scholarly journals Periodicity and Chaos Amidst Twisting and Folding in Two-Dimensional Maps

2018 ◽  
Vol 28 (04) ◽  
pp. 1830012
Author(s):  
Swier Garst ◽  
Alef E. Sterk
Keyword(s):  
Periodic Point ◽  
Period Doubling ◽  
Stable Fixed Point ◽  
The Third ◽  
Folded Structure ◽  
Real World Applications ◽  
Countably Infinite ◽  
Noninvertible Maps ◽  
Infinite Set ◽  
Critical Lines

We study the dynamics of three planar, noninvertible maps which rotate and fold the plane. Two maps are inspired by real-world applications whereas the third map is constructed to serve as a toy model for the other two maps. The dynamics of the three maps are remarkably similar. A stable fixed point bifurcates through a Hopf–Neĭmark–Sacker which leads to a countably infinite set of resonance tongues in the parameter plane of the map. Within a resonance tongue a periodic point can bifurcate through a period-doubling cascade. At the end of the cascade we detect Hénon-like attractors which are conjectured to be the closure of the unstable manifold of a saddle periodic point. These attractors have a folded structure which can be explained by means of the concept of critical lines. We also detect snap-back repellers which can either coexist with Hénon-like attractors or which can be formed when the saddle-point of a Hénon-like attractor becomes a source.

Monotone but not positive subsets of the Cantor space

1987 ◽  
Vol 52 (3) ◽  
pp. 817-818 ◽  
Author(s):  
Randall Dougherty
Keyword(s):  
Partial Ordering ◽  
Background Information ◽  
Countable Union ◽  
Abstract Form ◽  
Cantor Space ◽  
Countable Intersection ◽  
Power Set ◽  
Countably Infinite ◽  
Infinite Set

A subset of the Cantor space ω2 is called monotone iff it is closed upward under the partial ordering ≤ defined by x ≤ y iff x(n) ≤ y(n) for all n ∈ ω. A set is -positive (-positive) iff it is monotone and -positive set is a countable union of -positive sets; a -positive set is a countable intersection of -positive sets. (See Cenzer [2] for background information on these concepts.) It is clear that any -positive set is and monotone; the converse holds for n ≤ 2 [2] and was conjectured by Dyck to hold for greater n. In this note, we will disprove this conjecture by giving examples of monotone sets (for n ≥ 3) which are not even -positive.First we note a few isomorphisms. The space (ω2, ≤) is isomorphic to the space (ω2 ≥), so instead of monotone and positive sets we may construct hereditary and negative sets (the analogous notions with “closed upward” replaced by “closed downward”). Also, (ω2, ≤) is isomorphic to ((ω), ⊆), where denotes the power set operator, or to ((S), ⊆) for any countably infinite set S.In order to remove extraneous notation from the proofs, we state the results in an abstract form (whose generality is deceptive).

scholarly journals REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

2017 ◽  
Vol 82 (2) ◽  
pp. 576-589 ◽  
Author(s):  
KOSTAS HATZIKIRIAKOU ◽  
STEPHEN G. SIMPSON
Keyword(s):  
Theorem Proving ◽  
Chain Condition ◽  
Reverse Mathematics ◽  
Partial Ordering ◽  
Young Diagrams ◽  
Equivalence Proof ◽  
Image Position ◽  
Countably Infinite ◽  
Infinite Set ◽  
Ascending Chain Condition

AbstractLetSbe the group of finitely supported permutations of a countably infinite set. Let$K[S]$be the group algebra ofSover a fieldKof characteristic 0. According to a theorem of Formanek and Lawrence,$K[S]$satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over$RC{A_0}$(or even over$RCA_0^{\rm{*}}$) to the statement that${\omega ^\omega }$is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

How to Program an Infinite Abacus

1961 ◽  
Vol 4 (3) ◽  
pp. 295-302 ◽  
Author(s):  
Joachim Lambek
Keyword(s):  
Finite Number ◽  
Recursive Function ◽  
Computable Function ◽  
Unlimited Supply ◽  
Countably Infinite ◽  
Infinite Set

This is an expository note to show how an “infinite abacus” (to be defined presently) can be programmed to compute any computable (recursive) function. Our method is probably not new, at any rate, it was suggested by the ingenious technique of Melzak [2] and may be regarded as a modification of the latter.By an infinite abacus we shall understand a countably infinite set of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads etc.). The locations are distinguishable, the counters are not. The confirmed finitist need not worry about these two infinitudes: To compute any given computable function only a finite number of locations will be used, and this number does not depend on the argument (or arguments) of the function.

Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set

2017 ◽  
Vol 29 (4) ◽  
Author(s):  
Tiwadee Musunthia ◽  
Jörg Koppitz
Keyword(s):  
Natural Numbers ◽  
Maximal Subsemigroups ◽  
Countably Infinite ◽  
Infinite Set

AbstractIn this paper, we study the maximal subsemigroups of several semigroups of order-preserving transformations on the natural numbers and the integers, respectively. We determine all maximal subsemigroups of the monoid of all order-preserving injections on the set of natural numbers as well as on the set of integers. Further, we give all maximal subsemigroups of the monoid of all bijections on the integers. For the monoid of all order-preserving transformations on the natural numbers, we classify also all its maximal subsemigroups, containing a particular set of transformations.

scholarly journals RAID-6Plus: A Comprised Methodology for Extending RAID-6 Codes

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Ming-Zhu Deng ◽  
Nong Xiao ◽  
Song-Ping Yu ◽  
Fang Liu ◽  
Lingyu Zhu ◽  
...  
Keyword(s):  
Real World ◽  
Load Balance ◽  
Underlying Mechanism ◽  
Patterns Of Failure ◽  
The Third ◽  
Real World Applications ◽  
Failure Scenario ◽  
Reliability Performance ◽  
Occurrence Patterns

Existing RAID-6 code extensions assume that failures are independent and instantaneous, overlooking the underlying mechanism of multifailure occurrences. Also, the effect of reconstruction window is ignored. Additionally, these coding extensions have not been adapted to occurrence patterns of failure in real-world applications. As a result, the third parity drive is set to handle the triple-failure scenario; however, the lower level failure situations have been left unattended. Therefore, a new methodology of extending RAID-6 codes named RAID-6Plus with better compromise has been studied in this paper. RAID-6Plus (Deng et al., 2015) employs short combinations which can greatly reuse overlapped elements during reconstruction to remake the third parity drive. A sample extension code called RDP+ is given based on RDP. Moreover, we extended the study to present another extension example called X-code+ which has better update penalty and load balance. The analysis shows that RAID-6Plus is a balanced tradeoff of reliability, performance, and practicality. For instance, RDP+ could achieve speedups as high as 33.4% in comparison to the RTP with conventional rebuild, 11.9% in comparison to RTP with the optimal rebuild, 47.7% in comparison to STAR with conventional rebuild, and 26.2% for a single failure rebuild.

The extensions of the modal logic K5

1985 ◽  
Vol 50 (1) ◽  
pp. 102-109 ◽  
Author(s):  
Michael C. Nagle ◽  
S. K. Thomason
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Abstract Characterization ◽  
Propositional Modal Logic ◽  
Normal Logic ◽  
Countably Infinite ◽  
Infinite Set

Our purpose is to delineate the extensions (normal and otherwise) of the propositional modal logic K5. We associate with each logic extending K5 a finitary index, in such a way that properties of the logics (for example, inclusion, normality, and tabularity) become effectively decidable properties of the indices. In addition we obtain explicit finite axiomatizations of all the extensions of K5 and an abstract characterization of the lattice of such extensions.This paper refines and extends the Ph.D. thesis [2] of the first-named author, who wishes to acknowledge his debt to Brian F. Chellas for his considerable efforts in directing the research culminating in [2] and [3]. We also thank W. J. Blok and Gregory Cherlin for observations which greatly simplified the proofs of Theorem 3 and Corollary 10.By a logic we mean a set of formulas in the countably infinite set Var of propositional variables and the connectives ⊥, →, and □ (other connectives being used abbreviatively) which contains all the classical tautologies and is closed under detachment and substitution. A logic is classical if it is also closed under RE (from A↔B infer □A ↔□B) and normal if it is classical and contains □ ⊤ and □ (P → q) → (□p → □q). A logic is quasi-classical if it contains a classical logic and quasi-normal if it contains a normal logic. Thus a quasi-normal logic is normal if and only if it is classical, and if and only if it is closed under RN (from A infer □A).

On Outer-Commutator Words

1974 ◽  
Vol 26 (3) ◽  
pp. 608-620 ◽  
Author(s):  
Jeremy Wilson
Keyword(s):  
Arbitrary Group ◽  
Verbal Subgroup ◽  
Reduced Word ◽  
Image Position ◽  
Countably Infinite ◽  
Outer Commutator ◽  
Infinite Set

Let F be the group freely generated by the countably infinite set X = {x1, x2, . . . ,xi, . . . }. Let w(x1, x2, . . . , xn) be a reduced word representing an element of F and let G be an arbitrary group. Then V(w, G) will denote the setwhose elements will be called values of w in G. The subgroup of G generated by V(w, G) will be called the verbal subgroup of G with respect to w and be denoted by w(G).

Infinite-dimensional stochastic difference equations for particle systems and network flows

1989 ◽  
Vol 26 (02) ◽  
pp. 325-344
Author(s):  
R. W. R. Darling
Keyword(s):  
Neural Network ◽  
Flow Model ◽  
Network Flows ◽  
Particle Systems ◽  
Stochastic Flow ◽  
Random Vectors ◽  
Infinite Dimensional ◽  
Countably Infinite ◽  
Infinite Set ◽  
Number Of Births

Let V be a countably infinite set, and let {Xn, n = 0, 1, ·· ·} be random vectors in which satisfy Xn = AnXn – 1 + ζ n , for i.i.d. random matrices {An } and i.i.d. random vectors {ζ n }. Interpretation: site x in V is occupied by Xn (x) particles at time n; An describes random transport of existing particles, and ζ n (x) is the number of ‘births' at x. We give conditions for (1) convergence of the sequence {Xn } to equilibrium, and (2) a central limit theorem for n–1/2(X 1 + · ·· + Xn ), respectively. When the matrices {An } consist of 0's and 1's, these conditions are checked in two classes of examples: the ‘drip, stick and flow model' (a stochastic flow with births), and a neural network model.

Order arguments on the dimension of vector lattices

1981 ◽  
Vol 89 (1-2) ◽  
pp. 51-53
Author(s):  
John T. Annulis
Keyword(s):  
Complete Space ◽  
Infinite Dimensional ◽  
Vector Lattices ◽  
Countably Infinite ◽  
Infinite Set

SynopsisThe main result asserts that the base of an infinite dimensional Dedekind complete space with unit contains an infinite set of disjoint elements. From this result it can be shown that the dimension of Dedekind σ -complete spaces with unit is not countably infinite.

scholarly journals Ideal clones: Solution to a problem of Czédli and Heindorf

2010 ◽  
Vol 47 (4) ◽  
pp. 419-429
Author(s):  
Martin Goldstern ◽  
Michael Pinsker
Keyword(s):  
Affirmative Answer ◽  
Countably Infinite ◽  
Infinite Set ◽  
Lattice Of Clones

Given an infinite set X and an ideal I of subsets of X, the set of all finitary operations on X which map all (powers of) I-small sets to I-small sets is a clone. In [2], G. Czédli and L. Heindorf asked whether or not for two particular ideals I and J on a countably infinite set X, the corresponding ideal clones were a covering in the lattice of clones. We give an affirmative answer to this question.

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