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 Multiple solutions for critical Choquard-Kirchhoff type equations

2020 ◽  
Vol 10 (1) ◽  
pp. 400-419 ◽  
Author(s):  
Sihua Liang ◽  
Patrizia Pucci ◽  
Binlin Zhang
Keyword(s):  
Critical Exponent ◽  
Critical Exponents ◽  
Sobolev Inequality ◽  
Multiple Solutions ◽  
Concentration Compactness ◽  
Positive Real ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Concentration Compactness Principle ◽  
Compactness Principle

Abstract In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(ℝN), with r = 2∗/(2∗ − q) if 1 < q < 2* and r = ∞ if q ≥ 2∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

CRITICAL EXPONENTS FOR ANTIFERROMAGNETIC SPIN CHAINS OBTAINED FROM BOSONISATION

2012 ◽  
Vol 11 ◽  
pp. 183-190 ◽  
Author(s):  
MARCEL KOSSOW ◽  
PETER SCHUPP ◽  
STEFAN KETTEMANN
Keyword(s):  
Critical Exponent ◽  
Quantum Hall Effect ◽  
Critical Exponents ◽  
Spin Chain ◽  
Quantum Hall ◽  
Spin Chains ◽  
Special Focus ◽  
Heisenberg Spin ◽  
First Order ◽  
Antiferromagnetic Spin

The Heisenberg spin 1/2 chain is revisited in the perturbative RG approach with special focus on the transition of the critical exponents. We give a compact review that first order RG in the couplings is sufficient to derive the exact transition from ν = 1 to ν = 2/3, if the boson radius obtained in the bosonization procedure is replaced by the exact radius obtained in the Bethe approach. We explain the fact, that from the bosonization procedure alone, the critical exponent can not be derived correctly in the isotropic limit Jz → J. We further state that this fact is important if we consider to bosonize the antiferromagnetic super spin chain for the quantum Hall effect.

scholarly journals A Study of the Mutual Diffusion Coefficient in a Binary Liquid Mixture, Using a Laser Beating Technique

1983 ◽  
Vol 3 (1-6) ◽  
pp. 333-337
Author(s):  
F. M. El-Mekawey
Keyword(s):  
Diffusion Coefficient ◽  
Critical Temperature ◽  
Critical Exponent ◽  
Wide Temperature Range ◽  
Critical Exponents ◽  
Liquid Mixture ◽  
Binary Liquid Mixture ◽  
Mutual Diffusion Coefficient ◽  
Binary Liquid ◽  
Line Widths

Line-widths of light scattered by nitrobenzene-heptane system are measured over a wide temperature range including the vicinity of the critical temperature of separation. The critical exponents are measured. Noncoincident value of the critical exponent for the states of the system above and below the critical temperature of separation are obtained.

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.

New Characteristic Relations in Type-II and III Intermittency

1997 ◽  
Vol 07 (04) ◽  
pp. 831-836 ◽  
Author(s):  
M. O. Kim ◽  
Hoyun Lee ◽  
Chil-Min Kim ◽  
Hyun-Soo Pang ◽  
Eok-Kyun Lee ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Lower Bound ◽  
Probability Distribution ◽  
Critical Exponent ◽  
Simulation Study ◽  
Critical Exponents ◽  
Control Parameter ◽  
The Other ◽  
Square Root ◽  
Type Ii

We obtained new characteristic relations in Type-II and III intermittencies according to the reinjection probability distribution. When the reinjection probability distribution is fixed at the lower bound of reinjection, the critical exponents are -1, as is well known. However when the reinjection probability distribution is uniform, the critical exponent is -1/2, and when it is of form [Formula: see text], -3/4. On the other hand, if the square root of Δ, which represents the lower bound of reinjection, is much smaller than the control parameter ∊, i.e. ∊ ≫ Δ1/2, critical exponent is always -1, independent of the reinjection probability distribution. Those critical exponents are confirmed by numerical simulation study.

scholarly journals Application of Entropy Compression in Pattern Avoidance

10.37236/3038 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Pascal Ochem ◽  
Alexandre Pinlou
Keyword(s):  
Positive Answer ◽  
Pattern Avoidance ◽  
Algebraic Combinatorics ◽  
Combinatorics On Words ◽  
Letter Alphabet ◽  
Infinite Word ◽  
Entropy Compression

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $f$ of $w$ such that $f= h(p)$ where $h: \Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. We give a positive answer to Problem 3.3.2 in Lothaire's book "Algebraic combinatorics on words'", that is, every pattern with $k$ variables of length at least $2^k$ (resp. $3\times2^{k-1}$) is 3-avoidable (resp. 2-avoidable). This conjecture was first stated by Cassaigne in his thesis in 1994. This improves previous bounds due to Bell and Goh, and Rampersad.

scholarly journals On Repetition Thresholds of Caterpillars and Trees of Bounded Degree

10.37236/6793 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Borut Lužar ◽  
Pascal Ochem ◽  
Alexandre Pinlou
Keyword(s):  
Real Number ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Letter Alphabet ◽  
Graph Classes ◽  
Infinite Word

The repetition threshold is the smallest real number $\alpha$ such that there exists an infinite word over a $k$-letter alphabet that avoids repetition of exponent strictly greater than $\alpha$. This notion can be generalized to graph classes. In this paper, we completely determine the repetition thresholds for caterpillars and caterpillars of maximum degree $3$. Additionally, we present bounds for the repetition thresholds of trees with bounded maximum degrees.

The Critical Amplitude for Bifurcation Points of the Logistic Map

1996 ◽  
Vol 51 (12) ◽  
pp. 1170-1174
Author(s):  
Takashi Tsuchiya
Keyword(s):  
Critical Exponent ◽  
Critical Exponents ◽  
Amplitude Ratio ◽  
Logistic Map ◽  
Adjustable Parameter ◽  
Critical Slowing Down ◽  
Critical Amplitude ◽  
Bifurcation Points ◽  
Slowing Down

Abstract The phenomenon called critical slowing down is studied for some analytically tractable bifurcation points of the logistic map. The critical amplitude and the critical amplitude ratio are introduced to describe the critical behaviors more precisely when the critical exponents are identically unity for two critical behaviors which are to be compared. Since Hao predicted unity of the critical exponent for 1-d maps assuming exponential damping of the distance from the attractor, the assumption is checked. The result is that the shrinkage of the valid region of the assumption occurs as the adjustable parameter approaches one of the bifurcation points.

scholarly journals Avoiding conjugacy classes on the 5-letter alphabet

2020 ◽  
Vol 54 ◽  
pp. 2
Author(s):  
Golnaz Badkobeh ◽  
Pascal Ochem
Keyword(s):  
Conjugacy Class ◽  
Cyclic Permutation ◽  
Conjugacy Classes ◽  
Letter Alphabet ◽  
Infinite Word

We construct an infinite word w over the 5-letter alphabet such that for every factor f of w of length at least two, there exists a cyclic permutation of f that is not a factor of w. In other words, w does not contain a non-trivial conjugacy class. This proves the conjecture in Gamard et al. [Theoret. Comput. Sci. 726 (2018) 1–4].

scholarly journals Nonexistence results of solutions to systems of semilinear differential inequalities on the Heisenberg group

2004 ◽  
Vol 2004 (2) ◽  
pp. 155-164 ◽  
Author(s):  
Abdallah El Hamidi ◽  
Mokhtar Kirane
Keyword(s):  
Critical Exponent ◽  
Heisenberg Group ◽  
Critical Exponents ◽  
Differential Inequalities

We establish nonexistence results to systems of differential inequalities on the(2N+1)-Heisenberg group. The systems considered here are of the type(ESm). These nonexistence results hold forNless than critical exponents which depend onpiandγi,1≤i≤m. Our results improve the known estimates of the critical exponent.

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