Abstract:
There is a strong need for instruments which make it possible to evaluate students’ learning progress in a formative way and also in a way that can be accomplished in primary school settings (Klauer, 2006; Diehl & Hartke, 2007; Strathmann & Klauer, 2008; Walter, 2008; Diehl, Hartke & Knopp, 2009; Koch & Knopp, 2010). Instruments meeting these criteria can be considered as one crucial element of effectively preventing learning difficulties. In this article, the test “Rechenfische“ is discussed as one possibility for evaluating the learning progress in a formative way. It tests students’ knowledge in first-year arithmetic. First the test itself and the design of a study (N=1688) used to implement this test for the first time are described and, afterwards, findings are reported and discussed.

Abstract:
In this paper, we propose a novel low-complexity interference-aware receiver structure for multi-user MIMO that is based on the exploitation of the structure of residual interference. We show that multi-user MIMO can deliver its promised gains in modern wireless systems in spite of the limited channel state information at the transmitter (CSIT) only if users resort to intelligent interference-aware detection rather than the conventional single-user detection. As an example, we focus on the long term evolution (LTE) system and look at the two important characteristics of the LTE precoders, i.e., their low resolution and their applying equal gain transmission (EGT). We show that EGT is characterized by full diversity in the single-user MIMO transmission but it loses diversity in the case of multi-user MIMO transmission. Reflecting on these results, we propose a LTE codebook design based on two additional feedback bits of CSIT and show that this new codebook significantly outperforms the currently standardized LTE codebooks for multi-user MIMO transmission.

Abstract:
We track the trajectories of individual horocycles on the modular surface. Our tracking is constructive, and we thus \emph{effectively} establish topological transitivity and even line-transitivity for the horocyclic flow. We also describe homotopy class jumps that occur under continuous deformation of horocycles.

Abstract:
We characterize all logarithmic, holomorphic vector-valued modular forms which can be analytically continued to a region strictly larger than the upper half-plane.

Abstract:
We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $\rho: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that $\rho(T)$ has eigenvalues of absolute value 1. The main result is the construction of meromorphic matrix-valued Poincar\'e series associated to $\rho$ for all large enough weights. The component functions are logarithmic $q$-series, i.e., finite sums of products of $q$-series and powers of $\log q$. We derive several consequences, in particular we show that the space $\mathcal{H}(\rho)=\oplus_k \mathcal{H}(k, \rho)$ of all holomorphic logarithmic vector-valued modular forms associated to $\rho$ is a free module of rank $p$ over the ring of classical holomorphic modular forms on $SL_2(\mathbb{Z})$.

Abstract:
This paper studies certain horocyclic orbits on $\Gamma(1)\frontslash\mathcal{H}$. In the first instance we examine horocycles defined using the pencil of circles whose common point (in the words of the Nielsen-Fenchel manuscript is $\infty$. The orbits involved in this case are closed and long - judged by arc length between two points compared to the hyperbolic distance between them. We are concerned with tracking the paths of individual horocycles. Using Ford circles of Farey sequences we find lifts to the Standard Fundamental Region (SFR) and find points of these lifts making given angles with a horizontal. Next, we offer two methods, both involving continued fractions, of locating points with such angles whose lifts are near any given point in the SFR. This establishes in an effective manner a sort of transitivity, which necessarily involves infinitely many such horocycles. Next, we study the homotopy classes of horizontal horocycles as we descend to the real axis. We find these are stable during descent between encounters of the horizontal with elliptic fixed points. Such encounters change - complicate - the homotopy classes. We give these explicitly down to height $1/(2\sqrt{3})$. Finally we do an initial study of the open (infinite length) horocycle path with unit euclidean radius anchored at $\phi -1$, where $\phi$ is the Golden Mean. Enough information is adduced to suggest that this path is itself transitive. The methods resemble the Hardy-Littlewood Circle Method in a certain regard, albeit without the exponential sums.

Abstract:
This paper considers a communication network comprised of two nodes, which have no mutual direct communication links, communicating two-way with the aid of a common relay node (RN), also known as separated two-way relay (TWR) channel. We first recall a cut-set outer bound for the set of rates in the context of this network topology assuming full-duplex transmission capabilities. Then, we derive a new achievable rate region based on hash-and-forward (HF) relaying where the RN does not attempt to decode but instead hashes its received signal, and show that under certain channel conditions it coincides with Shannon's inner-bound for the two-way channel [1]. Moreover, for binary adder TWR channel with additive noise at the nodes and the RN we provide a detailed capacity achieving coding scheme based on structure codes.

Abstract:
This paper endeavors to track the trajectories of individual horocycles on \modsurf. It is far more common to study \emph{sets} of such trajectories, seeking some asymptotic behavior using an averaging process (see section \ref{previous}). Our work is only marginally related to these efforts. We begin by examining horocycles defined using the pencil of circles whose common point (in the words of the Nielsen-Fenchel manuscript \cite{wF}) is $\infty$. The orbits involved in this case are closed and long --- judged by arc length between two points compared to the hyperbolic distance between them. Using Ford circles of Farey sequences we find their lifts to the Standard Fundamental Region (SFR) and find points of these lifts making given angles with a horizontal. Next, we offer two algorithms, both involving continued fractions, of locating points whose angle with the horizontal is near any target angle and whose lifts are near any given point in the SFR. Next, we study the homotopy classes of horizontal horocycles as we descend to the real axis. We find these are stable during descent between encounters of the horizontal horocycle with elliptic fixed points. Such encounters change --- complicate --- the homotopy classes. We give these explicitly down to height $1/(2\sqrt{3})$. Finally we do an initial study of the open (infinite length) horocycle path with unit euclidean radius anchored at $\phi -1$, where $\phi$ is the Golden Mean. Enough information is adduced to suggest that this single doubly infinite path is transitive.