Journal articles: 'Polygonal space vector structures'

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Yan, Shiyao, Zequn Zhang, Xian Sun, Guangluan Xu, Shuchao Li, Qing Liu, Nayu Liu, and Shensi Wang. "PolygonE: Modeling N-ary Relational Data as Gyro-Polygons in Hyperbolic Space." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 4 (June 28, 2022): 4308–17. http://dx.doi.org/10.1609/aaai.v36i4.20351.

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N-ary relational knowledge base (KBs) embedding aims to map binary and beyond-binary facts into low-dimensional vector space simultaneously. Existing approaches typically decompose n-ary relational facts into subtuples (entity pairs, triples or quintuples, etc.), and they generally model n-ary relational KBs in Euclidean space. However, n-ary relational facts are semantically and structurally intact, decomposition leads to the loss of global information and undermines the semantical and structural integrity. Moreover, compared to the binary relational KBs, n-ary ones are characterized by more abundant and complicated hierarchy structures, which could not be well expressed in Euclidean space. To address the issues, we propose a gyro-polygon embedding approach to realize n-ary fact integrity keeping and hierarchy capturing, termed as PolygonE. Specifically, n-ary relational facts are modeled as gyro-polygons in the hyperbolic space, where we denote entities in facts as vertexes of gyro-polygons and relations as entity translocation operations. Importantly, we design a fact plausibility measuring strategy based on the vertex-gyrocentroid geodesic to optimize the relation-adjusted gyro-polygon. Extensive experiments demonstrate that PolygonE shows SOTA performance on all benchmark datasets, generalizability to binary data, and applicability to arbitrary arity fact. Finally, we also visualize the embedding to help comprehend PolygonE's awareness of hierarchies.

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Szczygieł, Jacek, and Maciej J. Mendecki. "Preliminary results of fractal analysis of the poligonal survey from cave: case study of Małołączniak area (Tatra Mts.)." Contemporary Trends in Geoscience 2, no. 1 (September 1, 2013): 95–100. http://dx.doi.org/10.2478/ctg-2014-0014.

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Abstract Traverses (polygons) from two caves have been tested: Wielka Śnieżna Cave System (2858 vectors) and Śnieżna Studnia cave (742). The box counting technique was applied to evaluate the fractal analysis of spatial orientation of traverses. The polygonal survey of Wielka Śnieżna Cave, Śnieżna Studnia and both caves merged together have a fractal geometry. It may be concluded that these caves are close to the full recognition of passages forming by a hydrological system and they could have close relation with geological structures. The usual explanation of fractal dimension D higher than 1 indicates that caves with such dimension fill more space than those with ideal dimension of 1.00 (for example a straight line), and the geological constraints limit the dimension to be lower than 2 (Verbovsek 2007). It may suggest that systems can be developped into more complicated passages in future. The fact that both caves merged together also have showed fractal geometry indicates that they are belonging to the same hydrological system. It was noticed that D-value of merged caves is slightly larger than individual cave. It can be explained by “occupying” more space in rock mass.

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CHETVERIKOV, B., O. KHINTSITSKY, and I. KALYNYCH. "Methodology of mapping of historical and cultural heritage objects by GIS technologies using archival cartographic and aerial materials." Modern achievements of geodesic science and industry 41, no. I (April 1, 2021): 97–103. http://dx.doi.org/10.33841/1819-1339-1-41-97-103.

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Aim. The purpose of the work is to process archival cartographic materials and remote sensing data for the interpretation of objects of historical and cultural heritage (OHCH) of Cherkasy, including those that have not been preserved. Method. One of the possible technological schemes for research is offered. According to her, the first step was to analyze the input data of the study, among which were: a map of Cherkasy in 1895 at a scale of 1:42000; German aerial image of 1944; a fragment of a space image of Cherkasy obtained from the GeoEye-1 satellite in 2018. Geometric correction of the input materials was performed in the Mercator projection and the WGS84 coordinate system, in which the transformed image was obtained. The next step was to vectorize the objects of historical and cultural heritage of Cherkasy, according to the list obtained on the city’s website. There are two types of objects: point and polygonal. When vectorizing polygonal objects, the historical boundaries were specified with the help of archival maps and aerial images. Special symbols have been developed for each of the types of historical and cultural heritage sites, according to the proposed classification. In addition, an attributive database of these objects was created, which had the following structure: number of the passport of object, the name of the object, the address of the OHCH, the number of the decision to take under protection, information about the OHCH. Also, the obtained vector data was exported to the exchange format with the extension kmz and an online version of the thematic map was created on the basis of the free GISFile resource. Results. As a result of the conducted researches, the thematic GIS of the objects of historical and cultural heritage of Cherkasy was created, which are plotted on the space image of high spatial resolution, obtained in 2018. An on-line version of the GIS of Cherkasy historical and cultural heritage sites has been created on the basis of the free GISFile cartographic service, with the possibility of analyzing the location of these objects and building optimal tourist routes. Scientific novelty. Possible algorithms for creating offline and on-line versions of thematic GIS are proposed. Practical value. The obtained results of mapping the objects of historical and cultural heritage of Cherkasy can be used by the structures of protection of objects of historical and cultural heritage of Cherkasy at the Ministry of Culture.

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Nagae, Sadahiko, and Chiaki Ohta. "Algorithm for Automatic Generation of Boundary Lines." Journal of Robotics and Mechatronics 2, no. 1 (February 20, 1990): 59–65. http://dx.doi.org/10.20965/jrm.1990.p0059.

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It may be supposed that the development of an algorithm capable of automatically generating boundary lines among parts for designing IC diagrams and automatic recognition of intervals among parts as well as mapping is widely desired in different industrial sectors. The most basic, effective solution conceivable will be image analysis using an ITV camera. The authors, however, tried to develop an algorithm capable of automatically generating boundary lines which would not interfere with one another using a great number of profiles comprising, in principle, polygons based on vector input data. It is not that this enables every boundary line between graphics to be automatically generated with good precision, but a result has been obtained which can appropriately be applied to roughly calculating the living space in a floor plan, and dividing structures for land use status drawing on a scale of about 1 to 2500. It can also be used for designing wiring drawings among parts. It is conceivable that the utilization of the results of this research will enable the algorithm to be used for graphic recognition by image processing. The following are the proposed problem, definitions, and a detailed description of the algorithm leading to a solution.

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Watson, Layne T., and C. Y. Wang. "Large deformations of rotating polygonal space structures." Computers & Mathematics with Applications 22, no. 9 (1991): 51–59. http://dx.doi.org/10.1016/0898-1221(91)90206-j.

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MILLETT, KENNETH C., MICHAEL PIATEK, and ERIC J. RAWDON. "POLYGONAL KNOT SPACE NEAR ROPELENGTH-MINIMIZED KNOTS." Journal of Knot Theory and Its Ramifications 17, no. 05 (May 2008): 601–31. http://dx.doi.org/10.1142/s0218216508006282.

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For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r < R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds on perturbation radii for which one can obtain small trefoil and figure-eight summands and use Monte Carlo simulations to estimate the relative probabilities of these structures as a function of the number of edges.

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Kuang, Wenhui, Shu Zhang, Xiaoyong Li, and Dengsheng Lu. "A 30 m resolution dataset of China's urban impervious surface area and green space, 2000–2018." Earth System Science Data 13, no. 1 (January 14, 2021): 63–82. http://dx.doi.org/10.5194/essd-13-63-2021.

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Abstract. Accurate and timely maps of urban underlying land properties at the national scale are of significance in improving habitat environment and achieving sustainable development goals. Urban impervious surface (UIS) and urban green space (UGS) are two core components for characterizing urban underlying environments. However, the UIS and UGS are often mosaicked in the urban landscape with complex structures and composites. The “hard classification” or binary single type cannot be used effectively to delineate spatially explicit urban land surface property. Although six mainstream datasets on global or national urban land use and land cover products with a 30 m spatial resolution have been developed, they only provide the binary pattern or dynamic of a single urban land type, which cannot effectively delineate the quantitative components or structure of intra-urban land cover. Here we propose a new mapping strategy to acquire the multitemporal and fractional information of the essential urban land cover types at a national scale through synergizing the advantage of both big data processing and human interpretation with the aid of geoknowledge. Firstly, the vector polygons of urban boundaries in 2000, 2005, 2010, 2015 and 2018 were extracted from China's Land Use/cover Dataset (CLUD) derived from Landsat images. Secondly, the national settlement and vegetation percentages were retrieved using a sub-pixel decomposition method through a random forest algorithm using the Google Earth Engine (GEE) platform. Finally, the products of China's UIS and UGS fractions (CLUD-Urban) at a 30 m resolution were developed in 2000, 2005, 2010, 2015 and 2018. We also compared our products with six existing mainstream datasets in terms of quality and accuracy. The assessment results showed that the CLUD-Urban product has higher accuracies in urban-boundary and urban-expansion detection than other products and in addition that the accurate UIS and UGS fractions were developed in each period. The overall accuracy of urban boundaries in 2000–2018 are over 92.65 %; and the correlation coefficient (R) and root mean square errors (RMSEs) of UIS and UGS fractions are 0.91 and 0.10 (UIS) and 0.89 and 0.11 (UGS), respectively. Our result indicates that 71 % of pixels of urban land were mosaicked by the UIS and UGS within cities in 2018; a single UIS classification may highly increase the mapping uncertainty. The high spatial heterogeneity of urban underlying covers was exhibited with average fractions of 68.21 % for UIS and 22.30 % for UGS in 2018 at a national scale. The UIS and UGS increased unprecedentedly with annual rates of 1605.56 and 627.78 km2 yr−1 in 2000–2018, driven by fast urbanization. The CLUD-Urban mapping can fill the knowledge gap in understanding impacts of the UIS and UGS patterns on ecosystem services and habitat environments and is valuable for detecting the hotspots of waterlogging and improving urban greening for planning and management practices. The datasets can be downloaded from https://doi.org/10.5281/zenodo.4034161 (Kuang et al., 2020a).

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Smith, Philip, and Eleni Panagiotou. "The second Vassiliev measure of uniform random walks and polygons in confined space." Journal of Physics A: Mathematical and Theoretical 55, no. 9 (February 3, 2022): 095601. http://dx.doi.org/10.1088/1751-8121/ac4abf.

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Abstract Biopolymers, like chromatin, are often confined in small volumes. Confinement has a great effect on polymer conformations, including polymer entanglement. Polymer chains and other filamentous structures can be represented by polygonal curves in three-space. In this manuscript, we examine the topological complexity of polygonal chains in three-space and in confinement as a function of their length. We model polygonal chains by equilateral random walks in three-space and by uniform random walks (URWs) in confinement. For the topological characterization, we use the second Vassiliev measure. This is an integer topological invariant for polygons and a continuous functions over the real numbers, as a function of the chain coordinates for open polygonal chains. For URWs in confined space, we prove that the average value of the Vassiliev measure in the space of configurations increases as O(n 2) with the length of the walks or polygons. We verify this result numerically and our numerical results also show that the mean value of the second Vassiliev measure of equilateral random walks in three-space increases as O(n). These results reveal the rate at which knotting of open curves and not simply entanglement are affected by confinement.

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Lakshminarayanan, Sanjay, R. S. Kanchan, P. N. Tekwani, and K. Gopakumar. "Multilevel inverter with 12-sided polygonal voltage space vector locations for induction motor drive." IEE Proceedings - Electric Power Applications 153, no. 3 (2006): 411. http://dx.doi.org/10.1049/ip-epa:20050444.

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Lakshminarayanan, S., G. Mondal, K. Gopakumar, N. S. Dinesh, and S. Figarado. "Eighteen-sided polygonal voltage space-vector-based PWM control for an induction motor drive." IET Electric Power Applications 2, no. 1 (January 1, 2008): 56–63. http://dx.doi.org/10.1049/iet-epa:20070269.

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Marialva, Eric Fabrício, Nágila F. Secundino, Fernando F. Fernandes, Helena R. C. Araújo, Claudia M. Ríos-Velásquez, Paulo F. P. Pimenta, and Felipe A. C. Pessoa. "Morphological aspects of immature stages of Migonemyia migonei (Diptera: Psychodidae, Phlebotominae), an important vector of Leishmaniosis in South America, described by scanning electron microscopy." PLOS ONE 15, no. 11 (November 12, 2020): e0242163. http://dx.doi.org/10.1371/journal.pone.0242163.

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We describe the immature stages of Migonemyia migonei, which is the vector of Leishmania (Viannia) braziliensis, the etiological agent of cutaneous leishmaniasis in South America, and a putative vector of Leishmania infantum chagasi. Scanning Electron Microscopy (SEM) was used to refine the description of the structures of the egg, all instar larvae, and the pupa. The eggs have polygonal cells on the egg exochorion, and differences between larval and pupal chaetotaxy have been highlighted. Different sensillary subtypes—trichoidea, basiconica, coelonica and campanoformia—were observed in the larval stages. The results presented herein contribute to the taxonomy of Mg. migonei and may contribute to future studies on the phylogeny of this important vector species.

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Gantes, C. J., R. D. Logcher, J. J. Connor, and Y. Rosenfeld. "Deployability Conditions for Curved and Flat, Polygonal and Trapezoidal Deployable Structures." International Journal of Space Structures 8, no. 1-2 (April 1993): 97–106. http://dx.doi.org/10.1177/0266351193008001-210.

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Prefabricated, deployable space frames that exhibit self-standing and stress-free states in both the deployed and collapsed configurations are investigated in this paper. This type of deployable structures shows considerable advantages as compared to previous designs that either required external stabilizing or had members with residual stresses in the deployed configuration. Following previous developments for flat deployable structures consisting of units with regular-polygon planviews, this study deals with flat structures made of trapezoidals, and curved structures assembled from regular-polygonal units. First, the general geometric constraints and deployability conditions for these units are formulated, and a methodology for using these constraints as geometric design criteria is presented. Furthermore, additional conditions for the assemblage of single units into larger structures are given. Then, structural analysis issues for this type of structures are discussed. The necessity of nonlinear analysis during deployment is emphasized. Finally, the above geometric design procedures are demonstrated with specific examples.

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Shojae Chaeikar, Saman, Azizah Abdul Manaf, Ala Abdulsalam Alarood, and Mazdak Zamani. "PFW: Polygonal Fuzzy Weighted—An SVM Kernel for the Classification of Overlapping Data Groups." Electronics 9, no. 4 (April 5, 2020): 615. http://dx.doi.org/10.3390/electronics9040615.

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Support vector machines are supervised learning models which are capable of classifying data and measuring regression by means of a learning algorithm. If data are linearly separable, a conventional linear kernel is used to classify them. Otherwise, the data are normally first transformed from input space to feature space, and then they are classified. However, carrying out this transformation is not always practical, and the process itself increases the cost of training and prediction. To address these problems, this paper puts forward an SVM kernel, called polygonal fuzzy weighted or PFW, which effectively classifies data without space transformation, even if the groups in question are not linearly separable and have overlapping areas. This kernel is based on Gaussian data distribution, standard deviation, the three-sigma rule and a polygonal fuzzy membership function. A comparison of our PFW, radial basis function (RBF) and conventional linear kernels in identical experimental conditions shows that PFW produces a minimum of 26% higher classification accuracy compared with the linear kernel, and it outperforms the RBF kernel in two-thirds of class labels, by a minimum of 3%. Moreover, Since PFW runs within the original feature space, it involves no additional computational cost.

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Gillette, Andrew, Alexander Rand, and Chandrajit Bajaj. "Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes." Computational Methods in Applied Mathematics 16, no. 4 (October 1, 2016): 667–83. http://dx.doi.org/10.1515/cmam-2016-0019.

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AbstractWe combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart–Thomas, and Brezzi–Douglas–Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space with regards to global continuity and that they reproduce the requisite polynomial differential forms described by finite element exterior calculus. We present a method to count the number of basis functions required to ensure these two key properties.

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Lehmkühler, Felix, Gerhard Grübel, and Christian Gutt. "Detecting orientational order in model systems by X-ray cross-correlation methods." Journal of Applied Crystallography 47, no. 4 (July 19, 2014): 1315–23. http://dx.doi.org/10.1107/s1600576714012424.

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The results of a computational X-ray cross-correlation analysis (XCCA) study on two-dimensional polygonal model structures are presented. This article shows how to detect and identify the orientational order of such systems, demonstrates how to eliminate the influence of the `computational box' on the XCCA results and develops new correlation functions that reflect the sample's orientational order only. For this purpose, the dependence of the correlation functions on the number of polygonal clusters and scattering vector magnitudeqis studied for various types of polygons, including mixtures of polygons and randomly placed particles. An order parameter that describes the orientational order within the sample is defined. Finally, the influence of detector noise and nonplanar wavefronts on the XCCA data is determined, both of which appear to affect the results significantly and have thus to be considered in real experiments.

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OZTURK, OZGUR, and HAKAN FERHATOSMANOGLU. "VECTOR SPACE INDEXING FOR BIOSEQUENCE SIMILARITY SEARCHES." International Journal on Artificial Intelligence Tools 14, no. 05 (October 2005): 811–26. http://dx.doi.org/10.1142/s0218213005002405.

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We present a multi-dimensional indexing approach for fast sequence similarity search in DNA and protein databases. In particular, we propose effective transformations of subsequences into numerical vector domains and build efficient index structures on the transformed vectors. We then define distance functions in the transformed domain and examine properties of these functions. We experimentally compared their (a) approximation quality for k-Nearest Neighbor (k-NN) queries and both (b) pruning ability and (c) approximation quality for ε-range queries. Results for k-NN queries, which we present here, show that our proposed distances FD2 and WD2 (i.e. Frequency and Wavelet Distance functions for 2-grams) perform significantly better than the others. We then develop effective index structures, based on R-trees and scalar quantization, on top of transformed vectors and distance functions. Promising results from the experiments on real biosequence data sets are presented.

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Raj R, Krishna, K. Gopakumar, Apurv Kumar Yadav, L. Umanand, Mariusz Malinowski, and Wojciech Jarzyna. "A Twelve Concentric Multilevel Twenty-Four Sided Polygonal Voltage Space Vector Structure for Variable Speed Drives." IEEE Transactions on Power Electronics 34, no. 10 (October 2019): 9906–15. http://dx.doi.org/10.1109/tpel.2019.2892329.

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Duggal, K. L. "Space time manifolds and contact structures." International Journal of Mathematics and Mathematical Sciences 13, no. 3 (1990): 545–53. http://dx.doi.org/10.1155/s0161171290000783.

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A new class of contact manifolds (carring a global non-vanishing timelike vector field) is introduced to establish a relation between spacetime manifolds and contact structures. We show that odd dimensional strongly causal (in particular, globally hyperbolic) spacetimes can carry a regular contact structure. As examples, we present a causal spacetime with a non regular contact structure and a physical model [Gödel Universe] of Homogeneous contact manifold. Finally, we construct a model of 4-dimensional spacetime of general relativity as a contact CR-submanifold.

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Goncharov, V. P., and V. I. Pavlov. "Some estimates on the space scales of vortex pairs emitted from river mouths." Nonlinear Processes in Geophysics 8, no. 1/2 (April 30, 2001): 1–7. http://dx.doi.org/10.5194/npg-8-1-2001.

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Abstract. Two-dimensional vortex pairs are frequently observed in geophysical conditions, for example, in a shelf zone of the ocean near river mouths. The main aims of the work are to estimate the space scales of such vortex structures, to analyze possible scenarios of vortex pair motion and to give the qualitative classification of their trajectories. We discuss some features of the motion of strong localized vorticity concentrations in a given flow in the presence of boundaries. The analyses are made in the framework of a 2D point vortex mo-del with an open polygonal boundary. Estimations are made for the characteristic parameters of dipole vortex structures emitted from river mouths into the open ocean.

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Ciaglia, Florio Maria, Alberto Ibort, and Giuseppe Marmo. "Geometrical structures for classical and quantum probability spaces." International Journal of Quantum Information 15, no. 08 (December 2017): 1740007. http://dx.doi.org/10.1142/s021974991740007x.

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On the affine space containing the space [Formula: see text] of quantum states of finite-dimensional systems, there are contravariant tensor fields by means of which it is possible to define Hamiltonian and gradient vector fields encoding the relevant geometrical properties of [Formula: see text]. Guided by Dirac’s analogy principle, we will use them as inspiration to define contravariant tensor fields, Hamiltonian and gradient vector fields on the affine space containing the space of fair probability distributions on a finite sample space and analyze their geometrical properties. Most of our considerations will be dealt with for the simple example of a three-level system.

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Malek, F., and A. Shafei Deh Abad. "Homogeneous Poisson structures." Bulletin of the Australian Mathematical Society 54, no. 2 (October 1996): 203–10. http://dx.doi.org/10.1017/s0004972700017664.

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In this paper we provide an algebraic definition for the Schouten product and give a decomposition for homogeneous Poisson structures in any n-dimensional vector space. A large class of n-homogeneous Poisson structures in ℝk is also characterised.

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Srivastava, J. K., and Narayan Prasad Pahari. "On Certain Topological Structures of Product Normed Space Valued Paranormed Space of Summable Sequences." Journal of Institute of Science and Technology 19, no. 1 (November 8, 2015): 25–29. http://dx.doi.org/10.3126/jist.v19i1.13822.

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The aim of this paper is to introduce and study a new vector valued sequence space S( ( Z, || ( . , . ) ||Z ), ?, u ) with terms from a product normed space as a generalization of sequence space studied by Srivastava and Pahari (2011) which is itself the generalization of the familiar absolutely summable sequence space l. We investigate its linear structure with respect to the co-ordinate wise vector operation and explore the conditions in terms of different u and ? so that a class is contained in another class of same kind and thereby derive the conditions of their equality. Finally we investigate the paranormed structure of S ( ( Z, || (. , . ) ||Z ), ?, u ) by endowing it with a suitable natural paranorm.Journal of Institute of Science and Technology, 2014, 19(1): 25-29

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Murali, V. "Ultrabornological structures." International Journal of Mathematics and Mathematical Sciences 14, no. 4 (1991): 723–30. http://dx.doi.org/10.1155/s0161171291000972.

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The purpose of this note is to extend Warner's idea of bornological structures to cover non-locally convex situations and to develop a framework unifying several variations on an ultrabornological theme (for example, ultrabornological spaces,o-ultrabornological spaces,i-ultrabornological algebras). We do this by first generalising the concept of a structure on a vector space as defined by Warner.

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Surendra Babu, N. N. V., and B. G. Fernandes. "Cascaded two-level inverter-based multilevel static VAr compensator using 12-sided polygonal voltage space vector modulation." IET Power Electronics 5, no. 8 (2012): 1500. http://dx.doi.org/10.1049/iet-pel.2012.0120.

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Mathew, K., Chintan Patel, K. Gopakumar, Anandarup Das, and Rijil Ramchand. "An asymmetric cascaded H-Bridge inverters for generating 12-sided polygonal space vector diagrams for Motor drives." EPE Journal 21, no. 1 (March 2011): 21–28. http://dx.doi.org/10.1080/09398368.2011.11463784.

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Schulte, Egon. "Polyhedra, complexes, nets and symmetry." Acta Crystallographica Section A Foundations and Advances 70, no. 3 (March 12, 2014): 203–16. http://dx.doi.org/10.1107/s2053273314000217.

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Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zigzag or helical. A polyhedron or complex isregularif its geometric symmetry group is transitive on the flags (incident vertex–edge–face triples). There are 48 regular polyhedra (18 finite polyhedra and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all infinite, which are not polyhedra. Their edge graphs are nets well known to crystallographers and they are identified explicitly. There are also six infinite families ofchiralapeirohedra, which have two orbits on the flags such that adjacent flags lie in different orbits.

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OIKONOMOU, V. K. "GRADED GEOMETRIC STRUCTURES UNDERLYING F-THEORY RELATED DEFECT THEORIES." International Journal of Modern Physics A 28, no. 21 (August 20, 2013): 1350106. http://dx.doi.org/10.1142/s0217751x13501066.

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In the context of F-theory, we study the related eight-dimensional super-Yang–Mills theory and reveal the underlying supersymmetric quantum mechanics algebra that the fermionic fields localized on the corresponding defect theory are related to. Particularly, the localized fermionic fields constitute a graded vector space, and in turn this graded space enriches the geometric structures that can be built on the initial eight-dimensional space. We construct the implied composite fiber bundles, which include the graded affine vector space and demonstrate that the composite sections of this fiber bundle are in one-to-one correspondence to the sections of the square root of the canonical bundle corresponding to the submanifold on which the zero modes are localized.

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German, Anna. "3D art cartography in Belarus: The historical development and achievements in the modern period." Abstracts of the ICA 1 (July 15, 2019): 1–2. http://dx.doi.org/10.5194/ica-abs-1-94-2019.

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Abstract. Three-dimensional images of the area were begun to create in the middle ages. The discoveries of new territories had led to the necessity of their mapping. On the perspective-panoramic maps of that time cities and fortresses were often depicted as they could be seen by an approaching traveler, usually from the most impressive side, from "bird's eye view". Cartographic images were performed manually, by the method of engraving on wood or on copper. One of the first panoramic maps of Belarusian cities was a cartographic image of the city of Grodno, presented in The Large Atlas of Cities called "Civitates Orbis Terrarum" ("Cities of the world"). The Atlas was published by the cartographer Georg Braun and engraver of maps Franz Hogenberg in Cologne in 1572–1617. The main method of presentation of the cities in it was a picture. A significant contribution to the development of cartography in Belarus was made by cartographer, artist, engraver and printer, the creator of the map of The Grand Duchy of Lithuania (1613) Tomasz Makowski. For many years Makowski were traveling all over the country and thoroughly studying its features as a topographer, historian, artist and geographer. He examined and sketched the most outstanding objects, as evidenced by his panoramic images of some cities. In particular, he created panoramic engravings of Nesvizh, Grodno, Brest and other cities of modern Belarus. Nowadays, the traditions of medieval art cartography in Belarus have received a new development in the works of artist-cartographer Ruben Atoyan. Initially, cartographic images were made manually with ink and watercolors. Panoramic maps of the main cities of Belarus, including Minsk, Grodno, Gomel, Mogilev, Mir, Nesvizh, Novogrudok, etc. were created in a such way. With the usage of computer technologies at the present stage, the technique of creating of artistic cartographic images has significantly changed. The majority of types of work which was performed previously by creating on paper, now is possible to implement using a personal computer, graphics tablet and vector and raster graphics programs. During this process – and it's very important – the handwritten style of images is kept and the ability of operative update of the content is provided. However, there are some types of artistic and cartographic work which easier and faster to accomplish manually than on the computer, for example, drawing of different types of trees, grasses, waters of rivers and lakes. In this case it's possible to speak about the combined handwriting-automated technology of artistic map image creation which includes elements made manually and the separated stages and elements which advisable to perform on the computer. The construction of the perspective image of an urban landscape according to the method developed by the author conditionally consists of several stages. The first step includes the selection of the projection direction and determination of the territory coverage by remote sensing images of the Earth and the city plan. The previously developed template of perspective (central) projection is used to build a perspective grid on the sheet of the drawing paper. Then the road network and configuration of the structures are drawn. The second stage is the construction of buildings’ frames of complex configuration (for example, architectural monuments) and drawing them in ink. The third stage includes the color design (with watercolor) of significant parkland, forests, and water areas. It was determined empirically that this type of work is more appropriate to perform manually. The handwritten component of this technique is completed by scanning of the image in high resolution. Further, the creation of a panoramic map is carried out with the usage of vector and raster graphics programs. The fourth stage involves the detailing of the facades, i.e. filling the frames with structural elements from the pre-created library of signs (different types of windows, doors, arches, balconies, etc.) (using the vector graphics program Adobe Illustrator). At the final, the fifth stage, the concluding color design of the panorama is realized according to the library of colors and textures (using the raster graphics program Adobe Photoshop). The similar steps are performed in graphic editors during the creating of a panoramic map using the automated technique. This method involves the usage of computer technologies only. Thus, at the first level of visualization the plan of the territory based on an aerial photograph of the city is created in the program Adobe Illustrator. Then this scheme is transformed into a perspective projection using the tool “Perspective Distort” in the Free transform toolset. The construction of all buildings’ frames (including structures of complex configuration) is also implemented in vector format with the tools "Pen" and "Line Segment" (the second level of visualization). At the third and fourth levels, the frames are filled with structural elements, and then the whole cartographic image is made in color according to the libraries of textures and colors. Operational updating of the content of panoramas created earlier is carried out mainly by the combined technique with the usage of Adobe Photoshop: new buildings which were drawn manually or on a computer are combined with the scanned original, the colors of facades and roofs are updated, park spaces and road junctions are added. For these purposes the following tools are used: "Polygonal Lasso", "Brush", "Clone Stamp", "Color Replacement ", – as well as the options "Curves", "Color Balance", "Hue/Saturation" of the section Image/Adjustment. The combined handwritten and automated method of constructing of three-dimensional cartographic images was tested by the author during the creating and updating of the panoramic maps of cities not only of Belarus, but also other world-class cities: Berlin, Moscow, Yerevan, Astana. Currently, the author is working on the practical usage of this method of creating of artistic cartographic images for the development of electronic multi-scale applications for the cities of the Republic of Belarus.

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Pahari, Narayan` Prasad. "On Certain Linear Structures of Bounded Vector - Valued Sequence Space on Product Normed Space." Journal of Advanced College of Engineering and Management 2 (November 29, 2016): 31. http://dx.doi.org/10.3126/jacem.v2i0.16096.

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Ohsaki, M., Tsuneyoshi Nakamura, and M. Kohiyama. "Shape Optimization of a Double-Layer Space Truss Described by a Parametric Surface." International Journal of Space Structures 12, no. 2 (June 1997): 109–19. http://dx.doi.org/10.1177/026635119701200205.

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A method is presented for finding the optimal configuration of a double-layer space truss described by a parametric surface. The number of design variables is drastically reduced, without sacrificing the smoothness of the upper layer surface, by using the control net parameters as design variables. The coordinates of the lower nodes are defined by using a vertical or normal offset vectors from the upper surface. Curvatures of the curve associated with the lower polygon are calculated for evaluating feasibility and regularity of the a double-layer lattice space truss. In the examples, a feasible optimum design, which minimizes compliance under constraints on structural volume, are successfully found by using the proposed method. It is shown that the optimal solutions strongly depend on the specified ratios of the cross-sectional areas of the members.

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Krishna, Raj R., K. Gopakumar, Mathews Boby, Apurv Kumar Yadav, Leopoldo Garcia Franquelo, and Sheldon S. Williamson. "Multilevel 24-Sided Polygonal Voltage-Space-Vector Structure Generation for an IM Drive Using a Single DC Source." IEEE Transactions on Industrial Electronics 66, no. 2 (February 2019): 1023–31. http://dx.doi.org/10.1109/tie.2018.2831189.

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Richter, Wolf-Dieter. "Three-Complex Numbers and Related Algebraic Structures." Symmetry 13, no. 2 (February 20, 2021): 342. http://dx.doi.org/10.3390/sym13020342.

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Three-complex numbers are introduced for using a geometric vector product in the three-dimensional Euclidean vector space R3 and proving its equivalence with a spherical coordinate product. Based upon the definitions of the geometric power and geometric exponential functions, some Euler-type trigonometric representations of three-complex numbers are derived. Further, a general l23−complex algebraic structure together with its matrix, polynomial and variable basis vector representations are considered. Then, the classes of lp3-complex numbers are introduced. As an application, Euler-type formulas are used to construct directional probability laws on the Euclidean unit sphere in R3.

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Karakaya, Vatan, and Hemen Dutta. "On some vector valued generalized difference modular sequence spaces." Filomat 25, no. 3 (2011): 15–27. http://dx.doi.org/10.2298/fil1103015k.

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In this paper we generalize the modular sequence space ?{Mk} by introducing the sequence space ?{Mk,p,q,s,?n/vm}. We give various properties relevant to algebraic and topological structures of this space and derived some other spaces.

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KHRENNIKOV, A. YU, V. M. SHELKOVICH, and O. G. SMOLYANOV. "LOCALLY CONVEX SPACES OF VECTOR-VALUED DISTRIBUTIONS WITH MULTIPLICATIVE STRUCTURES." Infinite Dimensional Analysis, Quantum Probability and Related Topics 05, no. 04 (December 2002): 483–502. http://dx.doi.org/10.1142/s0219025702001000.

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We construct an infinite-dimensional linear space [Formula: see text] of vector-valued distributions (generalized functions), or sequences, f*(x)=(fn(x)) finite from the left (i.e. fn(x)=0 for n<n0(f*)) whose components fn(x) belong to the linear span [Formula: see text] generated by the distributions δ(m-1)(x-ck), P((x-ck)-m), xm-1, where m=1, 2, …, ck ∈ ℝ, k = 1, …, s. The space of distributions [Formula: see text] can be realized as a subspace in [Formula: see text] This linear space [Formula: see text] has the structure of an associative and commutative algebra containing a unity element and free of zero divizors. The Schwartz counterexample does not hold in the algebra [Formula: see text]. Unlike the Colombeau algebra, whose elements have no explicit functional interpretation, elements of the algebra [Formula: see text] are infinite-dimensional Schwartz vector-valued distributions. This construction can be considered as a next step and a "model" on the way of constructing a nonlinear theory of distributions similar to that developed by L. Schwartz. The obtained results can be considerably generalized.

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Bao, Naping, and Yanyong Hong. "Algebraic Constructions for Novikov–Poisson Algebras." Mathematics 10, no. 17 (September 2, 2022): 3158. http://dx.doi.org/10.3390/math10173158.

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A Novikov–Poisson algebra (A,∘,·) is a vector space with a Novikov algebra structure (A,∘) and a commutative associative algebra structure (A,·) satisfying some compatibility conditions. Give a Novikov–Poisson algebra (A,∘,·) and a vector space V. A natural problem is how to construct and classify all Novikov–Poisson algebra structures on the vector space E=A⊕V such that (A,∘,·) is a subalgebra of E up to isomorphism whose restriction on A is the identity map. This problem is called extending structures problem. In this paper, we introduce the definition of a unified product for Novikov–Poisson algebras, and then construct an object GH2(V,A) to answer the extending structures problem. Note that unified product includes many interesting products such as bicrossed product, crossed product and so on. Moreover, the special case when dim(V)=1 is investigated in detail.

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Abdul Azeez, Najath, Anubrata Dey, K. Mathew, Jaison Mathew, and K. Gopakumar. "A Nearly Constant Switching Frequency Current Error Space Vector Based Hysteresis Controller for an IM Drive with 12-Sided Polygonal Voltage Space Vectors." EPE Journal 23, no. 4 (December 2013): 33–41. http://dx.doi.org/10.1080/09398368.2013.11463865.

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Wang, C. Y. "In-Plane Vibration of a Deployable Segmented Ring." International Journal of Structural Stability and Dynamics 20, no. 07 (July 2020): 2071007. http://dx.doi.org/10.1142/s0219455420710078.

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The in-plane vibrations of regular polygonal rings composed of rigid segments joined by torsional springs are studied for the first time. The nonlinear dynamical difference equations are formulated and solved by perturbation about the equilibrium state. As the number of segments increase, the frequencies, if aptly normalized, converge to the classical vibration frequencies of a continuous elastic ring. The vibration mode shapes are illustrated. The tiling of many identical polygons is discussed. Possible applications include the vibrations of space structures and graphene sheets.

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Lakshminarayanan, S., G. Mondal, P. N. Tekwani, K. K. Mohapatra, and K. Gopakumar. "Twelve-Sided Polygonal Voltage Space Vector Based Multilevel Inverter for an Induction Motor Drive With Common-Mode Voltage Elimination." IEEE Transactions on Industrial Electronics 54, no. 5 (October 2007): 2761–68. http://dx.doi.org/10.1109/tie.2007.899929.

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McRobie, Allan. "The geometry of structural equilibrium." Royal Society Open Science 4, no. 3 (March 2017): 160759. http://dx.doi.org/10.1098/rsos.160759.

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Building on a long tradition from Maxwell, Rankine, Klein and others, this paper puts forward a geometrical description of structural equilibrium which contains a procedure for the graphic analysis of stress resultants within general three-dimensional frames. The method is a natural generalization of Rankine’s reciprocal diagrams for three-dimensional trusses. The vertices and edges of dual abstract 4-polytopes are embedded within dual four-dimensional vector spaces, wherein the oriented area of generalized polygons give all six components (axial and shear forces with torsion and bending moments) of the stress resultants. The relevant quantities may be readily calculated using four-dimensional Clifford algebra. As well as giving access to frame analysis and design, the description resolves a number of long-standing problems with the incompleteness of Rankine’s description of three-dimensional trusses. Examples are given of how the procedure may be applied to structures of engineering interest, including an outline of a two-stage procedure for addressing the equilibrium of loaded gridshell rooves.

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CRASMAREANU, MIRCEA. "DIRAC STRUCTURES FROM LIE INTEGRABILITY." International Journal of Geometric Methods in Modern Physics 09, no. 04 (May 6, 2012): 1220005. http://dx.doi.org/10.1142/s0219887812200058.

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We prove that a pair (F = vector sub-bundle of TM, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable. Then a flat Ehresmann connection on a fiber bundle ξ yields two complementary, but not orthogonally, Dirac structures on the total space M of ξ. These Dirac structures are also Lagrangian sub-bundles with respect to the natural almost symplectic structure of the big tangent bundle of M. The tangent bundle in Riemannian geometry is discussed as particular case and the 3-dimensional Heisenberg space is illustrated as example. More generally, we study the Bianchi–Cartan–Vranceanu metrics and their Hopf bundles.

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WEICHERT, F., M. GASPAR, and M. WAGNER. "RADIAL-BASED SIGNAL-PROCESSING COMBINED WITH METHODS OF MACHINE LEARNING." International Journal of Pattern Recognition and Artificial Intelligence 27, no. 06 (September 2013): 1350018. http://dx.doi.org/10.1142/s0218001413500183.

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The present paper describes a novel approach to performing feature extraction and classification in possibly layered circular structures, as seen in two-dimensional cutting planes of three-dimensional tube-shaped objects. The algorithm can therefore be used to analyze histological specimens of blood vessels as well as intravascular ultrasound (IVUS) datasets. The approach uses a radial signal-based extraction of textural features in combination with methods of machine learning to integrate a priori domain knowledge. The algorithm in principle solves a two-dimensional classification problem that is reduced to parallel viable time series analysis. A multiscale approach hereby determines a feature vector for each analysis using either a Wavelet-transform (WT) or a S-transform (ST). The classification is done by methods of machine learning — here support vector machines. A modified marching squares algorithm extracts the polygonal segments for the two-dimensional classification. The accuracy is above 80% even in datasets with a considerable quantity of artifacts, while the mean accuracy is above 90%. The benefit of the approach therefore mainly lies in its robustness, efficient calculation, and the integration of domain knowledge.

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Abramczyk, Jacek. "Transformed Shell Structures Determined by Regular Networks as a Complex Material for Roofing." Materials 14, no. 13 (June 26, 2021): 3582. http://dx.doi.org/10.3390/ma14133582.

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The article presents a comprehensive extension of the proprietary basic method for shaping innovative systems of corrugated shell roof structures by means of a specific complex material that comprises regular transformable shell units limited by spatial quadrangles. The units are made up of nominally plane folded sheets transformed into shell shapes. The similar shell units are regularly and effectively arranged in the three-dimensional space in an orderly manner with a universal regular reference surface, polyhedral network, and polygonal network. The extended method leads to the increase in the variety of the designed complex shell roof forms and plane-walled elevation forms of buildings. For this purpose, the rules governing the creation of the continuous roof shell structures of many shells arranged in different unconventional visually attractive patterns and their discontinuous regular modifications are sought. To obtain several novel groups of similar unconventional parametric roof forms, single division coefficients and double division coefficients are used. The easy and intuitive modifications of the positions of the vertices belonging to the polygonal network on the side edges of the polyhedral network accomplished by means of a parametric algorithm allow one to adjust the geometry of the complete shell units to the geometric and material constraints related to the orthotropic properties of the transformed sheeting by means of these coefficients. The innovative approach to the shaping of the diverse unconventional roof structures requires the solving of many interdisciplinary problems in the field of mathematics, civil engineering, construction, morphology, architecture, mechanics, computer visualization, and programming.

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43

Yazi, M. "Ordered Structures and Projections." International Journal of Mathematics and Mathematical Sciences 2008 (2008): 1–6. http://dx.doi.org/10.1155/2008/783041.

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We associate a covering relation to the usual order relation defined in the set of all idempotent endomorphisms (projections) of a finite-dimensional vector space. A characterization is given of it. This characterization makes this order an order verifying the Jordan-Dedekind chain condition. We give also a property for certain finite families of this order. More precisely, the family of parts intervening in the linear representation of diagonalizable endomorphism, that is, the orthogonal families forming a decomposition of the identity endomorphism.

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Srivastava, Meenakshi, S. K. Singh, and S. Q. Abbas. "Applying metric space and pivot-based indexing on combined features of bio-images for fast execution of composite queries." International Journal of Engineering & Technology 7, no. 1 (January 29, 2018): 110. http://dx.doi.org/10.14419/ijet.v7i1.9009.

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Content based recovery of bio images requires index structures, which can retrieve the similar image objects in time proficient way. Conventional Structure/ sequence based recovery of bio-images (for example, protein structures) experiences, tedious online similarity check from huge web based databases. The general approach of image feature representations follows vector based portrayal. In present manuscript, visual highlights of 3D protein structures and their content highlights have been implemented in isolated metric space, rather than vector space which advances the similarity recovery. At long last, the Visual highlights and Content based highlights are consolidated in one metric space, through the component results of highlight and substance metric. Results have demonstrated that pivot based ordering/ indexing on Combined Index Metric can undoubtedly execute composite content construct queries with respect to bio images in time effective way.

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Miltiadou, Milto, Neill D. F. Campbell, Darren Cosker, and Michael G. Grant. "A Comparative Study about Data Structures Used for Efficient Management of Voxelised Full-Waveform Airborne LiDAR Data during 3D Polygonal Model Creation." Remote Sensing 13, no. 4 (February 4, 2021): 559. http://dx.doi.org/10.3390/rs13040559.

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In this paper, we investigate the performance of six data structures for managing voxelised full-waveform airborne LiDAR data during 3D polygonal model creation. While full-waveform LiDAR data has been available for over a decade, extraction of peak points is the most widely used approach of interpreting them. The increased information stored within the waveform data makes interpretation and handling difficult. It is, therefore, important to research which data structures are more appropriate for storing and interpreting the data. In this paper, we investigate the performance of six data structures while voxelising and interpreting full-waveform LiDAR data for 3D polygonal model creation. The data structures are tested in terms of time efficiency and memory consumption during run-time and are the following: (1) 1D-Array that guarantees coherent memory allocation, (2) Voxel Hashing, which uses a hash table for storing the intensity values (3) Octree (4) Integral Volumes that allows finding the sum of any cuboid area in constant time, (5) Octree Max/Min, which is an upgraded octree and (6) Integral Octree, which is proposed here and it is an attempt to combine the benefits of octrees and Integral Volumes. In this paper, it is shown that Integral Volumes is the more time efficient data structure but it requires the most memory allocation. Furthermore, 1D-Array and Integral Volumes require the allocation of coherent space in memory including the empty voxels, while Voxel Hashing and the octree related data structures do not require to allocate memory for empty voxels. These data structures, therefore, and as shown in the test conducted, allocate less memory. To sum up, there is a need to investigate how the LiDAR data are stored in memory. Each tested data structure has different benefits and downsides; therefore, each application should be examined individually.

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Hrushovski, Ehud, and James Loveys. "Strongly and co-strongly minimal abelian structures." Journal of Symbolic Logic 75, no. 2 (June 2010): 442–58. http://dx.doi.org/10.2178/jsl/1268917489.

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AbstractWe give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems:1. when the only finite 0-definable subgroup is {0}, or equivalently 0 is the only algebraic element (the co-strongly minimal case);2. when the theory of the structure is strongly minimal.In the first case, we identify the abelian structure as a “near-subspace” A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to acl(∅)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above, with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d ∈ D. the index of A ∩ dA, in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.

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Ohashi, Tadashi, Hajime Nobuhara, and Kaoru Hirota. "Hierarchical Concept Structures Based Data Retrieval/Mining by Fuzzy Document Ordering System." Journal of Advanced Computational Intelligence and Intelligent Informatics 8, no. 6 (November 20, 2004): 633–38. http://dx.doi.org/10.20965/jaciii.2004.p0633.

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A shrink operation for FOCUS (Fuzzy dOCUment ordering System) is proposed as a tool for changing user’s viewpoint in concept structures of documents. The FOCUS produces the user’s preference vector space and its concept hierarchical structure based on CSD (Concept System Dictionary) produced by EDR (Japan Electronic Dictionary Research Institute). Through document ordering experiments by 5 examinees using 20 given documents and 40 unknown ones extracted from IMDB (Internet Movie Data Base), the effectiveness of shrink operations, i.e., shrinkage of user’s preference vector space, is confirmed in the way that it avails to do fuzzy document matching in user’s rough point of view by using the hierarchical structure.

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Graben, Peter beim, Markus Huber, Werner Meyer, Ronald Römer, and Matthias Wolff. "Vector Symbolic Architectures for Context-Free Grammars." Cognitive Computation 14, no. 2 (December 24, 2021): 733–48. http://dx.doi.org/10.1007/s12559-021-09974-y.

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AbstractVector symbolic architectures (VSA) are a viable approach for the hyperdimensional representation of symbolic data, such as documents, syntactic structures, or semantic frames. We present a rigorous mathematical framework for the representation of phrase structure trees and parse trees of context-free grammars (CFG) in Fock space, i.e. infinite-dimensional Hilbert space as being used in quantum field theory. We define a novel normal form for CFG by means of term algebras. Using a recently developed software toolbox, called FockBox, we construct Fock space representations for the trees built up by a CFG left-corner (LC) parser. We prove a universal representation theorem for CFG term algebras in Fock space and illustrate our findings through a low-dimensional principal component projection of the LC parser state. Our approach could leverage the development of VSA for explainable artificial intelligence (XAI) by means of hyperdimensional deep neural computation.

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Catanese, Fabrizio, and Michael Dettweiler. "Vector bundles on curves coming from variation of Hodge structures." International Journal of Mathematics 27, no. 07 (June 2016): 1640001. http://dx.doi.org/10.1142/s0129167x16400012.

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Fujita’s second theorem for Kähler fibre spaces over a curve asserts, that the direct image [Formula: see text] of the relative dualizing sheaf splits as the direct sum [Formula: see text], where [Formula: see text] is ample and [Formula: see text] is unitary flat. We focus on our negative answer [F. Catanese and M. Dettweiler, Answer to a question by Fujita on variation of Hodge structures, to appear in Adv. Stud. Pure Math.] to a question by Fujita: is [Formula: see text] semiample? We give here an infinite series of counterexamples using hypergeometric integrals and we give a simple argument to show that the monodromy representation is infinite. Our counterexamples are surfaces of general type with positive index, explicitly given as abelian coverings with group [Formula: see text] of a Del Pezzo surface [Formula: see text] of degree 5 (branched on the union of the lines of [Formula: see text], which form a bianticanonical divisor), and endowed with a semistable fibration with only three singular fibres. The simplest such surfaces are the three ball quotients considered in [I. C. Bauer and F. Catanese, A volume maximizing canonical surface in 3-space, Comment. Math. Helv. 83(1) (2008) 387–406.], fibred over a curve of genus 2, and with fibres of genus 4. These examples are a larger class than the ones corresponding to Shimura curves in the moduli space of Abelian varieties.

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Stoica, Ovidiu Cristinel. "No-go results on emergent space and other structures." Journal of Physics: Conference Series 2533, no. 1 (June 1, 2023): 012027. http://dx.doi.org/10.1088/1742-6596/2533/1/012027.

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Abstract It is often claimed that, from the abstract quantum state vector and Hamiltonian, everything emerges uniquely (for example the 3D-space, a preferred factorization of the Hilbert space, and a preferred basis). This thesis, called Hilbert-Space Fundamentalism (HSF), is assumed for example in Carroll and Singh’s “Mad-dog Everettianism”, some proposals based on decoherence, proposals that aim to describe everything by the quantum structure alone, and proposals that spacetime emerges from a purely quantum theory of gravity. I show that whenever the 3D-space, a preferred factorization, or any other structure emerges like this, there are infinitely many solutions. This refutes Hilbert-Space Fundamentalism. Implications for various research programs aiming to show that certain structures emerge uniquely, and for various interpretations of Quantum Mechanics, are discussed.

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Journal articles on the topic 'Polygonal space vector structures'

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