Piecewise Monotone Research Articles

Energy Harvesting Transmitters That Heat Up: Throughput Maximization Under Temperature Constraints

Motivated by damage due to heating in sensor operation, we consider the throughput optimal offline data scheduling problem in an energy harvesting transmitter such that the resulting temperature increase remains below a critical level. We model the temperature dynamics of the transmitter as a linear system and determine the optimal transmit power policy under such temperature constraints as well as energy harvesting constraints over an AWGN channel. We first derive the structural properties of the solution for the general case with multiple energy arrivals. We show that the optimal power policy is piecewise monotone decreasing with possible jumps at the energy harvesting instants. We derive analytical expressions for the optimal solution in the single energy arrival case. We show that, in the single energy arrival case, the optimal power is monotone decreasing, the resulting temperature is monotone increasing, and both remain constant after the temperature hits the critical level. We then generalize the solution for the multiple energy arrival case.

Copulas related to piecewise monotone functions of the interval and associated processes

ABSTRACTIn this work, we derive the copulas related to vectors obtained from the so-called chaotic stochastic processes. These are defined by the iteration of certain piecewise monotone functions of the interval [0, 1] to some initial random variable. We study some of its properties and present some examples. Since often these types of copulas do not have closed formulas, we provide a general approximation method which converges uniformly to the true copula. Our results cover a wide class of processes, including the so-called Manneville–Pomeau processes. The general theory is applied to the parametric estimation in certain chaotic processes. A Monte Carlo simulation study is also presented.

Computing the topological entropy of continuous maps with at most three different kneading sequences with applications to Parrondo’s paradox

We introduce an algorithm to compute the topological entropy of piecewise monotone maps with at most three different kneading sequences, with prescribed accuracy. As an application, we compute the topological entropy of 3-periodic sequences of logistic maps, disproving a commutativity formula for topological entropy with three maps, and analyzing the dynamics Parrondo’s paradox in this setting.

The Milnor–Thurston Determinant and the Ruelle Transfer Operator

The topological entropy $h_{\rm top}$ of a continuous piecewise monotone interval map measures the exponential growth in the number of monotonicity intervals for iterates of the map. Milnor and Thurston showed that $\exp(-h_{\rm top})$ is the smallest zero of an analytic function, now coined the Milnor-Thurston determinant, that keeps track of relative positions of forward orbits of critical points. On the other hand $\exp(h_{\rm top})$ equals the spectral radius of a Ruelle transfer operator $L$, associated with the map. Iterates of $L$ keep track of inverse orbits of the map. For no obvious reason, a Fredholm determinant for the transfer operator has not only the same leading zero as the M-T determinant but all peripheral (those lying in the unit disk) zeros are the same. The purpose of this note is to show that on a suitable function space, the dual of the Ruelle transfer operator has a regularized determinant, identical to the Milnor-Thurston determinant, hereby providing a natural explanation for the above puzzle.

Kneading with weights

We generalize Milnor–Thurston's kneading theory to the setting of piecewise continuous and monotone interval maps with a weight associated to each branch. We define a weighted kneading determinant {\cal D}(t) and establish combinatorially two kneading identities, one with the cutting invariant and one with the dynamical zeta function. For the pressure \log \rho_1 of the weighted system, playing the role of entropy, we prove that {\cal D}(t) is non-zero when |t|<1/\rho_1 and has a zero at 1/\rho_1 . Furthermore, our map is semi-conjugate to every map in an analytic family \mathcal S_t, 0<t<1/\rho_1 of piecewise linear maps with slopes proportional to the prescribed weights and defined on a Cantor set. When the original map extends to a continuous map f , the family \mathcal S_t converges as t\rightarrow 1/\rho_1 to a continuous piecewise linear interval map \tilde{f} . Furthermore, f is semi-conjugate to \tilde{f} and the two maps have the same pressure.

A representation result for rate-independent systems

The representation formula for hysteresis operators acting on scalar-valued continuous input functions being piecewise monotone that was derived in Brokate and Sprekels (1996) [1] has been extended to results for hysteresis operators acting on vectorial input function, see Klein (2012, 2014) [2–5].In the current paper, the representation result is extended to rate-independent systems, as considered for example in Mielke (2005) [9]. The input functions are requested to be continuous and to be piecewise strictly monotaffine, i.e. to be piecewise the composition of a strictly monotone increasing function with an affine function, the strictly monotone function being applied first.

Nonlinear Variation of Constants Formula for Differential Equations with State-Dependent Delays

In this paper we consider a class of differential equations with state-dependent delays. We show differentiability of the solution with respect to the initial function and the initial time for each fixed time value assuming that the state-dependent time lag function is piecewise monotone increasing. Based on these results, we prove a nonlinear variation of constants formula for differential equations with state-dependent delay. As an application, we discuss asymptotic properties of perturbed nonlinear differential equations with state-dependent delays.

Semiconjugacy to a map of a constant slope

It is well known that a continuous piecewise monotoneinterval map with positive topological entropy is semiconjugate to amap of a constant slope and the same entropy, and if it isadditionally transitive then this semiconjugacy is actually aconjugacy. We generalize this result topiecewise continuous piecewise monotone interval maps, andas a consequence, get it also for piecewise monotone graph maps.We show that assigning to a continuous transitive piecewisemonotone map of positive entropy a map of constant slope conjugate toit defines an operator, and show that this operator is not continuous.

Pesin theory and equilibrium measures on the interval

We use Pesin theory to study possible equilibrium measures for piecewise monotone maps of the interval. The maps may have unbounded derivative.

Exact circle maps and KMS states

We describe the KMS states and the ground states for the gauge action on the C*-algebra of the oriented transformation groupoid of a continuous piecewise monotone and exact map of the circle.

On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions

In Brokate-Sprekels 1996, it is shown thathysteresis operators acting onscalar-valued, continuous, piecewise monotone input functionscan be represented by functionals acting on alternating strings. In a number of recent papers, this representation result is extended to hysteresis operators dealing with input functions in a general topological vector space. The input functions have to be continuous and piecewise monotaffine, i.e. being piecewise the composition of two functions such that the output of a monotone increasing function is used as input for an affine function.  &nbsp In the current paper, a representation result is formulated for hysteresis operators dealing with input functions being left-continuous and piecewise monotaffine and continuous. The operators are generated by functions acting on an admissible subset of the set of all strings of pairs of elements of the vector space.

No semiconjugacy to a map of constant slope

We study countably piecewise continuous, piecewise monotone interval maps. We establish a necessary and sufficient criterion for the existence of a non-decreasing semiconjugacy to a map of constant slope in terms of the existence of an eigenvector of an operator acting on a space of measures. Then we give sufficient conditions under which this criterion is not satisfied. Finally, we give examples of maps not semiconjugate to a map of constant slope via a non-decreasing map. Our examples are continuous and transitive.

Topological conjugacy of piecewise monotonic functions of nonmonotonicity height ≥1

The conjugacy problem is an important topic in the theory of dynamical systems and functional equations. In this paper, we investigate a class of piecewise monotone and continuous maps with nonmonotonicity height ≥1. We give a sufficient and necessary condition under which any two of these maps are topologically conjugate, and construct a topological conjugacy with an extension method if such a conjugacy exists.

Spaces of transitive interval maps

On a compact real interval, the spaces of all transitive maps, all piecewise monotone transitive maps and all piecewise linear transitive maps are considered with the uniform metric. It is proved that they are contractible and uniformly locally arcwise connected. Then the spaces of all piecewise monotone transitive maps with given number of pieces as well as various unions of such spaces are considered and their connectedness properties are studied.

On essential entropy-carrying sets

In this paper we study properties of essential entropy-carrying sets of a continuous map on a compact metric space. If f:X→X is continuous on a compact metric space X, then the intersection of all essential entropy-carrying sets of f may or may not be an essential entropy-carrying set of f. When this intersection is an essential entropy-carrying set we denote it by E(f), the least essential entropy-carrying set, otherwise we say that E(f) does not exist. We present an example where E(f) does not exist but also find a sufficient condition for E(f) to exist. If f is a piecewise monotone map, we show that E(f) exists and is the finite union of the entropy-carrying sets in the Nitecki Decomposition of the nonwandering set of f intersected with the closure of the periodic points of f. When E(f) exists we study how it relates to other entropy-carrying sets of f including subsets of itself.

Computing topological entropy for periodic sequences of unimodal maps

In this paper we introduce an algorithm which allows us to compute the topological entropy of a class of piecewise monotone continuous interval maps. The algorithm can be applied to a class of economic models called duopolies, and it can be useful to compute the topological entropy of periodic sequences of continuous maps which have been used in some population growth models.

Correlation range query for effective recommendations

Efficient correlation computation has been an active research area of data mining. Given a large dataset and a specified query item, we are interested in finding items in the dataset that are within certain range of correlation with the query item. Such a problem, known as the correlation range query, has been a common task in many application domains. In this paper, we identify piecewise monotone properties of the upper and lower bounds of the ? coefficient, and propose an efficient correlation range query algorithm, called CORAQ. The CORAQ algorithm effectively prunes many items without computing their actual correlation coefficients with the query item. CORAQ also attains completeness and correctness of the query results. Experiments with large benchmark datasets show that this algorithm is much faster than its brute-force alternative and scales well with large datasets. As case studies, real-world datasets from recommendation applications are analyzed to demonstrate that CORAQ can help effectively identify interesting items to recommend to users.

Circle maps and -algebras

AbstractWe consider a construction of${C}^{\ast } $-algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and Anantharaman-Delaroche for local homeomorphisms. Assuming that the map is surjective and not locally injective we give necessary and sufficient conditions for the simplicity of the${C}^{\ast } $-algebra and show that it is then a Kirchberg algebra. We provide tools for the calculation of the$K$-theory groups and turn them into an algorithmic method for Markov maps.

Distributions of Order Patterns of Interval Maps

A permutation σ describing the relative orders of the first n iterates of a point x under a self-map f of the interval I=[0,1] is called an order pattern. For fixed f and n, measuring the points x ∈ I (according to Lebesgue measure) that generate the order pattern σ gives a probability distribution μn(f) on the set of length n permutations. We study the distributions that arise this way for various classes of functions f.Our main results treat the class of measure-preserving functions. We obtain an exact description of the set of realizable distributions in this case: for each n this set is a union of open faces of the polytope of flows on a certain digraph, and a simple combinatorial criterion determines which faces are included. We also show that for general f, apart from an obvious compatibility condition, there is no restriction on the sequence {μn(f)}n=1,2,. . ..In addition, we give a necessary condition for f to have finite exclusion type, that is, for there to be finitely many order patterns that generate all order patterns not realized by f. Using entropy we show that if f is piecewise continuous, piecewise monotone, and either ergodic or with points of arbitrarily high period, then f cannot have finite exclusion type. This generalizes results of S. Elizalde.

Real polynomial iterative roots in the case of nonmonotonicity height ⩾ 2

It is known that a strictly piecewise monotone function with nonmonotonicity height ⩾ 2 on a compact interval has no iterative roots of order greater than the number of forts. An open question is: Does it have iterative roots of order less than or equal to the number of forts? An answer was given recently in the case of “equal to”. Since many theories of resultant and algebraic varieties can be applied to computation of polynomials, a special class of strictly piecewise monotone functions, in this paper we investigate the question in the case of “less than” for polynomials. For this purpose we extend the question from a compact interval to the whole real line and give a procedure of computation for real polynomial iterative roots. Applying the procedure together with the theory of discriminants, we find all real quartic polynomials of non-monotonicity height ⩾ 2 which have quadratic polynomial iterative roots of order 2 and answer the question.