits of their extremal black hole solutions and the spectrum generating extensions of their U-duality groups. For 5D, N=2 Maxwell-Einstein supergravity theories (MESGT) defined by Euclidean Jordan algebras, J, the spectrum generating symmetry groups are the conformal groups Conf(J) of J which are isomorphic to their U-duality groups in four dimensions. Similarly, the spectrum generating symmetry groups of 4D, N=2 MESGTs are the quasiconformal groups QConf(J) associated with J that are isomorphic to their U-duality groups in three dimensions. We then review the work on spectrum generating symmetries of spherically symmetric stationary 4D BPS black holes, based on the equivalence of their attractor equations and the equations for geodesic motion of a fiducial particle on the target spaces of corresponding 3D supergravity theories obtained by timelike reduction. We also discuss the connection between harmonic superspace formulation of 4D, N=2 sigma models coupled to supergravity and the minimal unitary representations of their isometry groups obtained by quantizing their quasiconformal realizations. We discuss the relevance of this connection to spectrum generating symmetries and conclude with a brief summary of more recent results.Comment: 55 pages; Latex fil

Spectrum Generating Conformal and Quasiconformal U-Duality Groups, Supergravity and Spherical Vectors

After reviewing the algebraic structures that underlie the geometries of N=2 Maxwell-Einstein supergravity theories (MESGT) in five and four dimensions with symmetric scalar manifolds, we give a unified realization of their three dimensional U-duality groups as spectrum generating quasiconformal groups. They are F_{4(4)}, E_{6(2)}, E_{7(-5)}, E_{8(-24)} and SO(n+2,4). Our formulation is covariant with respect to U-duality symmetry groups of corresponding five dimensional supergravity theories, which are SL(3,R), SL(3,C), SU*(6), E_{6(6)} and SO(n-1,1)X SO(1,1), respectively. We determine the spherical vectors of quasiconformal realizations of all these groups twisted by a unitary character. We also give their quadratic Casimir operators and determine their values. Our work lays the algebraic groundwork for constructing the unitary representations of these groups induced by their geometric quasiconformal actions, which include the quaternionic discrete series. For rank 2 cases, SU(2,1) and G_{2(2)}, corresponding to simple N=2 supergravity in four and five dimensions, this program was carried out in arXiv:0707.1669. We also discuss the corresponding algebraic structures underlying symmetries of matter coupled N=4 and N>4 supergravity theories. They lead to quasiconformal realizations of split real forms of U-duality groups as a straightforward extension of the quaternionic real forms.Comment: Section 4 is split with the addition of a subsection on quadratic Casimir operators; references added; typos corrected. Latex file; 53 page

Evaluating whole grain intervention study designs and reporting practices using evidence mapping methodology

Consumption of whole grains have been associated with reduced risk of chronic diseases in many observational studies; yet, results of intervention studies are mixed. We aimed to use evidence mapping to capture the methodological and reporting variability in whole grain intervention studies that may contribute to this inconsistency. We conducted a reproducible search in OVID Medline for whole grain human intervention studies (published 1946 to February 2018). After screening based on a priori criteria, we identified 202 publications describing a total of 213 unique trials. Over half (55%) were acute trials, lasting ≤1 day, 30% were moderate duration studies (up to 6 weeks) and 15% were of longer duration (more than 6 weeks). The majority of acute trials (75%) examined measures of glycaemia and/or insulinemia, while most of the longer trials included measures of cardiometabolic health (71%), appetite/satiety (57%) and weight/adiposity (56%). Among the moderate and long duration trials, there was a wide range of how whole grains were described but only 10 publications referenced an established definition. Only 55% of trials reported the actual amount of whole grains (in grams or servings), while 36% reported the amount of food/product and 9% did not report a dose at all. Of the interventions that provided a mixture of whole grains, less than half (46%) reported the distribution of the different grain types. Reporting of subject compliance also varied and only 22% used independent biomarkers of whole grain intake. This evidence map highlights the need to standardize both study protocols and reporting practices to support effective synthesis of study results and provide a stronger foundation to better inform nutrition scientists and public health policy