Phantom holomorphic projections arising from Sturm’s formula

A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger.     

New identities involving sums of the tails related to real quadratic fields

In previous work, the authors discovered new examples of q-hypergeometric series related to the arithmetic of $\mathbb {Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ . Building on this work, we construct in this paper sum of the tails identities for which some which some of these functions occur as error terms. As an application, we obtain formulas for the generating function of a certain zeta functions for real quadratic fields at negative integers.     

Optimization for discretized switched systems

Switching time optimization is a crucial topic in the optimal control of hybrid systems. Since it is rare that such problems can be solved analytically, the use of numerical discretization schemes for the integration of state and adjoint systems is indispensable. Thus, in this contribution, the switching time optimization problem is studied in a discretized formulation directly from the beginning. An analysis of the discretized problem reveals that smoothness of the original (continuous time) problem is lost, i.e. the problem will in general become nondifferentiable in discrete time. This has to be taken into account when deriving an adjoint-based formula for the optimization of the discretized problem. A hybrid double pendulum example is used for illustration of the results. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Synthesis and Characterization of Li‐containing Boron Carbide r‐Li~1B13C2

Transparent and nearly colorless single crystals of r‐LiB₁₃C₂ were obtained by reaction of boron with Li₂CO₃ in a Cu melt at 1250–1300 °C. The structure analysis [R3 m, a = 5.6535(1), c = 12.5320(2) Å, 421 independent reflections, 22 parameters, R₁ = 0.034, wR₂ = 0.093] revealed a crystal structure that can be described as a filling variant of rhombohedral B₁₃C₂. Li⁺ is located in a void above or below the linear CBC unit. The site occupation is close to 50 % resulting in an electron‐precise composition according to Wade's rules if a positive charge is given to the CBC entity: Li⁺(B₁₂)^2–(CBC)⁺. The displacement parameters of the CBC unit indicate disorder in the [001] direction, that relates to the short Li–C distance and the partial occupation of the Li⁺ site. The composition is confirmed by EELS measurements of single crystals. Band gap calculations give a value of 2.94 eV, which is in agreement to the crystals being colorless. The evaluation of the electron density by application of the QTAIM formalism as proposed by Bader modifies the assignment pictured above according to Wade's rules. In agreement to the electronegativities the carbon atoms carry a negative charge (–2.31/–2.42) and the effective charges are: Li^+0.81(B₁₂)^+2.02(CBC)^–2.83.     

Optimizing the Size of Platinum Nanoparticles for Enhanced Mass Activity in the Electrochemical Oxygen Reduction Reaction

High oxygen reduction (ORR) activity has been for many years considered as the key to many energy applications. Herein, by combining theory and experiment we prepare Pt nanoparticles with optimal size for the efficient ORR in proton‐exchange‐membrane fuel cells. Optimal nanoparticle sizes are predicted near 1, 2, and 3 nm by computational screening. To corroborate our computational results, we have addressed the challenge of approximately 1 nm sized Pt nanoparticle synthesis with a metal–organic framework (MOF) template approach. The electrocatalyst was characterized by HR‐TEM, XPS, and its ORR activity was measured using a rotating disk electrode setup. The observed mass activities (0.87±0.14 A mg_Pt^?1) are close to the computational prediction (0.99 A mg_Pt^?1). We report the highest to date mass activity among pure Pt catalysts for the ORR within similar size range. The specific and mass activities are twice as high as the Tanaka commercial Pt/C catalysis.