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Journal articles on the topic 'Dimensión de Frobenius-Perron'

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Published: 17 September 2021

Last updated: 9 February 2022

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1

Dong, Jingcheng, Sonia Natale, and Leandro Vendramin. "Frobenius property for fusion categories of small integral dimension." Journal of Algebra and Its Applications 14, no. 02 (October 19, 2014): 1550011. http://dx.doi.org/10.1142/s0219498815500115.

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Let k be an algebraically closed field of characteristic zero. In this paper, we prove that fusion categories of Frobenius–Perron dimensions 84 and 90 are of Frobenius type. Combining this with previous results in the literature, we obtain that every weakly integral fusion category of Frobenius–Perron dimension less than 120 is of Frobenius type.

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2

Drobot, Vladimir, and John Turner. "Hausdorff dimension and Perron-Frobenius theory." Illinois Journal of Mathematics 33, no. 1 (March 1989): 1–9. http://dx.doi.org/10.1215/ijm/1255988801.

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3

Metz, Volker. "Nonlinear Perron–Frobenius theory in finite dimensions." Nonlinear Analysis: Theory, Methods & Applications 62, no. 2 (July 2005): 225–44. http://dx.doi.org/10.1016/j.na.2005.02.116.

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4

Dong, Jingcheng, Libin Li, and Li Dai. "Integral almost square-free modular categories." Journal of Algebra and Its Applications 16, no. 06 (April 12, 2017): 1750104. http://dx.doi.org/10.1142/s0219498817501043.

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We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius–Perron dimension [Formula: see text], where [Formula: see text] is a prime number, [Formula: see text] is a square-free natural number and [Formula: see text]. We prove that, if [Formula: see text] or [Formula: see text] is prime with [Formula: see text], then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and Tucker. As an application, we prove that an integral modular category whose Frobenius–Perron dimension is odd and less than [Formula: see text] is group-theoretical.

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5

Dong, Jingcheng, and Henry Tucker. "Integral Modular Categories of Frobenius-Perron Dimension pq n." Algebras and Representation Theory 19, no. 1 (July 29, 2015): 33–46. http://dx.doi.org/10.1007/s10468-015-9560-9.

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6

Etingof, Pavel. "Frobenius-Perron Dimensions of Integral $\mathbb {Z}_{+}$-rings and Applications." Algebras and Representation Theory 23, no. 5 (November 7, 2019): 2059–78. http://dx.doi.org/10.1007/s10468-019-09924-1.

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7

Fox, Colin, Li-Jen Hsiao, and Jeong-Eun (Kate) Lee. "Solutions of the Multivariate Inverse Frobenius–Perron Problem." Entropy 23, no. 7 (June 30, 2021): 838. http://dx.doi.org/10.3390/e23070838.

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We address the inverse Frobenius–Perron problem: given a prescribed target distribution ρ, find a deterministic map M such that iterations of M tend to ρ in distribution. We show that all solutions may be written in terms of a factorization that combines the forward and inverse Rosenblatt transformations with a uniform map; that is, a map under which the uniform distribution on the d-dimensional hypercube is invariant. Indeed, every solution is equivalent to the choice of a uniform map. We motivate this factorization via one-dimensional examples, and then use the factorization to present solutions in one and two dimensions induced by a range of uniform maps.

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8

Bruillard, Paul, Cásar Galindo, Seung-Moon Hong, Yevgenia Kashina, Deepak Naidu, Sonia Natale, Julia Yael Plavnik, and Eric C. Rowell. "Classification of Integral Modular Categories of Frobenius–Perron Dimension pq4 and p2q2." Canadian Mathematical Bulletin 57, no. 4 (December 1, 2014): 721–34. http://dx.doi.org/10.4153/cmb-2013-042-6.

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AbstractWe classify integral modular categories of dimension pq4 and p2q2, where p and q are distinct primes. We show that such categories are always group-theoretical, except for categories of dimension 4q2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara–Yamagami categories and quantum groups. We show that a non-grouptheoretical integral modular category of dimension 4q2 is either equivalent to one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising froma certain quantum group.

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9

MAULDIN, R. D., and M. URBAŃSKI. "Parabolic iterated function systems." Ergodic Theory and Dynamical Systems 20, no. 5 (October 2000): 1423–47. http://dx.doi.org/10.1017/s0143385700000778.

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In this paper we introduce and explore conformal parabolic iterated function systems. We define and study topological pressure, Perron–Frobenius-type operators, semiconformal and conformal measures and the Hausdorff dimension of the limit set. With every parabolic system we associate an infinite hyperbolic conformal iterated function system and we employ it to study geometric and dynamical features (properly defined invariant measures for example) of the limit set.

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10

Duvall, P., and J. Keesling. "The dimension of the boundary of the Lévy Dragon." International Journal of Mathematics and Mathematical Sciences 20, no. 4 (1997): 627–32. http://dx.doi.org/10.1155/s0161171297000872.

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In this paper we describe the computations done by the authors in determining the dimension of the boundary of the Lévy Dragon. A general theory was developed for calculating the dimension of a self-similar tile and the theory was applied to this particular set. The computations were challenging. It seemed that a matrix which was215×215would have to be analyzed. It was possible to reduce the analysis to a752×752matrix. At last it was seen that ifλwas the largest eigenvalue of a certain734×734matrix, thendimH(K)=ln(λ)ln((2))Perron-Frobenius theory played an important role in analyzing this matrix.

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11

Natale, Sonia, and Edwin Pacheco Rodríguez. "Graphs attached to simple Frobenius-Perron dimensions of an integral fusion category." Monatshefte für Mathematik 179, no. 4 (January 18, 2015): 615–49. http://dx.doi.org/10.1007/s00605-015-0734-7.

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12

Jingheng, Zhou, Wang Yanhua, and Ding Jingru. "Frobenius-Perron dimension of representations of a class of $\mathbb{D}$-type quivers." SCIENTIA SINICA Mathematica 51, no. 5 (October 16, 2020): 673. http://dx.doi.org/10.1360/ssm-2020-0093.

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13

MAYER, VOLKER, MARIUSZ URBAŃSKI, and ANNA ZDUNIK. "Real analyticity for random dynamics of transcendental functions." Ergodic Theory and Dynamical Systems 40, no. 2 (August 10, 2018): 490–520. http://dx.doi.org/10.1017/etds.2018.42.

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Analyticity results of expected pressure and invariant densities in the context of random dynamics of transcendental functions are established. These are obtained by a refinement of work by Rugh [On the dimension of conformal repellors, randomness and parameter dependency. Ann. of Math. (2) 168(3) (2008), 695–748] leading to a simple approach to analyticity. We work under very mild dynamical assumptions. Just the iterates of the Perron–Frobenius operator are assumed to converge. We also provide Bowen’s formula expressing the almost sure Hausdorff dimension of the radial fiberwise Julia sets in terms of the zero of an expected pressure function. Our main application establishes real analyticity for the variation of this dimension for suitable hyperbolic random systems of entire or meromorphic functions.

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14

Natale, Sonia. "The core of a weakly group-theoretical braided fusion category." International Journal of Mathematics 29, no. 02 (February 2018): 1850012. http://dx.doi.org/10.1142/s0129167x1850012x.

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We show that the core of a weakly group-theoretical braided fusion category [Formula: see text] is equivalent as a braided fusion category to a tensor product [Formula: see text], where [Formula: see text] is a pointed weakly anisotropic braided fusion category, and [Formula: see text] or [Formula: see text] is an Ising braided category. In particular, if [Formula: see text] is integral, then its core is a pointed weakly anisotropic braided fusion category. As an application we give a characterization of the solvability of a weakly group-theoretical braided fusion category. We also prove that an integral modular category all of whose simple objects have Frobenius–Perron dimension at most 2 is necessarily group-theoretical.

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15

Falk, Richard, and Roger Nussbaum. "$C^m$ Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension: applications in $\mathbb R^1$." Journal of Fractal Geometry 5, no. 3 (July 6, 2018): 279–337. http://dx.doi.org/10.4171/jfg/62.

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16

Edie-Michell, Cain. "Classifying fusion categories $$\otimes $$-generated by an object of small Frobenius–Perron dimension." Selecta Mathematica 26, no. 2 (March 13, 2020). http://dx.doi.org/10.1007/s00029-020-0550-3.

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17

Etingof, Pavel, and Victor Ostrik. "On the Frobenius functor for symmetric tensor categories in positive characteristic." Journal für die reine und angewandte Mathematik (Crelles Journal), October 8, 2020. http://dx.doi.org/10.1515/crelle-2020-0033.

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AbstractWe develop a theory of Frobenius functors for symmetric tensor categories (STC) {\mathcal{C}} over a field {\boldsymbol{k}} of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor {F:\mathcal{C}\to\mathcal{C}\boxtimes{\rm Ver}_{p}}, where {{\rm Ver}_{p}} is the Verlinde category (the semisimplification of {\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k}}(\mathbb{Z}/p)}); a similar construction of the underlying additive functor appeared independently in [K. Coulembier, Tannakian categories in positive characteristic, preprint 2019]. This generalizes the usual Frobenius twist functor in modular representation theory and also the one defined in [V. Ostrik, On symmetric fusion categories in positive characteristic, Selecta Math. (N.S.) 26 2020, 3, Paper No. 36], where it is used to show that if {\mathcal{C}} is finite and semisimple, then it admits a fiber functor to {{\rm Ver}_{p}}. The main new feature is that when {\mathcal{C}} is not semisimple, F need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor {\mathcal{C}\to{\rm Ver}_{p}}. We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of F, and use it to show that for categories with finitely many simple objects F does not increase the Frobenius–Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which F is exact, and define the canonical maximal Frobenius exact subcategory {\mathcal{C}_{\rm ex}} inside any STC {\mathcal{C}} with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius–Perron dimension is preserved by F. One of our main results is that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to {{\rm Ver}_{p}}. This is the strongest currently available characteristic p version of Deligne’s theorem (stating that a STC of moderate growth in characteristic zero is the representation category of a supergroup). We also show that a sufficiently large power of F lands in {\mathcal{C}_{\rm ex}}. Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category (namely, one having a fiber functor into the category of representations of the triangular Hopf algebra {\boldsymbol{k}[d]/d^{2}} with d primitive and R-matrix {R=1\otimes 1+d\otimes d}), and show that a STC with Chevalley property is (almost) Frobenius exact. Finally, as a by-product, we resolve Question 2.15 of [P. Etingof and S. Gelaki, Exact sequences of tensor categories with respect to a module category, Adv. Math. 308 2017, 1187–1208].

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