The main thesis developed in this article is that the key feature of biological life is the a biological process can control and regulate other processes, and it maintains that ability over time. This control can happen hierarchically and/or reciprocally, and it takes place in three-dimensional space. This implies that the information that a biological process has to utilize is...
Calc. Var. ˛-Dirac-harmonic maps from closed surfaces Jürgen Jost 0 Jingyong Zhu 0 0 Max Planck Institute for Mathematics in the Sciences , Inselstrasse 22, 04103 Leipzig , Germany α-Dirac
We study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to a nonpositive curved manifold via the scheme of Sacks and Uhlenbeck. By choosing a suitable sequence of $$\alpha $$ -(Dirac-)harmonic maps from a sequence of suitable closed surfaces degenerating to a hyperbolic surface, we get the convergence and a cleaner energy identity under the...
In computer science, we can theoretically neatly separate transmission and processing of information, hardware and software, and programs and their inputs. This is much more intricate in biology. Nevertheless, I argue that Shannon’s concept of information is useful in biology, although its application is not as straightforward as many people think. In fact, the recently developed...
We offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ . With the same method, we also prove...
Many empirical networks incorporate higher order relations between elements and therefore are naturally modelled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs. In order to develop a systematic tool for the statistical analysis of such hypergraph, we propose a general definition of Ricci curvature on directed hypergraphs and explore the...
Relationships in real systems are often not binary, but of a higher order, and therefore cannot be faithfully modelled by graphs, but rather need hypergraphs. In this work, we systematically develop formal tools for analyzing the geometry and the dynamics of hypergraphs. In particular, we show that Ricci curvature concepts, inspired by the corresponding notions of Forman and...
In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis, the blow-up analysis usually strongly utilizes conformal invariance, which yields a Noether current from which strong estimates can be derived...
The relations, rather than the elements, constitute the structure of networks. We therefore develop a systematic approach to the analysis of networks, modelled as graphs or hypergraphs, that is based on structural properties of (hyper)edges, instead of vertices. For that purpose, we utilize so-called network curvatures. These curvatures quantify the local structural properties of...
Forman has developed a version of discrete Morse theory that can be understood in terms of arrow patterns on a (simplicial, polyhedral or cellular) complex without closed orbits, where each cell may either have no arrows, receive a single arrow from one of its facets, or conversely, send a single arrow into a cell of which it is a facet. By following arrows, one can then...
In this paper, we will study the partial regularity for stationary Dirac-harmonic maps with \(\lambda \)-curvature term. For a weakly stationary Dirac-harmonic map with \(\lambda \)-curvature term \((\phi ,\psi )\) from a smooth bounded open domain \(\Omega \subset {\mathbb {R}}^m\) with \(m\ge 2\) to a compact Riemannian manifold N, if \(\psi \in W^{1,p}(\Omega )\) for some \(p...
VT-harmonic maps generalize the standard harmonic maps, with respect to the structure of both domain and target. These can be manifolds with natural connections other than the Levi-Civita connection of Riemannian geometry, like Hermitian, affine or Weyl manifolds. The standard harmonic map semilinear elliptic system is augmented by a term coming from a vector field V on the...
We study the refined blow-up behaviour of a sequence of Dirac-harmonic maps from degenerating spin surfaces with uniformly bounded energy in the case that the domain surfaces converge to a spin surface with only Neveu–Schwarz type nodes. For Dirac-harmonic necks appearing near the nodes, we show that the limit of the map part of each neck is a geodesic in the target manifold...
With the tools of information geometry, we can express relations between marginals of a joint distribution in geometric terms. We develop this framework in the context of population genetics and use this to interpret the famous Ohta–Kimura formula (cf. Ohta and Kimura in Genet Res 13(01):47–55, 1969) and discuss its generalizations for linkage equilibria in Wright–Fisher models...
restructuring the University of Leipzig. In 1996, he could found the new Max Planck Institute for Mathematics in the Sciences in Leipzig, together with Jürgen Jost and Stefan Müller. He could then devote his ... Felix Finster: Positive Functionals Induced by Minimizers of Causal Variational Principles Jürgen Jost, Enno Keßler, Jürgen Tolksdorf, Ruijun Wu, and Miaomiao Zhu: From Harmonic Maps to the Nonlinear
Jürgen Jost 0 Eberhard Zeidler 0 B J. Jost 0 0 Max Planck Institute for Mathematics in the Sciences , Inselstr. 22, 04103 Leipzig , Germany Eberhard Zeidler passed away on Nov. 18, 2016 after a ... this then became also the guiding motto for its scientific work and atmosphere. The institute opened in 1996, with Eberhard Zeidler as a founding director, together with Jürgen Jost and Stefan Müller
In biological classification, a character is a property of a taxon that can distinguish it from other taxa. Characters are not independent, and the relations between characters can arise from structural constraints, developmental pathways or functional constraints. That has lead to famous controversies in the history of biology. In addition, a character as a tool of data analysis...
We analyze the effect of weak-noise-induced transitions on the dynamics of the FitzHugh–Nagumo neuron model in a bistable state consisting of a stable fixed point and a stable unforced limit cycle. Bifurcation and slow-fast analysis give conditions on the parameter space for the establishment of this bi-stability. In the parametric zone of bi-stability, weak-noise amplitudes may...