In the 1930's, Tarski introduced his plank problem at a time when the field Discrete Geometry was about to born. It is quite remarkable that Tarski's question and its variants continue to generate interest in the geometric and analytic aspects of coverings by planks in the present time as well. The paper is a survey type with a list of open research problems.

Abstract By linearly parameterizing the QCD Landau free energy near critical point in baryon chemical potential and temperature plane, we study fluctuations of chiral order parameter field (the $$\sigma $$ σ field) equilibrium case dynamical phase transition, respectively. setting system size to typical QGP f...

[Purpose] This study applied variable plank exercises on a stable surface consisting of a mat, on an unstable surface consisting of a dynamic cushion placed under the upper extremities and lower extremities. [Subjects] Forty-three healthy male and female subjects in their 20s voluntarily consented to participate in this study. [Methods] The subjects performed the plank exercises on the three su...

In this paper we develop a computationally efficient, two-population, time-dependent Fokker-Plank approach in the two dimensions of energy and angular momentum to study rates tidal disruption events (TDEs), extreme mass ratio inspirals (EMRIs) direct plunges occurring around massive black holes (MBHs) galactic nuclei. We test our code by exploring wide range astrophysically relevant parameter s...

Abstract Ball’s complex plank theorem states that if $$v_1,\dots ,v_n$$ v 1 , ⋯ n are unit vectors in $${\mathbb {C}}^d$$ C d , and $$t_1,\...

Molecular signal transmission in cell is very crucial for information exchange. How to understand its transmission mechanism has attracted many researchers. In this paper, we prove that signal transmission problem between neural tumor molecules and drug molecules can be achieved by synchronous control. To achieve our purpose, we derive the Fokker-Plank equation by using the Langevin equation an...

Proton beams in medical applications deal with relatively thick targets like the human head or trunk. Therefore, relatively small differences in the total proton stopping power given, for example, by the different models provided by GEANT4 can lead to significant disagreements in the final proton energy spectra when integrated along lengthy proton trajectories. This work presents proton energy ...

D. Imperati, M. Cross, K. Kruttwig, P. Galvosas, K. Ulrich, A. Pampel, J. Kärger, M. Hoehn, H. E. Möller, and M. Tittgemeyer Max-Plank Institute for Neurological Research, Cologne, Germany, Department of Computer Science, University of Milan, Milan, Italy, Interdisciplinary Center for Medical Research, Leipzig, Germany, Department of Physics, University of Leipzig, Leipzig, Germany, Max-Plank I...