The Max Planck Institute for Mathematics in the science in Leipzig was founded on March 1, 1996.At the institute scientists work on projects which apply mathematics in various areas of natural science, in particular physics, biology, chemistry and material science.Main research areas are Scientific computing , Pattern formation, energy landscapes and scaling laws Riemannian, Kählerian and algebraic geometry , neuronal networks ,The institute has an extensive visitors programme which had made Leipzig a main place for research in applied mathematics. Wikipedia.
Neukamm S.,Max Planck Institute for Mathematics in the Sciences
Archive for Rational Mechanics and Analysis | Year: 2012
We present a rigorous derivation of a homogenized, bending-torsion theory for inextensible rods from three-dimensional nonlinear elasticity in the spirit of Γ -convergence. We start with the elastic energy functional associated with a nonlinear composite material, which in a stress-free reference configuration occupies a thin cylindrical domain with thickness h ≪ 1. We consider composite materials that feature a periodic microstructure with period ε ≪ 1. We study the behavior as ε and h simultaneously converge to zero and prove that the energy (scaled by h -4) Γ -converges towards a non-convex, singular energy functional. The energy is finite only for configurations that correspond to pure bending and twisting of the rod. In this case, the energy is quadratic in curvature and torsion. Our derivation leads to a new relaxation formula that uniquely determines the homogenized coefficients. It turns out that their precise structure additionally depends on the ratio h/ε and, in particular, different relaxation formulas arise for h ≪ ε, ε ~ h and ε ≪ h. Although the initial elastic energy functional and the limiting functional are nonconvex, our analysis leads to a relaxation formula that is quadratic and involves only relaxation over a single cell. Moreover, we derive an explicit formula for isotropic materials in the cases h ≪ ε and h ≫ ε, and prove that the Γ -limits associated with homogenization and dimension reduction in general do not commute. © 2012 Springer-Verlag.
Rauh J.,Max Planck Institute for Mathematics in the Sciences
IEEE Transactions on Information Theory | Year: 2011
This paper investigates maximizers of the information divergence from an exponential family E. It is shown that the rI-projection of a maximizer P to E is a convex combination of P and a probability measure P- with disjoint support and the same value of the sufficient statistics A. This observation can be used to transform the original problem of maximizing D(·∥E) over the set of all probability measures into the maximization of a function Dr over a convex subset of ker A. The global maximizers of both problems correspond to each other. Furthermore, finding all local maximizers of D̄r yields all local maximizers of D(·∥E). This paper also proposes two algorithms to find the maximizers of D̄r and applies them to two examples, where the maximizers of D(·∥E) were not known before. © 2011 IEEE.
Bacak M.,Max Planck Institute for Mathematics in the Sciences
SIAM Journal on Optimization | Year: 2014
The geometric median as well as the Fréchet mean of points in a Hadamard space are important in both theory and applications. Surprisingly, no algorithms for their computation are hitherto known. To address this issue, we use a splitting version of the proximal point algorithm for minimizing a sum of convex functions and prove that this algorithm produces a sequence converging to a minimizer of the objective function, which extends a recent result of Bertsekas [Math. Program., 129(2011), pp. 163-195] into Hadamard spaces. The method is quite robust, and not only does it yield algorithms for the median and the mean, but also it applies to various other optimization problems. We, moreover, show that another algorithm for computing the Fréchet mean can be derived from the law of large numbers due to Sturm [Ann. Probab., 30(2002), pp. 1195-1222]. In applications, computing medians and means is probably most needed in tree space, which is an instance of a Hadamard space, invented by Billera, Holmes, and Vogtmann [Adv. in Appl. Math., 27(2001), pp. 733-767] as a tool for averaging phylogenetic trees. Since there now exists a polynomialtime algorithm for computing geodesics in tree space due to Owen and Provan [IEEE/ACM Trans. Comput. Biol. Bioinform., 8(2011), pp. 2-13], we obtain efficient algorithms for computing medians and means of trees, which can be directly used in practice. © 2014 Societ y for Industrial and Applied Mathematics.
Mlynarski W.,Max Planck Institute for Mathematics in the Sciences
PLoS Computational Biology | Year: 2015
In mammalian auditory cortex, sound source position is represented by a population of broadly tuned neurons whose firing is modulated by sounds located at all positions surrounding the animal. Peaks of their tuning curves are concentrated at lateral position, while their slopes are steepest at the interaural midline, allowing for the maximum localization accuracy in that area. These experimental observations contradict initial assumptions that the auditory space is represented as a topographic cortical map. It has been suggested that a “panoramic” code has evolved to match specific demands of the sound localization task. This work provides evidence suggesting that properties of spatial auditory neurons identified experimentally follow from a general design principle- learning a sparse, efficient representation of natural stimuli. Natural binaural sounds were recorded and served as input to a hierarchical sparse-coding model. In the first layer, left and right ear sounds were separately encoded by a population of complex-valued basis functions which separated phase and amplitude. Both parameters are known to carry information relevant for spatial hearing. Monaural input converged in the second layer, which learned a joint representation of amplitude and interaural phase difference. Spatial selectivity of each second-layer unit was measured by exposing the model to natural sound sources recorded at different positions. Obtained tuning curves match well tuning characteristics of neurons in the mammalian auditory cortex. This study connects neuronal coding of the auditory space with natural stimulus statistics and generates new experimental predictions. Moreover, results presented here suggest that cortical regions with seemingly different functions may implement the same computational strategy-efficient coding. © 2015 Wiktor Młynarski.
Elze T.,Max Planck Institute for Mathematics in the Sciences
PLoS ONE | Year: 2010
Background: In visual psychophysics, precise display timing, particularly for brief stimulus presentations, is often required. The aim of this study was to systematically review the commonly applied methods for the computation of stimulus durations in psychophysical experiments and to contrast them with the true luminance signals of stimuli on computer displays. Methodology/Principal Findings: In a first step, we systematically scanned the citation index Web of Science for studies with experiments with stimulus presentations for brief durations. Articles which appeared between 2003 and 2009 in three different journals were taken into account if they contained experiments with stimuli presented for less than 50 milliseconds. The 79 articles that matched these criteria were reviewed for their method of calculating stimulus durations. For those 75 studies where the method was either given or could be inferred, stimulus durations were calculated by the sum of frames (SOF) method. In a second step, we describe the luminance signal properties of the two monitor technologies which were used in the reviewed studies, namely cathode ray tube (CRT) and liquid crystal display (LCD) monitors. We show that SOF is inappropriate for brief stimulus presentations on both of these technologies. In extreme cases, SOF specifications and true stimulus durations are even unrelated. Furthermore, the luminance signals of the two monitor technologies are so fundamentally different that the duration of briefly presented stimuli cannot be calculated by a single method for both technologies. Statistics over stimulus durations given in the reviewed studies are discussed with respect to different duration calculation methods. Conclusions/Significance: The SOF method for duration specification which was clearly dominating in the reviewed studies leads to serious misspecifications particularly for brief stimulus presentations. We strongly discourage its use for brief stimulus presentations on CRT and LCD monitors. © 2010 Tobias Elze.