The Generalized Rank of Trace Languages

2019 ◽  
Vol 30 (01) ◽  
pp. 135-169
Author(s):  
Michal Kunc ◽  
Jan Meitner
Keyword(s):  
Infinite Sequence ◽  
Fixed Number ◽  
The Other ◽  
Direct Products ◽  
Commutative Monoids ◽  
Free Products ◽  
Free Monoids ◽  
Other Hand ◽  
Rational Series ◽  
Regular Sets

Given a partially commutative alphabet and a set of words [Formula: see text], the rank of [Formula: see text] expresses the amount of shuffling required to produce a word belonging to [Formula: see text] from two words whose concatenation belongs to the closure of [Formula: see text] with respect to the partial commutation. In this paper, the notion of rank is generalized from concatenations of two words to an arbitrary fixed number of words. In this way, an infinite sequence of non-negative integers and infinity is assigned to every set of words. It is proved that in the case of alphabets defining free commutative monoids, as well as in the more general case of direct products of free monoids, sequences of ranks of regular sets are exactly non-decreasing sequences that are eventually constant. On the other hand, by uncovering a relationship between rank sequences of regular sets and rational series over the min-plus semiring, it is shown that already for alphabets defining free products of free commutative monoids, rank sequences need not be eventually periodic.

On square-free vertex colorings of graphs

2007 ◽  
Vol 44 (3) ◽  
pp. 411-422 ◽  
Author(s):  
János Barát ◽  
Péter Varjú
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Path Following ◽  
Fixed Number ◽  
The Other ◽  
Graph Colorings ◽  
Outerplanar Graphs ◽  
Free Vertex ◽  
Other Hand ◽  
Color Sequence

A sequence of symbols a1 , a2 … is called square-free if it does not contain a subsequence of consecutive terms of the form x1 , …, xm , x1 , …, xm . A century ago Thue showed that there exist arbitrarily long square-free sequences using only three symbols. Sequences can be thought of as colors on the vertices of a path. Following the paper of Alon, Grytczuk, Hałuszczak and Riordan, we examine graph colorings for which the color sequence is square-free on any path. The main result is that the vertices of any k -tree have a coloring of this kind using O ( ck ) colors if c > 6. Alon et al. conjectured that a fixed number of colors suffices for any planar graph. We support this conjecture by showing that this number is at most 12 for outerplanar graphs. On the other hand we prove that some outerplanar graphs require at least 7 colors. Using this latter we construct planar graphs, for which at least 10 colors are necessary.

scholarly journals On the minimal ramification problem for ℓ-groups

2010 ◽  
Vol 146 (3) ◽  
pp. 599-606 ◽  
Author(s):  
Hershy Kisilevsky ◽  
Jack Sonn
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Fields ◽  
Symmetric Groups ◽  
Wreath Products ◽  
The Other ◽  
Classical Groups ◽  
Direct Products ◽  
Cyclic Groups ◽  
Other Hand

AbstractLet ℓ be a prime number. It is not known whether every finite ℓ-group of rank n≥1 can be realized as a Galois group over ${\Bbb Q}$ with no more than n ramified primes. We prove that this can be done for the (minimal) family of finite ℓ-groups which contains all the cyclic groups of ℓ-power order and is closed under direct products, (regular) wreath products and rank-preserving homomorphic images. This family contains the Sylow ℓ-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not ℓ. On the other hand, it does not contain all finite ℓ-groups.

scholarly journals The system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid

2019 ◽  
Vol 19 (07) ◽  
pp. 2050137 ◽  
Author(s):  
Felix Gotti
Keyword(s):  
Finite Rank ◽  
Convex Cone ◽  
The Other ◽  
Extremal System ◽  
Commutative Monoid ◽  
Commutative Monoids ◽  
Other Hand ◽  
Sets Of Lengths

Let [Formula: see text] be an atomic monoid. For [Formula: see text], let [Formula: see text] denote the set of all possible lengths of factorizations of [Formula: see text] into irreducibles. The system of sets of lengths of [Formula: see text] is the set [Formula: see text]. On the other hand, the elasticity of [Formula: see text], denoted by [Formula: see text], is the quotient [Formula: see text] and the elasticity of [Formula: see text] is the supremum of the set [Formula: see text]. The system of sets of lengths and the elasticity of [Formula: see text] both measure how far [Formula: see text] is from being half-factorial, i.e. [Formula: see text] for each [Formula: see text]. Let [Formula: see text] denote the collection comprising all submonoids of finite-rank free commutative monoids, and let [Formula: see text]. In this paper, we study the system of sets of lengths and the elasticity of monoids in [Formula: see text]. First, we construct for each [Formula: see text] a monoid in [Formula: see text] having extremal system of sets of lengths. It has been proved before that the system of sets of lengths does not characterize (up to isomorphism) monoids in [Formula: see text]. Here we use our construction to extend this result to [Formula: see text] for any [Formula: see text]. On the other hand, it has been recently conjectured that the elasticity of any monoid in [Formula: see text] is either rational or infinite. We conclude this paper by proving that this is indeed the case for monoids in [Formula: see text] and for any monoid in [Formula: see text] whose corresponding convex cone is polyhedral.

Dynamic Elections and Ideological Polarization

2017 ◽  
Vol 25 (4) ◽  
pp. 505-534 ◽  
Author(s):  
Salvatore Nunnari ◽  
Jan Zápal
Keyword(s):  
Infinite Sequence ◽  
The Other ◽  
Political Polarization ◽  
Political Actors ◽  
Other Hand ◽  
Ideological Polarization ◽  
Multiple Issues ◽  
Two Measures ◽  
Ideal Points ◽  
Empirical Implications

How does political polarization affect the welfare of the electorate? We analyze this question using a framework in which two policy and office motivated parties compete in an infinite sequence of elections. We propose two novel measures to describe the degree of conflict among agents: antagonism is the disagreement between parties; extremism is the disagreement between each party and the representative voter. These two measures do not coincide when parties care about multiple issues. We show that forward-looking parties have an incentive to implement policies favored by the representative voter, in an attempt to constrain future challengers. This incentive grows as antagonism increases. On the other hand, extremism decreases the electorate’s welfare. We discuss the methodological and empirical implications for the existing measures of political actors’ ideal points and for the debate on elite polarization.

scholarly journals On groups Gnk, braids and Brunnian braids

2016 ◽  
Vol 25 (13) ◽  
pp. 1650078
Author(s):  
S. Kim ◽  
V. O. Manturov
Keyword(s):  
Dynamical Systems ◽  
Braid Group ◽  
General Position ◽  
Knot Theory ◽  
The Other ◽  
Geometric Meaning ◽  
Free Products ◽  
Other Hand ◽  
Abstract Structure

In [V. O. Manturov, arXiv:1501.05208v1 ], the second author defined the [Formula: see text]-free braid group with [Formula: see text] strands [Formula: see text]. These groups appear naturally as groups describing dynamical systems of [Formula: see text] particles in some “general position”. Moreover, in [V. O. Manturov and I. M. Nikonov, J. Knot Theory Ramification 24 (2015) 1541009] the second author and Nikonov showed that [Formula: see text] is closely related to classical braids. The authors showed that there are homomorphisms from the pure braids group on [Formula: see text] strands to [Formula: see text] and [Formula: see text] and they defined homomorphisms from [Formula: see text] to the free products of [Formula: see text]. That is, there are invariants for pure free braids by [Formula: see text] and [Formula: see text]. On the other hand in [D. A. Fedoseev and V. O. Manturov, J. Knot Theory Ramification 24(13) (2015) 1541005, 12 pages] Fedoseev and the second author studied classical braids with addition structures: parity and points on each strands. The authors showed that the parity, which is an abstract structure, has geometric meaning — points on strands. In [S. Kim, arXiv:submit/1548032], the first author studied [Formula: see text] with parity and points. the author constructed a homomorphism from [Formula: see text] to the group [Formula: see text] with parity. In the present paper, we investigate the groups [Formula: see text] and extract new powerful invariants of classical braids from [Formula: see text]. In particular, these invariants allow one to distinguish the non-triviality of Brunnian braids.

On the paradox of grounded classes

1955 ◽  
Vol 20 (2) ◽  
pp. 140-140 ◽  
Author(s):  
Richard Montague
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Infinite Sequence ◽  
Axiom Of Choice ◽  
The Other ◽  
Other Hand ◽  
The Family ◽  
The One ◽  
The Axiom Of Choice ◽  
Regular Classes

Mr. Shen Yuting, in this Journal, vol. 18, no. 2 (June, 1953), stated a new paradox of intuitive set-theory. This paradox involves what Mr. Yuting calls the class of all grounded classes, that is, the family of all classes a for which there is no infinite sequence b such that … ϵ bn ϵ … ϵ b2ϵb1 ϵ a.Now it is possible to state this paradox without employing any complex set-theoretical notions (like those of a natural number or an infinite sequence). For let a class x be called regular if and only if (k)(x ϵ k ⊃ (∃y)(y ϵ k · ~(∃z)(z ϵ k · z ϵ y))). Let Reg be the class of all regular classes. I shall show that Reg is neither regular nor non-regular.Suppose, on the one hand, that Reg is regular. Then Reg ϵ Reg. Now Reg ϵ ẑ(z = Reg). Therefore, since Reg is regular, there is a y such that y ϵ ẑ(z = Reg) · ~(∃z)(z ϵ z(z = Reg) · z ϵ y). Hence ~(∃z)(z ϵ ẑ(z = Reg) · z ϵ Reg). But there is a z (namely Reg) such that z ϵ ẑ(z = Reg) · z ϵ Reg.On the other hand, suppose that Reg is not regular. Then, for some k, Reg ϵ k · [1] (y)(y ϵ k ⊃ (∃z)(z ϵ k · z ϵ y)). It follows that, for some z, z ϵ k · z ϵ Reg. But this implies that (ϵy)(y ϵ k · ~(ϵw)(w ϵ k · w ϵ y)), which contradicts [1].It can easily be shown, with the aid of the axiom of choice, that the regular classes are just Mr. Yuting's grounded classes.

On surjunctive monoids

2015 ◽  
Vol 25 (04) ◽  
pp. 567-606 ◽  
Author(s):  
Tullio Ceccherini-Silberstein ◽  
Michel Coornaert
Keyword(s):  
Cellular Automata ◽  
The Other ◽  
Finite Alphabet ◽  
Finitely Generated ◽  
Commutative Monoids ◽  
Residually Finite ◽  
Finite Monoids ◽  
Other Hand ◽  
Bicyclic Monoid

A monoid M is called surjunctive if every injective cellular automata with finite alphabet over M is surjective. We show that all finite monoids, all finitely generated commutative monoids, all cancellative commutative monoids, all residually finite monoids, all finitely generated linear monoids, and all cancellative one-sided amenable monoids are surjunctive. We also prove that every limit of marked surjunctive monoids is itself surjunctive. On the other hand, we show that the bicyclic monoid and, more generally, all monoids containing a submonoid isomorphic to the bicyclic monoid are non-surjunctive.

A study of cometary eruptions through OH radio-lines

1999 ◽  
Vol 173 ◽  
pp. 249-254
Author(s):  
A.M. Silva ◽  
R.D. Miró
Keyword(s):  
Short Duration ◽  
Growth Curves ◽  
The Other ◽  
Initial Mass ◽  
Radio Lines ◽  
Other Hand ◽  
Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

Life Beyond 1MeV

1980 ◽  
Vol 38 ◽  
pp. 30-33
Author(s):  
K.H. Westmacott
Keyword(s):  
Electron Microscopy ◽  
Past Experience ◽  
The Other ◽  
Good Case ◽  
Other Hand ◽  
The U.S

Life beyond 1MeV – like life after 40 – is not too different unless one takes advantage of past experience and is receptive to new opportunities. At first glance, the returns on performing electron microscopy at voltages greater than 1MeV diminish rather rapidly as the curves which describe the well-known advantages of HVEM often tend towards saturation. However, in a country with a significant HVEM capability, a good case can be made for investing in instruments with a range of maximum accelerating voltages. In this regard, the 1.5MeV KRATOS HVEM being installed in Berkeley will complement the other 650KeV, 1MeV, and 1.2MeV instruments currently operating in the U.S. One other consideration suggests that 1.5MeV is an optimum voltage machine – Its additional advantages may be purchased for not much more than a 1MeV instrument. On the other hand, the 3MeV HVEM's which seem to be operated at 2MeV maximum, are much more expensive.

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