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Invariant measures for bipermutative cellular automata
[article]

2003
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arXiv
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pre-print

A 'right-sided, nearest neighbour cellular automaton' (RNNCA) is a continuous transformation F:A^Z-->A^Z determined by a local rule f:A^0,1-->A so that, for any a in A^Z and any z in Z, F(a)_z = f(a_z,a_z+1) . We say that F is 'bipermutative' if, for any choice of a in A, the map g:A-->A defined by g(b) = f(a,b) is bijective, and also, for any choice of b in A, the map h:A-->A defined by h(a)=f(a,b) is bijective. We characterize the invariant measures of bipermutative RNNCA. First we introduce

arXiv:math/0306211v1
fatcat:bh5zq6k7x5hgtma4cbvnxcuxda
## more »

... he equivalent notion of a 'quasigroup CA', to expedite the construction of examples. Then we characterize F-invariant measures when A is a (nonabelian) group, and f(a,b) = a*b. Then we show that, if F is any bipermutative RNNCA, and mu is F-invariant, then F must be mu-almost everywhere K-to-1, for some constant K . We use this to characterize invariant measures when A^Z is a 'group shift' and F is an 'endomorphic CA'.##
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Ergodic Theory of Cellular Automata
[chapter]

2012
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Computational Complexity
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Proof: For (a), see (Hedlund, 1969, Theorem 5.9) or (Lind and

doi:10.1007/978-1-4614-1800-9_62
fatcat:6chdeuo5n5b3dj434scnmeplwq
*Marcus*, 1995, Corollary 8.1.20, p.271) . ... (b) The case M = Z is (Lind and*Marcus*, 1995, Corollary 8.1.20) (actually this holds for any sofic subshift); see also Fiorenzi (2000) . ...##
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Linear cellular automata, asymptotic randomization, and entropy
[article]

2002
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arXiv
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pre-print

If A=Z/2, then A^Z is a compact abelian group. A 'linear cellular automaton' is a shift-commuting endomorphism F of A^Z. If P is a probability measure on A^Z, then F 'asymptotically randomizes' P if F^j P converges to the Haar measure as j-->oo, for j in a subset of Cesaro density one. Via counterexamples, we show that nonzero entropy of P is neither necessary nor sufficient for asymptotic randomization.

arXiv:math/0210241v1
fatcat:75osk7mx2nchvefcypvgeo7ttm
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Pyramidal Democracy

2009
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Journal of Deliberative Democracy
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Tier 4
Tier 3
Tier 2
Tier 1
t o T i e r 5
b ac k to T ie r 3
This essay is available in Journal of Public Deliberation: https://www.publicdeliberation.net/jpd/vol5/iss1/art8

doi:10.16997/jdd.82
fatcat:fx27pjgyijffdbjsdvnbjkb7u4
*Pivato*...*Pivato*: Pyramidal Democracy See(Riker, 1982, Ch.7),(Austen-Smith and Banks, 1999, Ch.6), or(Mueller, 2003, §5.12.1) for a summary.17 See Nurmi and Uusi-Heikkilä (1985) , Wagner (1983 Wagner ( , 1984 ...##
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Embedding Bratteli-Vershik systems in cellular automata
[article]

2007
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arXiv
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pre-print

Many dynamical systems can be naturally represented as 'Bratteli-Vershik' (or 'adic') systems, which provide an appealing combinatorial description of their dynamics. If an adic system X satisfies two technical conditions ('focus' and 'bounded width') then we show how to represent X using a two-dimensional subshift of finite type Y; each 'row' in a Y-admissible configuration corresponds to an infinite path in the Bratteli diagram of X, and the vertical shift on Y corresponds to the 'successor'

arXiv:0710.3608v1
fatcat:hs3qjnfhzrhodo26zjopj6rq6m
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... ap of X. Any Y-admissible configuration can then be recoded as the spacetime diagram of a one-dimensional cellular automaton F; in this way X is 'embedded' in F (i.e. X is conjugate to a subsystem of F). With this technique, we can embed many odometers, Toeplitz systems, and constant-length substitution systems in one-dimensional cellular automata.##
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Statistical Utilitarianism
[chapter]

2016
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The Political Economy of Social Choices
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to maximizing the expected value of the utilitarian SWF, and also equivalent to ex ante Pareto efficiency, as Schmitz and Troger observe in footnote 13 of their paper.3 SeeNitzan (2009, Ch.11-12) or

doi:10.1007/978-3-319-40118-8_8
fatcat:lf5udt53nfapvhhx5ab4ooyrde
*Pivato*...##
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Module Shifts and Measure Rigidity in Linear Cellular Automata
[article]

2007
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arXiv
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pre-print

Suppose R is a finite commutative ring of prime characteristic, A is a finite R-module, M:=Z^D x N^E, and F is an R-linear cellular automaton on A^M. If mu is an F-invariant measure which is multiply shift-mixing in a certain way, then we show that mu must be the Haar measure on a coset of some submodule shift of A^M. Under certain conditions, this means mu must be the uniform Bernoulli measure on A^M.

arXiv:0707.1408v1
fatcat:ceelgqmponer3nskdtxuplnkka
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Positive expansiveness versus network dimension in symbolic dynamical systems
[article]

2009
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arXiv
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pre-print

A 'symbolic dynamical system' is a continuous transformation F:X-->X of a closed perfect subset X of A^V, where A is a finite set and V is countable. (Examples include subshifts, odometers, cellular automata, and automaton networks.) The function F induces a directed graph structure on V, whose geometry reveals information about the dynamical system (X,F). The 'dimension' dim(V) is an exponent describing the growth rate of balls in the digraph as a function of their radius. We show: if X has

arXiv:0907.2935v1
fatcat:43xdh4bkt5d6zn6mv3ak5r7ygy
## more »

... itive entropy and dim(V)>1, and the system (A^V,X,F) satisfies minimal symmetry and mixing conditions, then (X,F) cannot be positively expansive; this generalizes a well-known result of Shereshevsky about multidimensional cellular automata. We also construct a counterexample to a version of this result without the symmetry condition. Finally, we show that network dimension is invariant under topological conjugacies which are Holder-continuous.##
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Spectral domain boundaries in cellular automata
[article]

2007
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arXiv
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pre-print

Let L:=Z^D be a D-dimensional lattice. Let A^L be the Cantor space of L-indexed configurations in a finite alphabet A, with the natural L-action by shifts. A 'cellular automaton' is a continuous, shift-commuting self-map F:A^L-->A^L. An 'F-invariant subshift' is a closed, F-invariant and shift-invariant subset X of A^L. Suppose x is an element of A^L that is X-admissible everywhere except for some small region of L which we call a 'defect'. Such defects are analogous to 'domain boundaries' in a

arXiv:math/0507091v2
fatcat:cel6ehgjhjdfxg356shjjhv26a
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... crystalline solid. It has been empirically observed that these defects persist under iteration of F, and often propagate like 'particles' which coalesce or annihilate on contact. We use spectral theory to explain the persistence of some defects under F, and partly explain the outcomes of their collisions.##
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The median rule in judgement aggregation

2021
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Economic Theory
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There exists Q ∈ (0, 1) ∩ Q such that [−Q, Q] K ⊂ conv(X rk A ) (Nehring and

doi:10.1007/s00199-021-01348-7
fatcat:lbji5megmvduvdbyaphzoeoifi
*Pivato*2011, Example 3.3) . ... (Or see Lemma D.7 from Nehring and*Pivato*(2019) .) Thus, it remains to show that F α satisfies Reinforcement. ...##
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Social preference under twofold uncertainty

2019
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Economic Theory
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See Mongin and

doi:10.1007/s00199-019-01237-0
fatcat:lb5t25umkvgdxg63ovks4ksfuy
*Pivato*(2016) for a review, and Fleurbaey and Mongin (2016) for a new defence of Harsanyi's position. ... See Mongin and*Pivato*(2015) . Part (a) follows from Theorem 1(c,d), and part (b) from Corollary 1(c,d). ... ., Karni's (2014) review of SEU theories. 6 For a recent extension of Anscombe and Aumann's theorem, see Mongin and*Pivato*(2015) . ...##
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Building a Stationary Stochastic Process From a Finite-dimensional Marginal
[article]

2001
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arXiv
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pre-print

If A is a finite alphabet, Z^D is a D-dimensional lattice, U is a subset of Z^D, and mu_U is a probability measure on A^U that "looks like" the marginal projection of a stationary random field on A^(Z^D), then can we "extend" mu_U to such a field? Under what conditions can we make this extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying classical work on this problem when D = 1, we provide some sufficient conditions and some necessary conditions for mu_U to be extendible for

arXiv:math/0108081v1
fatcat:2i4f3oghdzbzxgwffic5vlb4eq
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... D > 1, and show that, in general, the problem is not formally decidable.##
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Variable-population voting rules

2013
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Journal of Mathematical Economics
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groups are useful for representing infinite-horizon intertemporal preferences, non-probabilistic uncertainty, and preferences where some decision variables have lexicographical priority over others (

doi:10.1016/j.jmateco.2013.02.001
fatcat:rruhfqyn6nh6dh442ce3ly5wce
*Pivato*...##
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Voting rules as statistical estimators

2012
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Social Choice and Welfare
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Thus, a result of

doi:10.1007/s00355-011-0619-1
fatcat:z5327izgpnhjnj3nkv7ul2lduq
*Pivato*(2011) implies that there is some r > 0 and some function t : V−→R such that S(v, x) = r S(v, x) + t(v) for all v ∈ V and x ∈ X . ...##
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Epistemic democracy with correlated voters

2017
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Journal of Mathematical Economics
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*Pivato*(2013a, Theorem 2) proved a generalization of Myerson's result which keeps neutrality but relaxes the Continuity condition, by allowing the score vectors to take "infinitesimal" values. ... Furthermore, in these contexts, the "nonasymptotic" part of the CJT can be refined: under certain conditions, the output of the voting rule is a maximum likelihood estimator of the correct answer (see

*Pivato*...

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