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Journal articles on the topic 'Spectral polytopes'

Author: Grafiati

Published: 30 June 2021

Last updated: 2 February 2022

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1

Greenfeld, Rachel, and Nir Lev. "Fuglede’s spectral set conjecture for convex polytopes." Analysis & PDE 10, no. 6 (July 14, 2017): 1497–538. http://dx.doi.org/10.2140/apde.2017.10.1497.

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2

KIRAT, IBRAHIM, and ILKER KOCYIGIT. "REMARKS ON SELF-AFFINE FRACTALS WITH POLYTOPE CONVEX HULLS." Fractals 18, no. 04 (December 2010): 483–98. http://dx.doi.org/10.1142/s0218348x1000510x.

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Suppose that the set [Formula: see text] of real n × n matrices has joint spectral radius less than 1. Then for any digit set D = {d1, …, dq} ⊂ ℝn, there exists a unique non-empty compact set [Formula: see text] satisfying [Formula: see text], which is typically a fractal set. We use the infinite digit expansions of the points of F to give simple necessary and sufficient conditions for the convex hull of F to be a polytope. Additionally, we present a technique to determine the vertices of such polytopes. These answer some of the related questions of Strichartz and Wang, and also enable us to approximate the Lebesgue measure of such self-affine sets. To show the use of our results, we also give several examples including the Levy dragon and the Heighway dragon.

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3

Saldanha, Nicolau C., and Carlos Tomei. "Spectra of regular polytopes." Discrete & Computational Geometry 7, no. 4 (April 1992): 403–14. http://dx.doi.org/10.1007/bf02187851.

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4

Saldanha, Nicolau C., and Carlos Tomei. "Spectra of semi-regular polytopes." Boletim da Sociedade Brasileira de Matem�tica 29, no. 1 (March 1998): 25–51. http://dx.doi.org/10.1007/bf01245867.

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5

Douai, Antoine. "Global spectra, polytopes and stacky invariants." Mathematische Zeitschrift 288, no. 3-4 (August 22, 2017): 889–913. http://dx.doi.org/10.1007/s00209-017-1918-8.

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6

Jungers, Raphaël M., Antonio Cicone, and Nicola Guglielmi. "Lifted Polytope Methods for Computing the Joint Spectral Radius." SIAM Journal on Matrix Analysis and Applications 35, no. 2 (January 2014): 391–410. http://dx.doi.org/10.1137/130907811.

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7

Paluri, Nataraj S. V., and Jayesh J. Barve. "A Reliable Algorithm to Compute the Spectral Set of a Polytope of Polynomials to Prescribed Accuracy." Journal of Dynamic Systems, Measurement, and Control 128, no. 2 (March 28, 2005): 400–405. http://dx.doi.org/10.1115/1.2196416.

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We propose an algorithm to compute the spectral set of a polytope of polynomials. The proposed algorithm offers several key guarantees that are not available with existing techniques. It guarantees that the generated spectral set: (i) contains all the actual points, (ii) is computed to a prescribed accuracy, (iii) is computed reliably in face of all kinds of computational errors, and (iv) is computed in a finite number of algorithmic iterations. A further merit is that the computational complexity of the proposed algorithm is O(n) in contrast to O(n2) for existing techniques, where n is the degree of the polynomial. The algorithm is demonstrated on a few examples.

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8

Cerone, V. "A fast technique for the generation of the spectral set of a polytope of polynomials." Automatica 33, no. 2 (February 1997): 277–80. http://dx.doi.org/10.1016/s0005-1098(96)00196-3.

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9

Khatsymovsky, V. M. "Spectrum of area in the Faddeev formulation of gravity." Modern Physics Letters A 31, no. 19 (June 21, 2016): 1650114. http://dx.doi.org/10.1142/s0217732316501145.

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Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual general relativity (GR). Earlier, we have proposed first-order representation of the minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like four-simplices or, say, cuboids into which [Formula: see text] can be decomposed, an analogue of the Cartan–Weyl connection-type form of the Hilbert–Einstein action in the usual continuum GR. In the Hamiltonian formalism, the tetrad bilinears are canonically conjugate to the orthogonal connection matrices. We evaluate the spectrum of the elementary areas, functions of the tetrad bilinears. The spectrum is discrete and proportional to the Faddeev analog [Formula: see text] of the Barbero–Immirzi parameter [Formula: see text]. The possibility of the tetrad and metric discontinuities in the Faddeev gravity allows to consider any surface as consisting of a set of virtually independent elementary areas and its spectrum being the sum of the elementary spectra. Requiring consistency of the black hole entropy calculations known in the literature we are able to estimate [Formula: see text].

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10

Banigan, James R., Anindita Gayen, and Nathaniel J. Traaseth. "Correlating lipid bilayer fluidity with sensitivity and resolution of polytopic membrane protein spectra by solid-state NMR spectroscopy." Biochimica et Biophysica Acta (BBA) - Biomembranes 1848, no. 1 (January 2015): 334–41. http://dx.doi.org/10.1016/j.bbamem.2014.05.003.

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11

Wunsch, Patrick, and Walter G. Zumft. "Functional Domains of NosR, a Novel Transmembrane Iron-Sulfur Flavoprotein Necessary for Nitrous Oxide Respiration." Journal of Bacteriology 187, no. 6 (March 15, 2005): 1992–2001. http://dx.doi.org/10.1128/jb.187.6.1992-2001.2005.

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ABSTRACT Bacterial nitrous oxide (N2O) respiration depends on the polytopic membrane protein NosR for the expression of N2O reductase from the nosZ gene. We constructed His-tagged NosR and purified it from detergent-solubilized membranes of Pseudomonas stutzeri ATCC 14405. NosR is an iron-sulfur flavoprotein with redox centers positioned at opposite sides of the cytoplasmic membrane. The flavin cofactor is presumably bound covalently to an invariant threonine residue of the periplasmic domain. NosR also features conserved CX3CP motifs, located C-terminally of the transmembrane helices TM4 and TM6. We genetically manipulated nosR with respect to these different domains and putative functional centers and expressed recombinant derivatives in a nosR null mutant, MK418nosR::Tn5. NosR's function was studied by its effects on N2O respiration, NosZ synthesis, and the properties of purified NosZ proteins. Although all recombinant NosR proteins allowed the synthesis of NosZ, a loss of N2O respiration was observed upon deletion of most of the periplasmic domain or of the C-terminal parts beyond TM2 or upon modification of the cysteine residues in a highly conserved motif, CGWLCP, following TM4. Nonetheless, NosZ purified from the recombinant NosR background exhibited in vitro catalytic activity. Certain NosR derivatives caused an increase in NosZ of the spectral contribution from a modified catalytic Cu site. In addition to its role in nosZ expression, NosR supports in vivo N2O respiration. We also discuss its putative functions in electron donation and redox activation.

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12

Stadler, Adrian-Mihail, Fausto Puntoriero, Francesco Nastasi, Sebastiano Campagna, and Jean-Marie Lehn. "RuIIMultinuclear Metallosupramolecular Rack-Type Architectures of Polytopic Hydrazone-Based Ligands: Synthesis, Structural Features, Absorption Spectra, Redox Behavior, and Near-Infrared Luminescence." Chemistry - A European Journal 16, no. 19 (April 8, 2010): 5645–60. http://dx.doi.org/10.1002/chem.200900632.

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13

Loiseau, Frédérique, Cinzia Di Pietro, Scolastica Serroni, Sebastiano Campagna, Antonino Licciardello, Amedea Manfredi, Gianluca Pozzi, and Silvio Quici. "A New Polytopic Bis-diazacrown-ether-polypyridine Ligand and Its Complexes with Zn(II) Salts and Mononuclear and Dendritic Ru(II) Precursors. Synthesis, Absorption Spectra, Redox Behavior, and Luminescence Properties." Inorganic Chemistry 40, no. 27 (December 2001): 6901–9. http://dx.doi.org/10.1021/ic010338+.

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14

Lev, Nir, and Bochen Liu. "Spectrality of Polytopes and Equidecomposability by Translations." International Mathematics Research Notices, October 31, 2019. http://dx.doi.org/10.1093/imrn/rnz191.

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Abstract Let $A$ be a polytope in ${\mathbb{R}}^d$ (not necessarily convex or connected). We say that $A$ is spectral if the space $L^2(A)$ has an orthogonal basis consisting of exponential functions. A result due to Kolountzakis and Papadimitrakis (2002) asserts that if $A$ is a spectral polytope, then the total area of the $(d-1)$-dimensional faces of $A$ on which the outward normal is pointing at a given direction, must coincide with the total area of those $(d-1)$-dimensional faces on which the outward normal is pointing at the opposite direction. In this paper, we prove an extension of this result to faces of all dimensions between $1$ and $d-1$. As a consequence we obtain that any spectral polytope $A$ can be dissected into a finite number of smaller polytopes, which can be rearranged using translations to form a cube.

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15

Wei, Xiaoqi, and Guo-Wei Wei. "Homotopy continuation for the spectra of persistent Laplacians." Foundations of Data Science, 2021, 0. http://dx.doi.org/10.3934/fods.2021017.

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<p style='text-indent:20px;'>The <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-persistent <inline-formula><tex-math id="M2">\begin{document}$ q $\end{document}</tex-math></inline-formula>-combinatorial Laplacian defined for a pair of simplicial complexes is a generalization of the <inline-formula><tex-math id="M3">\begin{document}$ q $\end{document}</tex-math></inline-formula>-combinatorial Laplacian. Given a filtration, the spectra of persistent combinatorial Laplacians not only recover the persistent Betti numbers of persistent homology but also provide extra multiscale geometrical information of the data. Paired with machine learning algorithms, the persistent Laplacian has many potential applications in data science. Seeking different ways to find the spectrum of an operator is an active research topic, becoming interesting when ideas are originated from multiple fields. In this work, we explore an alternative approach for the spectrum of persistent Laplacians. As the eigenvalues of a persistent Laplacian matrix are the roots of its characteristic polynomial, one may attempt to find the roots of the characteristic polynomial by homotopy continuation, and thus resolving the spectrum of the corresponding persistent Laplacian. We consider a set of simple polytopes and small molecules to prove the principle that algebraic topology, combinatorial graph, and algebraic geometry can be integrated to understand the shape of data.</p>

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16

Oehlmann, Paul-Konstantin, and Thorsten Schimannek. "GV-spectroscopy for F-theory on genus-one fibrations." Journal of High Energy Physics 2020, no. 9 (September 2020). http://dx.doi.org/10.1007/jhep09(2020)066.

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Abstract We present a novel technique to obtain base independent expressions for the matter loci of fibrations of complete intersection Calabi-Yau onefolds in toric ambient spaces. These can be used to systematically construct elliptically and genus one fibered Calabi-Yau d-folds that lead to desired gauge groups and spectra in F-theory. The technique, which we refer to as GV-spectroscopy, is based on the calculation of fiber Gopakumar-Vafa invariants using the Batyrev-Borisov construction of mirror pairs and application of the so-called Frobenius method to the data of a parametrized auxiliary polytope. In particular for fibers that generically lead to multiple sections, only multi-sections or that are complete intersections in higher codimension, our technique is vastly more efficient than classical approaches. As an application we study two Higgs chains of six-dimensional supergravities that are engineered by fibrations of codimension two complete intersection fibers. Both chains end on a vacuum with G = ℤ4 that is engineered by fibrations of bi-quadrics in ℙ3. We use the detailed knowledge of the structure of the reducible fibers that we obtain from GV-spectroscopy to comment on the corresponding Tate-Shafarevich group. We also show that for all fibers the six-dimensional supergravity anomalies including the discrete anomalies generically cancel.

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