Entropy dimension of shifts of finite type on free groups

Jung-Chao Ban;  ; Chih-Hung Chang;  ;

doi:10.3934/math.2020329

Free groups and finite-type invariants of pure braids

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004101005333 ◽

2002 ◽

Vol 132 (1) ◽

pp. 117-130 ◽

Cited By ~ 4

Author(s):

JACOB MOSTOVOY ◽

SIMON WILLERTON

Keyword(s):

Finite Type ◽

Free Group ◽

Free Groups ◽

Braid Groups ◽

Point Of View ◽

Magnus Expansion ◽

Vassiliev Invariants ◽

Integer Coefficients ◽

Finite Type Invariants ◽

Theoretic Point

In this paper finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus expansion of a free group and a calculation of numbers of independent invariants of each type for all pure braid groups.

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Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2021143 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Silvère Gangloff

Keyword(s):

Finite Type ◽

Symbolic Dynamics ◽

Recent Trend ◽

Higher Dimensions ◽

Topological Invariant ◽

Entropy Dimension ◽

Zero Entropy ◽

Subshifts Of Finite Type ◽

Present Text ◽

Made In

<p style='text-indent:20px;'>In this text I study the asymptotics of the complexity function of <i>minimal</i> multidimensional subshifts of finite type through their entropy dimension, a topological invariant that has been introduced in order to study zero entropy dynamical systems. Following a recent trend in symbolic dynamics I approach this using concepts from computability theory. In particular it is known [<xref ref-type="bibr" rid="b12">12</xref>] that the possible values of entropy dimension for d-dimensional subshifts of finite type are the <inline-formula><tex-math id="M1">\begin{document}$ \Delta_2 $\end{document}</tex-math></inline-formula>-computable numbers in <inline-formula><tex-math id="M2">\begin{document}$ [0, d] $\end{document}</tex-math></inline-formula>. The kind of constructions that underlies this result is however quite complex and minimality has been considered thus far as hard to achieve with it. In this text I prove that this is possible and use the construction principles that I developped in order to prove (in principle) that for all <inline-formula><tex-math id="M3">\begin{document}$ d \ge 2 $\end{document}</tex-math></inline-formula> the possible values for entropy dimensions of <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional SFT are the <inline-formula><tex-math id="M5">\begin{document}$ \Delta_2 $\end{document}</tex-math></inline-formula>-computable numbers in <inline-formula><tex-math id="M6">\begin{document}$ [0, d-1] $\end{document}</tex-math></inline-formula>. In the present text I prove formally this result for <inline-formula><tex-math id="M7">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>. Although the result for other dimensions does not follow directly, it is enough to understand this construction to see that it is possible to reproduce it in higher dimensions (I chose dimension three for optimality in terms of exposition). The case <inline-formula><tex-math id="M8">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula> requires some substantial changes to be made in order to adapt the construction that are not discussed here.</p>

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Sulphoamidic Phtalocyaninic Pigments and Dyes(II)

Revista de Chimie ◽

10.37358/rc.08.5.1820 ◽

2008 ◽

Vol 59 (5) ◽

Author(s):

Elena Stingaciu ◽

Corneliu Minca ◽

Ion Sebe

Keyword(s):

Ir Spectra ◽

Aromatic Amines ◽

Free Groups ◽

Cyanuric Chloride ◽

Layer Chromatography ◽

Phenylene Diamine

This work concerns the synthesis of pigments and phtalocyanine dyes obtained through the sulphonation of copper phtalocyanine and amidation with some aliphatic and aromatic amines (lauryl-amine, i-propyl-amine, hexadecyl-amine, stearyl-amine and acetyl-p-phenylene-diamine) with good properties for the electrotechnic utilisation and for toner materials. The pigments with amino free groups are transformed by condensation with cyanuric chloride in phtalocyanine pigments with different tinctorial properties. The dyes were analyzed through the layer chromatography and were characterized on the IR spectra bases and tinctorial tests.

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Generating the inverse limit of free groups

Journal of Algebra ◽

10.1016/j.jalgebra.2021.02.017 ◽

2021 ◽

Vol 578 ◽

pp. 371-401

Author(s):

Gregory R. Conner ◽

Wolfgang Herfort ◽

Curtis A. Kent ◽

Petar Pavešić

Keyword(s):

Inverse Limit ◽

Free Groups

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Minimal partition-free groups

Ricerche di Matematica ◽

10.1007/s11587-021-00579-z ◽

2021 ◽

Author(s):

Afsane Bahri ◽

Zeinab Akhlaghi ◽

Behrooz Khosravi

Keyword(s):

Free Groups

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Freely indecomposable almost free groups with free abelianization

Journal of Group Theory ◽

10.1515/jgth-2019-0102 ◽

2020 ◽

Vol 23 (3) ◽

pp. 531-543

Author(s):

Samuel M. Corson

Keyword(s):

Free Groups ◽

Uncountable Cardinals

AbstractFor certain uncountable cardinals κ, we produce a group of cardinality κ which is freely indecomposable, strongly κ-free, and whose abelianization is free abelian of rank κ. The construction takes place in Gödel’s constructible universe L. This strengthens an earlier result of Eklof and Mekler.

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