Chapter

Elements of Symbolic Dynamics

January 1997

DOI:10.1007/978-94-015-8897-3_2

In book: Dynamics of One-Dimensional Maps (pp.35-53)

Authors:

Aleksandr Sharkovsky

National Academy of Sciences of Ukraine

Symbolic dynamics is a part of the general theory of dynamical systems dealing with cascades generated by shifts in various spaces of sequences

\Theta = \left( {...{\theta _{ - 2}},{\theta _{ - 1}},{\theta _0},{\theta _1},{\theta _2},...} \right)\quad or\quad \Theta = \left( {{\theta _0},{\theta _1},{\theta _2},...} \right),

where θn are letters of an alphabet A = {θ1, θ2, ..., θm } The methods of symbolic dynamics are now widely applied to the investigation of various types of dynamical systems.

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Preprint

Pure semisimplicity conjecture and Artin problem for dimension sequences

September 2019

Jan Šaroch

Inspired by a recent paper due to Jos\'{e} Luis Garc\'{i}a, we revisit the attempt of Daniel Simson to construct a counterexample to the pure semisimplicity conjecture. Using compactness, we show that the existence of such counterexample would readily follow from the very existence of certain (countable set of) hereditary artinian rings of finite representation type. The existence of such rings ... [Show full abstract] is then proved to be equivalent to the existence of special types of embeddings, which we call tight, of division rings into simple artinian rings. Using the tools by Aidan Schofield from 1980s, we can show that such an embedding

F\hookrightarrow M_n(G)

exists provided that

n<5

. As a byproduct, we obtain a division ring extension

G\subseteq F

such that the bimodule

{}_GF_F

has the right dimension sequence (1,2,2,2,1,4). Finally, we formulate Conjecture A, which asserts that a particular type of adjunction of an element to a division ring can be made, and demonstrate that its validity would be sufficient to prove the existence of tight embeddings in general, and hence to disprove the pure semisimplicity conjecture.

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\left\{ {c_k } \right\}_{k = 0}^n

is a sequence of linear bounded functionals such that c k (x l ) = δ kl ,

R_{n,r} (x) = \sum\limits_{k = 0}^n {\left( {1 - \left( {\frac{k}{n + 1}} \right)^r } \right)ck(x)x_k }

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\Phi :X \to \mathbb{R}_{ + }

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Frank K. Hwang

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A_m

and

B_n

of a linearly-ordered set S. The problem is to determine the linear ordering of their union (i.e., to “merge”

A_m

and

B_n

) by means of a sequence of pairwise comparisons between an element of

A_m

and an element of

B_n

. An algorithm to merge

A_m

and

B_n

is called M-optimal if it minimizes the maximum ... [Show full abstract] number of comparisons required where the maximum is taken over all possible ordering of

A_m \cup B_n

. Let

f_m (k)

denote the largest n such that

A_m

and

B_n

, can be merged in k comparisons by an M-optimal algorithm. The determination of

f_3 (k)

has been an open problem for a long time. In this paper we give a constructive proof that

f_3 (1) = 0,\quad f_3 (2) = 1,\quad f_3 (3) = 1,\quad f_3 (4) = 2,

f_3 (5) = 3,\quad f_3 (6) = 4,\quad f_3 (7) = 6,\quad f_3 (8) = 8,

and for

r \geqq 3

,

f_3 (3r) = \left[ {\frac{{43}}{7}2^{r - 2} } \right] - 2,

f_3 (3r + 1) = \left[ {\frac{{107}}{7}2^{r - 3} } \right] - 2,

f_3 (3r + 2) = \left[ {\frac{{17 \cdot 2^r - 6}}{7}} \right] - 1,

where [x] denotes the integral part of x.

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Chapter

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January 1994

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January 1999

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Gordon Blower

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