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Mikhail Zaicev - One of the best experts on this subject based on the ideXlab platform.
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Abelian gradings on upper block triangular matrices
Canadian Mathematical Bulletin, 2012Co-Authors: A. Valenti, Mikhail ZaicevAbstract:Let be an arbitrary Finite Abelian Group. We describe all possible -gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero.
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Abelian gradings on upper-triangular matrices
Archiv der Mathematik, 2003Co-Authors: A. Valenti, Mikhail ZaicevAbstract:Let G be an arbitrary Finite Abelian Group. We describe all possible G-gradings on an upper-triangular matrix algebra over an algebraically closed field of characteristic zero.
Nathaniel Stapleton - One of the best experts on this subject based on the ideXlab platform.
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the balmer spectrum of the equivariant homotopy category of a Finite Abelian Group
Inventiones Mathematicae, 2019Co-Authors: Tobias Barthel, Niko Naumann, Justin Noel, Thomas Nikolaus, Markus Hausmann, Nathaniel StapletonAbstract:For a Finite Abelian Group A, we determine the Balmer spectrum of \({\mathrm {Sp}}_A^{\omega }\), the compact objects in genuine A-spectra. This generalizes the case \(A={\mathbb {Z}}/p{\mathbb {Z}}\) due to Balmer and Sanders (Invent Math 208(1):283–326, 2017), by establishing (a corrected version of) their \(\hbox {log}_p\)-conjecture for Abelian Groups. We also work out the consequences for the chromatic type of fixed-points and establish a generalization of Kuhn’s blue-shift theorem for Tate-constructions (Kuhn in Invent Math 157(2):345–370, 2004).
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the balmer spectrum of the equivariant homotopy category of a Finite Abelian Group
arXiv: Algebraic Topology, 2017Co-Authors: Tobias Barthel, Niko Naumann, Justin Noel, Thomas Nikolaus, Markus Hausmann, Nathaniel StapletonAbstract:For a Finite Abelian Group $A$, we determine the Balmer spectrum of $\mathrm{Sp}_A^{\omega}$, the compact objects in genuine $A$-spectra. This generalizes the case $A=\mathbb{Z}/p\mathbb{Z}$ due to Balmer and Sanders \cite{Balmer-Sanders}, by establishing (a corrected version of) their log$_p$-conjecture for Abelian Groups. We work out the consequences for the chromatic type of fixed-points. We also establish a generalization of Kuhn's blue-shift theorem for Tate-constructions \cite{kuhn}.
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the balmer spectrum of the equivariant homotopy category of a Finite Abelian Group
arXiv: Algebraic Topology, 2017Co-Authors: Tobias Barthel, Niko Naumann, Justin Noel, Thomas Nikolaus, Markus Hausmann, Nathaniel StapletonAbstract:For a Finite Abelian Group $A$, we determine the Balmer spectrum of $\mathrm{Sp}_A^{\omega}$, the compact objects in genuine $A$-spectra. This generalizes the case $A=\mathbb{Z}/p\mathbb{Z}$ due to Balmer and Sanders \cite{Balmer-Sanders}, by establishing (a corrected version of) their log$_p$-conjecture for Abelian Groups. We also work out the consequences for the chromatic type of fixed-points and establish a generalization of Kuhn's blue-shift theorem for Tate-constructions \cite{kuhn}.
Luciano Margara - One of the best experts on this subject based on the ideXlab platform.
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decidable characterizations of dynamical properties for additive cellular automata over a Finite Abelian Group with applications to data encryption
Information Sciences, 2021Co-Authors: Alberto Dennunzio, Enrico Formenti, Darij Grinberg, Luciano MargaraAbstract:Abstract Additive cellular automata over a Finite Abelian Group are a wide class of cellular automata (CA) that are able to exhibit the complex behaviors of general CA and are often exploited for designing applications in different practical contexts. We provide decidable characterizations for Additive CA of the following important properties defining complex behaviors of complex systems: injectivity , surjectivity, equicontinuity, sensitivity to the initial conditions, topological transitivity , and ergodicity . Since such properties describe the main features required by real systems, the decision algorithms from our decidability results are then important tools for designing proper applications based on Additive CA. Indeed, we describe how our results can be exploited in some emblematic applications of cryptosystems , a paradigmatic and nowadays crucial applicative domain in which Additive CA are extensively used. We deal with methods for data encryption and, namely, we propose some strong modifications to the existing schemes in order to increase their security level and make attacks much harder.
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dynamical behavior of additive cellular automata over Finite Abelian Groups
Theoretical Computer Science, 2020Co-Authors: Alberto Dennunzio, Enrico Formenti, Darij Grinberg, Luciano MargaraAbstract:Abstract We study the dynamical behavior of additive D-dimensional ( D ≥ 1 ) cellular automata where the alphabet is any Finite Abelian Group. This class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list [38] , [44] , [41] . Among our major contributions, there is the proof that topologically transitive additive D-dimensional cellular automata over a Finite Abelian Group are ergodic. This result represents a solid bridge between the world of measure theory and that of topology and greatly extends previous results obtained in [12] , [44] for linear CA over Z / m Z , i.e., additive CA in which the alphabet is the cyclic Group Z / m Z and the local rules are linear combinations with coefficients in Z / m Z . In our scenario, the alphabet is any Finite Abelian Group and the global rule is any additive map. This class of CA strictly contains the class of linear CA over ( Z / m Z ) n , i.e., with the local rule defined by n × n matrices with elements in Z / m Z which, in turn, strictly contains the class of linear CA over Z / m Z . In order to further emphasize that Finite Abelian Groups are more expressive than Z / m Z we prove that, contrary to what happens in Z / m Z , there exist additive CA over suitable Finite Abelian Groups which are roots (with arbitrarily large indices) of the shift map. As a relevant consequence of our results, we have that, for additive D-dimensional CA over a Finite Abelian Group, ergodic mixing, weak ergodic mixing, ergodicity, topological mixing, weak topological mixing, topological total transitivity and topological transitivity are all equivalent properties. As a corollary, we see that invertible transitive additive CA are isomorphic to Bernoulli shifts. Furthermore, we prove that surjectivity implies openness for additive D-dimensional CA over a Finite Abelian Group. Hence, we get that topological transitivity is equivalent to the well-known Devaney notion of chaos when D = 1 . Moreover, we provide a first characterization of strong transitivity for additive CA which we suspect to be true also for the general case.
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From Linear to Additive Cellular Automata
2020Co-Authors: Alberto Dennunzio, Enrico Formenti, Darij Grinberg, Luciano MargaraAbstract:This paper proves the decidability of several important properties of additive cellular automata over Finite Abelian Groups. First of all, we prove that equicontinuity and sensitivity to initial conditions are decidable for a nontrivial subclass of additive cellular automata, namely, the linear cellular automata over \u207f, where is the ring \u2124/m\u2124. The proof of this last result has required to prove a general result on the powers of matrices over a commutative ring which is of interest in its own. Then, we extend the decidability result concerning sensitivity and equicontinuity to the whole class of additive cellular automata over a Finite Abelian Group and for such a class we also prove the decidability of topological transitivity and all the properties (as, for instance, ergodicity) that are equivalent to it. Finally, a decidable characterization of injectivity and surjectivity for additive cellular automata over a Finite Abelian Group is provided in terms of injectivity and surjectivity of an associated linear cellular automata over \u207f
A. Valenti - One of the best experts on this subject based on the ideXlab platform.
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Abelian gradings on upper block triangular matrices
Canadian Mathematical Bulletin, 2012Co-Authors: A. Valenti, Mikhail ZaicevAbstract:Let be an arbitrary Finite Abelian Group. We describe all possible -gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero.
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Abelian gradings on upper-triangular matrices
Archiv der Mathematik, 2003Co-Authors: A. Valenti, Mikhail ZaicevAbstract:Let G be an arbitrary Finite Abelian Group. We describe all possible G-gradings on an upper-triangular matrix algebra over an algebraically closed field of characteristic zero.
Tobias Barthel - One of the best experts on this subject based on the ideXlab platform.
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the balmer spectrum of the equivariant homotopy category of a Finite Abelian Group
Inventiones Mathematicae, 2019Co-Authors: Tobias Barthel, Niko Naumann, Justin Noel, Thomas Nikolaus, Markus Hausmann, Nathaniel StapletonAbstract:For a Finite Abelian Group A, we determine the Balmer spectrum of \({\mathrm {Sp}}_A^{\omega }\), the compact objects in genuine A-spectra. This generalizes the case \(A={\mathbb {Z}}/p{\mathbb {Z}}\) due to Balmer and Sanders (Invent Math 208(1):283–326, 2017), by establishing (a corrected version of) their \(\hbox {log}_p\)-conjecture for Abelian Groups. We also work out the consequences for the chromatic type of fixed-points and establish a generalization of Kuhn’s blue-shift theorem for Tate-constructions (Kuhn in Invent Math 157(2):345–370, 2004).
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the balmer spectrum of the equivariant homotopy category of a Finite Abelian Group
arXiv: Algebraic Topology, 2017Co-Authors: Tobias Barthel, Niko Naumann, Justin Noel, Thomas Nikolaus, Markus Hausmann, Nathaniel StapletonAbstract:For a Finite Abelian Group $A$, we determine the Balmer spectrum of $\mathrm{Sp}_A^{\omega}$, the compact objects in genuine $A$-spectra. This generalizes the case $A=\mathbb{Z}/p\mathbb{Z}$ due to Balmer and Sanders \cite{Balmer-Sanders}, by establishing (a corrected version of) their log$_p$-conjecture for Abelian Groups. We work out the consequences for the chromatic type of fixed-points. We also establish a generalization of Kuhn's blue-shift theorem for Tate-constructions \cite{kuhn}.
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the balmer spectrum of the equivariant homotopy category of a Finite Abelian Group
arXiv: Algebraic Topology, 2017Co-Authors: Tobias Barthel, Niko Naumann, Justin Noel, Thomas Nikolaus, Markus Hausmann, Nathaniel StapletonAbstract:For a Finite Abelian Group $A$, we determine the Balmer spectrum of $\mathrm{Sp}_A^{\omega}$, the compact objects in genuine $A$-spectra. This generalizes the case $A=\mathbb{Z}/p\mathbb{Z}$ due to Balmer and Sanders \cite{Balmer-Sanders}, by establishing (a corrected version of) their log$_p$-conjecture for Abelian Groups. We also work out the consequences for the chromatic type of fixed-points and establish a generalization of Kuhn's blue-shift theorem for Tate-constructions \cite{kuhn}.
