scholarly journals A general renormalization procedure on the one-dimensional lattice and decay of correlations

2019 ◽  
Vol 19 (01) ◽  
pp. 1950003
Author(s):  
Artur O. Lopes
Keyword(s):  
Fixed Point ◽  
Final Section ◽  
Polynomial Decay ◽  
Renormalization Procedure ◽  
Real Numbers ◽  
One Dimensional ◽  
Point Potential ◽  
Renormalization Operator ◽  
The One ◽  
Natural Way

We present a general form of renormalization operator [Formula: see text] acting on potentials [Formula: see text]. We exhibit the analytical expression of the fixed point potential [Formula: see text] for such operator [Formula: see text]. This potential can be expressed in a natural way in terms of a certain integral over the Hausdorff probability on a Cantor type set on the interval [0,1]. This result generalizes a previous one by Baraviera, Leplaideur and Lopes where the fixed point potential [Formula: see text] was of Hofbauer type. For the potentials of Hofbauer type (a well-known case of phase transition) the decay is like [Formula: see text], [Formula: see text]. Among other things we present the estimation of the decay of correlation of the equilibrium probability associated to the fixed potential [Formula: see text] of our general renormalization procedure. In some cases we get polynomial decay like [Formula: see text], [Formula: see text], and in others a decay faster than [Formula: see text], when [Formula: see text]. The potentials [Formula: see text] we consider here are elements of the so-called family of Walters’ potentials on [Formula: see text] which generalizes a family of potentials considered initially by Hofbauer. For these potentials some explicit expressions for the eigenfunctions are known. In a final section we also show that given any choice [Formula: see text] of real numbers varying with [Formula: see text] there exists a potential [Formula: see text] on the class defined by Walters which has a invariant probability with such numbers as the coefficients of correlation (for a certain explicit observable function).

A Pappus Type Theorem in the Affine Group

1968 ◽  
Vol 11 (4) ◽  
pp. 547-554
Author(s):  
R. Paré
Keyword(s):  
Projective Plane ◽  
Final Section ◽  
Type Theorem ◽  
Affine Group ◽  
Real Numbers ◽  
One Dimensional ◽  
Master's Thesis ◽  
Geometrical Concepts ◽  
The One ◽  
The Relationship

In [3] H. Schwerdtfeger embedded the one-dimensional affine group over the real numbers in the projective plane. The relationship between group-theoretical properties and geometrical concepts was studied.In this paper the methods of [3] are used to prove Pappus' theorem. In the final section we give a similar theorem for (4n+2)-gons.This paper is a generalization of part of my master's thesis, written under the direction of Professor H. Schwerdtfeger.

scholarly journals Chaotic period doubling

2009 ◽  
Vol 29 (2) ◽  
pp. 381-418 ◽  
Author(s):  
V. V. M. S. CHANDRAMOULI ◽  
M. MARTENS ◽  
W. DE MELO ◽  
C. P. TRESSER
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Chaotic Dynamics ◽  
A Priori ◽  
Period Doubling ◽  
Small Scale ◽  
Unimodal Maps ◽  
One Dimensional ◽  
Renormalization Operator ◽  
Renormalization Theory

AbstractThe period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that is also hyperbolic among generic smooth-enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that, in the space ofC2+αunimodal maps, forα>0, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space ofC2unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to geta prioribounds. In this smoother class, calledC2+∣⋅∣, the failure of hyperbolicity is tamer than inC2. Things get much worse with just a bit less smoothness thanC2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space ofC1+Lipunimodal maps has infinite topological entropy.

scholarly journals Marcinkiewicz-type strong law of large numbers for double arrays of pairwise independent random variables

1999 ◽  
Vol 22 (1) ◽  
pp. 171-177 ◽  
Author(s):  
Dug Hun Hong ◽  
Seok Yoon Hwang
Keyword(s):  
Law Of Large Numbers ◽  
Double Sequence ◽  
Random Variables ◽  
Independent Random Variables ◽  
Real Numbers ◽  
Strong Law ◽  
One Dimensional ◽  
Large Numbers ◽  
Pairwise Independent Random Variables ◽  
The One

Let {Xij}be a double sequence of pairwise independent random variables. If P{|Xmn|≥t}≤P{|X|≥t}for all nonnegative real numbers tandE|X|p(log+|X|)3<∞, for1<p<2, then we prove that∑i=1m∑j=1n(Xij−EXij)(mn)1/p→0    a.s.   as  m∨n→∞.                                     (0.1)Under the weak condition ofE|X|plog+|X|<∞, it converges to 0inL1. And the results can be generalized to anr-dimensional array of random variables under the conditionsE|X|p(log+|X|)r+1<∞,E|X|p(log+|X|)r−1<∞, respectively, thus, extending Choi and Sung's result [1] of the one-dimensional case.

Projective Geometry in the One-Dimensional Affine Group

1964 ◽  
Vol 16 ◽  
pp. 683-700 ◽  
Author(s):  
Hans Schwerdtfeger
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Simple Group ◽  
Final Section ◽  
Theoretical Interpretation ◽  
Special Group ◽  
One Dimensional ◽  
The One ◽  
Straight Lines ◽  
Geometrical Investigation

The idea of considering the set of the elements of a group as a space, provided with a topology, measure, or metric, connected somehow with the group operation, has been used often in the work of E. Cartan and others. In the present paper we shall study a very special group whose space can be embedded naturally into a projective plane and where the straight lines have a very simple group-theoretical interpretation. It remains to be seen whether this geometrical embedding in a projective space can be extended to other classes of groups and whether the method could become an instrument of geometrical investigation, like co-ordinates or reflections. In the final section it is shown how a geometrical theorem may lead to relations within the group.

RELATION BETWEEN A CLASS OF QUASIPERIODIC LATTICES GENERATED BY THE SUBSTITUTION METHOD AND THE GENERALIZED CONTINUED FRACTION EXPANSION

1996 ◽  
Vol 10 (17) ◽  
pp. 2081-2101
Author(s):  
TOSHIO YOSHIKAWA ◽  
KAZUMOTO IGUCHI
Keyword(s):  
Continued Fraction ◽  
Positive Real Number ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Positive Real ◽  
Periodic Sequences ◽  
One Dimensional ◽  
Finite Period ◽  
The One ◽  
Positive Integers

The continued fraction expansion for a positive real number is generalized to that for a set of positive real numbers. For arbitrary integer n≥2, this generalized continued fraction expansion generates (n−1) sequences of positive integers {ak}, {bk}, … , {yk} from a given set of (n−1) positive real numbers α, β, …ψ. The sequences {ak}, {bk}, … ,{yk} determine a sequence of substitutions Sk: A → Aak Bbk…Yyk Z, B → A, C → B,…,Z → Y, which constructs a one-dimensional quasiperiodic lattice with n elements A, B, … , Z. If {ak}, {bk}, … , {yk} are infinite periodic sequences with an identical period, then the ratio between the numbers of n elements A, B, … , Z in the lattice becomes a : β : … : ψ : 1. Thereby the correspondence is established between all the sets of (n−1) positive real numbers represented by a periodic generalized continued fraction expansion and all the one-dimensional quasiperiodic lattices with n elements generated by a sequence of substitutions with a finite period.

scholarly journals Optimal Polynomial Decay to Coupled Wave Equations and Its Numerical Properties

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
R. F. C. Lobato ◽  
S. M. S. Cordeiro ◽  
M. L. Santos ◽  
D. S. Almeida Júnior
Keyword(s):  
Decay Rate ◽  
Spectral Methods ◽  
Finite Differences ◽  
Wave Equations ◽  
Coupled System ◽  
Polynomial Decay ◽  
One Dimensional ◽  
Coupled Wave Equations ◽  
Optimal Polynomial ◽  
The One

In this work we consider a coupled system of two weakly dissipative wave equations. We show that the solution of this system decays polynomially and the decay rate is optimal. Computational experiments are conducted in the one-dimensional case in order to show that the energies properties are preserved. In particular, we use finite differences and also spectral methods.

scholarly journals Inverse multiparameter eigenvalue problems for matrices II

1986 ◽  
Vol 29 (3) ◽  
pp. 343-348 ◽  
Author(s):  
Patrick J. Browne ◽  
B. D. Sleeman
Keyword(s):  
Fixed Point ◽  
Eigenvalue Problems ◽  
Symmetric Matrix ◽  
Sufficient Conditions ◽  
Degree Theory ◽  
Main Tool ◽  
Real Numbers ◽  
Inverse Eigenvalue Problems ◽  
Inverse Eigenvalue ◽  
The One

This is a sequel to our previous paper [4] where we initiated a study of inverse eigenvalue problems for matrices in the multiparameter setting. The one parameter version of the problem under consideration asks for conditions on a given n × n symmetric matrix A and on n given real numbers s1≦s2≦…≦sn under which a diagonal matrix V can be found so that A + V has sl,…,sn as its eigenvalues. Our motivation for this problem and our method of attack on it in [4]p comes chiefly from the work of Hadeler [5] in which sufficient conditions were given for existence of the desired diagonal V. Hadeler's approach in [5] relied heavily on the Brouwer fixed point theorem and this was also our main tool in [4]. Subsequently, using properties of topological degree, Hadeler [6] gave somewhat different conditions for the existence of the diagonal V. It is our desire here to follow this lead and to use degree theory to give some results extending those in [6] to the multiparameter case.

One-dimensional Large-U Hubbard Model in Strong Coupling: Charge and Spin Separation

1991 ◽  
Vol 05 (10) ◽  
pp. 1801-1807
Author(s):  
Z. Y. Weng ◽  
D. N. Sheng ◽  
C. S. Ting
Keyword(s):  
Fixed Point ◽  
Hubbard Model ◽  
Weak Coupling ◽  
Composite Particle ◽  
Electron Hole ◽  
Integral Formalism ◽  
One Dimensional ◽  
Coupling Case ◽  
Non Local ◽  
The One

A path-integral formalism of the Hubbard model is used to study the one-dimensional large-U case. It is shown that the bare electron (hole) becomes a composite particle of two decoupled excitations, holon and spinon, together with the non-local string fields. Various correlation functions are analytically derived. The results strongly suggest a U*=∞ fixed point of Hubbard model which is distinct from the weak coupling case.

EXTENSIONS OF THE NOTION OF CHAOTIC AREA IN SECOND-ORDER ENDOMORPHISMS

1995 ◽  
Vol 05 (03) ◽  
pp. 751-777 ◽  
Author(s):  
A. BARUGOLA ◽  
J.C. CATHALA ◽  
C. MIRA
Keyword(s):  
Fixed Point ◽  
Finite Number ◽  
Periodic Point ◽  
Critical Points ◽  
Unstable Manifold ◽  
Two Dimensional ◽  
One Dimensional ◽  
Critical Curves ◽  
The One ◽  
Invariant Domains

Properties of chaotic areas (i.e. invariant domains of points positively stable in the Poisson’s sense) of non-invertible maps of the plane are studied by using the method of critical curves (two-dimensional extension of the notion of critical points in the one-dimensional case). The classical situation is that of a chaotic area bounded by a finite number of critical curves segments. This paper considers another class of chaotic areas bounded by the union of critical curves segments and segments of the unstable manifold of a saddle fixed point, or that of saddle cycle (periodic point). Different configurations are examined, as their bifurcations when a map parameter varies.

scholarly journals Triple positive solutions for the one-dimensional $p$-laplacian

2006 ◽  
Vol 37 (1) ◽  
pp. 15-25
Author(s):  
Zhanbing Bai ◽  
Mingfu Ma ◽  
Xiangqian Liang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Nonlinear Term ◽  
Sufficient Conditions ◽  
One Dimensional ◽  
First Order ◽  
Triple Positive Solutions ◽  
The One

We consider the boundary value problem: $ \left(\varphi_p(x'(t))\right)'+ q(t)f(t, x(t), x'(t))=0, p>1, t \in [0, 1] $, with $ x(0)=x(1)=0 $, or $ x(0)=x'(1)=0 $. Using a fixed point theorem due to Avery and Peterson, sufficient conditions are obtained that guarantee the existence of at least three positive solutions. The emphasis here is the nonlinear term $ f $ is involved with the first order derivative. An example is also included to illustrate the importance of the results obtained

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