arXiv subjects
Publications and source records attributed to Martin Winter.
We present a new, explicit and very geometric construction for the iterated clique graphs of the hexagonal lattice $\mathrm{Hex}$ which makes apparent its clique-divergence and sheds light on some previous observations, such as the boundedness of the degrees and clique sizes of $k^n \mathrm{Hex}$ as $n\to\infty$.
Training a sparse neural network from scratch requires optimizing connections at the same time as the weights themselves. Typically, the weights are redistributed after a predefined number of weight updates, removing a fraction of the parameters of each layer and inserting them at different locations in the same layers. The density of each layer is determined using heuristics, often purely based on the size of the parameter tensor. While the connections per layer are optimized multiple times during training, the density of each layer remains constant. This leaves great unrealized potential, especially in scenarios with a high sparsity of 90% and more. We propose Global Gradient-based Redistribution, a technique which distributes weights across all layers - adding more weights to the layers that need them most. Our evaluation shows that our approach is less prone to unbalanced weight distribution at initialization than previous work and that it is able to find better performing sparse subnetworks at very high sparsity levels.
Lithium salts with low coordinating anions like bis(trifluoromethanesulfonyl)imide (TFSI) have been the state-of-the-art for PEO-based 'dry' polymer electrolytes for three decades. Plasticizing PEO with TFSI-based ionic liquids (ILs) to form ternary solid polymer electrolytes (TSPEs) increases conductivity and Li$^+$ diffusivity. However, the Li$^+$ transport mechanism is unaffected compared to their 'dry' counterpart and essentially coupled to the dynamics of the polymer host matrix, which limits Li$^+$ transport improvement. Thus, a paradigm shift is hereby suggested: The utilization of more coordinating anions such as trifluoromethanesulfonyl-N-cyanoamide (TFSAM), able to compete with PEO for Li$^+$ solvation to accelerate the Li$^+$ transport and reach higher Li$^+$ transference number. The Li-TFSAM interaction in binary and ternary TFSAM-based electrolytes was probed by experimental methods and discussed in the context of recent computational results. In PEO-based TSPEs, TFSAM drastically accelerates the Li$^+$ transport (increased Li$^+$ transference number by 600$\%$ and Li$^+$ conductivity by 200-300$\%$) and computer simulations reveal that lithium dynamics are effectively re-coupled from polymer to anion dynamics. Finally, this concept of coordinating anions in TSPEs was successfully applied in LFP$||$Li metal cells leading to enhanced capacity retention (86$\%$ after 300 cycles) and an improved rate performance at 2C.
A general (convex) polytope $P\subset\mathbb R^d$ and its edge-graph $G_P$ can have very distinct symmetry properties. We construct a coloring (of the vertices and edges) of the edge-graph so that the combinatorial symmetry group of the colored edge-graph is isomorphic (in a natural way) to $\mathrm{Aut}_{\mathrm{GL}}(P)$, the group of linear symmetries of the polytope. We also construct an analogous coloring for $\mathrm{Aut}_{\mathrm{O}}(P)$, the group of orthogonal symmetries of $P$.
In 3-dimensional Euclidean space there exist two exceptional polyhedra, the rhombic dodecahedron and the rhombic triacontahedron, the only known polytopes (besides polygons) that are edge-transitive without being vertex-transitive. We show that these polyhedra do not have higher-dimensional analogues, that is, that in dimension $d\ge 4$, edge-transitivity of convex polytopes implies vertex-transitivity. More generally, we give a classification of all convex polytopes which at the same time have all edges of the same length, an edge in-sphere and a bipartite edge-graph. We show that any such polytope in dimension $d\ge 4$ is vertex-transitive.
Open access to research data is increasingly important for accelerating research. Grant authorities therefore request detailed plans for how data is managed in the projects they finance. We have recently developed such a plan for the EU-H2020 BIG-MAP project - a cross-disciplinary project targeting disruptive battery-material discoveries. Essential for reaching the goal is extensive sharing of research data across scales, disciplines and stakeholders, not limited to BIG-MAP and the European BATTERY 2030+ initiative but within the entire battery community. The key challenges faced in developing the data management plan for such a large and complex project were to generate an overview of the enormous amount of data that will be produced, to build an understanding of the data flow within the project and to agree on a roadmap for making all data FAIR. This paper describes the process we followed and how we structured the plan.
We investigate point arrangements $v_i\in\mathbb R^d,i\in \{1,...,n \}$ with certain prescribed symmetries. The arrangement space of $v$ is the column span of the matrix in which the $v_i$ are the rows. We characterize properties of $v$ in terms of the arrangement space, e.g. we characterize whether an arrangement possesses certain symmetries or whether it can be continuously deformed into another arrangement while preserving symmetry in the process. We show that whether a symmetric arrangement can be continuously deformed into its mirror image depends non-trivially on several factors, e.g. the decomposition of its representation into irreducible constituents, and whether we are in even or odd dimensions.
Starting from a finite simple graph $G$, for each eigenvalue $θ$ of its adjacency matrix one can construct a convex polytope $P_G(θ)$, the so called $θ$-eigenpolytop of $G$. For some polytopes this technique can be used to reconstruct the polytopes from its edge-graph. Such polytopes (we shall call them spectral) are still badly understood. We give an overview of the literature for eigenpolytopes and spectral polytopes. We introduce a geometric condition by which to prove that a given polytope is spectral (more exactly, $θ_2$-spectral). We apply this criterion to the edge-transitive polytopes. We show that every edge-transitive polytope is $θ_2$-spectral, is uniquely determined by this graph, and realizes all its symmetries. We give a complete classification of distance-transitive polytopes.
A realization of a graph $G=(V,E)$ is a map $v\colon V\to\Bbb R^d$ that assigns to each vertex a point in $d$-dimensional Euclidean space. We study graph realizations from the perspective of representation theory (expressing certain symmetries), spectral graph theory (satisfying certain self-stress conditions) and rigidity theory (admitting deformations that do not alter the symmetry properties). We explore the connections between these perspectives, with a focus on realizations of highly symmetric graphs (arc-transitive/distance-transitive) and the question of how much symmetry is necessary to ensure that a realization is balanced, spectral, rigid etc. We include many examples to give a broad overview of the possibilities and restrictions of symmetric and spectral graph realizations.
Energy density limitations of layered oxides with different Ni contents, i.e., of the conventional cathode materials in Li-ion batteries, are investigated across the first discharge cycle using advanced spectroscopy and state-of-the-art diffraction. For the first time unambiguous experimental evidence is provided, that redox reactions in NCMs proceed via a reversible oxidation of Ni and a hybridization with O, and not, as widely assumed, via pure cationic or more recently discussed, pure anionic redox processes. Once Ni-O hybrid states are formed, the sites cannot be further oxidized. Instead, irreversible reactions set in which lead to a structural collapse and thus, the lack of ionic Ni limits the reversible capacity. Moreover, the degree of hybridization, which varies with the Ni content, triggers the electronic structure and the operation potential of the cathodes. With an increasing amount of Ni, the covalent character of the materials increases and the potential decreases.
We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a $Γ$-permutahedron for some finite reflection group $Γ\subset\mathrm{O}(\mathbb R^d)$. The same holds true for zonotopes in which all vertices are on a common sphere, and all edges are of the same length (which we call homogeneous zonotopes). The classification of these then follows from the classification of finite reflection groups. We proof that root systems can be characterized as those centrally symmetric sets of vectors, for which all intersections with half-spaces, that contain exactly half the vectors, are congruent. We provide a further sufficient condition for a centrally symmetric set being a root system.
Motivated by the search for reduced polytopes, we consider the following question: For which polytopes exists a vertex-facet assignment, that is, a matching between vertices and non-incident facets, so that the matching covers either all vertices, or all facets? We provide general conditions for the existence of such an assignment. We conclude that such exist for all simple and simplicial polytopes, as well as all polytopes of dimension $d\le 6$. We construct counterexample in all dimensions $d\ge 7$.
In recent years, mesh subdivision---the process of forging smooth free-form surfaces from coarse polygonal meshes---has become an indispensable production instrument. Although subdivision performance is crucial during simulation, animation and rendering, state-of-the-art approaches still rely on serial implementations for complex parts of the subdivision process. Therefore, they often fail to harness the power of modern parallel devices, like the graphics processing unit (GPU), for large parts of the algorithm and must resort to time-consuming serial preprocessing. In this paper, we show that a complete parallelization of the subdivision process for modern architectures is possible. Building on sparse matrix linear algebra, we show how to structure the complete subdivision process into a sequence of algebra operations. By restructuring and grouping these operations, we adapt the process for different use cases, such as regular subdivision of dynamic meshes, uniform subdivision for immutable topology, and feature-adaptive subdivision for efficient rendering of animated models. As the same machinery is used for all use cases, identical subdivision results are achieved in all parts of the production pipeline. As a second contribution, we show how these linear algebra formulations can effectively be translated into efficient GPU kernels. Applying our strategies to $\sqrt{3}$, Loop and Catmull-Clark subdivision shows significant speedups of our approach compared to state-of-the-art solutions, while we completely avoid serial preprocessing.