IA Scholar Query: On r-Simple k-Path and Related Problems Parameterized by k/r.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 30 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Conjunctive queries for logic-based information extraction
https://scholar.archive.org/work/wd2pb3qomzeb7lqcc3av3fepqq
This thesis offers two logic-based approaches to conjunctive queries in the context of information extraction. The first and main approach is the introduction of conjunctive query fragments of the logics FC and FC[REG], denoted as FC-CQ and FC[REG]-CQ respectively. FC is a first-order logic based on word equations, where the semantics are defined by limiting the universe to the factors of some finite input word. FC[REG] is FC extended with regular constraints. Our first results consider the comparative expressive power of FC[REG]-CQ in relation to document spanners (a formal framework for the query language AQL), and various fragments of FC[REG]-CQ – some of which coincide with well-known language generators, such as patterns and regular expressions. Then, we look at decision problems. We show that many decision problems for FC-CQ and FC[REG]-CQ (such as equivalence and regularity) are undecidable. The model checking problem for FC-CQ and FC[REG]-CQ is NP-complete even if the FC-CQ is acyclic – under the definition of acyclicity where each word equation in an FC-CQ is an atom. This leads us to look at the "decomposition" of an FC word equation into a conjunction of binary word equations (i.e., of the form x =˙ y · z). If a query consists of only binary word equations and the query is acyclic, then model checking is tractable and we can enumerate results efficiently. We give an algorithm that decomposes an FC-CQ into an acyclic FC-CQ consisting of binary word equations in polynomial time, or determines that this is not possible. The second approach is to consider the dynamic complexity of FC. This uses the common way of encoding words in a relational structure using a universe with a linear order along with symbol predicates. Then, each element of the universe can carry a symbol if the predicate for said symbol holds for that element. Instead of the "usual way" (looking at first-order logic over these structures), we study the dynamic complexity, where symbols can be modified. As each of these modifications only c [...]Sam M Thompsonwork_wd2pb3qomzeb7lqcc3av3fepqqWed, 30 Nov 2022 00:00:00 GMTARC - Actor Residual Critic for Adversarial Imitation Learning
https://scholar.archive.org/work/tnt2juu75re3nlychm6yabuc3q
Adversarial Imitation Learning (AIL) is a class of popular state-of-the-art Imitation Learning algorithms commonly used in robotics. In AIL, an artificial adversary's misclassification is used as a reward signal that is optimized by any standard Reinforcement Learning (RL) algorithm. Unlike most RL settings, the reward in AIL is differentiable but current model-free RL algorithms do not make use of this property to train a policy. The reward is AIL is also shaped since it comes from an adversary. We leverage the differentiability property of the shaped AIL reward function and formulate a class of Actor Residual Critic (ARC) RL algorithms. ARC algorithms draw a parallel to the standard Actor-Critic (AC) algorithms in RL literature and uses a residual critic, C function (instead of the standard Q function) to approximate only the discounted future return (excluding the immediate reward). ARC algorithms have similar convergence properties as the standard AC algorithms with the additional advantage that the gradient through the immediate reward is exact. For the discrete (tabular) case with finite states, actions, and known dynamics, we prove that policy iteration with C function converges to an optimal policy. In the continuous case with function approximation and unknown dynamics, we experimentally show that ARC aided AIL outperforms standard AIL in simulated continuous-control and real robotic manipulation tasks. ARC algorithms are simple to implement and can be incorporated into any existing AIL implementation with an AC algorithm. Video and link to code are available at: https://sites.google.com/view/actor-residual-critic.Ankur Deka, Changliu Liu, Katia Sycarawork_tnt2juu75re3nlychm6yabuc3qWed, 30 Nov 2022 00:00:00 GMTOpen r-spin theory II: The analogue of Witten's conjecture for r-spin disks
https://scholar.archive.org/work/auaxfqsmjjeblnehr7hxl65l24
We conclude the construction of r-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define open r-spin intersection numbers, and we prove that their generating function is closely related to the wave function of the rth Gelfand-Dickey integrable hierarchy. This provides an analogue of Witten's r-spin conjecture in the open setting and a first step toward the construction of an open version of Fan-Jarvis-Ruan-Witten theory. As an unexpected consequence, we establish a mysterious relationship between open r-spin theory and an extension of Witten's closed theory.Alexandr Buryak and Emily Clader and Ran J. Tesslerwork_auaxfqsmjjeblnehr7hxl65l24Wed, 30 Nov 2022 00:00:00 GMTAnalytical solutions of topological surface states in a series of lattice models
https://scholar.archive.org/work/kgm6uh5apvadpboisrcz7s3k2u
We derive the analytical solutions of surface states in a series of lattice models for three-dimensional topological insulators and their nontopological counterparts based on an ansatz. A restriction on the spin-flip matrices in nearest-neighbor hopping characterizes the series. This restriction affords the ansatz and favors analytical solvability of surface-state eigenvectors. Despite the restriction, the series retains sufficient designability to describe various types of surface states. We also describe how it can serve as a tractable tool for elucidating unique phenomena on topological surfaces.Masaru Onodawork_kgm6uh5apvadpboisrcz7s3k2uWed, 30 Nov 2022 00:00:00 GMTGluing theories of contact instantons and of pseudoholomoprhic curves in SFT
https://scholar.archive.org/work/5dyvmd7oerg7xncznqt6gmtgki
We develop the gluing theory of contact instantons in the context of open strings and in the context of closed strings with vanishing charge, for example in the symplectization context. This is one of the key ingredients for the study of (virtually) smooth moduli space of (bordered) contact instantons needed for the construction of contact instanton Floer cohomology and more generally for the construction of Fukaya-type category of Legendrian submanifolds in contact manifold (M,ξ). As an application, we apply the gluing theorem to give the construction of (cylindrical) Legendrian contact instanton homology that enters in our solution to Sandon's question for the nondegenerate case. We also apply this gluing theory to that of moduli spaces of holomorphic buildings arising in Symplectic Field Theory (SFT) by canonically lifting the former to that of the latter.Yong-Geun Ohwork_5dyvmd7oerg7xncznqt6gmtgkiWed, 30 Nov 2022 00:00:00 GMTDeterministic and Random Perturbations of the Kepler Problem
https://scholar.archive.org/work/pzo77qhwhzfdhclgaednpm3dyu
We investigate perturbations in the Kepler problem. We offer an overview of the dynamical system using Newtonian, Lagrangian and Hamiltonian Mechanics to build a foundation for analyzing perturbations. We consider the effects of a deterministic perturbation in the form of a first order relativistic correction which change bounded orbits from standard to precessing ellipses. We also consider the effects of stochastic perturbations with certain potentials and evaluate the analytical results of mean exit times using Monte Carlo simulations.Jesse Diminowork_pzo77qhwhzfdhclgaednpm3dyuTue, 29 Nov 2022 00:00:00 GMTFaster Algorithms for Sparse ILP and Hypergraph Multi-Packing/Multi-Cover Problems
https://scholar.archive.org/work/mwuzhie7xbf3bjuxjuabzyaye4
In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in P ∩ Z^n, assuming that P is a polyhedron, defined by systems A x ≤ b or Ax = b, x ≥ 0 with a sparse matrix A. We develop algorithms for these problems that outperform state of the art ILP and counting algorithms on sparse instances with bounded elements. We use known and new methods to develop new exponential algorithms for Edge/Vertex Multi-Packing/Multi-Cover Problems on graphs and hypergraphs. This framework consists of many different problems, such as the Stable Multi-set, Vertex Multi-cover, Dominating Multi-set, Set Multi-cover, Multi-set Multi-cover, and Hypergraph Multi-matching problems, which are natural generalizations of the standard Stable Set, Vertex Cover, Dominating Set, Set Cover, and Maximal Matching problems.Dmitry Gribanov, Dmitry Malyshev, Nikolai Zolotykhwork_mwuzhie7xbf3bjuxjuabzyaye4Tue, 29 Nov 2022 00:00:00 GMTCausal identification for continuous-time stochastic processes
https://scholar.archive.org/work/oxk3wqqmcnfhrmuuxdvjftv7jq
Many real-world processes are trajectories that may be regarded as continuous-time "functional data". Examples include patients' biomarker concentrations, environmental pollutant levels, and prices of stocks. Corresponding advances in data collection have yielded near continuous-time measurements, from e.g. physiological monitors, wearable digital devices, and environmental sensors. Statistical methodology for estimating the causal effect of a time-varying treatment, measured discretely in time, is well developed. But discrete-time methods like the g-formula, structural nested models, and marginal structural models do not generalize easily to continuous time, due to the entanglement of uncountably infinite variables. Moreover, researchers have shown that the choice of discretization time scale can seriously affect the quality of causal inferences about the effects of an intervention. In this paper, we establish causal identification results for continuous-time treatment-outcome relationships for general cadlag stochastic processes under continuous-time confounding, through orthogonalization and weighting. We use three concrete running examples to demonstrate the plausibility of our identification assumptions, as well as their connections to the discrete-time g methods literature.Jinghao Sun, Forrest W. Crawfordwork_oxk3wqqmcnfhrmuuxdvjftv7jqTue, 29 Nov 2022 00:00:00 GMTMaximum cut on interval graphs of interval count four is NP-complete
https://scholar.archive.org/work/s24r6zqdebdv7gmbhl5cfktyju
The computational complexity of the MaxCut problem restricted to interval graphs has been open since the 80's, being one of the problems proposed by Johnson on his Ongoing Guide to NP-completeness, and has been settled as NP-complete only recently by Adhikary, Bose, Mukherjee and Roy. On the other hand, many flawed proofs of polynomiality for MaxCut on the more restrictive class of unit/proper interval graphs (or graphs with interval count 1) have been presented along the years, and the classification of the problem is still unknown. In this paper, we present the first NP-completeness proof for MaxCut when restricted to interval graphs with bounded interval count, namely graphs with interval count 4.Celina M. H. de Figueiredo, Alexsander A. de Melo, Fabiano S. Oliveira, Ana Silvawork_s24r6zqdebdv7gmbhl5cfktyjuTue, 29 Nov 2022 00:00:00 GMTThe Complexity of Infinite-Horizon General-Sum Stochastic Games
https://scholar.archive.org/work/rzuknccxqfgzlntwc7xkouqibi
We study the complexity of computing stationary Nash equilibrium (NE) in n-player infinite-horizon general-sum stochastic games. We focus on the problem of computing NE in such stochastic games when each player is restricted to choosing a stationary policy and rewards are discounted. First, we prove that computing such NE is in PPAD (in addition to clearly being PPAD-hard). Second, we consider turn-based specializations of such games where at each state there is at most a single player that can take actions and show that these (seemingly-simpler) games remain PPAD-hard. Third, we show that under further structural assumptions on the rewards computing NE in such turn-based games is possible in polynomial time. Towards achieving these results we establish structural facts about stochastic games of broader utility, including monotonicity of utilities under single-state single-action changes and reductions to settings where each player controls a single state.Yujia Jin, Vidya Muthukumar, Aaron Sidfordwork_rzuknccxqfgzlntwc7xkouqibiTue, 29 Nov 2022 00:00:00 GMTHomotopic Policy Mirror Descent: Policy Convergence, Implicit Regularization, and Improved Sample Complexity
https://scholar.archive.org/work/3t65mqbk5nfqxfp4nkugkgxmla
We propose a new policy gradient method, named homotopic policy mirror descent (HPMD), for solving discounted, infinite horizon MDPs with finite state and action spaces. HPMD performs a mirror descent type policy update with an additional diminishing regularization term, and possesses several computational properties that seem to be new in the literature. We first establish the global linear convergence of HPMD instantiated with Kullback-Leibler divergence, for both the optimality gap, and a weighted distance to the set of optimal policies. Then local superlinear convergence is obtained for both quantities without any assumption. With local acceleration and diminishing regularization, we establish the first result among policy gradient methods on certifying and characterizing the limiting policy, by showing, with a non-asymptotic characterization, that the last-iterate policy converges to the unique optimal policy with the maximal entropy. We then extend all the aforementioned results to HPMD instantiated with a broad class of decomposable Bregman divergences, demonstrating the generality of the these computational properties. As a by product, we discover the finite-time exact convergence for some commonly used Bregman divergences, implying the continuing convergence of HPMD to the limiting policy even if the current policy is already optimal. Finally, we develop a stochastic version of HPMD and establish similar convergence properties. By exploiting the local acceleration, we show that for small optimality gap, a better than 𝒪̃(|𝒮| |𝒜| / ϵ^2) sample complexity holds with high probability, when assuming a generative model for policy evaluation.Yan Li, Guanghui Lan, Tuo Zhaowork_3t65mqbk5nfqxfp4nkugkgxmlaTue, 29 Nov 2022 00:00:00 GMTTopological Matter and Fractional Entangled Geometry
https://scholar.archive.org/work/ijlpsyes2jbe7nryynyciyaaea
Here, we review our progress on a geometrical approach of quantum physics and topological crystals starting from nature, electrodynamics of planets and linking with Dirac magnetic monopoles and gauge fields. The Bloch sphere of a quantum spin-1/2 particle can also acquire an integer topological charge in the presence of a radial magnetic field. We show that the global topological properties are revealed from the poles of the surface allowing a correspondence between smooth fields, metric and quantum distance. The information is transported from each pole to the equatorial plane on a thin Dirac string. We develop the theory, "the quantum topometry" in space and time, and present applications on transport from a Newtonian approach, on a quantized photo-electric effect from circular dichroism of light towards topological band structures of crystals. The occurrence of robust edge modes related to the topological lattice models are revealed analytically when deforming the sphere or ellipse onto a cylinder. The topological properties of the quantum Hall effect, the quantum anomalous Hall effect and the quantum spin Hall effect on the honeycomb lattice can be measured locally in the Brillouin zone from the light-matter coupling. The formalism allows us to include interaction effects from the momentum space. Interactions may also result in fractional entangled geometry within the curved space. We develop a relation between entangled wavefunction in quantum mechanics, coherent superposition of geometries, a way to one-half topological numbers and Majorana fermions. We show realizations in topological matter. We present a relation between axion electrodynamics, topological insulators on a surface of a cube and the two-spheres' model via the meron.Karyn Le Hurwork_ijlpsyes2jbe7nryynyciyaaeaTue, 29 Nov 2022 00:00:00 GMTHuman Joint Kinematics Diffusion-Refinement for Stochastic Motion Prediction
https://scholar.archive.org/work/fwshc3vvmreuzo6vjhppvofo24
Stochastic human motion prediction aims to forecast multiple plausible future motions given a single pose sequence from the past. Most previous works focus on designing elaborate losses to improve the accuracy, while the diversity is typically characterized by randomly sampling a set of latent variables from the latent prior, which is then decoded into possible motions. This joint training of sampling and decoding, however, suffers from posterior collapse as the learned latent variables tend to be ignored by a strong decoder, leading to limited diversity. Alternatively, inspired by the diffusion process in nonequilibrium thermodynamics, we propose MotionDiff, a diffusion probabilistic model to treat the kinematics of human joints as heated particles, which will diffuse from original states to a noise distribution. This process offers a natural way to obtain the "whitened" latents without any trainable parameters, and human motion prediction can be regarded as the reverse diffusion process that converts the noise distribution into realistic future motions conditioned on the observed sequence. Specifically, MotionDiff consists of two parts: a spatial-temporal transformer-based diffusion network to generate diverse yet plausible motions, and a graph convolutional network to further refine the outputs. Experimental results on two datasets demonstrate that our model yields the competitive performance in terms of both accuracy and diversity.Dong Wei, Huaijiang Sun, Bin Li, Jianfeng Lu, Weiqing Li, Xiaoning Sun, Shengxiang Huwork_fwshc3vvmreuzo6vjhppvofo24Mon, 28 Nov 2022 00:00:00 GMTConnecting the Dots: Floorplan Reconstruction Using Two-Level Queries
https://scholar.archive.org/work/i54di3mkprantplf4amyvuwbyq
We address 2D floorplan reconstruction from 3D scans. Existing approaches typically employ heuristically designed multi-stage pipelines. Instead, we formulate floorplan reconstruction as a single-stage structured prediction task: find a variable-size set of polygons, which in turn are variable-length sequences of ordered vertices. To solve it we develop a novel Transformer architecture that generates polygons of multiple rooms in parallel, in a holistic manner without hand-crafted intermediate stages. The model features two-level queries for polygons and corners, and includes polygon matching to make the network end-to-end trainable. Our method achieves a new state-of-the-art for two challenging datasets, Structured3D and SceneCAD, along with significantly faster inference than previous methods. Moreover, it can readily be extended to predict additional information, i.e., semantic room types and architectural elements like doors and windows. Our code and models will be available at: https://github.com/ywyue/RoomFormer.Yuanwen Yue, Theodora Kontogianni, Konrad Schindler, Francis Engelmannwork_i54di3mkprantplf4amyvuwbyqMon, 28 Nov 2022 00:00:00 GMTTurbulence as Clebsch Confinement
https://scholar.archive.org/work/qrlmjshh65cfddfvw4x3lbhb44
We argue that in the strong turbulence phase, as opposed to the weak one, the Clebsch variables compactify to the sphere S_2 and are not observable as wave excitations. Various topologically nontrivial configurations of this confined Clebsch field are responsible for vortex sheets. Stability equations (CVS) for closed vortex surfaces (bubbles of Clebsch field) are derived and investigated. The exact non-compact solution for the stable vortex sheet family is presented. Compact solutions are proven not to exist by De Lellis and Brué. Asymptotic conservation of anomalous dissipation on stable vortex surfaces in the turbulent limit is discovered. We derive an exact formula for this anomalous dissipation as a surface integral of the square of velocity gap times the square root of minus local normal strain. Topologically stable time-dependent solutions, which we call Kelvinons, are introduced. They have a conserved velocity circulation around static loop; this makes them responsible for asymptotic PDF tails of velocity circulation, perfectly matching numerical simulations. The loop equation for circulation PDF as functional of the loop shape is derived and studied. This equation is exactly equivalent to the Schrödinger equation in loop space, with viscosity ν playing the role of Planck's constant. This equivalence opens the way for direct numerical simulation of turbulence on quantum computers. Kelvinons are fixed points of the loop equation at turbulent limit ν→ 0. Area law and the asymptotic scaling law for mean circulation at a large area are derived. The representation of the solution of the loop equation in terms of a singular stochastic equation for momentum loop trajectory is presented.Alexander Migdalwork_qrlmjshh65cfddfvw4x3lbhb44Mon, 28 Nov 2022 00:00:00 GMTA Nonparametric Framework for Online Stochastic Matching with Correlated Arrivals
https://scholar.archive.org/work/2kvfyovgyndbzcd3jfu2h62m7q
The design of online policies for stochastic matching and revenue management settings is usually bound by the Bayesian prior that the demand process is formed by a fixed-length sequence of queries with unknown types, each drawn independently. This assumption of serial independence implies that the demand of each type, i.e., the number of queries of a given type, has low variance and is approximately Poisson-distributed. Thus, matching policies are often based on "fluid" LPs that only use the expectations of these distributions. This paper explores alternative stochastic models for online matching that allow for nonparametric, higher variance demand distributions. We propose two new models, \Indep and \Correl, that relax the serial independence assumption in different ways by combining a nonparametric distribution for the demand with standard assumptions on the arrival patterns -- adversarial or random-order. In our \Indep model, the demand for each type follows an arbitrary distribution, while being mutually independent across different types. In our \Correl model, the total demand follows an arbitrary distribution, and conditional on the sequence length, the type of each query is drawn independently. In both settings, we show that the fluid LP relaxation based on only expected demands can be an arbitrarily bad benchmark for algorithm design. We develop tighter LP relaxations for the \Indep and \Correl models that leverage the exact distribution of the demand, leading to matching algorithms that achieve constant-factor performance guarantees under adversarial and random-order arrivals. More broadly, our paper provides a data-driven framework for expressing demand uncertainty (i.e., variance and correlations) in online stochastic matching models.Ali Aouad, Will Mawork_2kvfyovgyndbzcd3jfu2h62m7qMon, 28 Nov 2022 00:00:00 GMTPolynomial-Time Data Reduction for Weighted Problems Beyond Additive Goal Functions
https://scholar.archive.org/work/2mfmgkycwjapbbwi5jrsuvwwvu
Dealing with NP-hard problems, kernelization is a fundamental notion for polynomial-time data reduction with performance guarantees: in polynomial time, a problem instance is reduced to an equivalent instance with size upper-bounded by a function of a parameter chosen in advance. Kernelization for weighted problems particularly requires to also shrink weights. Marx and V\'egh [ACM Trans. Algorithms 2015] and Etscheid et al. [J. Comput. Syst. Sci. 2017] used a technique of Frank and Tardos [Combinatorica 1987] to obtain polynomial-size kernels for weighted problems, mostly with additive goal functions. We characterize the function types that the technique is applicable to, which turns out to contain many non-additive functions. Using this insight, we systematically obtain kernelization results for natural problems in graph partitioning, network design, facility location, scheduling, vehicle routing, and computational social choice, thereby improving and generalizing results from the literature.Matthias Bentert and René van Bevern and Till Fluschnik and André Nichterlein and Rolf Niedermeierwork_2mfmgkycwjapbbwi5jrsuvwwvuMon, 28 Nov 2022 00:00:00 GMTEnsure Differential Privacy and Convergence Accuracy in Consensus Tracking and Aggregative Games with Coupling Constraints
https://scholar.archive.org/work/vrh6tpiuqbe5nfbyxu4swuzeum
We address differential privacy for fully distributed aggregative games with shared coupling constraints. By co-designing the generalized Nash equilibrium (GNE) seeking mechanism and the differential-privacy noise injection mechanism, we propose the first GNE seeking algorithm that can ensure both provable convergence to the GNE and rigorous epsilon-differential privacy, even with the number of iterations tending to infinity. As a basis of the co-design, we also propose a new consensus-tracking algorithm that can achieve rigorous epsilon-differential privacy while maintaining accurate tracking performance, which, to our knowledge, has not been achieved before. To facilitate the convergence analysis, we also establish a general convergence result for stochastically-perturbed nonstationary fixed-point iteration processes, which lie at the core of numerous optimization and variational problems. Numerical simulation results confirm the effectiveness of the proposed approach.Yongqiang Wangwork_vrh6tpiuqbe5nfbyxu4swuzeumMon, 28 Nov 2022 00:00:00 GMTStatistical Shape Analysis of Shape Graphs with Applications to Retinal Blood-Vessel Networks
https://scholar.archive.org/work/sbrte3ncajg65lzvbo2xihs27q
This paper provides theoretical and computational developments in statistical shape analysis of shape graphs, and demonstrates them using analysis of complex data from retinal blood-vessel (RBV) networks. The shape graphs are represented by a set of nodes and edges (planar articulated curves) connecting some of these nodes. The goals are to utilize shapes of edges and connectivities and locations of nodes to: (1) characterize full shapes, (2) quantify shape differences, and (3) model statistical variability. We develop a mathematical representation, elastic Riemannian shape metrics, and associated tools for such statistical analysis. Specifically, we derive tools for shape graph registration, geodesics, summaries, and shape modeling. Geodesics are convenient for visualizing optimal deformations, and PCA helps in dimension reduction and statistical modeling. One key challenge here is comparisons of shape graphs with vastly different complexities (in number of nodes and edges). This paper introduces a novel multi-scale representation of shape graphs to handle this challenge. Using the notions of (1) "effective resistance" to cluster nodes and (2) elastic shape averaging of edge curves, one can reduce shape graph complexity while maintaining overall structures. This way, we can compare shape graphs by bringing them to similar complexity. We demonstrate these ideas on Retinal Blood Vessel (RBV) networks taken from the STARE and DRIVE databases.Aditi Basu Bal, Xiaoyang Guo, Tom Needham, Anuj Srivastavawork_sbrte3ncajg65lzvbo2xihs27qMon, 28 Nov 2022 00:00:00 GMTAn 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 four-element, rank 3 chirotope, aka affine oriented matroid.Timothy F. Havelwork_wq4vgu5fmfdtrmsnbtmke6m3kqMon, 28 Nov 2022 00:00:00 GMT