The irregular set for maps with the specification property has full topological pressure

Let (X, d) be a compact metric space, f : X ? X be a continuous map with the specification property and ? : X ? ? a continuous function. We consider the set of points for which the Birkhoff average of ? does not exist (which we call the irregular set for ?) and show that this set is either empty or carries full topological pressure (in the sense of Pesin and Pitskel see, for example [Y.B. Pesin, Dimension Theory in Dimensional Systems: Contemporary Views and Applications, University of Chicago Press, Chicago, 1997]). We formulate various equivalent natural conditions on ? that completely describe when the latter situation holds and give examples of interesting systems to which our results apply but were not previously known. As an application, we show that for a suspension flow over a continuous map with specification, the irregular set carries full topological entropy.     

Topological entropy of continuous transitive maps on the real line

The topological entropy of a continuous transitive map f : ? → ? is studied. We find lower bounds for the topological entropy of f and prove that transitive real maps can have finite entropy.     

An Approximation for the Entropy Measuring in the General Structure of<i>SGSP</i>₃

In this article, we calculate various topological invariants such as symmetric division degree index, redefined Zagreb index, VL index, first and second exponential Zagreb index, first and second multiplicative exponential Zagreb indices, symmetric division degree entropy, redefined Zagreb entropy, VL entropy, first and second exponential Zagreb entropies, multiplicative exponential Zagreb entropy. We take the chemical compound named Proanthocyanidins, which is a very useful polyphenol in human’s diet. They are very beneficial for one’s health. These chemical compounds are extracted from grape seeds. They are tremendously anti-inflammatory. A subdivision form of this compound is presented in this article. The compound named subdivided grape seed Proanthocyanidins is abbreviated as SGSP₃. This network SGSP₃, is converted and modeled into its mathematical graphical formation with the support of the latest mathematical tools. We have also developed many closed formulas for the measurement of entropy for the general chemical structure of the subdivided grape seed Proanthocyanidins network. The achieved outcomes can be correlated with the chemical version of SGSP₃ to get a better understanding of its biological as well as physical features.     

Frontiers and symmetries of dynamical systems

The frontiers of boundedness ? _b of the orbits of dynamical systems X defined on ? ⁿ are studied. When X is completely integrable some topological properties of ? _b are found and, in certain cases, ? _b is localized with the help of symmetries of X. Several examples in dimensions 2 and 3 are provided. In case the number of known first integrals of the vector field X is less than n ? 1, an interesting connection of ? _b with the frontier of boundedness of the level-sets of the first integrals of X is proved. This result also applies to Hamiltonian systems.     

Some properties of topological pressure of a semigroup of continuous maps

Based on the work of Ma and Wu,[9 Ma D, Wu M. Topological pressure and topological entropy of a semigroup of maps. Discrete Contin Dyn Syst. 2011;31:545–557.[Crossref], [Web of Science ®] , [Google Scholar]] some equivalent definitions of topological pressure of a semigroup of continuous maps are given and several of their basic properties are provided by using separated sets and spanning sets. We also answer an open problem of Bi? and Urbański.[19 Bi? A, Urbański M. Some remarks on topological entropy of a semigroup of continuous maps. Cubo. 2006;8:63–71. [Google Scholar]] That is, letting f_i, i = 2, … , k, be homeomorphisms acting on a compact metric space, G₁ = {id_X, f₂, … , f_k}, G^{? 1}₁ = {id_X, f^{? 1}₂, …, f_k^{? 1}} and letting G and G^?1 denote the semigroups generated by G₁ and G^{? 1}₁, respectively, we give an example showing that the topological entropy of G does not equal the topological entropy of G^?1.     

Different forms of entropy dimension for zero entropy systems

The aim of this paper is to introduce the lower s-topological entropy to distinguish zero entropy systems. That this quantity is an invariant factor under topological conjugacy and a power rule is shown. Some examples are given to show that the lower entropy dimension can attain any value in (0, 1), and are different with the upper one and the entropy dimension in the sense of Bowen. A counterexample is used to indicate that the product rule does not hold, and the lower s-topological entropy of the subsystem for the non-wandering set can be strictly less than that of the system when 0 < s < 1. Finally, this study also constructs a dynamical system to show that the transitive system with zero entropy dimension may not be minimal.     

Regular and Inverse interval Estimation of Individual Observations Using Uniform Confidence Bands

Bowden and Graybill construct uniform confidence bands under the familiar simple linear regression model, and use those bands to write joint confidence intervals for expected Y given X = X_o for all X_o in a specified finite range X_a ≤ X_o ≤ X_b . Their approach is extended here to provide joint confidence intervals for individual observations on Y given X, and also for individual X given Y (the inverse interval estimation problem), in both cases using uniform confidence bands constructed for X_a ≤ X ≤ X_b . We show that Bowden and Graybill's tables can be used in these applications, and give supplementary tables to cover additional cases which more commonly arise in these new contexts. Asymptotic results are also given, allowing the user to generate additional “table” values.

The relative efficiencies of the uniform and Scheffé procedures for inverse interval estimation of individual X given Y are extensively compared. It seems that one need not pay a price in efficiency for the convenience of the uniform procedure.     

Topological Quantum Phase Transition and Superconductivity Induced by Pressure in the Bismuth Tellurohalide BiTeI

A pressure‐induced topological quantum phase transition has been theoretically predicted for the semiconductor bismuth tellurohalide BiTeI with giant Rashba spin splitting. In this work, evolution of the electrical transport properties in BiTeI and BiTeBr is investigated under high pressure. The pressure‐dependent resistivity in a wide temperature range passes through a minimum at around 3 GPa, indicating the predicted topological quantum phase transition in BiTeI. Superconductivity is observed in both BiTeI and BiTeBr, while resistivity at higher temperatures still exhibits semiconducting behavior. Theoretical calculations suggest that superconductivity may develop from the multivalley semiconductor phase. The superconducting transition temperature, T_c, increases with applied pressure and reaches a maximum value of 5.2 K at 23.5 GPa for BiTeI (4.8 K at 31.7 GPa for BiTeBr), followed by a slow decrease. The results demonstrate that BiTeX (X = I, Br) compounds with nontrivial topology of electronic states display new ground states upon compression.     

Dynamical intricacy and average sample complexity

We propose a new way to measure the balance between freedom and coherence in a dynamical system and a new measure of its internal variability. Based on the concept of entropy and ideas from neuroscience and information theory, we define intricacy and average sample complexity for topological and measure-preserving dynamical systems. We establish basic properties of these quantities, show that their suprema over covers or partitions equal the ordinary entropies, compute them for many shifts of finite type, and indicate natural directions for further research.     

A topological classification of D-dimensional cellular automata

K?rka proposed a topological classification for cellular automata (CAs) on the one-dimensional full shift space [D. Lind and B. Marcus. An Introduction to Symbolic Dynamics and Coding. New York: Cambridge University Press, 1995]. and we extended part of this classification to higher dimensional subshift spaces in [R.H. Gilman. Periodic behaviour of linear automata, in Dynamical Systems: Proceedings of the Special Year Held at the University of Maryland, College Park, 1986–1987, J.C. Alexander, ed., Lecture Notes in Mathematics, 1342, Berlin: Springer, pp. 216–219, 1988]. Here, we focus on expansiveness, showing that there exist subshift spaces in all dimensions on which there expansive CAs. This counters a result of Shereshevsky that there are no expansive CAs on full shift spaces in dimension greater than one [J. Milnor. On the entropy geometry of cellular automata. Complex Syst. 2 : 357–386, 1988]. We also supplement portions of [R.H. Gilman. Periodic behaviour of linear automata, in Dynamical systems: Proceedings of the Special Year Held at the University of Maryland, College Park, 1986–1987 J.C. Alexander, ed., Lecture Notes in Mathematics, 1342, Berlin: Springer, pp. 216–219, 1988] in order to give a topological classification of CAs on D-dimensional subshift spaces.     

Maximum entropy and discretization of probability distributions

Using gaussian quadrature we can find m concentrations of probability that replace the density function of a random variable X and match 2m - 1 of its moments. This reduces a probabilistic analysis to m deterministic ones. Even small values of m provide excellent accuracy in many practical circumstances. When fewer than 2m - 1 moments are known there is arbitrariness in the choice of the concentrations, which is overcome by resorting to the maximum entropy formalism. Its use is here systematized for the case in which αXb and we know N moments of the density of X, so that calculation of N - 1 integrals suffices for finding the density function and any number of its moments. The approach is illustrated for m = 2 and 3, N = 2, 3, α = 0, B = ∞ and graphs are provided for finding the equivalent concentrations.     

Two-level modified simulated annealing based approach for solving facility layout problem

In this paper, we are considering the quadratic assignment model (QAP) of the facility layout problem (FLP) which is known to be NP-hard. We relax the integer constraints of the QAP and solve it on a commercially available package called LINGO 8. In the optimal solution so obtained, X_ij s take real values between zero and one. We identify promising X_ij s having a value strictly greater than 0.5 in the optimal solution and set them to one. We add the constraints (X_ij ?=?1) associated with promising X_ij s into the QAP (with integer restrictions) and resolve using LINGO 8. In all the cases attempted we obtained a superior feasible solution to the QAP which was further improved by the proposed modified simulated annealing (MSA) procedure. An encouraging comparative performance of this procedure is thus reported.     

A good property of the maximum likelihood estimator in a restricted normal model

Summary We consider a normal model with known diagonal covariance matrix and a vector of means constrained to belong to a polyhedral cone. The standard estimatorsX (unrestricted MLE) andX ^* (restricted MLE) are compared for estimation of several components of the parameter simultaneously. We show thatX ^* is preferred toX under several conditions.     

Intermittent estimation of stationary time series

Let {X _n } _n=0 ^∞ be a stationary real-valued time series with unknown distribution. Our goal is to estimate the conditional expectation ofX _n+1 based on the observations,X _i , 0≤i≤n in a strongly consistent way. Bailey and Ryabko proved that this is not possible even for ergodic binary time series if one estimates at all values ofn. We propose a very simple algorithm which will make prediction infinitely often at carefully selected stopping times chosen by our rule. We show that under certain conditions our procedure is strongly (pointwise) consistent, andL ₂ consistent without any condition. An upper bound on the growth of the stopping times is also presented in this paper.     

Isomers of the C84 fullerene: A theoretical consideration within energetic,structural, and topological approaches

Currently, there is no common conception on the stability and achievability of fullerene isomers under the arc-discharge synthesis conditions. Different approaches operate with energetic, structural, and topological parameters of the fullerene molecules to explain why some fullerene isomers are more preferable than the others. In the present work, we have selected the most appropriate approaches based on topological roundness, information entropy, nuclear volumes, and sphericities and compared their predictions with the relevant experimental data on the C₈₄ fullerene isomers (obeying the isolated pentagon rule) and density functional theory estimates of their stability. We have found that the molecules of most stable (and most abundant) C₈₄ isomers have the minimal extremal roundness, maximal sphericity, and largest volume and vice versa. In the case of the information entropy, no correlation is observed. Interestingly, the found correlation between the volume and the stability of the C₈₄ isomers is unexpectedly inverse, i.e. more stable C₈₄ isomers have larger inner cavities inside, though traditionally large volume is associated with instability of hollow framework molecules. We assume that the large volumes allows enhancing the sphericity of the fullerene cages as we found the last one favoring the stability and the unstable C₈₄ fullerene cages having low volumes are far from the spherical shape. We think that the results obtained may be extrapolated to other fullerene isomeric sets and contribute to the understanding the grounds of the interconnection in the triad “topology – structure – energy” underlying structural chemistry.     

A compact adjacency‐based topological data structure for finite element mesh representation

This paper presents a novel compact adjacency‐based topological data structure for finite element mesh representation. The proposed data structure is designed to support, under the same framework, both two‐ and three‐dimensional meshes, with any type of elements defined by templates of ordered nodes. When compared to other proposals, our data structure reduces the required storage space while being ‘complete’, in the sense that it preserves the ability to retrieve all topological adjacency relationships in constant time or in time proportional to the number of retrieved entities. Element and node are the only entities explicitly represented. Other topological entities, which include facet, edge, and vertex, are implicitly represented. In order to simplify accessing topological adjacency relationships, we also define and implicitly represent oriented entities, associated to the use of facets, edges, and vertices by an element. All implicit entities are represented by concrete types, being handled as values, which avoid usual problems encountered in other reduced data structures when performing operations such as entity enumeration and attribute attachment. We also extend the data structure with the use of ‘reverse indices’, which improves performance for extracting adjacency relationships while maintaining storage space within reasonable limits. The data structure effectiveness is demonstrated by two different applications: for supporting fragmentation simulation and for supporting volume rendering algorithms. Copyright © 2005 John Wiley & Sons, Ltd.     

The Berlekamp-Massey algorithm and linear recurring sequences over a factorial domain

We present an extended polynomial remainder sequence algorithm XPRS for R[X] whereR is a domain. From this we derive a Berlekamp-Massey algorithm BM/R overR. We show that if () is a linear recurring sequence in a factorial domainU, then the characteristic polynomials for () form aprincipal ideal which is generated by a primitive minimal polynomial. Moreover, this generator ismonic when U[[X]] is factorial (for example, whenU is Z orK[X ₁,X₂,...,X_n] whereK is a field). From XPRS we derive an algorithm MINPOL for determining the minimal polynomial of () when an upper bound on the degree of some characteristic polynomial and sufficiently many initial terms of () are known. We also show how to obtain a Berlekamp-Massey type minimal polynomial algorithm from BM/U and state BM_MINPOL/K explicitly with a further refinement. Examples are given forU=Z, GF(2)[Y].A preliminary version of this paper [7] was presented at the Third Int. Coll. on Coding Theory and Applications, Toulon, Nov. 1988.     

A Selection Procedure Using a Screening Variate

Consider N objects on which two correlated measurements X and Y can be made and where the measurement of X is easier or less expensive to make than the one of Y. Suppose that the probability that the Y measurement meets a certain specification is γ. We present a method, which uses the conditional probabilities that Y will meet the specification given the observed value of the x's, for choosing a maximal subset of m out of the N objects, based on the observed X measurements, so that there is a high probability that a large proportion (Π ≥ γ) of the selected subset will meet the desired specification. The procedure is useful for helping to meet a guaranteed proportion of satisfactory items. It compares favorably with one suggested by Owen, Li, and Chou (1981).