scholarly journals Finite-Support Capacity-Approaching Distributions for AWGN Channels

2021 ◽  
Author(s):  
Derek Xiao ◽  
Linfang Wang ◽  
Dan Song ◽  
Richard D. Wesel
Keyword(s):  
Finite Support

Phase Type Distributions with Finite Support

2014 ◽  
Vol 30 (4) ◽  
pp. 576-597 ◽  
Author(s):  
V. Ramaswami ◽  
N. C. Viswanath
Keyword(s):  
Finite Support ◽  
Phase Type ◽  
Phase Type Distributions

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

scholarly journals Transitive action on finite points of a full shift and a finitary Ryan’s theorem

2017 ◽  
Vol 39 (06) ◽  
pp. 1637-1667 ◽  
Author(s):  
VILLE SALO
Keyword(s):  
Automorphism Group ◽  
Commutator Subgroup ◽  
Turing Machines ◽  
Finite Support ◽  
Transitive Action ◽  
Group Invariant ◽  
Finite Set ◽  
Transitivity Property ◽  
Full Shift ◽  
Subshifts Of Finite Type

We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to shift-commutation. As a corollary, we obtain that there is a finite set of automorphisms whose centralizer is $\mathbb{Z}$ (the shift group), giving a finitary version of Ryan’s theorem (on the four-symbol full shift), suggesting an automorphism group invariant for mixing subshifts of finite type (SFTs). We show that any such set of automorphisms must generate an infinite group, and also show that there is also a group with this transitivity property that is a subgroup of the commutator subgroup and whose elements can be written as compositions of involutions. We ask many related questions and prove some easy transitivity results for the group of reversible Turing machines, topological full groups and Thompson’s  $V$ .

scholarly journals Simultaneously vanishing higher derived limits

2021 ◽  
Vol 9 ◽  
Author(s):  
Jeffrey Bergfalk ◽  
Chris Lambie-Hanson
Keyword(s):  
Euclidean Space ◽  
Metric Spaces ◽  
Inverse System ◽  
Finite Support ◽  
Partial Functions ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Strong Homology ◽  
Finite Support Iteration ◽  
Weakly Compact Cardinal

Abstract In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $\textbf {A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim ^n\textbf {A}$ (the nth derived limit of $\textbf {A}$ ) vanishes for every $n>0$ . Since that time, the question of whether it is consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for every $n>0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for all $n>0$ . We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to $\lim ^n\textbf {A}=0$ will hold for each $n>0$ . This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions $\mathbb {N}^2\to \mathbb {Z}$ which are indexed in turn by n-tuples of functions $f:\mathbb {N}\to \mathbb {N}$ . The triviality and coherence in question here generalise the classical and well-studied case of $n=1$ .

Cyclic operators with finite support

2012 ◽  
Vol 193 (1) ◽  
pp. 131-167 ◽  
Author(s):  
F. Bayart ◽  
G. Costakis
Keyword(s):  
Finite Support ◽  
Cyclic Operators

Products of some special compact spaces and restricted forms of AC

2010 ◽  
Vol 75 (3) ◽  
pp. 996-1006 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis
Keyword(s):  
Set Theory ◽  
Closed Subset ◽  
Choice Function ◽  
Ordinal Number ◽  
Axiom Of Choice ◽  
Finite Support ◽  
Compact Spaces ◽  
The Axiom Of Choice ◽  
Infinite Set

AbstractWe establish the following results:1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent:(a) The Tychonoff product of ∣α∣ many non-empty finite discrete subsets of I is compact.(b) The union of ∣α∣ many non-empty finite subsets of I is well orderable.2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1]Iwhich consists offunctions with finite support is compact, is not provable in ZF set theory.3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF0 (i.e., ZF minus the Axiom of Regularity).4. The statement: For every set I, every ℵ0-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ0many non-empty finite discrete subsets of I is compact, is not provable in ZF0.

Rigidity of measures invariant under the action of a multiplicative semigroup of polynomial growth on 𝕋

2009 ◽  
Vol 30 (1) ◽  
pp. 151-157 ◽  
Author(s):  
MANFRED EINSIEDLER ◽  
ALEXANDER FISH
Keyword(s):  
Probability Measure ◽  
Polynomial Growth ◽  
Borel Probability Measure ◽  
Finite Support ◽  
Multiplicative Semigroup ◽  
Logarithmic Density ◽  
Circle Group ◽  
Borel Probability

AbstractWe prove that if a Borel probability measure on the circle group is invariant under the action of a ‘large’ multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then the measure is either Lebesgue or has finite support.

The Discrete Optimal Assignment Problem

2016 ◽  
Author(s):  
Alfred Galichon
Keyword(s):  
Linear Programming ◽  
Assignment Problem ◽  
Linear Programming Problem ◽  
Matrix Algebra ◽  
Probability Distributions ◽  
Marginal Probability ◽  
Finite Support ◽  
Complementary Slackness ◽  
Finite Dimensional ◽  
Finite Dimensional Case

This chapter considers the finite-dimensional case, which is the case when the marginal probability distributions are discrete with finite support. In this case, the Monge–Kantorovich problem becomes a finite-dimensional linear programming problem; the primal and the dual solutions are related by complementary slackness, which is interpreted in terms of stability. The solutions can be conveniently computed by linear programming solvers, and the chapter shows how this is done using some matrix algebra and Gurobi.

scholarly journals On pseudo-distributive near-rings

1985 ◽  
Vol 28 (2) ◽  
pp. 133-141 ◽  
Author(s):  
Gordon Mason
Keyword(s):  
Group Ring ◽  
Finite Support ◽  
First Case ◽  
Image Position

If G is a group and N a ring, the elements of the group ring NG can be thought of either as formal sums or as functions Φ:G→Nwith finite support. If N is a nearring, problems arise in trying to construct a group near-ring either way. In the first case, Meldrum [7] was abl to exploit properties of distributively generated near-rings (N, S) to build free (N,S)-products and hence a near-ring analogue of a group ring. For the latter case, Heatherly and Ligh [3] observed that the set of functions could be made into a near-ring under multiplication given by provided N satisfiesfor all ai,bin∈N and k∈Z+. Such near-rings are called pseudo-distributive. In fact these are precisely the conditions under which the set Nk of k x k matrices over N is also a near-ring and then both NG and Nk are pseudo-distributive.

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