scholarly journals Cone exchange transformations and boundedness of orbits

2009 ◽  
Vol 30 (5) ◽  
pp. 1311-1330 ◽  
Author(s):  
PETER ASHWIN ◽  
AREK GOETZ
Keyword(s):  
Sufficient Conditions ◽  
Point Of View ◽  
Unique Ergodicity ◽  
Interval Exchange ◽  
Ergodic Properties ◽  
Interval Exchange Transformations ◽  
Piecewise Isometries ◽  
Almost All ◽  
Necessary And Sufficient ◽  
Negative Flux

AbstractWe introduce a class of two-dimensional piecewise isometries on the plane that we refer to as cone exchange transformations (CETs). These are generalizations of interval exchange transformations (IETs) to 2D unbounded domains. We show for a typical CET that boundedness of orbits is determined by ergodic properties of an associated IET and a quantity we refer to as the ‘flux at infinity’. In particular we show, under an assumption of unique ergodicity of the associated IET, that a positive flux at infinity implies unboundedness of almost all orbits outside some bounded region, while a negative flux at infinity implies boundedness of all orbits. We also discuss some examples of CETs for which the flux is zero and/or we do not have unique ergodicity of the associated IET; in these cases (which are of great interest from the point of view of applications such as dual billiards) it remains an outstanding problem to find computable necessary and sufficient conditions for boundedness of orbits.

scholarly journals Biquasitriangularity and spectral continuity

1985 ◽  
Vol 26 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Extension Property ◽  
Point Of View ◽  
Single Valued Extension Property ◽  
Continuity Theorem ◽  
Spectral Continuity ◽  
Necessary And Sufficient ◽  
Continuity Of The Spectrum

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

scholarly journals A Highly D‑efficient Spring Balance Weighing Design for an Even Number of Objects

2019 ◽  
Vol 5 (344) ◽  
pp. 17-27
Author(s):  
Małgorzata Graczyk ◽  
Bronisław Ceranka
Keyword(s):  
Optimal Design ◽  
Sufficient Conditions ◽  
Optimality Criteria ◽  
Point Of View ◽  
Necessary And Sufficient Conditions ◽  
Optimal Designs ◽  
Efficient Design ◽  
Spring Balance ◽  
Experimental Errors ◽  
Necessary And Sufficient

The problem of determining unknown measurements of objects in the model of spring balance weighing designs is presented. These designs are considered under the assumption that experimental errors are uncorrelated and that they have the same variances. The relations between the parameters of weighing designs are deliberated from the point of view of optimality criteria. In the paper, designs in which the product of the variances of estimators is possibly the smallest one, i.e. D‑optimal designs, are studied. A highly D‑efficient design in classes in which a D‑optimal design does not exist are determined. The necessary and sufficient conditions under which a highly efficient design exists and methods of its construction, along with relevant examples, are introduced.

scholarly journals Invariant measures with bounded variation densities for piecewise area preserving maps

2012 ◽  
Vol 33 (2) ◽  
pp. 624-642 ◽  
Author(s):  
YIWEI ZHANG ◽  
CONGPING LIN
Keyword(s):  
Bounded Variation ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Topological Transitivity ◽  
Borel Measures ◽  
Interval Exchange Transformations ◽  
Area Preserving Maps ◽  
Piecewise Isometries ◽  
Area Preserving ◽  
Absolutely Continuous Invariant

AbstractWe investigate the properties of absolutely continuous invariant probability measures (ACIPs), especially those measures with bounded variation densities, for piecewise area preserving maps (PAPs) on ℝd. This class of maps unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where Lebesgue measure is locally preserved. Using a functional analytic approach, we first explore the relationship between topological transitivity and uniqueness of ACIPs, and then give an approach to construct invariant measures with bounded variation densities for PWIs. Our results ‘partially’ answer one of the fundamental questions posed in [13]—to determine all invariant non-atomic probability Borel measures in piecewise rotations. When restricting PAPs to interval exchange transformations (IETs), our results imply that for non-uniquely ergodic IETs with two or more ACIPs, these ACIPs have very irregular densities, i.e. they have unbounded variation.

scholarly journals Regular and Irregular Performance Variation of Module String and Occurred Conditions for Potential Induced Degradation-Affected Crystalline Silicon Photovoltaic Power Plants

Energies ◽  
2019 ◽  
Vol 12 (22) ◽  
pp. 4230 ◽  
Author(s):  
Jingsheng Huang ◽  
Yaojie Sun ◽  
He Wang ◽  
Junjun Zhang
Keyword(s):  
Power Plants ◽  
Crystalline Silicon ◽  
Sufficient Conditions ◽  
Ethyl Vinyl ◽  
Climate Conditions ◽  
Photovoltaic Power ◽  
Performance Variation ◽  
Current Voltage ◽  
Almost All ◽  
Necessary And Sufficient

Potential induced degradation (PID) leads to power degradation, and reduces durability and reliability of solar modules. However, this problem has not been thoroughly solved so far. The results from interlaboratory and field study show contradictory fault phenomenon for PID. In this paper, PID of crystalline silicon photovoltaic power plants distributed in various climate conditions was investigated. These photovoltaic power plants consist of two types of crystalline silicon solar modules, which cover almost all kinds of front glass, ethyl vinyl acetate (EVA) and backsheet available commercially. It was found that only a few of power plants were affected by PID. By measuring current voltage characteristics of PID-affected solar modules, the real faults phenomenon was uncovered and classified into regular and irregular power degradation in a module string. The results obtained in this work show that the negative potential caused by high system voltage and stacking faults are necessary and sufficient conditions for PID occurrence for the first time. The anomalous power degradation is related to the stacking fault, which appears randomly during the crystal growth.

scholarly journals On the Option Pricing by the Binomial Model

2021 ◽  
Author(s):  
Nikos Halidias
Keyword(s):  
Sufficient Conditions ◽  
Fair Value ◽  
Point Of View ◽  
Binomial Model ◽  
No Arbitrage ◽  
Mathematical Arguments ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Probabilistic Point ◽  
Absence Of Arbitrage

In this note we study the binomial model applied to European, American and Bermudan type of derivatives. Our aim is to give the necessary and sufficient conditions under which we can define a fair value via replicating portfolios for any derivative using simple mathematical arguments and without using no arbitrage techniques. Giving suitable definitions we are able to define rigorously the fair value of any derivative without using concepts from probability theory or stochastic analysis therefore is suitable for students or young researchers. It will be clear in our analysis that if $e^{r \delta} \notin [d,u]$ then we can not define a fair value by any means for any derivative while if $d \leq e^{r \delta} \leq u$ we can. Therefore the definition of the fair value of a derivative is not so closely related with the absence of arbitrage. In the usual probabilistic point of view we assume that $d < e^{r \delta} < u$ in order to define the fair value but it is not clear what we can (or we can not) do in the cases where $e^{r \delta} \leq d$ or $e^{r \delta} \geq u$.

Deviation for interval exchange transformations

1997 ◽  
Vol 17 (6) ◽  
pp. 1477-1499 ◽  
Author(s):  
ANTON ZORICH
Keyword(s):  
Ergodic Theorem ◽  
Gauss Map ◽  
Interval Exchange ◽  
Actual Number ◽  
Interval Exchange Transformation ◽  
Exchange Transformation ◽  
Interval Exchange Transformations ◽  
Starting Point ◽  
Almost All ◽  
Uniquely Ergodic

Consider a long piece of a trajectory $x, T(x), T(T(x)), \ldots, T^{n-1}(x)$ of an interval exchange transformation $T$. A generic interval exchange transformation is uniquely ergodic. Hence, the ergodic theorem predicts that the number $\chi_i(x,n)$ of visits of our trajectory to the $i$th subinterval would be approximately $\lambda_i n$. Here $\lambda_i$ is the length of the corresponding subinterval of our unit interval $X$. In this paper we give an estimate for the deviation of the actual number of visits to the $i$th subinterval $X_i$ from one predicted by the ergodic theorem.We prove that for almost all interval exchange transformations the following bound is valid: $$ \max_{\ssty x\in X \atop \ssty 1\le i\le m} \limsup_{n\to +\infty} \frac {\log | \chi_i(x,n) -\lambda_in|}{\log n} = \frac{\theta_2}{\theta_1} < 1. $$ Roughly speaking the error term is bounded by $n^{\theta_2/\theta_1}$. The numbers $0\le \theta_2 < \theta_1$ depend only on the permutation $\pi$ corresponding to the interval exchange transformation (actually, only on the Rauzy class of the permutation). In the case of interval exchange of two intervals we obviously have $\theta_2=0$. In the case of exchange of three and more intervals the numbers $\theta_1, \theta_2$ are the two top Lyapunov exponents related to the corresponding generalized Gauss map on the space of interval exchange transformations.The limit above ‘converges to the bound’ uniformly for all $x\in X$ in the following sense. For any $\varepsilon >0$ the ratio of logarithms would be less than $\theta_2(\pi)/\theta_1(\pi)+\varepsilon $ for all $n\ge N(\varepsilon)$, where $N(\varepsilon)$ does not depend on the starting point $x\in X$.

scholarly journals XVI. Functions of positive and negative type, and their connection the theory of integral equations

1909 ◽  
Vol 209 (441-458) ◽  
pp. 415-446 ◽  
Keyword(s):  
Continuous Function ◽  
Early Stage ◽  
Sufficient Conditions ◽  
Slight Modification ◽  
Negative Type ◽  
Original Plan ◽  
Definition Of ◽  
Almost All ◽  
Necessary And Sufficient

The present memoir is the outcome of an attempt to obtain the conditions under which a given symmetric and continuous function k ( s, t ) is definite, in the sense of Hilbert. At an early stage, however, it was found that the class of definite functions was too restricted to allow the determination of necessary and sufficient conditions in terms of the determinants of § 10. The discovery that this could be done for functions of positive or negative type, and the fact that almost all the theorems which are true of definite functions are, with slight modification, true of these, led finally to the abandonment of the original plan in favour of a discussion of the properties of functions belonging to the wider classes. The first part of the memoir is devoted to the definition of various terms employed, and to the re-statement of the consequences which follow from Hilbert’s theorem.

Nonstationary Theory of the Thermal Explosion for Monomolecular Exothermic Reactions

2012 ◽  
Vol 521 ◽  
pp. 61-77
Author(s):  
V.Y. Filimonov
Keyword(s):  
Thermal Explosion ◽  
Sufficient Conditions ◽  
Point Of View ◽  
New Approach ◽  
Exothermic Reactions ◽  
Nonstationary Theory ◽  
Characteristic Modes ◽  
Monomolecular Reactions ◽  
Necessary And Sufficient ◽  
Qualitative Changes

A new approach to the consideration of the thermal explosion macrokinetic features for monomolecular reactions in homogeneous systems based on a strict accounting of burnout during the reaction process is proposed. It is established, that the qualitative changes of the phase trajectory structure (phase portrait) on the plane: heating rate-temperature determine the characteristic modes of reaction. This approach makes it possible to go beyond the Semenov theory and allows us to consider the variety of the reaction modes. From this point of view, the theory of Semenov is a special case which is valid only for reactions of zero order. The phase trajectories analyze on the parametrical plane Semenov criterion – Todes criterion gives an opportunity to define the regions of the thermal explosion degeneration, the transition regions and the region of the thermal explosion realization. With the use of such consideration, the necessary and sufficient conditions of the thermal explosion are found.

scholarly journals Certain results on invariant submanifolds of an almost Kenmotsu $$(\kappa ,\mu ,\nu )$$-space

2021 ◽  
Author(s):  
Mehmet Atc̣eken
Keyword(s):  
Sufficient Conditions ◽  
Riemannian Curvature ◽  
Invariant Submanifold ◽  
Totally Geodesic ◽  
Ambient Manifold ◽  
Riemannian Curvature Tensor ◽  
Nullity Distribution ◽  
Invariant Submanifolds ◽  
Almost All ◽  
Necessary And Sufficient

AbstractIn the present paper, we study invariant submanifolds of almost Kenmotsu structures whose Riemannian curvature tensor has $$(\kappa ,\mu ,\nu )$$ ( κ , μ , ν ) -nullity distribution. Since the geometry of an invariant submanifold inherits almost all properties of the ambient manifold, we research how the functions $$\kappa ,\mu $$ κ , μ and $$\nu $$ ν behave on the submanifold. In this connection, necessary and sufficient conditions are investigated for an invariant submanifold of an almost Kenmotsu $$(\kappa ,\mu ,\nu )$$ ( κ , μ , ν ) -space to be totally geodesic under some conditions.

scholarly journals On uniform exponential trisplitting for cocycles of linear operators in Banach spaces

2018 ◽  
Vol 56 (2) ◽  
pp. 81-103
Author(s):  
Larisa Elena Biriş ◽  
Claudia Luminiţa Mihiţ ◽  
Traian Ceauşu ◽  
Ioan-Lucian Popa
Keyword(s):  
Banach Spaces ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Type A ◽  
Point Of View ◽  
Necessary And Sufficient Conditions ◽  
Linear Operators ◽  
Skew Product ◽  
Exponential Trichotomy ◽  
Necessary And Sufficient

AbstractThe aim of this paper is to study the concept of uniform exponential trisplitting for skew-product semiflow in Banach spaces. This concept is a generalisation of the well-known concept of uniform exponential trichotomy. We obtain necessary and sufficient conditions for this concept of Datko’s type. a character-isation in terms of Lyapunov functions is provided. The results are obtained from the point of view of the projector families, i.e. invariant and strongly invariant.

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