Interpolated sequences and critical LL-values of modular forms

Recently, Zagier expressed an interpolated version of the Ap\'ery numbers for $\zeta(3)$ in terms of a critical $L$-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier's six sporadic sequences are essentially critical $L$-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown's cellular integrals and critical $L$-values of modular forms of odd weight.Comment: 23 pages, to appear in Proceedings for the KMPB conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theor

Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant

Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler's constant. In the final section, we evaluate more general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral -- over some polytopes that are higher-dimensional analogs of $T$. This leads to a relation between certain multiple polylogarithm values and multiple zeta values.Comment: 19 pages, to appear in Mat Zametki. Ver 2.: Added Remark 3 on a Chen (Drinfeld-Kontsevich) iterated integral; simplified Proposition 2; gave reference for (19); corrected [16]; fixed typ

Study on Solar KANG Heating System for Cold Areas

AbstractThe current rural traditional heated kang cannot meet people's increasing requirements of comfort and environmental protection. This paper propose solar kang heating system in cold regions. System performance and heating effect were analyzed. We selected two typical rooms. One was set in traditional kang, and the other one was solar Kang type. Using temperature recording instrument and 64 roads inspection instrument and other instruments, we test the indoor temperature and the kang surface temperature of two rooms. Solar kang thermal resistance, heat storage, heat dissipation and heating effect were analyzed and compared. The results of the study show this system have the smaller fluctuation, more comfort while alleviating the kang surface overheat or super-cooling problem. It satisfied the requirements of indoor thermal comfort. The warming rate is 5.17°C/h, and the cooling rate is 3.01°C/h. These are slower than traditional Huokang speed. It improved the heat storage capacity of kang body with surface heat dissipation 1237W. Average temperature of the solar kang heating room was improved 3.28°C. It gets the smaller indoor temperature fluctuation. PMV values are concentrated about -0.5, and this basically meet the requirements of the user comfort

Gauge Theory Wilson Loops and Conformal Toda Field Theory

The partition function of a family of four dimensional N=2 gauge theories has been recently related to correlation functions of two dimensional conformal Toda field theories. For SU(2) gauge theories, the associated two dimensional theory is A_1 conformal Toda field theory, i.e. Liouville theory. For this case the relation has been extended showing that the expectation value of gauge theory loop operators can be reproduced in Liouville theory inserting in the correlators the monodromy of chiral degenerate fields. In this paper we study Wilson loops in SU(N) gauge theories in the fundamental and anti-fundamental representation of the gauge group and show that they are associated to monodromies of a certain chiral degenerate operator of A_{N-1} Toda field theory. The orientation of the curve along which the monodromy is evaluated selects between fundamental and anti-fundamental representation. The analysis is performed using properties of the monodromy group of the generalized hypergeometric equation, the differential equation satisfied by a class of four point functions relevant for our computation.Comment: 17 pages, 3 figures; references added

Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.Comment: 39 page

Representations of sl(2,?) in category O and master symmetries

We show that the indecomposable sl(2,?)-modules in the Bernstein-Gelfand-Gelfand category O naturally arise for homogeneous integrable nonlinear evolution systems. We then develop a new approach called the O scheme to construct master symmetries for such integrable systems. This method naturally allows computing the hierarchy of time-dependent symmetries. We finally illustrate the method using both classical and new examples. We compare our approach to the known existing methods used to construct master symmetries. For new integrable equations such as a Benjamin-Ono-type equation, a new integrable Davey-Stewartson-type equation, and two different versions of (2+1)-dimensional generalized Volterra chains, we generate their conserved densities using their master symmetries

Super congruences and Euler numbers

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $\pi^2$, $\pi^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=\pi^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$

Inequalities in education and national income are associated with poorer diet: pooled analysis of individual participant data across 12 European countries

Background: Malnutrition linked to noncommunicable diseases presents major health problems across Europe. The World Health Organisation encourages countries to conduct national dietary surveys to obtain data to inform public health policies designed to prevent noncommunicable diseases. Methods: Data on 27334 participants aged 19-64y were harmonised and pooled across national dietary survey datasets from 12 countries across the WHO European Region. Weighted mean nutrient intakes were age-standardised using the Eurostat 2013 European Standard Population. Associations between country-level Gross Domestic Product (GDP) and key nutrients and nutrient densities were investigated using linear regression. The potential mitigating influence of participant-level educational status was explored. Findings: Higher GDP was positively associated with total sugar intake (5·0% energy for each 10% increase in GDP, 95% CI 0·6, 9·3). Scandinavian countries had the highest vitamin D intakes. Participants with higher educational status had better nutritional intakes, particularly within lower GDP countries. A 10% higher GDP was associated with lower total fat intakes (-0·2% energy, 95% CI -0·3, -0·1) and higher daily total folate intakes (14µg, 95% CI 12, 16) in higher educated individuals. Interpretation: Lower income countries and lower education groups had poorer diet, particularly for micronutrients. We demonstrate for the first time that higher educational status appeared to have a mitigating effect on poorer diet in lower income countries. It illustrates the feasibility and value of harmonising national dietary survey data to inform European policy regarding access to healthy diets, particularly in disadvantaged groups. It specifically highlights the need for strong policies supporting nutritional intakes, prioritising lower education groups and lower income countries

Effects of ecosystem protection on scallop populations within a community-led temperate marine reserve

This study investigated the effects of a newly established, fully protected marine reserve on benthic habitats and two commercially valuable species of scallop in Lamlash Bay, Isle of Arran, United Kingdom. Annual dive surveys from 2010 to 2013 showed the abundance of juvenile scallops to be significantly greater within the marine reserve than outside. Generalised linear models revealed this trend to be significantly related to the greater presence of macroalgae and hydroids growing within the boundaries of the reserve. These results suggest that structurally complex habitats growing within the reserve have substantially increased spat settlement and/or survival. The density of adult king scallops declined threefold with increasing distance from the boundaries of the reserve, indicating possible evidence of spillover or reduced fishing effort directly outside and around the marine reserve. However, there was no difference in the mean density of adult scallops between the reserve and outside. Finally, the mean age, size, and reproductive and exploitable biomass of king scallops were all significantly greater within the reserve. In contrast to king scallops, the population dynamics of queen scallops (Aequipecten opercularis) fluctuated randomly over the survey period and showed little difference between the reserve and outside. Overall, this study is consistent with the hypothesis that marine reserves can encourage the recovery of seafloor habitats, which, in turn, can benefit populations of commercially exploited species, emphasising the importance of marine reserves in the ecosystem-based management of fisheries

Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros

of cyclotomic polynomials with interlacing zero