On the orthogonal projections of Dirichlet–Voronoi cells of lattices

2011 ◽  
Vol 52 (2) ◽  
pp. 487-493
Author(s):  
A. Végh
Keyword(s):  
Orthogonal Projections ◽  
Voronoi Cells

scholarly journals Further steps on the reconstruction of convex polyominoes from orthogonal projections

2021 ◽  
Author(s):  
Paolo Dulio ◽  
Andrea Frosini ◽  
Simone Rinaldi ◽  
Lama Tarsissi ◽  
Laurent Vuillon
Keyword(s):  
Discrete Geometry ◽  
Convex Subset ◽  
Convex Sets ◽  
A Priori ◽  
Orthogonal Projections ◽  
Reconstruction Process ◽  
Combinatorial Properties ◽  
The Family ◽  
Discrete Counterpart ◽  
Interior Part

AbstractA remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean convex sets, and combine the constraints of convexity and connectedness. In this paper we study the problem of their reconstruction from orthogonal projections, relying on the approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of its contour, without losing its convexity. To reach this goal we consider convexity in terms of certain combinatorial properties of the boundary word encoding the polyomino. So, we first show some conditions that allow us to extend the kernel maintaining the convexity. Then, we provide examples where the addition of one or two points causes a loss of convexity, which can be restored by adding other points, whose number and positions cannot be determined a priori.

On farthest Bregman Voronoi cells

Optimization ◽  
2021 ◽  
pp. 1-11
Author(s):  
Juan Enrique Martínez-Legaz ◽  
Maryam Tamadoni Jahromi ◽  
Eskandar Naraghirad
Keyword(s):  
Voronoi Cells

scholarly journals Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization

2021 ◽  
Vol 51 (3) ◽  
Author(s):  
Maurice A. de Gosson
Keyword(s):  
Uncertainty Principle ◽  
Convex Geometry ◽  
Geometric Approach ◽  
Point Of View ◽  
Reconstruction Problem ◽  
Orthogonal Projections ◽  
Topological Version ◽  
Mahler Conjecture ◽  
Quantum Indeterminacy ◽  
Dual Quantum

AbstractWe define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke–Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho–Stark principle from the point of view of quantum polarity.

scholarly journals Nuclear Magnetic Resonance Metabolomics with Double Pulsed-Field-Gradient Echo and Automatized Solvent Suppression Spectroscopy for Multivariate Data Matrix Applied in Novel Wine and Juice Discriminant Analysis

Molecules ◽  
2021 ◽  
Vol 26 (14) ◽  
pp. 4146
Author(s):  
José Enrique Herbert-Pucheta ◽  
José Daniel Lozada-Ramírez ◽  
Ana E. Ortega-Regules ◽  
Luis Ricardo Hernández ◽  
Cecilia Anaya de Parrodi
Keyword(s):  
Statistical Analysis ◽  
Discriminant Analysis ◽  
Field Gradient ◽  
Multivariate Statistical Analysis ◽  
Data Matrix ◽  
Cabernet Sauvignon ◽  
Gradient Echo ◽  
Multivariate Statistical ◽  
Orthogonal Projections ◽  
Pulsed Field

The quality of foods has led researchers to use various analytical methods to determine the amounts of principal food constituents; some of them are the NMR techniques with a multivariate statistical analysis (NMR-MSA). The present work introduces a set of NMR-MSA novelties. First, the use of a double pulsed-field-gradient echo (DPFGE) experiment with a refocusing band-selective uniform response pure-phase selective pulse for the selective excitation of a 5–10-ppm range of wine samples reveals novel broad 1H resonances. Second, an NMR-MSA foodomics approach to discriminate between wine samples produced from the same Cabernet Sauvignon variety fermented with different yeast strains proposed for large-scale alcohol reductions. Third a comparative study between a nonsupervised Principal Component Analysis (PCA), supervised standard partial (PLS-DA), and sparse (sPLS-DA) least squares discriminant analysis, as well as orthogonal projections to a latent structures discriminant analysis (OPLS-DA), for obtaining holistic fingerprints. The MSA discriminated between different Cabernet Sauvignon fermentation schemes and juice varieties (apple, apricot, and orange) or juice authentications (puree, nectar, concentrated, and commercial juice fruit drinks). The new pulse sequence DPFGE demonstrated an enhanced sensitivity in the aromatic zone of wine samples, allowing a better application of different unsupervised and supervised multivariate statistical analysis approaches.

scholarly journals AB0185 PROSPECTIVE PROFILE OF URINE METABOLOME IN RHEUMATOID ARTHRITIS

2020 ◽  
Vol 79 (Suppl 1) ◽  
pp. 1392.2-1392
Author(s):  
M. De Oliveira ◽  
P. V. Alabarse ◽  
M. Farinon ◽  
R. Cavalheiro Do Espírito Santo ◽  
R. Xavier
Keyword(s):  
Rheumatoid Arthritis ◽  
Skeletal Muscle ◽  
Disease Activity ◽  
Pearson Correlation ◽  
Statistical Significance ◽  
Resonance Spectroscopy ◽  
Metabolome Analysis ◽  
Orthogonal Projections ◽  
Urine Metabolome

Background:Rheumatoid arthritis (RA) is a chronic inflammatory disease characterized by increased mortality and associated with metabolic disorders. Since the metabolomic profile is known to vary in response to different inflammatory conditions, metabolome analysis could substantially improve diagnosis and prognosis of RA.Objectives:To analyze the urine metabolome profile in RA patients and correlate it with disease activity changes over 12 monthsMethods:Seventy-nine RA patients, according to ACR/EULAR 2010 classification criteria, between 40 and 70 years old, were recruited and followed for 12 months. Metabolome analysis was performed by Nuclear Magnetic Resonance spectroscopy (NMR), resulting in the identification of 93 metabolites in urine collected at the baseline and after 12 months. Frequency analysis, Pearson Correlation and Multivariate data analysis with orthogonal projections to latent structures (OPLS) method were performed and a statistical significance was considered as p<0.05.Results:The study population was characterized by the majority of women (86.7%), mean age of 56 years old, around 80% with positive anti-CCP or Rheumatoid Factor. During the one year of follow-up, there was no substantial variation in the DAS28 measurement (baseline: 3.8, after 12 months: 4.0). There was no significant correlation between the metabolome pattern and DAS28 score (p>0.05) over time. However, multivariate analysis (OPLS-DA) demonstrated an adequate differentiation of the population with 0.92 of accuracy (Q2: 0.72 and R2: 0.89).There was a significant increase of L-cysteine, choline, L-Phenylalanin, creatine, L-histidine, oxalacetic acid and xanthine, and a decrease of L-threonine, taurine, butyric and gluconic acid (p<0.05) during the follow-up, metabolites that are involved in the skeletal muscle metabolism.Conclusion:The observed biomarkers indicate,as expected, that the RA metabolic profile is associated with inflammation injury and skeletal muscle amino acid metabolism. Correlations with disease activity changes was compromised by the stable disease status during the 12 months. More studies evaluating correlations with skeletal muscle function and mass are underway.Acknowledgments:Disclosure of interest: Marianne Oliveira: None declared, Rafaela Santo: None declared, Mirian Farinon: None declared, Ricardo Xavier Consultant of: Abbvie, Pfizer, Novartis, Janssen, Lilly, RocheDisclosure of Interests:Marianne de Oliveira: None declared, Paulo Vinicius Alabarse: None declared, Mirian Farinon: None declared, Rafaela Cavalheiro do Espírito Santo: None declared, Ricardo Xavier Consultant of: AbbVie, Pfizer, Novartis, Janssen, Eli Lilly, Roche

scholarly journals Tetrahedra of varying density and their applications

2021 ◽  
Author(s):  
Dennis R. Bukenberger ◽  
Hendrik P. A. Lensch
Keyword(s):  
Tetrahedral Mesh ◽  
Rotational Inertia ◽  
Exact Mass ◽  
Accurate Analysis ◽  
Exact Integration ◽  
Voronoi Cells ◽  
Current State ◽  
Chebyshev Norm ◽  
Exact Calculations ◽  
Vector Matrix

Abstract We propose concepts to utilize basic mathematical principles for computing the exact mass properties of objects with varying densities. For objects given as 3D triangle meshes, the method is analytically accurate and at the same time faster than any established approximation method. Our concept is based on tetrahedra as underlying primitives, which allows for the object’s actual mesh surface to be incorporated in the computation. The density within a tetrahedron is allowed to vary linearly, i.e., arbitrary density fields can be approximated by specifying the density at all vertices of a tetrahedral mesh. Involved integrals are formulated in closed form and can be evaluated by simple, easily parallelized, vector-matrix multiplications. The ability to compute exact masses and centroids for objects of varying density enables novel or more exact solutions to several interesting problems: besides the accurate analysis of objects under given density fields, this includes the synthesis of parameterized density functions for the make-it-stand challenge or manufacturing of objects with controlled rotational inertia. In addition, based on the tetrahedralization of Voronoi cells we introduce a precise method to solve $$L_{2|\infty }$$ L 2 | ∞ Lloyd relaxations by exact integration of the Chebyshev norm. In the context of additive manufacturing research, objects of varying density are a prominent topic. However, current state-of-the-art algorithms are still based on voxelizations, which produce rather crude approximations of masses and mass centers of 3D objects. Many existing frameworks will benefit by replacing approximations with fast and exact calculations. Graphic abstract

Markov paths on the Poisson-Delaunay graph with applications to routeing in mobile networks

2000 ◽  
Vol 32 (01) ◽  
pp. 1-18 ◽  
Author(s):  
F. Baccelli ◽  
K. Tchoumatchenko ◽  
S. Zuyev
Keyword(s):  
Communication Networks ◽  
Mobile Networks ◽  
Path Length ◽  
Euclidean Distance ◽  
Voronoi Tessellation ◽  
Voronoi Cells ◽  
Ergodic Properties ◽  
Potential Applications ◽  
The Mean ◽  
Mobile Communication Networks

Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path. We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

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