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.     

Shintani lifts and fractional derivatives for harmonic weak Maass forms

In this paper, we construct Shintani lifts from integral weight weakly holomorphic modular forms to half-integral weight weakly holomorphic modular forms. Although defined by different methods, these coincide with the classical Shintani lifts when restricted to the space of cusp forms. As a side effect, this gives the coefficients of the classical Shintani lifts as new cycle integrals. This yields new formulas for the L-values of Hecke eigenforms. When restricted to the space of weakly holomorphic modular forms orthogonal to cusp forms, the Shintani lifts introduce a definition of weakly holomorphic Hecke eigenforms. Along the way, auxiliary lifts are constructed from the space of harmonic weak Maass forms which yield a “fractional derivative” from the space of half-integral weight harmonic weak Maass forms to half-integral weight weakly holomorphic modular forms. This fractional derivative complements the usual ξ-operator introduced by Bruinier and Funke.     

Introducing oblique norms into multiple criteria programming

We propose to use block norms to generate nondominated solutions of multiple criteria programs and introduce the new concept of the oblique norm that is specially tailored to handle general problems. We prove the equivalence of finding the properly nondominated solutions of a multiple criteria program and solving its scalarization by means of oblique norms.     

Erratum to ``On a conjecture of Koike on identities between Thompson series and Rogers-Ramanujan functions'

This note gives a correction to formula (A.18) of Appendix A.

     

Studying Dynamical Principles of Human Locomotion using Inverse Optimal Control

Human movement, as for example human gait, can be considered as an optimal realization of some given task. However, the criterion for which the naturally performed human motion is optimal, is generally not known. In this article we formulate an inverse optimal control problem to study the relevance of four different optimization criteria in human locomotion. As a walking model we use an actuated three dimensional spring loaded inverted pendulum (3D-SLIP), which is able to mirror the typical shape of the center of mass trajectory in human gait. Using a direct all-at-once approach, the weighting of the optimization criteria and the position of the footsteps are optimized in such a way, that the center of mass trajectory of the resulting optimal state fits real motion capture data as good as possible. Numerical experiments show, that whereas the so called capture point seems to have a great impact on human walking, minimization of the vertical center of mass movement does not show any relevance at all. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms

In this paper we construct a lifting map from a vector space of generalized Jacobi cusp forms to a certain subspace of elliptic cusp forms and vice versa such that both mappings are adjoint with respect to the Petersson scalar products.     

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.     

Search for the two-photon decay of a light penetrating particle from a 590 MeV proton beam dump

A search was performed for the two-photon decay of a light penetrating particle (e.g. an axion) produced in the dump of the 590 MeV proton beam of the Swiss Institute for Nuclear Research (SIN). The apparatus was situated behind the beam dump and 8 m of iron-concrete shielding. The detector consisted of two counter triggered optical spark chambers placed 2m apart. Photons were converted in a lead foil at the entrance of either spark chamber. From connecting the conversion points in the two modules the primary particle direction is determined to 1 degree (rms). The apparatus was exposed to 518 coulomb of protons. No two-photon decay signal was found in excess of the background, which was measured during almost 1000 h with beam off, and which was well reproduced by Monte Carlo simulations of cosmic ray gamma radiation. This null result places restrictive limits on the production and the decay of axions and similar particles.