Searching for Apery-Style Miracles [Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm]

Roger Apery's seminal method for proving irrationality is "turned on its head" and taught to computers, enabling a one second redux of the original proof of zeta(3), and many new irrationality proofs of many new constants, alas, none of them is both famous and not-yet-proved-irrational.Comment: 16 pages. Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, May 2014, and this arxiv.org. Accompanied my Maple package NesApery, available from http://www.math.rutgers.edu/~zeilberg/tokhniot/NesAper

Lattice Green functions in all dimensions

We give a systematic treatment of lattice Green functions (LGF) on the $d$-dimensional diamond, simple cubic, body-centred cubic and face-centred cubic lattices for arbitrary dimensionality $d \ge 2$ for the first three lattices, and for $2 \le d \le 5$ for the hyper-fcc lattice. We show that there is a close connection between the LGF of the $d$-dimensional hypercubic lattice and that of the $(d-1)$-dimensional diamond lattice. We give constant-term formulations of LGFs for all lattices and dimensions. Through a still under-developed connection with Mahler measures, we point out an unexpected connection between the coefficients of the s.c., b.c.c. and diamond LGFs and some Ramanujan-type formulae for $1/\pi.$Comment: 30 page

Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals

Lattice statistical mechanics, often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in a general mathematical framework, be too complex, or could not be defined. Considering several Picard-Fuchs systems of two-variables "above" Calabi-Yau ODEs, associated with double hypergeometric series, we show that holonomic functions are actually a good framework for actually finding the singular manifolds. We, then, analyse the singular algebraic varieties of the n-fold integrals $\chi^{(n)}$, corresponding to the decomposition of the magnetic susceptibility of the anisotropic square Ising model. We revisit a set of Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find a first set of non-Nickelian singularities for $\chi^{(3)}$ and $\chi^{(4)}$, that also turns out to be rational or ellipic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model. We address, from a birational viewpoint, the emergence of families of elliptic curves, and of Calabi-Yau manifolds on such problems. We discuss the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility.Comment: 36 page

On the asymptotics of higher-dimensional partitions

We conjecture that the asymptotic behavior of the numbers of solid (three-dimensional) partitions is identical to the asymptotics of the three-dimensional MacMahon numbers. Evidence is provided by an exact enumeration of solid partitions of all integers <=68 whose numbers are reproduced with surprising accuracy using the asymptotic formula (with one free parameter) and better accuracy on increasing the number of free parameters. We also conjecture that similar behavior holds for higher-dimensional partitions and provide some preliminary evidence for four and five-dimensional partitions.Comment: 30 pages, 8 tables, 4 figures (v2) New data (63-68) for solid partitions added; (v3) published version, new subsection providing an unbiased estimate of the leading for the leading coefficient added, some tables delete

A Note on Computations of D-brane Superpotential

We develop some computational methods for the integrals over the 3-chains on the compact Calabi-Yau 3-folds that plays a prominent role in the analysis of the topological B-model in the context of the open mirror symmetry. We discuss such 3-chain integrals in two approaches. In the first approach, we provide a systematic algorithm to obtain the inhomogeneous Picard-Fuchs equations. In the second approach, we discuss the analytic continuation of the period integral to compute the 3-chain integral directly. The latter direct integration method is applicable for both on-shell and off-shell formalisms.Comment: 61 pages, 5 figures; v2: typos corrected, minor changes, references adde

The accuracy of the MMSE in detecting cognitive impairment when administered by general practitioners: A prospective observational study

<p>Abstract</p> <p>Background</p> <p>The Mini-Mental State Examination (MMSE) has contributed to detecting cognitive impairment, yet few studies have evaluated its accuracy when used by general practitioners (GP) in an actual public-health setting.</p> <p>Objectives</p> <p>We evaluated the accuracy of MMSE scores obtained by GPs by comparing them to scores obtained by Alzheimer's Evaluation Units (UVA).</p> <p>Methods</p> <p>The study was observational in design and involved 59 voluntary GPs who, after having undergone training, administered the MMSE to patients with symptoms of cognitive disturbances. Individuals who scored ≤ 24 (adjusted by age and educational level) were referred to Alzheimer's Evaluation Units (UVA) for diagnosis (including the MMSE). UVAs were unblinded to the MMSE score of the GP. To measure interrater agreement, the weighted Kappa statistic was calculated. To evaluate factors associated with the magnitude of the difference between paired scores, a linear regression model was applied. To quantify the accuracy in discriminating no cognitive impairment from any cognitive impairment and from Alzheimer's disease (AD), the ROC curves (AUC) were calculated.</p> <p>Results</p> <p>For the 317 patients, the mean score obtained by GPs was significantly lower (15.8 vs. 17.4 for the UVAs; p < 0.01). However, overall concordance was good (Kappa = 0.86). Only the diagnosis made by the UVA was associated with the difference between paired scores: the adjusted mean difference was 3.1 for no cognitive impairment and 3.8 for mild cognitive impairment. The AUC of the scores for GPs was 0.80 (95%CI: 0.75–0.86) for discriminating between no impairment and any impairment and 0.89 (95%CI: 0.84–0.94) for distinguishing patients with AD, though the UVA scores discriminated better.</p> <p>Conclusion</p> <p>In a public-health setting involving patients with symptoms of cognitive disturbances, the MMSE used by the GPs was sufficiently accurate to detect patients with cognitive impairment, particularly those with dementia.</p

The Carbohydrate-Binding Site in Galectin-3 Is Preorganized To Recognize a Sugarlike Framework of Oxygens: Ultra-High-Resolution Structures and Water Dynamics

The recognition of carbohydrates by proteins is a fundamental aspect of communication within and between living cells. Understanding the molecular basis of carbohydrate-protein interactions is a prerequisite for the rational design of synthetic ligands. Here we report the high- to ultrahigh-resolution crystal structures of the carbohydrate recognition domain of galectin-3 (Gal3C) in the ligand-free state (1.08 angstrom at 100 K, 1.25 angstrom at 298 K) and in complex with lactose (0.86 angstrom) or glycerol (0.9 angstrom). These structures reveal striking similarities in the positions of water and carbohydrate oxygen atoms in all three states, indicating that the binding site of Gal3C is preorganized to coordinate oxygen atoms in an arrangement that is nearly optimal for the recognition of beta-galactosides. Deuterium nuclear magnetic resonance (NMR) relaxation dispersion experiments and molecular dynamics simulations demonstrate that all water molecules in the lactose-binding site exchange with bulk water on a time scale of nanoseconds or shorter. Nevertheless, molecular dynamics simulations identify transient water binding at sites that agree well with those observed by crystallography, indicating that the energy landscape of the binding site is maintained in solution. All heavy atoms of glycerol are positioned like the corresponding atoms of lactose in the Gal3C complexes. However, binding of glycerol to Gal3C is insignificant in solution at room temperature, as monitored by NMR spectroscopy or isothermal titration calorimetry under conditions where lactose binding is readily detected. These observations make a case for protein cryo-crystallography as a valuable screening method in fragment-based drug discovery and further suggest that identification of water sites might inform inhibitor design