On the accuracy of solving confluent Prony systems

In this paper we consider several nonlinear systems of algebraic equations which can be called "Prony-type". These systems arise in various reconstruction problems in several branches of theoretical and applied mathematics, such as frequency estimation and nonlinear Fourier inversion. Consequently, the question of stability of solution with respect to errors in the right-hand side becomes critical for the success of any particular application. We investigate the question of "maximal possible accuracy" of solving Prony-type systems, putting stress on the "local" behavior which approximates situations with low absolute measurement error. The accuracy estimates are formulated in very simple geometric terms, shedding some light on the structure of the problem. Numerical tests suggest that "global" solution techniques such as Prony's algorithm and ESPRIT method are suboptimal when compared to this theoretical "best local" behavior

Natural equilibrium states for multimodal maps

This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials $-t \log|Df|$, for the largest possible interval of parameters $t$. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained

Phase transitions for suspension flows

This paper is devoted to study thermodynamic formalism for suspension flows defined over countable alphabets. We are mostly interested in the regularity properties of the pressure function. We establish conditions for the pressure function to be real analytic or to exhibit a phase transition. We also construct an example of a potential for which the pressure has countably many phase transitions.Comment: Example 5.2 expanded. Typos corrected. Section 6.1 superced the note "Thermodynamic formalism for the positive geodesic flow on the modular surface" arXiv:1009.462

On conformal measures and harmonic functions for group extensions

We prove a Perron-Frobenius-Ruelle theorem for group extensions of topological Markov chains based on a construction of $\sigma$-finite conformal measures and give applications to the construction of harmonic functions.Comment: To appear in Proceedings of "New Trends in Onedimensional Dynamics, celebrating the 70th birthday of Welington de Melo

Renormalisation-induced phase transitions for unimodal maps

The thermodynamical formalism is studied for renormalisable maps of the interval and the natural potential $-t \log|Df|$. Multiple and indeed infinitely many phase transitions at positive $t$ can occur for some quadratic maps. All unimodal quadratic maps with positive topological entropy exhibit a phase transition in the negative spectrum.Comment: 14 pages, 2 figures. Revised following comments of referees. First page is blan

Equilibrium states and invariant measures for random dynamical systems

Random dynamical systems with countably many maps which admit countable Markov partitions on complete metric spaces such that the resulting Markov systems are uniformly continuous and contractive are considered. A non-degeneracy and a consistency conditions for such systems, which admit some proper Markov partitions of connected spaces, are introduced, and further sufficient conditions for them are provided. It is shown that every uniformly continuous Markov system associated with a continuous random dynamical system is consistent if it has a dominating Markov chain. A necessary and sufficient condition for the existence of an invariant Borel probability measure for such a non-degenerate system with a dominating Markov chain and a finite (16) is given. The condition is also sufficient if the non-degeneracy is weakened with the consistency condition. A further sufficient condition for the existence of an invariant measure for such a consistent system which involves only the properties of the dominating Markov chain is provided. In particular, it implies that every such a consistent system with a finite Markov partition and a finite (16) has an invariant Borel probability measure. A bijective map between these measures and equilibrium states associated with such a system is established in the non-degenerate case. Some properties of the map and the measures are given.Comment: The article is published in DCDS-A, but without the 3rd paragraph on page 4 (the complete removal of the paragraph became the condition for the publication in the DCDS-A after the reviewer ran out of the citation suggestions collected in the paragraph

The Analyticity of a Generalized Ruelle's Operator

In this work we propose a generalization of the concept of Ruelle operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle operator, that generalizes both the Ruelle operator proposed in [BCLMS] and the Perron Frobenius operator defined in [Bowen]. We suppose the alphabet is given by a compact metric space, and consider a general a-priori measure to define the operator. We also consider the case where the set of symbols that can follow a given symbol of the alphabet depends on such symbol, which is an extension of the original concept of transition matrices from the theory of subshifts of finite type. We prove the analyticity of the Ruelle operator and present some examples

Secondary dentin formation mechanism: The effect of attrition

Human dentin consists of a primary layer produced during tooth formation in early child-hood and a second layer which first forms upon tooth eruption and continues throughout life, termed secondary dentin (SD). The effect of attrition on SD formation was considered to be confined to the area subjacent to attrition facets. However, due to a lack of three‐dimensional methodologies to demonstrate the structure of the SD, this association could not be determined. Therefore, in the current study, we aimed to explore the thickening pattern of the SD in relation to the amount of occlusal and interproximal attrition. A total of 30 premolars (50–60 years of age) with varying attrition rates were evaluated using micro‐computerized tomography. The results revealed thickening of the SD below the cementoenamel junction (CEJ), mostly in the mesial and distal aspects of the root (p < 0.05). The pattern of thickening under the tooth cervix, rather than in proximity to attrition facets, was consistent regardless of the attrition level. The amount of SD thickening mildly corre-lated with occlusal attrition (r = 0.577, p < 0.05) and not with interproximal attrition. The thickening of the SD below the CEJ coincided with previous finite element models, suggesting that this area is mostly subjected to stress due to occlusal loadings. Therefore, we suggest that the SD formation might serve as a compensatory mechanism aimed to strengthen tooth structure against deflection caused by mechanical loading. Our study suggests that occlusal forces may play a significant role in SD formation

Moment inversion problem for piecewise D-finite functions

We consider the problem of exact reconstruction of univariate functions with jump discontinuities at unknown positions from their moments. These functions are assumed to satisfy an a priori unknown linear homogeneous differential equation with polynomial coefficients on each continuity interval. Therefore, they may be specified by a finite amount of information. This reconstruction problem has practical importance in Signal Processing and other applications. It is somewhat of a ``folklore'' that the sequence of the moments of such ``piecewise D-finite''functions satisfies a linear recurrence relation of bounded order and degree. We derive this recurrence relation explicitly. It turns out that the coefficients of the differential operator which annihilates every piece of the function, as well as the locations of the discontinuities, appear in this recurrence in a precisely controlled manner. This leads to the formulation of a generic algorithm for reconstructing a piecewise D-finite function from its moments. We investigate the conditions for solvability of the resulting linear systems in the general case, as well as analyze a few particular examples. We provide results of numerical simulations for several types of signals, which test the sensitivity of the proposed algorithm to noise

Managing Injuries of the Neck Trial (MINT) : design of a randomised controlled trial of treatments for whiplash associated disorders

Background: A substantial proportion of patients with whiplash injuries develop chronic symptoms. However, the best treatment of acute injuries to prevent long-term problems is uncertain. A stepped care treatment pathway has been proposed, in which patients are given advice and education at their initial visit to the emergency department (ED), followed by review at three weeks and physiotherapy for those with persisting symptoms. MINT is a two-stage randomised controlled trial to evaluate two components of such a pathway: 1. use of The Whiplash Book versus usual advice when patients first attend the emergency department; 2. referral to physiotherapy versus reinforcement of advice for patients with continuing symptoms at three weeks. Methods: Evaluation of the Whiplash Book versus usual advice uses a cluster randomised design in emergency departments of eight NHS Trusts. Eligible patients are identified by clinicians in participating emergency departments and are sent a study questionnaire within a week of their ED attendance. Three thousand participants will be included. Patients with persisting symptoms three weeks after their ED attendance are eligible to join an individually randomised study of physiotherapy versus reinforcement of the advice given in ED. Six hundred participants will be randomised. Follow-up is at 4, 8 and 12 months after their ED attendance. Primary outcome is the Neck Disability Index (NDI), and secondary outcomes include quality of life and time to return to work and normal activities. An economic evaluation is being carried out. Conclusion: This paper describes the protocol and operational aspects of a complex intervention trial based in NHS emergency and physiotherapy departments, evaluating two components of a stepped-care approach to the treatment of whiplash injuries. The trial uses two randomisations, with the first stage being cluster randomised and the second individually randomised