IA Scholar Query: Note on Geometric Graphs.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 31 Dec 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Computing Graph Neural Networks: A Survey from Algorithms to Accelerators
https://scholar.archive.org/work/7uww2lnxrbdpnnyvzsanojgnba
Graph Neural Networks (GNNs) have exploded onto the machine learning scene in recent years owing to their capability to model and learn from graph-structured data. Such an ability has strong implications in a wide variety of fields whose data are inherently relational, for which conventional neural networks do not perform well. Indeed, as recent reviews can attest, research in the area of GNNs has grown rapidly and has lead to the development of a variety of GNN algorithm variants as well as to the exploration of ground-breaking applications in chemistry, neurology, electronics, or communication networks, among others. At the current stage research, however, the efficient processing of GNNs is still an open challenge for several reasons. Besides of their novelty, GNNs are hard to compute due to their dependence on the input graph, their combination of dense and very sparse operations, or the need to scale to huge graphs in some applications. In this context, this article aims to make two main contributions. On the one hand, a review of the field of GNNs is presented from the perspective of computing. This includes a brief tutorial on the GNN fundamentals, an overview of the evolution of the field in the last decade, and a summary of operations carried out in the multiple phases of different GNN algorithm variants. On the other hand, an in-depth analysis of current software and hardware acceleration schemes is provided, from which a hardware-software, graph-aware, and communication-centric vision for GNN accelerators is distilled.Sergi Abadal, Akshay Jain, Robert Guirado, Jorge López-Alonso, Eduard Alarcónwork_7uww2lnxrbdpnnyvzsanojgnbaSat, 31 Dec 2022 00:00:00 GMTArea-Optimal Simple Polygonalizations: The CG Challenge 2019
https://scholar.archive.org/work/ecyawviv5fapxkan7qzuh7u7fe
We give an overview of theoretical and practical aspects of finding a simple polygon of minimum ( Min-Area ) or maximum ( Max-Area ) possible area for a given set of n points in the plane. Both problems are known to be NP -hard and were the subject of the 2019 Computational Geometry Challenge, which presented the quest of finding good solutions to more than 200 instances, ranging from n = 10 all the way to n = 1, 000, 000.Erik D. Demaine, Sndor P. Fekete, Phillip Keldenich, Dominik Krupke, Joseph S. B. Mitchellwork_ecyawviv5fapxkan7qzuh7u7feSat, 31 Dec 2022 00:00:00 GMTAn Algorithmic Study of Fully Dynamic Independent Sets for Map Labeling
https://scholar.archive.org/work/by4kwstrpzgk3fvpxqnu3yoeiq
Map labeling is a classical problem in cartography and geographic information systems that asks to place labels for area, line, and point features, with the goal to select and place the maximum number of independent (i.e., overlap-free) labels. A practically interesting case is point labeling with axis-parallel rectangular labels of common size. In a fully dynamic setting, at each timestep, either a new label appears or an existing label disappears. Then, the challenge is to maintain a maximum cardinality subset of pairwise independent labels with sublinear update time. Motivated by this, we study the maximal independent set ( MIS ) and maximum independent set ( Max-IS ) problems on fully dynamic (insertion/deletion model) sets of axis-parallel rectangles of two types: (i) uniform height and width and (ii) uniform height and arbitrary width; both settings can be modeled as rectangle intersection graphs. We present the first deterministic algorithm for maintaining an MIS (and thus a 4-approximate Max-IS ) of a dynamic set of uniform rectangles with polylogarithmic update time. This breaks the natural barrier of \( \Omega (\Delta) \) update time (where \( \Delta \) is the maximum degree in the graph) for vertex updates presented by Assadi et al. (STOC 2018). We continue by investigating Max-IS and provide a series of deterministic dynamic approximation schemes. For uniform rectangles, we first give an algorithm that maintains a 4-approximate Max-IS with \( O(1) \) update time. In a subsequent algorithm, we establish the trade-off between approximation quality \( 2(1+\frac{1}{k}) \) and update time \( O(k^2\log n) \) , for \( k\in \mathbb {N} \) . We conclude with an algorithm that maintains a 2-approximate Max-IS for dynamic sets of unit-height and arbitrary-width rectangles with \( O(\log ^2 n + \omega \log n) \) update time, where \( \omega \) is the maximum size of an independent set of rectangles stabbed by any horizontal line. We implement our algorithms and report the results of an experimental comparison exploring the trade-off between solution quality and update time for synthetic and real-world map labeling instances. We made several major observations in our empirical study. First, the original approximations are well above their respective worst-case ratios. Second, in comparison with the static approaches, the dynamic approaches show a significant speedup in practice. Third, the approximation algorithms show their predicted relative behavior. The better the solution quality, the worse the update times. Fourth, a simple greedy augmentation to the approximate solutions of the algorithms boost the solution sizes significantly in practice.Sujoy Bhore, Guangping Li, Martin Nöllenburgwork_by4kwstrpzgk3fvpxqnu3yoeiqSat, 31 Dec 2022 00:00:00 GMTCHARACTERIZATION OF BIPOLAR ULTRAMETRIC SPACES AND SOME FIXED POINT THEOREMS
https://scholar.archive.org/work/necafcxgone3hfe577crgpggym
Ultrametricity condition on bipolar metric spaces is considered and a geometric characterization of bipolar ultrametric spaces is given. Also embedding a bipolar ultrametric space into a pseudo-ultrametric space is discussed and some conditions are found to be able to embed them into an ultrametric space. Finally, some fixed point theorems on bipolar ultrametric spaces are proven.Selim ÇETİN, Utku GÜRDALwork_necafcxgone3hfe577crgpggymSat, 31 Dec 2022 00:00:00 GMT6 Between Knowledge and Truth
https://scholar.archive.org/work/dq336s3bdveyvlbhuldhaiid5u
work_dq336s3bdveyvlbhuldhaiid5uSat, 31 Dec 2022 00:00:00 GMTA Weighted Topp-Leone G Family of Distributions: Properties, Applications for Modelling Reliability Data and Different Method of Estimation
https://scholar.archive.org/work/cg2xea6uurh6rojfultrgojmz4
Based on the Topp-Leone distribution, we propose a new family of continuous distributions with one shape parameter called the weighted Topp-Leone family (WTL-G). We study some basic properties including quantile function, asymptotic, mixture for cdf and pdf, various entropies and order statistics .Then we study Lindley case as special case with more details. The maximum likelihood estimates of parameters are compared with various methods of estimations by conducting a simulation study. Finally, three real data sets are illustration the purposes.Majid HASHEMPOURwork_cg2xea6uurh6rojfultrgojmz4Sat, 31 Dec 2022 00:00:00 GMTApproximation properties of the fractional q-integral of Riemann-Liouville integral type Szasz-Mirakyan-Kantorovich operators
https://scholar.archive.org/work/kg2g5mrgqbh5noaz3uhi7thmga
In the present paper, we introduce the fractional q-integral of Riemann-Liouville integral type Szász-Mirakyan-Kantorovich operators. Korovkin-type approximation theorem is given and the order of convergence of these operators are obtained by using Lipschitz-type maximal functions, second order modulus of smoothness and Peetre's K-functional. Weighted approximation properties of these operators in terms of modulus of continuity have been investigated. Then, for these operators, we give a Voronovskaya-type theorem. Moreover, bivariate fractional q- integral Riemann-Liouville fractional integral type Szász-Mirakyan-Kantorovich operators are constructed. The last section is devoted to detailed graphical representation and error estimation results for these operators.Mustafa KARAwork_kg2g5mrgqbh5noaz3uhi7thmgaFri, 30 Dec 2022 00:00:00 GMTZipper Fractal Functions with Variable Scalings
https://scholar.archive.org/work/yneivrwij5hyrjwu4eu6vdu7hq
Zipper fractal interpolation function (ZFIF) is a generalization of fractal interpolation function through an improved version of iterated function system by using a binary parameter called a signature. The signature allows the horizontal scalings to be negative. ZFIFs have a complex geometric structure, and they can be non-differentiable on a dense subset of an interval I. In this paper, we construct k-times continuously differentiable ZFIFs with variable scaling functions on I. Some properties like the positivity, monotonicity, and convexity of a zipper fractal function and the one-sided approximation for a continuous function by a zipper fractal function are studied. The existence of Schauder basis of zipper fractal functions for the space of k-times continuously differentiable functions and the space of p-integrable functions for p ∈ [1,∞) are studied. We introduce the zipper versions of full Müntz theorem for continuous function and p-integrable functions on I for p ∈ [1,∞).. VİJAY, A. K. B. CHANDwork_yneivrwij5hyrjwu4eu6vdu7hqFri, 30 Dec 2022 00:00:00 GMTEarly cephalopod evolution clarified through Bayesian phylogenetic inference
https://scholar.archive.org/work/tw43rrpbfnht3oezkik4aculc4
Despite the excellent fossil record of cephalopods, their early evolution is poorly understood. Different, partly incompatible phylogenetic hypotheses have been proposed in the past, which reflected individual author's opinions on the importance of certain characters but were not based on thorough cladistic analyses. At the same time, methods of phylogenetic inference have undergone substantial improvements. For fossil datasets, which typically only include morphological data, Bayesian inference and in particular the introduction of the fossilized birth-death model have opened new possibilities. Nevertheless, many tree topologies recovered from these new methods reflect large uncertainties, which have led to discussions on how to best summarize the information contained in the posterior set of trees. Results We present a large, newly compiled morphological character matrix of Cambrian and Ordovician cephalopods to conduct a comprehensive phylogenetic analysis and resolve existing controversies. Our results recover three major monophyletic groups, which correspond to the previously recognized Endoceratoidea, Multiceratoidea, and Orthoceratoidea, though comprising slightly different taxa. In addition, many Cambrian and Early Ordovician representatives of the Ellesmerocerida and Plectronocerida were recovered near the root. The Ellesmerocerida is para-and polyphyletic, with some of its members recovered among the Multiceratoidea and early Endoceratoidea. These relationships are robust against modifications of the dataset. While our trees initially seem to reflect large uncertainties, these are mainly a consequence of the way clade support is measured. We show that clade posterior probabilities and tree similarity metrics often underestimate congruence between trees, especially if wildcard taxa are involved. Conclusions Our results provide important insights into the earliest evolution of cephalopods and clarify evolutionary pathways. We provide a classification scheme that is based on a robust phylogenetic analysis. Moreover, we provide some general insights on the application of Bayesian phylogenetic inference on morphological datasets. We support earlier findings that quartet similarity metrics should be preferred over the Robinson-Foulds distance when higher-level phylogenetic relationships are of interest and propose that using a posteriori pruned maximum clade credibility trees help in assessing support for phylogenetic relationships among a set of relevant taxa, because they provide clade support values that better reflect the phylogenetic signal.Alexander Pohle, Björn Kröger, Rachel C M Warnock, Andy H King, David H Evans, Martina Aubrechtová, Marcela Cichowolski, Xiang Fang, Christian Klugwork_tw43rrpbfnht3oezkik4aculc4Thu, 01 Dec 2022 00:00:00 GMTIdentification of Tool Wear During Cast Iron Drilling Using Machine Learning Methods
https://scholar.archive.org/work/wkyky3kpgnczrgb3nid4geuiby
The paper concerns the monitoring of the tool condition on the basis of vibration acceleration signals. The cutting edge condition is determined by wear on the flank surface of the drill. As tools, a twist drills made of cemented carbide were used. A gray cast iron plate EN-GJL-250 was used as the workpiece. Based on the signals, appropriate measures correlated with the wear of the drill were developed. By using binary decision trees CART (Classification and Regression Tree) with two data partitioning methods (Gini index and Cross-entropy), the original number of measures was limited to the most common and those that provide the smallest error in the tool condition classification. Comparing the results for the best trees built with different measures of partition quality in nodes for all available data indicated a better performance of the Gini index. The applied solution allows for high accuracy of the tool classification. The solution is to be used in industry.Paweł Twardowski, Maciej Tabaszewski, Agata Felusiak-Czyryca, Piotr Kieruj, Martyna Wiciak-Pikuła, Jakub Czyżyckiwork_wkyky3kpgnczrgb3nid4geuibyThu, 01 Dec 2022 00:00:00 GMTMath and the Mouse: Explorations of Mathematics and Science in Walt Disney World
https://scholar.archive.org/work/jtkp5zcinzb2pmrpff2wt6233q
Math and the Mouse is an intensive, collaborative, project-driven, study away course that runs during the three-week May Experience term at Furman University and has many of the attributes of a course-based undergraduate research experience in mathematics. We take twelve students to Orlando, Florida to study the behind-the-scenes mathematics employed to make Walt Disney World operate efficiently. Students learn techniques of mathematical modeling (mostly resource allocation, logistics, and scheduling models), statistical analysis (mostly probability, clustering, data collection, and hypothesis testing), and flow management (queuing theory and some beginning flow dynamics) in an applied setting. Through planned course modules, collaborative activities, conversations with guest speakers, and three group projects, one of which is of the students' choosing, this academic experience provides an engaged learning experience that shows how material from eleven academic courses comes together in connection with real-world applications.Elizabeth L. Bouzarth, John M. Harris, Kevin R. Hutsonwork_jtkp5zcinzb2pmrpff2wt6233qThu, 01 Dec 2022 00:00:00 GMTOn duality in convex optimization of second-order differential inclusions with periodic boundary conditions
https://scholar.archive.org/work/ujhbc6k565gg5jowbvp4ijny54
The present paper is devoted to the duality theory for the convex optimal control problem of second-order differential inclusions with periodic boundary conditions. First, we use an auxiliary problem with second-order discrete-approximate inclusions and focus on formulating sufficient conditions of optimality for the differential problem. Then, we concentrate on the duality that exists in periodic boundary conditions to establish a dual problem for the differential problem and prove that Euler-Lagrange inclusions are duality relations for both primal and dual problems. Finally, we consider an example of the duality for the second-order linear optimal control problem.Sevilay DEMİR SAĞLAM, Elimhan MAHMUDOVwork_ujhbc6k565gg5jowbvp4ijny54Thu, 01 Dec 2022 00:00:00 GMTTowards a predictive multi-phase model for alpine mass movements and process cascades
https://scholar.archive.org/work/ckh43oqgjjanjgntl3spacckuy
Alpine mass movements can generate process cascades involving different materials including rock, ice, snow, and water. Numerical modelling is an essential tool for the quantification of natural hazards. Yet, state-of-the-art operational models are based on parameter back-calculation and thus reach their limits when facing unprecedented or complex events. Here, we advance our predictive capabilities for mass movements and process cascades on the basis of a three-dimensional numerical model, coupling fundamental conservation laws to finite strain elastoplasticity. In this framework, model parameters have a true physical meaning and can be evaluated from material testing, thus conferring to the model a strong predictive nature. Through its hybrid Eulerian-Lagrangian character, our approach naturally reproduces fractures and collisions, erosion/deposition phenomena, and multi-phase interactions, which finally grant accurate simulations of complex dynamics. Four benchmark simulations demonstrate the physical detail of the model and its applicability to real-world full-scale events, including various materials and ranging through five orders of magnitude in volume. In the future, our model can support risk-management strategies through predictions of the impact of potentially catastrophic cascading mass movements at vulnerable sites.Alessandro Cicoira, L Blatny, X Li, B Trottet, J Gaumework_ckh43oqgjjanjgntl3spacckuyThu, 01 Dec 2022 00:00:00 GMTOn maximum degree (signless) Laplacian matrix of a graph
https://scholar.archive.org/work/sxrjmwjohvgevg6e7yty7am3sy
Let G be a simple graph on n vertices and v1, v2,...,vn be the vértices of G. We denote the degree of a vertex vi in G by dG(vi) = di. The maximum degree matrix of G, denoted by M(G), is the real symmetric matrix with its ijth entry equal to max{di, dj} if the vertices vi and vj are adjacent in G, 0 otherwise. In analogous to the definitions of Laplacian matrix and signless Laplacian matrix of a graph, we consider Laplacian and signless Laplacian for the maximum degree matrix, called the maximum degree Laplacian matrix and the maximum degree signless Laplacian matrix, respectively. Also, we introduce maximum degree Laplacian energy and maximum degree signless Laplacian energy of a graph. Then we determine the maximum degree (signless) Laplacian energy of some graphs in terms of ordinary energy, and (signless) Laplacian energy. We compute the máximum degree (signless) Laplacian spectra of some graph compositions. A lower and upper bound for the largest eigenvalue of the maximum degree (signless) Laplacian matrix is established and also we determine an upper bound for the second smallest eigenvalue of maximum degree Laplacian matrix in terms of vertex connectivity. We also determine bounds for the maximum degree (signless) Laplacian energy in terms of first Zagreb index.R Rangarajan, V. D. Raghu, B. R. Rakshithwork_sxrjmwjohvgevg6e7yty7am3syThu, 01 Dec 2022 00:00:00 GMTTeeth out of proportion: Smaller horse and cattle breeds have comparatively larger teeth
https://scholar.archive.org/work/y3ktsliny5de5e5umcxzdt4gby
There are different descriptions of allometric relationships between important components of the mammalian skull. Craniofacial evolutionary allometry describes a pattern of increasing facial cranium in larger skulls. Another body of literature describes disproportionately larger teeth in smaller species or specimens, matching anecdotal observations with dental problems in dwarf breeds whose teeth appear "too large for their skulls." We test the scaling of tooth row length with body size and skull length in a data set comprising 114 domestic horses (representing 40 breeds) and in another data set of 316 domestic cattle (of >60 breeds). We demonstrate that smaller skulls have a relatively longer tooth row in both horses and cattle; larger specimens have relatively shorter tooth rows. Whereas in horses, larger skulls have a relatively longer diastema, the distance of the mesial maxillary premolar to the premaxilla was proportional to cranium length in cattle. While the reasons for these patterns remain to be detected, they support the hypothesis that tooth size might be less "evolvable," in terms of time required for changes, than body size. The pattern may affect (i) the selective breeding for dwarf breeds by setting minimum constraints for skull size, as described previously for domestic horses with the same data set; (ii) the susceptibility of small breeds for dental problems; and (iii) differences in chewing efficiency between breeds of different sizes. The findings support the existing concept that scaling of tooth to body size across taxa becomes more isometric the longer these taxa are separated in evolutionary time.Marcus Clauss, Laura Heck, Kristof Veitschegger, Madeleine Geigerwork_y3ktsliny5de5e5umcxzdt4gbyThu, 01 Dec 2022 00:00:00 GMTAtmospheric dispersion of chemical, biological, and radiological hazardous pollutants: Informing risk assessment for public safety
https://scholar.archive.org/work/pyxngevtjzd5nlo44lelq6vqzi
Modern society is confronted with emerging threats from chemical, biological, and radiological (CBR) hazardous substances, which are intensively utilized in the chemical, medical, and energy industries. The atmospheric dispersion of released CBR hazardous pollutants can influence a large percentage of the population owing to their rapid process with extensive spatial coverage. It is important to comprehensively understand the behaviors of the released CBR pollutants in the atmosphere to fully evaluate the risks and protect public safety. In this study, we reviewed the advancements in the atmospheric transport of CBR pollutants, including the urban atmospheric boundary layer, unique concepts, and models for CBR pollutants. We underlined the development of innovative methodologies (e.g., inverse estimation and data assimilation methods) for the atmospheric transport of accidentally released CBR pollutants to reduce uncertainties in emissions and accumulated errors during dispersion by combining numerical models with monitoring data. Finally, we introduced progress in quantitative risk assessment, including exposure assessment and dose-response relationships for CBR hazardous pollutants. A framework, source, assimilation, fundamentals, exposure, and risk (SAFER), has been proposed to integrate the key components in the risk assessment of airborne CBR hazardous pollutants. These methods and models can contribute to effective risk preparedness, prevention, evidence-based policymaking, and emergency response to airborne CBR pollutants.Xiaole Zhang, Jing Wangwork_pyxngevtjzd5nlo44lelq6vqziThu, 01 Dec 2022 00:00:00 GMTINVESTIGATION OF FRACTURE PREVENTION EFFECTS OF COMPOSITE PATCHES REPAIR ON CRACKED ALUMINUM PLATES EXPERIMENTAL AND NUMERICAL STUDY
https://scholar.archive.org/work/fs7gqyp7nbdujbsrpsrbn3j5ou
Nowadays, composite materials are extensively applicable in various aspects including automotive, marine and most importantly aeronautical engineering because of their excellent characteristics. Innovative maintenance using materials with extraordinary features has led to contributing of composites with the goal of extending their lives services. Glass-epoxy composites are lifesaver, inexpensive and highly practical for repairing the damaged components of aeronautical structures. The present study has employed composite patching technique to repair 50 centrally cracked rectangular aluminum plates subjected uniaxial tensile loading experiment. The 600 KN Santam testing machine used for performing tensile tests. Then, the force-displacement linear graph for each sample was plotted and monitored by its user's software. Next, the Abaqus finite element software simulated the same testing conditions on developed 3D- models using XFEM method. Finally, the numerical and experimental results indicated 0.0266 error percentage between the real and simulated tensile loading tests. In addition, specimens repaired by composite patches whose glass fibers orientations angles are 0,90 proved to have the best optimum performance while those that were repaired by patches with fiber orientations angle -45, +45 had the maximum displacement extension before failure.Mohammadjavad RANJBARANwork_fs7gqyp7nbdujbsrpsrbn3j5ouThu, 01 Dec 2022 00:00:00 GMTOne-dimensional nanospace confinement effects on the chemical properties of organic molecules in carbon nanotubes: Quantum chemical calculation analyses
https://scholar.archive.org/work/lysvbeszhjg25jqaec5vy2kof4
We performed dispersion-corrected density functional theory (DFT) calculations to investigate the energetically stable structures of armchair (m,m) carbon nanotubes containing π-conjugated molecules, such as methyl-terminated thiophene oligomers, p,p′-dimethyaminonitorostilbene (DANS) molecules, and triiodobenzene. The stability of such tube-based host-guest structures was analyzed by using their stabilization energy, which consisted of three terms. These analyses found that the long-range interaction energy between guest and a host, mainly originating from π-π and CH-π interactions, and the deformation energy of the guest molecule are what mostly control the stability of the tube-based host-guest structures considered in this paper. The long-range interactions mainly cause nanospace confinement effects in tube-based host-guest structures. To strengthen the long-range attraction interactions, a π-conjugated molecule and its aggregates inside a tube are substantially deformed, which costs energy compared to the corresponding optimized structures without surrounding tubes. Accordingly, we found a substantial impact of nanotube confinement in determining the structure of a guest and its aggregates. These structures play a dominant role in electronic properties (e.g. optoelectronic properties in thiophene oligomers and nonlinear second-order nonlinear properties in DANS molecules), and therefore nanospace confinement effects can be used to change the electronic properties of π-conjugated molecules inside a nanotube by changing its diameter.Takashi Yumurawork_lysvbeszhjg25jqaec5vy2kof4Thu, 01 Dec 2022 00:00:00 GMTDynamics of the Geometric Phase in Inhomogeneous Quantum Spin Chains
https://scholar.archive.org/work/nwc35ji3hnec5myfkr2zkez5j4
The dynamics of the geometric phase are studied in inhomogeneous quantum spin chains after a quench. Analytic expressions of the Pancharatnam geometric phase (PGP) are derived, for both the period-two quantum Ising chain (QIC) and the disordered QIC. In the period-two QIC, due to the period modulation, PGP changes with time at the boundary of the Brillouin zone, and consequently, the winding number based on PGP is not quantized anymore. Therefore, PGP and its winding number are not topological. Nevertheless, they have non-analytic singularities at the critical times of the dynamical quantum phase transitions (DQPTs). This relation between PGP and DQPT is further confirmed in the disordered QIC by decomposing PGP for each quasi-particle mode. The critical time of DQPT induced by weak disorder is also accompanied by the non-analytic singularity of PGP. From these observations and analysis, a simple mathematical theory is suggested to explain the non-analytic behavior of PGP at the critical time of DQPT, which clarifies that PGP needs not to be topological in the general case.Kaiyuan Cao, Shuxiang Yang, Yayun Hu, Guangwen Yangwork_nwc35ji3hnec5myfkr2zkez5j4Wed, 30 Nov 2022 00:00:00 GMTThe hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets
https://scholar.archive.org/work/qmv6ipdowbdfrjaivt6ca2mrpu
Kontsevich's graph flows are – universally for all finite-dimensional affine Poisson manifolds – infinitesimal symmetries of the spaces of Poisson brackets. We show that the previously known tetrahedral flow and the recently obtained pentagon-wheel flow preserve the class of Nambu-determinant Poisson bi-vectors P=[[ ϱ(x) ∂_x∧∂_y∧∂_z,a]] on ℝ^3∋x=(x,y,z) and P=[[ [[ϱ(y) ∂_x^1∧...∧∂_x^4,a_1]],a_2]] on ℝ^4∋y, including the general case ϱ≢1. We detect that the Poisson bracket evolution Ṗ = Q_γ(P^⊗^# Vert(γ)) is trivial in the second Poisson cohomology, Q_γ = [[ P, X⃗([ϱ],[a]) ]], for the Nambu-determinant bi-vectors P(ϱ,[a]) on ℝ^3. For the global Casimirs 𝐚 = (a_1,...,a_d-2) and inverse density ϱ on ℝ^d, we analyse the combinatorics of their evolution induced by the Kontsevich graph flows, namely ϱ̇ = ϱ̇([ϱ], [𝐚]) and 𝐚̇ = 𝐚̇([ϱ],[𝐚]) with differential-polynomial right-hand sides. Besides the anticipated collapse of these formulas by using the Civita symbols (three for the tetrahedron γ_3 and five for the pentagon-wheel graph cocycle γ_5), as dictated by the behaviour ϱ(𝐱') = ϱ(𝐱) ·∂𝐱' / ∂𝐱 of the inverse density ϱ under reparametrizations 𝐱⇄𝐱', we discover another, so far hidden discrete symmetry in the construction of these evolution equations.Ricardo Buring, Dimitri Lipper, Arthemy V. Kiselevwork_qmv6ipdowbdfrjaivt6ca2mrpuWed, 30 Nov 2022 00:00:00 GMT