Traces, high powers and one level density for families of curves over finite fields

AbstractThe zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix ΘC. We develop and present a new technique to compute the expected value of tr(ΘCn) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [Rud10] and Chinis [Chi16]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [BDF+16] and [Zha]. We extend [BDF+16] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L-functions L(1/2 + it, χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers.</jats:p

Short-Interval Averages of Sums of Fourier Coefficients of Cusp Forms

Let $f$ be a weight $k$ holomorphic cusp form of level one, and let $S_f(n)$ denote the sum of the first $n$ Fourier coefficients of $f$. In analogy with Dirichlet's divisor problem, it is conjectured that $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + \epsilon}$. Understanding and bounding $S_f(X)$ has been a very active area of research. The current best bound for individual $S_f(X)$ is $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{3}} (\log X)^{-0.1185}$ from Wu. Chandrasekharan and Narasimhan showed that the Classical Conjecture for $S_f(X)$ holds on average over intervals of length $X$. Jutila improved this result to show that the Classical Conjecture for $S_f(X)$ holds on average over short intervals of length $X^{\frac{3}{4} + \epsilon}$. Building on the results and analytic information about $\sum \lvert S_f(n) \rvert^2 n^{-(s + k - 1)}$ from our recent work, we further improve these results to show that the Classical Conjecture for $S_f(X)$ holds on average over short intervals of length $X^{\frac{2}{3}}(\log X)^{\frac{1}{6}}$.Comment: To Appear in the Journal of Number Theor

Models and Algorithms for Production Planning and Scheduling in Foundries – Current State and Development Perspectives

Mathematical programming, constraint programming and computational intelligence techniques, presented in the literature in the field of operations research and production management, are generally inadequate for planning real-life production process. These methods are in fact dedicated to solving the standard problems such as shop floor scheduling or lot-sizing, or their simple combinations such as scheduling with batching. Whereas many real-world production planning problems require the simultaneous solution of several problems (in addition to task scheduling and lot-sizing, the problems such as cutting, workforce scheduling, packing and transport issues), including the problems that are difficult to structure. The article presents examples and classification of production planning and scheduling systems in the foundry industry described in the literature, and also outlines the possible development directions of models and algorithms used in such systems

The correlation between the energy gap and the pseudogap temperature in cuprates: the YCBCZO and LSHCO case

The paper analyzes the influence of the hole density, the out-of-plane or in-plane disorder, and the isotopic oxygen mass on the zero temperature energy gap ($2\Delta\left(0\right)$) for $\rm{Y}_{1-x}\rm{Ca}_{x}\rm{Ba}_2\left(\rm{Cu}_{1-y}\rm{Zn}_{y}\right)_{3}\rm{O}_{7-\delta}$ (YCBCZO) and $\rm{La}_{1.96-x}\rm{Sr}_{x}\rm{Ho}_{0.04}\rm{CuO}_{4}$ (LSHCO) superconductors. It has been found that the energy gap is visibly correlated with the value of the pseudogap temperature ($T^{\star}$). On the other hand, no correlation between $2\Delta\left(0\right)$ and the critical temperature ($T_{C}$) has been found. The above results mean that the value of the dimensionless ratio $2\Delta\left(0\right)/k_{B}T_{C}$ can vary very strongly together with the chemical composition, while the parameter $2\Delta\left(0\right)/k_{B}T^{\star}$ does not change significantly. In the paper, the analytical formula which binds the zero temperature energy gap and the pseudogap temperature has been also presented.Comment: 7 pages, 4 figures, 3 table

Characteristics of the Eliashberg formalism on the example of high-pressure superconducting state in phosphor

The work describes the properties of the high-pressure superconducting state in phosphor: $p\in\{20, 30, 40, 70\}$ GPa. The calculations were performed in the framework of the Eliashberg formalism, which is the natural generalization of the BCS theory. The exceptional attention was paid to the accurate presentation of the used analysis scheme. With respect to the superconducting state in phosphor it was shown that: (i) the observed not-high values of the critical temperature ($\left[T_{C}\right]_{p=30{\rm GPa}}^{\rm max}=8.45$ K) result not only from the low values of the electron - phonon coupling constant, but also from the very strong depairing Coulomb interactions, (ii) the inconsiderable strong - coupling and retardation effects force the dimensionless ratios $R_{\Delta}$, $R_{C}$, and $R_{H}$ - related to the critical temperature, the order parameter, the specific heat and the thermodynamic critical field - to take the values close to the BCS predictions.Comment: 6 pages, 6 figure

Large magnetic circular dichroism in resonant inelastic x-ray scattering at the Mn L-edge of Mn-Zn ferrite

We report resonant inelastic x-ray scattering (RIXS) excited by circularly polarized x-rays on Mn-Zn ferrite at the Mn L2,3-resonances. We demonstrate that crystal field excitations, as expected for localized systems, dominate the RIXS spectra and thus their dichroic asymmetry cannot be interpreted in terms of spin-resolved partial density of states, which has been the standard approach for RIXS dichroism. We observe large dichroic RIXS at the L2-resonance which we attribute to the absence of metallic core hole screening in the insulating Mn-ferrite. On the other hand, reduced L3-RIXS dichroism is interpreted as an effect of longer scattering time that enables spin-lattice core hole relaxation via magnons and phonons occurring on a femtosecond time scale.Comment: 7 pages, 2 figures, http://link.aps.org/doi/10.1103/PhysRevB.74.17240