IA Scholar Query: The almost simple cubical polytopes.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 28 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440An Extension of Heron's Formula to Tetrahedra, and the Projective Nature of Its Zeros
https://scholar.archive.org/work/wq4vgu5fmfdtrmsnbtmke6m3kq
A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four faces and three medial parallelograms, which will be referred to herein as "interior faces." Geometrically, these rational functions are the areas of the triangles into which the exterior faces are divided by the points at which the tetrahedron's in-sphere touches those faces. This leads to a conjecture as to how the formula extends to n-dimensional simplices for all n > 3. Remarkably, for n = 3 the zeros of the polynomial constitute a five-dimensional semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices separated by infinite distances, but with generically well-defined distance ratios. These unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to ℤ_2^4, wherein four-point configurations in the affine plane constitute a distinguished three-dimensional subset. The paper closes by noting that the algebraic structure of the zeros in the affine plane naturally defines the associated 4-element, rank-3 chirotope, aka affine oriented matroid.Timothy F. Havelwork_wq4vgu5fmfdtrmsnbtmke6m3kqWed, 28 Sep 2022 00:00:00 GMTOn Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
https://scholar.archive.org/work/5giksvcv7readlihzldqrfeuwq
We concentrate on a treatment of a Higgs-Coulomb duality as an absence of manifest phase transition between ordered and disordered phases of 2d 𝒩=(2,2) theories. We consider these examples of QFTs in the Schrödinger picture and identify Hilbert spaces of BPS states with morphisms in triangulated categories of D-brane boundary conditions. As a result of Higgs-Coulomb duality D-brane categories on IR vacuum moduli spaces are equivalent, this resembles an analog of homological mirror symmetry. Following construction ideas behind the Gaiotto-Moore-Witten algebra of the infrared one is able to introduce interface defects in these theories and associate them to D-brane parallel transport functors. We concentrate on surveying simple examples, analytic when possible calculations, numerical estimates and simple physical picture behind curtains of geometric objects. Categorification of hypergeometric series analytic continuation is derived as an Atiyah flop of the conifold. Finally we arrive to an interpretation of the braid group action on the derived category of coherent sheaves on cotangent bundles to flag varieties as a categorification of Berry connection on the Fayet-Illiopolous parameter space of a sigma-model with a quiver variety target space.Dmitry Galakhovwork_5giksvcv7readlihzldqrfeuwqMon, 26 Sep 2022 00:00:00 GMTContinuous finite elements satisfying a strong discrete Miranda–Talenti identity
https://scholar.archive.org/work/mpiqkwpkcfb5ddpki2s475nhiq
This article introduces continuous H^2-nonconforming finite elements in two and three space dimensions which satisfy a strong discrete Miranda–Talenti inequality in the sense that the global L^2 norm of the piecewise Hessian is bounded by the L^2 norm of the piecewise Laplacian. The construction is based on globally continuous finite element functions with C^1 continuity on the vertices (2D) or edges (3D). As an application, these finite elements are used to approximate uniformly elliptic equations in non-divergence form under the Cordes condition without additional stabilization terms. For the biharmonic equation in three dimensions, the proposed methods has less degrees of freedom than existing nonconforming schemes of the same order. Numerical results in two and three dimensions confirm the practical feasibility of the proposed schemes.Dietmar Gallistl, Shudan Tianwork_mpiqkwpkcfb5ddpki2s475nhiqMon, 26 Sep 2022 00:00:00 GMTEfficient Numerical Simulation of Soil-Tool Interaction
https://scholar.archive.org/work/abcs2uszmrdtdgmywcpj2vtuoa
The simulation of soil-tool interaction forces using the Discrete Element Method (DEM) is widely established. In addition to an acceptable prediction quality, the efficient simulation of granular material on high performance clusters with modern parallelization strategies for the industrial applications is indispensable. Although, for relevant problem sizes such simulations are so far not real-time capable. Further on, the inclusion of the human-machine interaction at a driving simulator combined with soil-tool simulation poses many interesting research questions. We therefore strive for sufficient performance and consider alternative models and algorithms to achieve real-time capability. First, we discuss different types of particle models regarding force accuracy and efficiency. The pros and cons are pointed out and the suitability for real-time applications is discussed. Second, we present two machine learning algorithms which are real-time capable and allow force predictions in real-time. The application at the in-house excavator simulator is discussed and the capability is shown using relevant numerical examples.Jonathan Jahnke, Fraunhofer-Gesellschaftwork_abcs2uszmrdtdgmywcpj2vtuoaWed, 21 Sep 2022 00:00:00 GMTSurface area and volume of excursion sets observed on point cloud based polytopic tessellations
https://scholar.archive.org/work/gsxxiifmlve5nci2yhjjfl5lxq
The excursion set of a C^2 smooth random field carries relevant information in its various geometric measures. From a computational viewpoint, one never has access to the continuous observation of the excursion set, but rather to observations at discrete points in space. It has been reported that for specific regular lattices of points in dimensions 2 and 3, the usual estimate of the surface area of the excursions remains biased even when the lattice becomes dense in the domain of observation. In the present work, under the key assumptions of stationarity and isotropy, we demonstrate that this limiting bias is invariant to the locations of the observation points. Indeed, we identify an explicit formula for the bias, showing that it only depends on the spatial dimension d. This enables us to define an unbiased estimator for the surface area of excursion sets that are approximated by general tessellations of polytopes in ℝ^d, including Poisson-Voronoi tessellations. We also establish a joint central limit theorem for the surface area and volume estimates of excursion sets observed over hypercubic lattices.Ryan Cotsakis, Elena Di Bernardino, Céline Duvalwork_gsxxiifmlve5nci2yhjjfl5lxqWed, 21 Sep 2022 00:00:00 GMTTopology of complements of skeletons
https://scholar.archive.org/work/527ni5n3sna73brgtmh4kvd2xa
Given a polytopal complex X, we examine the topological complement of its k-skeleton. We construct a long exact sequence relating the homologies of the skeleton complements and links of faces in X, and using this long exact sequence, we obtain characterisations of Cohen-Macaulay and Leray complexes, stacked balls, and neighbourly spheres in terms of their skeleton complements. We also apply these results to CAT(0) cubical complexes, and find new similarities between such a complex and an associated simplicial complex, the crossing complex.Rowan Rowlandswork_527ni5n3sna73brgtmh4kvd2xaSun, 18 Sep 2022 00:00:00 GMTToric polar maps and characteristic classes
https://scholar.archive.org/work/tdcph5w67rb7vhb3q7k6uqsx34
Given a hypersurface in the complex projective space, we prove that the degree of its toric polar map is given by the signed topological Euler characteristic of a distinguished open set, namely the complement of the union of the hypersurface and the coordinate hyperplanes. In addition, we prove that if the hypersurface is in general position or is nondegenerate with respect to its Newton polytope, then the coefficients of the Chern-Schwartz-MacPherson class of the distinguished open set agree, up to sign, with the multidegrees of the toric polar map. In the latter case, we also recover the multidegrees from mixed volumes. For plane curves, a precise formula for the degree of the toric polar map is obtained in terms of local invariants. Finally, we construct families, in arbitrary dimension, of irreducible hypersurfaces whose toric polar map is birational.Thiago Fassarella, Nivaldo Medeiros, Rodrigo Salomãowork_tdcph5w67rb7vhb3q7k6uqsx34Fri, 16 Sep 2022 00:00:00 GMTMathematical programming for stable control and safe operation of gas transport networks
https://scholar.archive.org/work/jbrpzy7fcrf4hccjg7q4udzdzm
The fight against climate change makes extreme but inevitable changes in the energy sector necessary. These in turn lead to novel and complex challenges for the transmission system operators (TSOs) of gas transport networks. In this thesis, we consider four different planning problems emerging from real-world operations and present mathematical programming models and solution approaches for all of them. Due to regulatory requirements and side effects of renewable energy production, controlling today's gas networks with their involved topologies is becoming increasingly difficult. Based on the network station modeling concept for approximating the technical capabilities of complex subnetworks, e.g., compressor stations, we introduce a tri-level MIP model to determine important global control decisions. Its goal is to avoid changes in the network elements' settings while deviations from future inflow pressures as well as supplies and demands are minimized. A sequential linear programming inspired post-processing routine is run to derive physically accurate solutions w.r.t. the transient gas flow in pipelines. Computational experiments based on real-world data show that meaningful solutions are quickly and reliably determined. Therefore, the algorithmic approach is used within KOMPASS, a decision support system for the transient network control that we developed together with the Open Grid Europe GmbH (OGE), one of Europe's largest natural gas TSOs. Anticipating future use cases, we adapt the aforementioned algorithmic approach for hydrogen transport. We investigate whether the natural gas infrastructure can be repurposed and how the network control changes when energy-equivalent amounts of hydrogen are transported. Besides proving the need for purpose-built compressors, we observe that, due to the reduced linepack, the network control becomes more dynamic, compression energy increases by 440% on average, and stricter regulatory rules regarding the balancing of supply and demand become necessary. Extreme load flows [...]Kai Hoppmann-Baum, Technische Universität Berlin, Thorsten Kochwork_jbrpzy7fcrf4hccjg7q4udzdzmWed, 14 Sep 2022 00:00:00 GMTSmall Transformers Compute Universal Metric Embeddings
https://scholar.archive.org/work/x4pqf4thszadvn667xhnsqe75u
We study representations of data from an arbitrary metric space 𝒳 in the space of univariate Gaussian mixtures with a transport metric (Delon and Desolneux 2020). We derive embedding guarantees for feature maps implemented by small neural networks called probabilistic transformers. Our guarantees are of memorization type: we prove that a probabilistic transformer of depth about nlog(n) and width about n^2 can bi-Hölder embed any n-point dataset from 𝒳 with low metric distortion, thus avoiding the curse of dimensionality. We further derive probabilistic bi-Lipschitz guarantees which trade off the amount of distortion and the probability that a randomly chosen pair of points embeds with that distortion. If 𝒳's geometry is sufficiently regular, we obtain stronger, bi-Lipschitz guarantees for all points in the dataset. As applications we derive neural embedding guarantees for datasets from Riemannian manifolds, metric trees, and certain types of combinatorial graphs.Anastasis Kratsios, Valentin Debarnot, Ivan Dokmanićwork_x4pqf4thszadvn667xhnsqe75uWed, 14 Sep 2022 00:00:00 GMTGenerating MCMC proposals by randomly rotating the regular simplex
https://scholar.archive.org/work/ubhedjukrzbx7acztvfsdfkagi
We present the simplicial sampler, a class of parallel MCMC methods that generate and choose from multiple proposals at each iteration. The algorithm's multiproposal randomly rotates a simplex connected to the current Markov chain state in a way that inherently preserves symmetry between proposals. As a result, the simplicial sampler leads to a simplified acceptance step: it simply chooses from among the simplex nodes with probability proportional to their target density values. We also investigate a multivariate Gaussian-based symmetric multiproposal mechanism and prove that it also enjoys the same simplified acceptance step. This insight leads to significant theoretical and practical speedups. While both algorithms enjoy natural parallelizability, we show that conventional implementations are sufficient to confer efficiency gains across an array of dimensions and a number of target distributions.Andrew J. Holbrookwork_ubhedjukrzbx7acztvfsdfkagiTue, 13 Sep 2022 00:00:00 GMTUniqueness of the Gibbs measure for the 4-state anti-ferromagnetic Potts model on the regular tree
https://scholar.archive.org/work/nwnvdchg2vht5geevyb2kp4iwm
We show that the $4$ -state anti-ferromagnetic Potts model with interaction parameter $w\in (0,1)$ on the infinite $(d+1)$ -regular tree has a unique Gibbs measure if $w\geq 1-\dfrac{4}{d+1_{_{\;}}}$ for all $d\geq 4$ . This is tight since it is known that there are multiple Gibbs measures when $0\leq w\lt 1-\dfrac{4}{d+1}$ and $d\geq 4$ . We moreover give a new proof of the uniqueness of the Gibbs measure for the $3$ -state Potts model on the $(d+1)$ -regular tree for $w\geq 1-\dfrac{3}{d+1}$ when $d\geq 3$ and for $w\in (0,1)$ when $d=2$ .David de Boer, Pjotr Buys, Guus Regtswork_nwnvdchg2vht5geevyb2kp4iwmWed, 07 Sep 2022 00:00:00 GMTA Polyhedral Study for the Cubic Formulation of the Unconstrained Traveling Tournament Problem
https://scholar.archive.org/work/kbp5iwsumfes7blps2roqyaoji
We consider the unconstrained traveling tournament problem, a sports timetabling problem that minimizes traveling of teams. Since its introduction about 20 years ago, most research was devoted to modeling and reformulation approaches. In this paper we carry out a polyhedral study for the cubic integer programming formulation by establishing the dimension of the integer hull as well as of faces induced by model inequalities. Moreover, we introduce a new class of inequalities and show that they are facet-defining. Finally, we evaluate the impact of these inequalities on the linear programming bounds.Marije Siemann, Matthias Walterwork_kbp5iwsumfes7blps2roqyaojiWed, 07 Sep 2022 00:00:00 GMTReduced symplectic homology and the secondary continuation map
https://scholar.archive.org/work/lgdfnyvdxjedfjs4zlgwqj4zfy
We introduce the notion of a Weinstein domain with strongly R-essential skeleton and study a reduced version of symplectic homology in this context. In the open string case we introduce the notion of a strongly R-essential Lagrangian submanifold and study a reduced version of wrapped Floer homology. These reduced homologies provide a common domain of definition for the pair-of-pants product and for pair-of-pants secondary coproducts, which combine into the structure of a unital infinitesimal anti-symmetric bialgebra. The coproducts on reduced homology depend on choices, and this dependence is controlled by secondary continuation maps. Reduced homology and cohomology provide splittings of Rabinowitz Floer homology which are compatible with the products and the coproducts. These splittings also depend on choices, in the same way as the coproducts. We provide sufficient conditions under which the secondary continuation maps vanish, giving rise to canonical coproducts and splittings.Kai Cieliebak, Alexandru Oanceawork_lgdfnyvdxjedfjs4zlgwqj4zfyWed, 07 Sep 2022 00:00:00 GMTUnderstanding the Behavior of Belief Propagation
https://scholar.archive.org/work/hcvsy4klsnaatckaysmgy5yk6m
Probabilistic graphical models are a powerful concept for modeling high-dimensional distributions. Besides modeling distributions, probabilistic graphical models also provide an elegant framework for performing statistical inference; because of the high-dimensional nature, however, one must often use approximate methods for this purpose. Belief propagation performs approximate inference, is efficient, and looks back on a long success-story. Yet, in most cases, belief propagation lacks any performance and convergence guarantees. Many realistic problems are presented by graphical models with loops, however, in which case belief propagation is neither guaranteed to provide accurate estimates nor that it converges at all. This thesis investigates how the model parameters influence the performance of belief propagation. We are particularly interested in their influence on (i) the number of fixed points, (ii) the convergence properties, and (iii) the approximation quality.Christian Knollwork_hcvsy4klsnaatckaysmgy5yk6mMon, 05 Sep 2022 00:00:00 GMTWild Patterns Reloaded: A Survey of Machine Learning Security against Training Data Poisoning
https://scholar.archive.org/work/3l4cmaz2pfbqrjyjuahh4domsq
The success of machine learning is fueled by the increasing availability of computing power and large training datasets. The training data is used to learn new models or update existing ones, assuming that it is sufficiently representative of the data that will be encountered at test time. This assumption is challenged by the threat of poisoning, an attack that manipulates the training data to compromise the model's performance at test time. Although poisoning has been acknowledged as a relevant threat in industry applications, and a variety of different attacks and defenses have been proposed so far, a complete systematization and critical review of the field is still missing. In this survey, we provide a comprehensive systematization of poisoning attacks and defenses in machine learning, reviewing more than 100 papers published in the field in the last 15 years. We start by categorizing the current threat models and attacks, and then organize existing defenses accordingly. While we focus mostly on computer-vision applications, we argue that our systematization also encompasses state-of-the-art attacks and defenses for other data modalities. Finally, we discuss existing resources for research in poisoning, and shed light on the current limitations and open research questions in this research field.Antonio Emanuele Cinà, Kathrin Grosse, Ambra Demontis, Sebastiano Vascon, Werner Zellinger, Bernhard A. Moser, Alina Oprea, Battista Biggio, Marcello Pelillo, Fabio Roliwork_3l4cmaz2pfbqrjyjuahh4domsqThu, 01 Sep 2022 00:00:00 GMTCrystal gravity
https://scholar.archive.org/work/qrylz2b2wbai3lfjd3ah2h5c44
We address a subject that could have been analyzed century ago: how does the universe of general relativity look like when it would have been filled with solid matter? Solids break spontaneously the translations and rotations of space itself. Only rather recently it was realized in various context that the order parameter of the solid has a relation to Einsteins dynamical space time which is similar to the role of a Higgs field in a Yang-Mills gauge theory. Such a "crystal gravity" is therefore like the Higgs phase of gravity. The usual Higgs phases are characterized by a special phenomenology. A case in point is superconductivity exhibiting phenomena like the Type II phase, characterized by the emergence of an Abrikosov lattice of quantized magnetic fluxes absorbing the external magnetic field. What to expect in the gravitational setting? The theory of elasticity is the universal effective field theory associated with the breaking of space translations and rotations having a similar status as the phase action describing a neutral superfluid. A geometrical formulation appeared in its long history, similar in structure to general relativity, which greatly facilitates the marriage of both theories. With as main limitation that we focus entirely on stationary circumstances -- the dynamical theory is greatly complicated by the lack of Lorentz invariance -- we will present a first exploration of a remarkably rich and often simple physics of "Higgsed gravity".Jan Zaanen, Floris Balm, Aron Beekmanwork_qrylz2b2wbai3lfjd3ah2h5c44Tue, 30 Aug 2022 00:00:00 GMTEnumerative Geometry of Del Pezzo Surfaces
https://scholar.archive.org/work/5pksyvyg5vb3lorctbmfkeugwu
We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin. We also include some explicit calculations for the projective plane, which confirm some folklore conjecture in this case.Yu-Shen Linwork_5pksyvyg5vb3lorctbmfkeugwuTue, 30 Aug 2022 00:00:00 GMTLifted edges as connectivity priors for multicut and disjoint paths
https://scholar.archive.org/work/edizj43isvflhhihrsapdwjlhu
This work studies graph decompositions and their representation by 0/1 labeling of edges. We study two problems. The first is multicut (MC) which represents decompositions of undirected graphs (clustering of nodes into connected components). The second is disjoint paths (DP) in directed acyclic graphs where the clusters correspond to nodedisjoint paths. Unlike an alternative representation by node labeling, the number of clusters is not part of the input but is fully determined by the costs of edges. I would like to thank all my co-authors for a pleasant and constructive cooperation. Besides my supervisor Paul Swoboda, I would like to name especially Roberto Henschel, Timo Kaiser, Bjoern Andres, and Jan-Hendrik Lange for their major contribution to the shared publications that are part of this thesis. The publications could not be realized without their part of the work. I would like to thank Bjoern Andres for his supervision and help during the work on our common paper. I would like to mention also Michal Rolinek who helped us with our latest publication. I would like to thank Jiles Vreeken, Marcel Schulz and Markus List who cooperated with me on a research project that is not part of this thesis. I am very grateful to Bernt Schiele, the director of our department, who provided me with good working conditions, fully supported me in combining my working duties with family, and found a solution in the difficult stage of my PhD study by finding a new supervisor. Also, other people at MPI and Saarland University helped me to organize my work and family life and helped me with administrative issues.Andrea Hornakova, Universität Des Saarlandeswork_edizj43isvflhhihrsapdwjlhuMon, 29 Aug 2022 00:00:00 GMTCanonical models of toric hypersurfaces
https://scholar.archive.org/work/ivwi3oye35aehclewx5a4so6hy
Let Z be a nondegenerate hypersurface in d-dimensional torus (ℂ^*)^d defined by a Laurent polynomial f with a d-dimensional Newton polytope P. The subset F(P) ⊂ P consisting of all points in P having integral distance at least 1 to all integral supporting hyperplanes of P is called the Fine interior of P. If F(P) ≠∅ we construct a unique projective model Z of Z having at worst canonical singularities and obtain minimal models Ẑ of Z by crepant morphisms Ẑ→Z. We show that the Kodaira dimension κ =κ(Z) equals min{ d-1, F(P) } and the general fibers in the Iitaka fibration of the canonical model Z arenondegenerate (d-1-κ)-dimensional toric hypersurfaces of Kodaira dimension 0. Using F(P), we obtain a simple combinatorial formula for the intersection number (K_Z)^d-1.Victor V. Batyrevwork_ivwi3oye35aehclewx5a4so6hySun, 28 Aug 2022 00:00:00 GMTTowards Safe and Sample-efficient Learning for Autonomous Energy Systems
https://scholar.archive.org/work/7xwksh3dqvc3viwutdvvwukfjm
Given the dire consequences of climate change, there are growing incentives to curb carbon emissions by reducing energy consumption and increasing the penetration of renewable energy generation, along with other measures to jointly combat this global challenge. In this thesis, we focus on learning-based controls to 1) reduce the energy consumption in buildings, and 2) to facilitate the integration of distributed energy resources (DERs). Recently, there is increasing interest in applying reinforcement learning (RL) for energy systems operation given that 1) high-fidelity models for these system are resourceintensive to develop and not commonly available, 2) energy systems are heterogeneous and the solution for one system may not be transferable to others, and 3) some of these systems are undergoing transitions and thus the control should be adaptive and future-proof. While RL is a promising solution, real-world applications of RL agents are numbered due to the facts that 1) RL agents generally take a long time to learn and reach acceptable performance and that 2) the actions by RL agents may not satisfy safety constraints posed by the underlying physical systems or the functional requirements. Thus, RL agents should learn safely and sample-efficiently to be practical for real-world energy systems. Firstly, we address the challenge of sample complexity in Research Question 1, grounded in the application of improving energy efficiency in building operations. We expedite the learning process by warm-starting the RL agent with expert demonstrations and by incorporating domain knowledge on building thermodynamics in its policy. We validate that our proposed agent, Gnu-RL, can be deployed on real-world testbeds with satisfactory initial performance, and improve energy efficiency over time. In a notable experiment, Gnu-RL was deployed to operate a real-world testbed for three weeks, wherein it saved 16.7% of cooling demand compared to the existing controller while maintaining better thermal comfort. In comparison to exis [...]Bingqing Chenwork_7xwksh3dqvc3viwutdvvwukfjmFri, 26 Aug 2022 00:00:00 GMT