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Journal articles on the topic 'Invariant cone'

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Author: Grafiati

Published: 11 March 2023

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1

Kasigwa, Michael, and Michael Tsatsomeros. "Eventual Cone Invariance." Electronic Journal of Linear Algebra 32 (February 6, 2017): 204–16. http://dx.doi.org/10.13001/1081-3810.3484.

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Eventually nonnegative matrices are square matrices whose powers become and remain (entrywise) nonnegative. Using classical Perron-Frobenius theory for cone preserving maps, this notion is generalized to matrices whose powers eventually leave a proper cone K â R^n invariant, that is, A^mK â K for all sufficiently large m. Also studied are the related notions of eventual cone invariance by the matrix exponential, as well as other generalizations of M-matrix and dynamical system notions.

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2

Parrilo, P. A., and S. Khatri. "On cone-invariant linear matrix inequalities." IEEE Transactions on Automatic Control 45, no. 8 (2000): 1558–63. http://dx.doi.org/10.1109/9.871772.

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3

Abbas, Mujahid, and Pasquale Vetro. "Invariant approximation results in‎ ‎cone metric spaces." Annals of Functional Analysis 2, no. 2 (2011): 101–13. http://dx.doi.org/10.15352/afa/1399900199.

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4

Westland, Stephen, and Caterina Ripamonti. "Invariant cone-excitation ratios may predict transparency." Journal of the Optical Society of America A 17, no. 2 (February 1, 2000): 255. http://dx.doi.org/10.1364/josaa.17.000255.

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5

Chodos, Alan. "Tachyons as a Consequence of Light-Cone Reflection Symmetry." Symmetry 14, no. 9 (September 19, 2022): 1947. http://dx.doi.org/10.3390/sym14091947.

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We introduce a new symmetry, light-cone reflection (LCR), which interchanges timelike and spacelike intervals. Our motivation is to provide a reason, based on symmetry, why tachyons might exist, with emphasis on application to neutrinos. We show that LCR, combined with translations, leads to a much larger symmetry. We construct an LCR-invariant Lagrangian and discuss some of its properties. In a simple example, we find complete symmetry in the spectrum between tachyons and ordinary particles. We also show that the theory allows for the introduction of a further gauge invariance related to chiral symmetry.

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6

Malesza, Wiktor, and Witold Respondek. "Linear cone-invariant control systems and their equivalence." International Journal of Control 91, no. 8 (June 21, 2017): 1818–34. http://dx.doi.org/10.1080/00207179.2017.1333153.

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7

Hisabia, Aritra Narayan, and Manideepa Saha. "On Properties of Semipositive Cones and Simplicial Cones." Electronic Journal of Linear Algebra 36, no. 36 (December 3, 2020): 764–72. http://dx.doi.org/10.13001/ela.2020.5553.

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For a given nonsingular $n\times n$ matrix $A$, the cone $S_{A}=\{x:Ax\geq 0\}$ , and its subcone $K_A$ lying on the positive orthant, called as semipositive cone, are considered. If the interior of the semipositive cone $K_A$ is not empty, then $A$ is named as semipositive matrix. It is known that $K_A$ is a proper polyhedral cone. In this paper, it is proved that $S_{A}$ is a simplicial cone and properties of its extremals are analyzed. An one-one relation between simplicial cones and invertible matrices is established. For a proper cone $K$ in $\mathbb{R}^n$, $\pi(K)$ denotes the collection of $n\times n$ matrices that leave $K$ invariant. For a given minimally semipositive matrix (no column-deleted submatrix is semipositive) $A$, it is shown that the invariant cone $\pi(K_A)$ is a simplicial cone.

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8

BRISUDOVA, MARTINA. "SMALL x DIVERGENCES IN THE SIMILARITY RG APPROACH TO LF QCD." Modern Physics Letters A 17, no. 02 (January 20, 2002): 59–81. http://dx.doi.org/10.1142/s0217732302006308.

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We study small x divergences in boost invariant similarity renormalization group approach to light-front QCD in a heavy quark–antiquark state. With the boost invariance maintained, the infrared divergences do not cancel out in the physical states, contrary to previous studies where boost invariance was violated by a choice of a renormalization scale. This may be an indication that the zero mode, or nontrivial light-cone vacuum structure, might be important for recovering full Lorentz invariance.

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9

Kosheleva, Olga, and Vladik Kreinovich. "ON GEOMETRY OF FINSLER CAUSALITY: FOR CONVEX CONES, THERE IS NO AFFINE-INVARIANT LINEAR ORDER (SIMILAR TO COMPARING VOLUMES)." Mathematical Structures and Modeling, no. 1 (May 30, 2020): 49–55. http://dx.doi.org/10.24147/2222-8772.2020.1.49-55.

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Some physicists suggest that to more adequately describe the causal structure of space-time, it is necessary to go beyond the usual pseudoRiemannian causality, to a more general Finsler causality. In this general case, the set of all the events which can be influenced by a given event is, locally, a generic convex cone, and not necessarily a pseudo-Reimannian-style quadratic cone. Since all current observations support pseudo-Riemannian causality, Finsler causality cones should be close to quadratic ones. It is therefore desirable to approximate a general convex cone by a quadratic one. This can be done if we select a hyperplane, and approximate intersections of cones and this hyperplane. In the hyperplane, we need to approximate a convex body by an ellipsoid. This can be done in an affine-invariant way, e.g., by selecting, among all ellipsoids containing the body, the one with the smallest volume; since volume is affine-covariant, this selection is affine-invariant. However, this selection may depend on the choice of the hyperplane. It is therefore desirable to directly approximate the convex cone describing Finsler causality with the quadratic cone, ideally in an affine-invariant way. We prove, however, that on the set of convex cones, there is no affine-covariant characteristic like volume. So, any approximation is necessarily not affine-invariant.

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10

HATZINIKITAS, AGAPITOS, and IOANNIS SMYRNAKIS. "CLOSED BOSONIC STRING PARTITION FUNCTION IN TIME INDEPENDENT EXACT pp-WAVE BACKGROUND." International Journal of Modern Physics A 21, no. 05 (February 20, 2006): 995–1013. http://dx.doi.org/10.1142/s0217751x06025493.

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The modular invariance of the one-loop partition function of the closed bosonic string in four dimensions in the presence of certain homogeneous exact pp -wave backgrounds is studied. In the absence of an axion field, the partition function is found to be modular invariant and equal to the free field partition function. The partition function remains unchanged also in the presence of a fixed axion field. However, in this case, the covariant form of the action suggests summation over all possible twists generated by the axion field. This is shown to modify the partition function. In the light-cone gauge, the axion field generates twists only in the worldsheet σ-direction, so the resulting partition function is not modular invariant, hence wrong. To obtain the correct partition function one needs to sum over twists in the t-direction as well, as suggested by the covariant form of the action away from the light-cone gauge.

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11

Alexeev, Valery, Angela Gibney, and David Swinarski. "Higher-Level Conformal Blocks Divisors on." Proceedings of the Edinburgh Mathematical Society 57, no. 1 (January 16, 2014): 7–30. http://dx.doi.org/10.1017/s0013091513000941.

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AbstractWe study a family of semi-ample divisors on the moduli space of n-pointed genus 0 curves given by higher-level conformal blocks. We derive formulae for their intersections with a basis of 1-cycles, show that they form a basis for the Sn-invariant Picard group, and generate a full-dimensional subcone of the Sn-invariant nef cone. We find their position in the nef cone and study their associated morphisms.

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12

Legendre, Eveline. "Localizing the Donaldson–Futaki invariant." International Journal of Mathematics 32, no. 08 (June 22, 2021): 2150055. http://dx.doi.org/10.1142/s0129167x21500555.

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We use the equivariant localization formula to prove that the Donaldson–Futaki invariant of a compact smooth (Kähler) test configuration coincides with the Futaki invariant of the induced action on the central fiber when this fiber is smooth or have orbifold singularities. We also localize the Donaldson–Futaki invariant of the deformation to the normal cone.

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13

Protasov, V. Yu. "When do several linear operators share an invariant cone?" Linear Algebra and its Applications 433, no. 4 (October 2010): 781–89. http://dx.doi.org/10.1016/j.laa.2010.04.006.

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14

CRUZ, ANDERSON, GIOVANE FERREIRA, and PAULO VARANDAS. "Volume lemmas and large deviations for partially hyperbolic endomorphisms." Ergodic Theory and Dynamical Systems 41, no. 1 (September 24, 2019): 213–40. http://dx.doi.org/10.1017/etds.2019.63.

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We consider partially hyperbolic attractors for non-singular endomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We prove volume lemmas for both Lebesgue measure on the topological basin of the attractor and the SRB measure supported on the attractor. As a consequence, under a mild assumption we prove exponential large-deviation bounds for the convergence of Birkhoff averages associated to continuous observables with respect to the SRB measure.

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15

Nascimento, S. M. C., and D. H. Foster. "Misinterpreting Changes of Illuminant on Complex Mondrian Patterns." Perception 25, no. 1_suppl (August 1996): 111. http://dx.doi.org/10.1068/v96l0410.

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Ratios of cone excitations from different surfaces of the same coloured scene are almost invariant under illuminance changes, and might provide the cue by which the visual system discriminates illuminant from non-illuminant changes in coloured scenes. Previous work with pairs of surfaces showed that observers were able to detect small, naturally occurring, violations in these ratios (Nascimento and Foster, 1995 Perception24 Supplement, 60 – 61). In the present study, sensitivity to violations was assessed with more complex, Mondrian patterns. In a two-interval forced-choice experiment, two colour transformations of the same Mondrian pattern were compared by observers. The patterns comprised 7 × 7 coloured patches. Each patch was a simulation of a Munsell surface, and the whole pattern was illuminated by a Planckian illuminant of variable colour temperature. In one of the intervals only the colour temperature of the illuminant changed; in the other, the same colour-temperature change was made, but, in addition, the spectral reflectances of the surfaces were adjusted such that all cone ratios were exactly preserved for the three classes of cone. The task was to identify which of the intervals contained the pure illuminant change. Observers could reliably discriminate the intervals but systematically interpreted colour changes with invariant cone-ratios as being illuminant changes, with a probability that increased as the degree of violation of invariance increased. Performance depended mainly on long-wavelength-sensitive cones, less on medium-wavelength-sensitive cones, and little or not at all on short-wavelength-sensitive cones or luminance signals. Cone-excitation ratios, although sometimes unreliable, appear to be the dominant cue for deciding on the nature of colour changes.

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16

LOPUSHANSKY, OLEH, and SERGII SHARYN. "Operators commuting with multi-parameter shift semigroups." Carpathian Journal of Mathematics 30, no. 2 (2014): 217–24. http://dx.doi.org/10.37193/cjm.2014.02.07.

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Using operators of cross-correlation with ultradistributions supported by a positive cone, we describe a commutative algebra of shift-invariant continuous linear operators, commuting with contraction multi-parameter semigroups over a Banach space. Thereby, we generalize classic Schwartz’s and Hormander’s theorems on shift-invariant operators.

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17

Wünsche, Alfred. "Optic Axes and Elliptic Cone Equation in Coordinate-Invariant Treatment." Journal of Modern Physics 13, no. 06 (2022): 1001–43. http://dx.doi.org/10.4236/jmp.2022.136057.

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18

Tam, Bit-Shun, and Hans Schneider. "On the invariant faces associated with a cone-preserving map." Transactions of the American Mathematical Society 353, no. 1 (July 12, 2000): 209–45. http://dx.doi.org/10.1090/s0002-9947-00-02597-6.

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19

Jungers, Raphaël M. "On asymptotic properties of matrix semigroups with an invariant cone." Linear Algebra and its Applications 437, no. 5 (September 2012): 1205–14. http://dx.doi.org/10.1016/j.laa.2012.04.006.

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20

De Leenheer, Patrick. "Stability of diffusively coupled linear systems with an invariant cone." Linear Algebra and its Applications 580 (November 2019): 396–416. http://dx.doi.org/10.1016/j.laa.2019.06.024.

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21

Kook, W. "Edge-rooted forests and the α-invariant of cone graphs." Discrete Applied Mathematics 155, no. 8 (April 2007): 1071–75. http://dx.doi.org/10.1016/j.dam.2006.11.002.

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22

Dante DeBlassie, R. "The cone of positive harmonic functions for scale-invariant diffusions." Stochastics and Stochastic Reports 75, no. 4 (August 2003): 181–203. http://dx.doi.org/10.1080/1045112031000120649.

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23

Wilson, P. M. H. "Elliptic ruled surfaces on Calabi–Yau threefolds." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 1 (July 1992): 45–52. http://dx.doi.org/10.1017/s0305004100070742.

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In [5], we studied the behaviour of the Kähler cone of Calabi–Yau threefolds under deformations. We saw that the Kähler cone is locally constant in a smooth family of Calabi–Yau threefolds, unless some of the threefolds Xb contain elliptic ruled surfaces. Moreover, if X is a Calabi–Yau threefold containing an elliptic ruled surface, then the Kähler cone is not invariant under a generic small deformation.

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24

CRUZ, ANDERSON, and PAULO VARANDAS. "SRB measures for partially hyperbolic attractors of local diffeomorphisms." Ergodic Theory and Dynamical Systems 40, no. 6 (October 17, 2018): 1545–93. http://dx.doi.org/10.1017/etds.2018.115.

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We contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak$^{\ast }$ topology) and that their entropy varies continuously with respect to the local diffeomorphism.

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25

Fiedler, Leander, and Pieter Naaijkens. "Haag duality for Kitaev’s quantum double model for abelian groups." Reviews in Mathematical Physics 27, no. 09 (October 2015): 1550021. http://dx.doi.org/10.1142/s0129055x1550021x.

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We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.

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26

DEHGHANI, M. "A NEW PHYSICAL STATE FOR DE SITTER LINEAR GRAVITY." International Journal of Modern Physics A 26, no. 02 (January 20, 2011): 301–15. http://dx.doi.org/10.1142/s0217751x11051251.

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Based on conformal invariance and using Dirac's six-cone formalism, a new conformally invariant physical field equation for de Sitter (dS) linear gravity has been obtained, which corresponds to one of the unitary irreducible representations of the dS group and is denoted by [Formula: see text] in the sense of discrete series. Using ambient space notations, it has been shown that the solution to this new field equation can be written as the multiplication of a generalized symmetric polarization tensor of rank 2 and a massless conformally coupled scalar field in dS space–time. The physical tensor two-point function has been calculated in terms of the conformally coupled scalar two-point function in the ambient space formalism. It has been expressed in terms of dS intrinsic coordinates from its ambient space counterpart, which is dS-invariant and free of any divergences.

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27

Zhou, Jinchuan, and Jein-Shan Chen. "The Vector-Valued Functions Associated with Circular Cones." Abstract and Applied Analysis 2014 (2014): 1–21. http://dx.doi.org/10.1155/2014/603542.

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The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. LetLθdenote the circular cone inRn. For a functionffromRtoR, one can define a corresponding vector-valued functionfLθonRnby applyingfto the spectral values of the spectral decomposition ofx∈Rnwith respect toLθ. In this paper, we study properties that this vector-valued function inherits fromf, including Hölder continuity,B-subdifferentiability,ρ-order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.

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28

Achar, Pramod N., Anthony Henderson, and Benjamin F. Jones. "Normality of orbit closures in the enhanced nilpotent cone." Nagoya Mathematical Journal 203 (September 2011): 1–45. http://dx.doi.org/10.1215/00277630-1331854.

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AbstractWe continue the study of the closures of GL(V)-orbits in the enhanced nilpotent cone V × N begun by the first two authors. We prove that each closure is an invariant-theoretic quotient of a suitably defined enhanced quiver variety. We conjecture, and prove in special cases, that these enhanced quiver varieties are normal complete intersections, implying that the enhanced nilpotent orbit closures are also normal.

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29

Achar, Pramod N., Anthony Henderson, and Benjamin F. Jones. "Normality of orbit closures in the enhanced nilpotent cone." Nagoya Mathematical Journal 203 (September 2011): 1–45. http://dx.doi.org/10.1017/s0027763000010308.

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AbstractWe continue the study of the closures of GL(V)-orbits in the enhanced nilpotent coneV × Nbegun by the first two authors. We prove that each closure is an invariant-theoretic quotient of a suitably defined enhanced quiver variety. We conjecture, and prove in special cases, that these enhanced quiver varieties are normal complete intersections, implying that the enhanced nilpotent orbit closures are also normal.

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30

Huan, Song-Mei, and Xiao-Song Yang. "On the Number of Invariant Cones and Existence of Periodic Orbits in 3-dim Discontinuous Piecewise Linear Systems." International Journal of Bifurcation and Chaos 26, no. 03 (March 2016): 1650043. http://dx.doi.org/10.1142/s0218127416500437.

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For a family of discontinuous 3-dim homogeneous piecewise linear dynamical systems with two zones, we investigate the number of invariant cones and the existence of periodic orbits as a spatial relationship between the invariant manifolds of the subsystem changes. By studying the number of real roots of a quadratic equation induced by slopes of half straight lines starting from the origin in required domain, we obtain complete results on the number and stability of invariant cones. Especially, we prove that the maximum number of invariant cones is two, and obtain complete parameter regions on which there exist one or two invariant cones, on which one or two fake cones (corresponding to real roots of the quadratic equation that are not in the required domain) appear and on which an invariant cone will be foliated by periodic orbits.

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31

Revyakov, Mikhail I. "Probability of hitting a random vector in a polyhedral cone: Majorization aspect." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 9, no. 3 (2022): 506–16. http://dx.doi.org/10.21638/spbu01.2022.311.

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The article presents conditions under which the probability of a linear combination of random vectors falling into a polyhedral cone is a Schur-concave function of the coefficients of the combination. It is required that the cone contains the point 0, its edges are parallel to the coordinate axes, and the distribution density of vectors is a logarithmically concave sign-invariant function.

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32

J. Capiński, Maciej, and Piotr Zgliczyński. "Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds." Discrete & Continuous Dynamical Systems - A 30, no. 3 (2011): 641–70. http://dx.doi.org/10.3934/dcds.2011.30.641.

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33

Wu, Y. H., and Z. Y. Hu. "The invariant representations of a quadric cone and a twisted cubic." IEEE Transactions on Pattern Analysis and Machine Intelligence 25, no. 10 (October 2003): 1329–32. http://dx.doi.org/10.1109/tpami.2003.1233907.

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34

Oukil, W., Ph Thieullen, and A. Kessi. "Invariant cone and synchronization state stability of the mean field models." Dynamical Systems 34, no. 3 (November 28, 2018): 422–33. http://dx.doi.org/10.1080/14689367.2018.1547683.

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35

Burns, Keith, and Marlies Gerber. "Continuous invariant cone families and ergodicity of flows in dimension three." Ergodic Theory and Dynamical Systems 9, no. 1 (March 1989): 19–25. http://dx.doi.org/10.1017/s014338570000479x.

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AbstractIt is shown that a C2 flow on a compact three-dimensional manifold that preserves a smooth measure and has a continuous family of cones satisfying a certain invariance property must be ergodic.

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36

Tarbouriech, S., and C. Burgat. "Positively invariant sets for constrained continuous-time systems with cone properties." IEEE Transactions on Automatic Control 39, no. 2 (1994): 401–5. http://dx.doi.org/10.1109/9.272344.

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37

Acerbi, C., and A. Bassetto. "Renormalization of gauge-invariant composite operators in the light-cone gauge." Physical Review D 49, no. 2 (January 15, 1994): 1067–76. http://dx.doi.org/10.1103/physrevd.49.1067.

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38

Guo, Fangcheng, Guanghan Li, and Chuanxi Wu. "Mean Curvature Type Flow with Perpendicular Neumann Boundary Condition inside a Convex Cone." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/315768.

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We investigate the evolution of hypersurfaces with perpendicular Neumann boundary condition under mean curvature type flow, where the boundary manifold is a convex cone. We find that the volume enclosed by the cone and the evolving hypersurface is invariant. By maximal principle, we prove that the solutions of this flow exist for all time and converge to some part of a sphere exponentially asttends to infinity.

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39

KULSHRESHTHA, USHA. "LIGHT-FRONT HAMILTONIAN AND PATH INTEGRAL QUANTIZATION OF VECTOR SCHWINGER MODEL WITH A PHOTON MASS TERM." Modern Physics Letters A 27, no. 27 (August 23, 2012): 1250157. http://dx.doi.org/10.1142/s021773231250157x.

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Vector Schwinger model with a mass term for the photon, describing 2D electrodynamics with massless fermions, studied by us recently [U. Kulshreshtha, Mod. Phys. Lett. A22, 2993 (2007); U. Kulshreshtha and D. S. Kulshreshtha, Int. J. Mod. Phys. A22, 6183 (2007); U. Kulshreshtha, PoS LC2008, 008 (2008)], represents a new class of models. This theory becomes gauge-invariant when studied on the light-front. This is in contrast to the instant-form theory which is gauge-non-invariant. In this work, we study the light-front Hamiltonian and path integral quantization of this theory under appropriate light-cone gauge-fixing. The discretized light-cone quantization of the theory where we wish to make contact with the experimentally observational aspects of the theory would be presented in a separate paper.

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40

Wang, Weiqiang. "Resolution of Singularities of Null Cones." Canadian Mathematical Bulletin 44, no. 4 (December 1, 2001): 491–503. http://dx.doi.org/10.4153/cmb-2001-049-6.

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AbstractWe give canonical resolutions of singularities of several cone varieties arising from invariant theory. We establish a connection between our resolutions and resolutions of singularities of closure of conjugacy classes in classical Lie algebras.

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41

van der Mark, Martin B., and John G. Williamson. "Relativistic Inversion, Invariance and Inter-Action." Symmetry 13, no. 7 (June 23, 2021): 1117. http://dx.doi.org/10.3390/sym13071117.

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A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, leads to a natural extension incorporating rest mass and spin, and allows an integration with relativistic quantum mechanics. Although the algebra is not a division algebra, it parallels reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter.

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42

Marcum, Howard J. "Cone length of the exterior join." Glasgow Mathematical Journal 40, no. 3 (September 1998): 445–61. http://dx.doi.org/10.1017/s001708950003278x.

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The cone length Cl(f) of a map f: X → Y is defined to be the least number of attaching maps possible in a conic (or iterated mapping cone) structure for f. Cone length is a homotopy invariant in the sense that if φ: X → X and ρ: Y → Y are homotopy equivalences then Cl (ρ°f°φ) = Cl(f). Furthermore Cl(f) depends only on the homotopy class of f. It was shown by Ganea [8] that the cone length of the map * → X coincides with the strong Lusternik-Schnirelmann category of X as a space (see Proposition 1.6 below). Recent work of Cornea ([3]–[6]) is much concerned with cone length and its role in critical point theory. For example, let f be a smooth real valued function on a manifold triad (M; V0, V1) with V0 ≠ θ. Under certain conditions, if f has only “reasonable” critical points then it must have at least Cl(V0↪M) of them (see [6]).

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43

HU, YI. "RELATIVE GEOMETRIC INVARIANT THEORY AND UNIVERSAL MODULI SPACES." International Journal of Mathematics 07, no. 02 (April 1996): 151–81. http://dx.doi.org/10.1142/s0129167x96000098.

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We expose in detail the principle that the relative geometric invariant theory of equivariant morphisms is related to the GIT for linearizations near the boundary of the G-effective ample cone. We then apply this principle to construct and reconstruct various universal moduli spaces. In particular, we constructed the universal moduli space over [Formula: see text] of Simpson’s p-semistable coherent sheaves and a canonical rational morphism from the universal Hilbert scheme over [Formula: see text] to a compactified universal Picard.

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44

Bélanger, Alain, and Erik G. F. Thomas. "Positive Forms on Nuclear *-Algebras and Their Integral Representations." Canadian Journal of Mathematics 42, no. 3 (June 1, 1990): 410–69. http://dx.doi.org/10.4153/cjm-1990-023-3.

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Abstract.The main result of this paper establishes the existence and uniqueness of integral representations of KMS functionals on nuclear *- algebras. Our first result is about representations of *-algebras by means of operators having a common dense domain in a Hilbert space. We show, under certain regularity conditions, that (Powers) self-adjoint representations of a nuclear *-algebra, which admit a direct integral decomposition, disintegrate into representations which are almost all self-adjoint. We then define and study the class of self-derivative algebras. All algebras with an identity are in this class and many other examples are given. We show that if is a self-derivative algebra with an equicontinuous approximate identity, the cone of all positive forms on is isomorphic to the cone of all positive invariant kernels on These in turn correspond bijectively to the invariant Hilbert subspaces of the dual space This shows that if is a nuclear -space, the positive cone of has bounded order intervals, which implies that each positive form on has an integral representation in terms of the extreme generators of the cone. Given a continuous exponentially bounded one-parameter group of *-automorphisms of we can define the subcone of all invariant positive forms satisfying the KMS condition. Central functionals can be viewed as KMS functionals with respect to a trivial group action. Assuming that is a self-derivative algebra with an equicontinuous approximate identity, we show that the face generated by a self-adjoint KMS functional is a lattice. If is moreover a nuclear *-algebra the previous results together imply that each self-adjoint KMS functional has a unique integral representation by means of extreme KMS functionals almost all of which are self-adjoint.

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NOURI-MOGHADAM, M., C. R. R. SMITH, and J. G. TAYLOR. "REPARAMETRISATION — INVARIANT CLOSED STRING FIELD THEORY: FOLIATION STRUCTURE AND PHYSICAL SPECTRUM." Modern Physics Letters A 02, no. 11 (November 1987): 887–92. http://dx.doi.org/10.1142/s0217732387001129.

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The foliation structure and physical spectrum of a reparametrisation-invariant field theory of closed strings are analyzed to show (a) independence of the action of the foliation (b) that the spectrum is that of the usual light cone gauge formulation.

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Levasseur, Thierry. "Invariant distributions supported on the nilpotent cone of a semisimple Lie algebra." Transactions of the American Mathematical Society 353, no. 10 (June 1, 2001): 4189–202. http://dx.doi.org/10.1090/s0002-9947-01-02851-3.

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Mokler, C. "Invariant convex subcones of the Tits cone of a linear Coxeter group." Journal of Pure and Applied Algebra 222, no. 6 (June 2018): 1405–77. http://dx.doi.org/10.1016/j.jpaa.2017.07.006.

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TARBOURIECH, S., and C. BURGAT. "Positively invariant sets for continuous-time systems with the cone-preserving property." International Journal of Systems Science 24, no. 6 (June 1993): 1037–47. http://dx.doi.org/10.1080/00207729308949542.

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Ben-Dayan, I., M. Gasperini, G. Marozzi, F. Nugier, and G. Veneziano. "Backreaction on the luminosity-redshift relation from gauge invariant light-cone averaging." Journal of Cosmology and Astroparticle Physics 2012, no. 04 (April 24, 2012): 036. http://dx.doi.org/10.1088/1475-7516/2012/04/036.

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Jacobson, T., R. P. Woodard, and N. C. Tsamis. "The light-cone gauge M−i generator and invariant string field theory." Physics Letters B 176, no. 3-4 (August 1986): 387–90. http://dx.doi.org/10.1016/0370-2693(86)90182-6.

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