scholarly journals For each $\alpha > 2$ there is an Infinite Binary Word with Critical Exponent $\alpha$

10.37236/909 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
James D. Currie ◽  
Narad Rampersad
Keyword(s):  
Critical Exponent ◽  
Rational Numbers ◽  
Alpha Power ◽  
Binary Word ◽  
Infinite Word ◽  
Open Question

The critical exponent of an infinite word ${\bf w}$ is the supremum of all rational numbers $\alpha$ such that ${\bf w}$ contains an $\alpha$-power. We resolve an open question of Krieger and Shallit by showing that for each $\alpha > 2$ there is an infinite binary word with critical exponent $\alpha$.

scholarly journals On the Constructibility of Real 5th Roots of Rational Numbers with Marked Ruler and Compass

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Elliot Benjamin
Keyword(s):  
Rational Numbers ◽  
Real Numbers ◽  
Real Roots ◽  
Open Question

We demonstrate that there are infinitely many real numbers constructible by marked ruler and compass which are unique real roots of irreducible quintic polynomials over the field of rational numbers. This result can be viewed as a generalization of the historical open question of the constructibility by marked ruler and compass of real 5th roots of rational numbers. We obtain our results through marked ruler and compass constructions involving the intersection of conchoids and circles, and the application of number theoretic divisibility criteria.

scholarly journals THE CRITICAL EXPONENT IS COMPUTABLE FOR AUTOMATIC SEQUENCES

2012 ◽  
Vol 23 (08) ◽  
pp. 1611-1626 ◽  
Author(s):  
LUKE SCHAEFFER ◽  
JEFFREY SHALLIT
Keyword(s):  
Critical Exponent ◽  
Rational Number ◽  
Automatic Sequences ◽  
Infinite Word ◽  
Automatic Sequence

The critical exponent of an infinite word is defined to be the supremum of the exponent of each of its factors. For k-automatic sequences, we show that this critical exponent is always either a rational number or infinite, and its value is computable. Our results also apply to variants of the critical exponent, such as the initial critical exponent of Berthé, Holton, and Zamboni and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes or recovers previous results of Krieger and others, and is applicable to other situations; e.g., the computation of the optimal recurrence constant for a linearly recurrent k-automatic sequence.

scholarly journals Extremal Infinite Overlap-Free Binary Words

10.37236/1365 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Jean-Paul Allouche ◽  
James Currie ◽  
Jeffrey Shallit
Keyword(s):  
Fixed Point ◽  
Binary Word ◽  
Infinite Word

Let $\overline{\bf t}$ be the infinite fixed point, starting with $1$, of the morphism $\mu: 0 \rightarrow 01$, $1 \rightarrow 10$. An infinite word over $\lbrace 0, 1 \rbrace$ is said to be overlap-free if it contains no factor of the form $axaxa$, where $a \in \lbrace 0,1 \rbrace$ and $x \in \lbrace 0,1 \rbrace^*$. We prove that the lexicographically least infinite overlap-free binary word beginning with any specified prefix, if it exists, has a suffix which is a suffix of $\overline{\bf t}$. In particular, the lexicographically least infinite overlap-free binary word is $001001 \overline{\bf t}$.

scholarly journals Critical Exponents of Words over 3 Letters

10.37236/612 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Elise Vaslet
Keyword(s):  
Critical Exponent ◽  
Critical Exponents ◽  
Letter Alphabet ◽  
Infinite Word

For all $\alpha \geq RT(3)$ (where $RT(3) = 7/4$ is the repetition threshold for the $3$-letter alphabet), there exists an infinite word over 3 letters whose critical exponent is $\alpha$.

scholarly journals On the critical exponent of generalized Thue-Morse words

2007 ◽  
Vol Vol. 9 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Alexandre B. Massé ◽  
Srečko Brlek ◽  
Amy Glen ◽  
Sébastien Labbé
Keyword(s):  
Critical Exponent ◽  
Rational Numbers ◽  
International Audience

Automata, Logic and Semantics International audience For certain generalized Thue-Morse words t, we compute the critical exponent, i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.

A NEW COMPLEXITY FUNCTION FOR WORDS BASED ON PERIODICITY

2013 ◽  
Vol 23 (04) ◽  
pp. 963-987 ◽  
Author(s):  
FILIPPO MIGNOSI ◽  
ANTONIO RESTIVO
Keyword(s):  
The Other ◽  
Factorization Theorem ◽  
Complexity Measures ◽  
Infinite Words ◽  
Average Value ◽  
Complexity Function ◽  
Binary Word ◽  
Infinite Word

Motivated by the extension of the critical factorization theorem to infinite words, we study the (local) periodicity function, i.e. the function that, for any position in a word, gives the size of the shortest square centered in that position. We prove that this function characterizes any binary word up to exchange of letters. We then introduce a new complexity function for words (the periodicity complexity) that, for any position in the word, gives the average value of the periodicity function up to that position. The new complexity function is independent from the other commonly used complexity measures as, for instance, the factor complexity. Indeed, whereas any infinite word with bounded factor complexity is periodic, we will show a recurrent non-periodic word with bounded periodicity complexity. Further, we will prove that the periodicity complexity function grows as Θ( log n) in the case of the Fibonacci infinite word and that it grows as Θ(n) in the case of the Thue–Morse word. Finally, we will show examples of infinite recurrent words with arbitrary high periodicity complexity.

Mars: was There an Ancient Eden?

1997 ◽  
Vol 161 ◽  
pp. 203-218 ◽  
Author(s):  
Tobias C. Owen
Keyword(s):  
Positive Answer ◽  
Biogenic Elements ◽  
Habitable Zones ◽  
The Face ◽  
The Origin Of Life ◽  
Early Mars ◽  
Favorable Conditions ◽  
Open Question ◽  
Rocky Planets ◽  
Impact Erosion

AbstractThe clear evidence of water erosion on the surface of Mars suggests an early climate much more clement than the present one. Using a model for the origin of inner planet atmospheres by icy planetesimal impact, it is possible to reconstruct the original volatile inventory on Mars, starting from the thin atmosphere we observe today. Evidence for cometary impact can be found in the present abundances and isotope ratios of gases in the atmosphere and in SNC meteorites. If we invoke impact erosion to account for the present excess of129Xe, we predict an early inventory equivalent to at least 7.5 bars of CO2. This reservoir of volatiles is adequate to produce a substantial greenhouse effect, provided there is some small addition of SO2(volcanoes) or reduced gases (cometary impact). Thus it seems likely that conditions on early Mars were suitable for the origin of life – biogenic elements and liquid water were present at favorable conditions of pressure and temperature. Whether life began on Mars remains an open question, receiving hints of a positive answer from recent work on one of the Martian meteorites. The implications for habitable zones around other stars include the need to have rocky planets with sufficient mass to preserve atmospheres in the face of intensive early bombardment.

Aberrant Sensorimotor Integration in Musicians' Cramp Patients

2003 ◽  
Vol 17 (4) ◽  
pp. 195-202 ◽  
Author(s):  
Vanessa K. Lim ◽  
John L. Bradshaw ◽  
Michael E.R. Nicholls ◽  
Ian J. Kirk ◽  
Jeff P. Hamm ◽  
...  
Keyword(s):  
Movement Disorder ◽  
Sensorimotor Cortex ◽  
Sensorimotor Integration ◽  
Group Interaction ◽  
Focal Dystonia ◽  
The Other ◽  
Alpha Power ◽  
Significant Group ◽  
Group Interactions

AbstractSimple tapping and complex movements (Luria finger apposition task) were performed unimanually and bimanually by two groups of professional guitarists while EEG was recorded from electrodes over the sensorimotor cortex. One group had a task-specific movement disorder (focal dystonia or musicians' cramp), while the other group did not (controls). There were no significant group interactions in the task-related power (TRPow) within the alpha range of 8-10Hz (mu1). In contrast, there was a significant group interaction within the alpha range of 10-12Hz (mu2); these latter frequencies are associated with task-specific sensorimotor integration. The significant group interaction included task (simple and complex) by hand (left, right, and both) by electrodes (10 electrodes over the sensorimotor areas). In the rest conditions, the alpha power (10-12Hz) was comparable between the groups; during movement, however, compared to the controls, patients demonstrated the greatest TRPow (10-12Hz) over all conditions. This was particularly evident when patients used their affected hand and suggests that patients with musicians' cramp have impaired task-specific sensorimotor integration.

On the Semantic Content of Grammatical Gender and Its Impact on the Representation of Human Referents

2010 ◽  
Vol 57 (5) ◽  
pp. 367-375 ◽  
Author(s):  
Lisa Irmen ◽  
Julia Kurovskaja
Keyword(s):  
Semantic Content ◽  
Grammatical Gender ◽  
Differential Effects ◽  
Gender Information ◽  
Masculine Role ◽  
Feminine Role ◽  
And Gender ◽  
Gender Categories ◽  
Open Question

Grammatical gender has been shown to provide natural gender information about human referents. However, due to formal and conceptual differences between masculine and feminine forms, it remains an open question whether these gender categories influence the processing of person information to the same degree. Experiment 1 compared the semantic content of masculine and feminine grammatical gender by combining masculine and feminine role names with either gender congruent or incongruent referents (e.g., Dieser Lehrer [masc.]/Diese Lehrerin [fem.] ist mein Mann/meine Frau; This teacher is my husband/my wife). Participants rated sentences in terms of correctness and customariness. In Experiment 2, in addition to ratings reading times were recorded to assess processing more directly. Both experiments were run in German. Sentences with grammatically feminine role names and gender incongruent referents were rated as less correct and less customary than those with masculine forms and incongruent referents. Combining a masculine role name with an incongruent referent slowed down reading to a greater extent than combining a feminine role name with an incongruent referent. Results thus specify the differential effects of masculine and feminine grammatical gender in denoting human referents.

Die Europäische Union: eine neue Supermacht mit imperialen Ambitionen?

2005 ◽  
Vol 35 (139) ◽  
pp. 247-266 ◽  
Author(s):  
Hans-Jürgen Bieling
Keyword(s):  
World War Ii ◽  
Economic Crisis ◽  
Europäische Union ◽  
World War ◽  
Political Sovereignty ◽  
Military Conflicts ◽  
Imperial Power ◽  
International Role ◽  
Open Question ◽  
The Eu

Recent theoretical conceptions of imperialism may be useful correctives against idealising and harmonising views of international interdependency and co-operation. Analytically, however, they are not necessarily helpful. In terms of the EU, they do not really comprehend its particular international role. Despite improved financial and military capacities, the EU represents not yet an imperial power. Instead, it still pursues a rather hegemonic foreign policy approach due to internal economic restrictions, fragmented political sovereignty and the historical experiences of beneficial economic and political co-operation after World War II. Eventually, however, it remains an open question, whether the multilateralist, law-based and co-operative posture of the EU will prevail even under conditions of economic crisis and further military conflicts in the adjacent neighbourhood.

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