Groupoid Approach to Ergodic Dynamical System of Commutative von Neumann Algebra

Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L∞(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L2(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .

无穷个增生映射和的半隐式算法、Ergodic收敛及Curvature系统（英文）

在实一致凸且q一致光滑Banach空间中,构造无穷个m增生映射和μ_i逆强增生映射和的公共零点的半隐式迭代算法.证明ergodic收敛性.与近期研究成果相比,限定条件更弱.此外,还研究了一类curvature系统并证明其解恰好是无穷个m增生映射和μ_i逆强增生映射和的公共零点,进而验证了迭代算法的有效性.

Ergodic capacity analysis for device-to-device communication underlaying cellular networks

The ergodic capacity of device-to-device （D2D） communication underlaying cellular networks is analyzed. First,the D2D communication model is introduced and the interference during uplink period and downlink period is analyzed.In a D2D communication system,since it is very difficult to obtain the instantaneous channel state information （CSI）,assume that only the transmitters know the statistical CSI and the channel coefficient follows an independent complex Gaussian distribution.Based on the assumptions,for the uplink period,the signal to interference plus noise ratio （SINR）of the D2D user equipments（DUEs）is expressed. Then the cumulative distribution function （CDF ） and probability distribution function （PDF）formulae of the SINR of the DUEs are presented.Based on the SINR formulae during the uplink period,the ergodic capacity formula of the uplink period is derived. Subsequently, using the same methods,the ergodic capacity formula of the downlink period is derived.The simulation results show that the DUEs can still obtain a high ergodic capacity even in the case of a large number of DUEs.This result can be applied to the design and optimization of D2D communications.

Maximal Ergodic定理的证明

讨论了Maximal Ergod ic定理:设(X,Β,μ)是一个有限测度空间,T是X到X上的保测变换,f是X上的可测函数,那么就有∫{f*>0}fdμ≥0.我们利用数学归纳法重新证明了这个经典定理。

Public Key Cryptography Based on Ergodic Matrices over Finite Field

A new public key encryption scheme is proposed in this paper, which is based on a hard problem over ergodic matrices. The security of this scheme is equal to the MQ-problem： multivariate quadratic equations over finite fields. This problem has been shown to be NP-complete and can＇t be solved with polynomial time algorithm.

Hierarchical Content Caching in Fog Radio Access Networks:Ergodic Rate and Transmit Latency

In order to alleviate capacity constraints on the fronthaul and decrease the transmit latency, a hierarchical content caching paradigm is applied in the fog radio access networks(F-RANs). In particular, a specific cluster of remote radio heads is formed through a common centralized cloud at the baseband unit pool, while the local content is directly delivered at fog access points with edge cache and distributed radio signal processing capability. Focusing on a downlink F-RAN, the explicit expressions of ergodic rate for the hierarchical paradigm is derived. Meanwhile, both the waiting delay and latency ratio for users requiring a single content are exploited. According to the evaluation results of ergodic rate on waiting delay, the transmit latency can be effectively reduced through improving the capacity of both fronthaul and radio access links. Moreover, to fully explore the potential of hierarchical content caching, the transmit latency for users requiring multiple content objects is optimized as well in three content transmission cases with different radio access links. The simulation results verify the accuracy of the analysis, further show the latency decreases significantly due to the hierarchical paradigm.

A self-adapting image encryption algorithm based on spatiotemporal chaos and ergodic matrix

To ensure the security of a digital image, a new self-adapting encryption algorithm based on the spatiotemporal chaos and ergodic matrix is proposed in this paper. First, the plain-image is divided into different blocks of the same size, and each block is sorted in ascending order to obtain the corresponding standard ergodic matrix. Then each block is encrypted by the spatiotemporal chaotic system and shuffled according to the standard ergodic matrix. Finally, all modules are rearranged to acquire the final encrypted image. In particular, the plain-image information is used in the initial conditions of the spatiotemporal chaos and the ergodic matrices, so different plain-images will be encrypted to obtain different cipherimages. Theoretical analysis and simulation results indicate that the performance and security of the proposed encryption scheme can encrypt the image effectively and resist various typical attacks.

An ergodic algorithm for long-term coverage of elliptical orbits

This paper deals with the coverage analysis problem of elliptical orbits. An algorithm based on ergodic theory, for long-term coverage of elliptical orbits, is proposed. The differential form of the invariant measure is constructed via the perturbation on mean orbital elements resulted from the J2 term of non-spherical shape of the earth. A rigorous proof for this is then given. Different from the case of circular orbits, here the flow and its space of the dynamical system are defined on a physical space, and the real-value function is defined as the characteristic function on station mask. Therefore, the long-term coverage is reduced to a double integral via Birkhoff-Khinchin theorem. The numerical implementation indicates that the ergodic algorithm developed is available for a wide range of eccentricities.

THE ERGODICITY OF STOCHASTIC GENERALIZED POROUS MEDIA EQUATIONS WITH LVY JUMP

In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of（pt）t≥0 to equilibrium uniform on any bounded subset in H.

Ergodic Rate Analysis on Applying Antenna Selection in D2D Communication Underlaying Cellular Networks

By reusing the spectrum of a cellular network, device-to-device(D2D) communications is known to greatly improve the spectral efficiency bypassing the base station(BS) of the cellular network. Antenna selection is the most cost efficient scheme for interference management, which is crucial to D2D systems. This paper investigates the achievable rate performance of the D2D communication underlaying the cellular network where a multiple-antenna base station with antenna selection scheme is deployed. We derive an exact closed-form expression of the ergodic achievable rate. Also, using Jensen's inequality, two pairs of upper and lower bounds of the rate are derived and we validate the tightness of the two sets of bounds. Based on the bounds obtained, we analyze the ergodic achievable rate in noise-limited scenario, interference-limited high SNR scenario and larger-scale antenna systems. Our analysis shows that the presence of D2D users could be counter-productive if the SNR at cellular UE is high. Further analysis shows that the relationship between the ergodic rate and the number of antennas it positive, but keeps decreasing as the antenna number increasing. These show the inefficiency of antenna selection in D2D interference management.

A one-time one-key encryption algorithm based on the ergodicity of chaos

In this paper, we propose a new one-time one-key encryption algorithm based on the ergodicity of a skew tent chaotic map. We divide the chaotic trajectory into sub-intervals and map them to integers, and use this scheme to encrypt plaintext and obtain ciphertext. In this algorithm, the plaintext information in the key is used, so different plaintexts or different total numbers of plaintext letters will encrypt different ciphertexts. Simulation results show that the performance and the security of the proposed encryption algorithm can encrypt plaintext effectively and resist various typical attacks.

Laplace Transformation and Ergodic Potential Kernel for q-Process in Random Environment

This paper introduces some concepts such as q- process in random environment, Laplace transformation, ergodic potential kernel, error function and some basic lemmas.We study the continuity and Laplace transformation of random transition function. Finally, we give the sufficient condition for the existence of ergodic potential kernel for homogeneous q- processes in random environments.

V-uniform ergodicity for fluid queues

In this paper, we show that a positive recurrent ?uid queue is automatically V-uniformly ergodic for some function V ≥ 1 but never uniformly ergodic. This reveals a similarity of ergodicity between a ?uid queue and a quasi-birth-and-death process. As a byproduct of V-uniform ergodicity, we derive computable bounds on the exponential moments of the busy period.

The Convergence Rate of Ergodic Limits for Regularized Resolvent Families

This paper is concerned with the convergence rates of ergodic limits and approximation for regularized resolvent families for a linear Volterra integral equation. The results contain C 0-semigroups, cosine operator functions and α-times integrated resolvent family as special cases.

Ergodic Hypothesis and Equilibrium Statistical Mechanics in the Quantum Mechanical World View

In this paper, we study and answer the following fundamental problems concerning classical equilibrium statistical mechanics: 1): Is the principle of equal a priori probabilities indispensable for equilibrium statistical mechanics? 2): Is the ergodic hypothesis related to equilibrium statistical mechanics? Note that these problems are not yet answered, since there are several opinions for the formulation of equilibrium statistical mechanics. In order to answer the above questions, we first introduce measurement theory (i.e., the theory of quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics. And we propose the measurement theoretical foundation of equili-brium statistical mechanics, and further, answer the above 1) and 2), that is, 1) is “No”, but, 2) is “Yes”.

Ergodic sensitivity analysis of one-dimensional chaotic maps

Sensitivity analysis in chaotic dynamical systems is a challenging task from a computational point of view.In this work,we present a numerical investigation of a novel approach,known as the space-split sensitivity or S3 algorithm.The S3 algorithm is an ergodic-averaging method to differentiate statistics in ergodic,chaotic systems,rigorously based on the theory of hyperbolic dynamics.We illustrate S3 on one-dimensional chaotic maps,revealing its computational advantage over na?ve finite difference computations of the same statistical response.In addition,we provide an intuitive explanation of the key components of the S3 algorithm,including the density gradient function.

Exploiting Virtual Elasticity of Manufacturing Systems to Respect OTD—Part 2: Post-Optimality Conditions for the Cases of Ergodic and Non-Ergodic Order Rate with Deterministic Product-Mix

Respecting the on-time delivery (OTD) for manufacturing orders is mandatory. However, for non-JIT Batch & Queue Push-manufacturing systems, the compliance of OTD is not intrinsically guaranteed. As an OTD related manufacturing theory is largely missing it is crucial to understand and formalize the necessary conditions of OTD compliance for complex production environments for maximum exploitation of the production capacity. This paper evaluates the conditions of post-optimality while being OTD compliant for production systems, which are characterized by stochastic order rate and a deterministic product-mix. Instead of applying discrete event simulation to explore the real case-by-case order scheduling optimization for OTD compliance, a Cartesian approach is followed. This enables to define theoretically the solution space of order backlog for OTD, which contributes to developing further manufacturing theory. At the base stands the recently defined new concept of virtual manufacturing elasticity by reducing lead-time to increase virtually production capacity. The result has led to defining additional two corollaries to the OTD theorem, which sets up basic OTD theory. Apart from defining the post-optimal requirements to guarantee for orders at least a weak solution for OTD compliance, this paper reveals that for a deterministic product-mix a non-ergodic order arrival rate can be rescheduled into an ergodic order input rate to the shopfloor if the virtual elasticity T is large enough, hence the importance of having fast and flexible production lines.

Exploiting Virtual Elasticity of Production Systems for Respecting OTD-Part 1: Post-Optimality Conditions for Ergodic Order Arrivals in Fixed Capacity Regimes

Respecting the on-time-delivery (OTD) for manufacturing orders is mandatory. This depends, however, on the probability distribution of incoming order rate. The case of non-equal distribution, such as aggregated arrivals, may compromise the observance of on-time supplies for some orders. The purpose of this paper is to evaluate the conditions of post-optimality for stochastic order rate governed production systems in order to observe OTD. Instead of a heuristic or a simulative exploration, a Cartesian-based approach is applied to developing the necessary and sufficient mathematical condition to solve the problem statement. The research result demonstrates that increasing speed of throughput reveals a latent capacity, which allows arrival orders above capacity limits to be backlog-buffered and rescheduled for OTD, exploiting the virtual manufacturing elasticity inherent to all production systems to increase OTD reliability of non JIT-based production systems.

Does Amplify-and-Forward Cooperative Relay Improve OFDM/OFDMA Ergodic Capacity？

The ergodic information rate for Orthogonal Frequency-Division Multiplexing / Orthogonal Frequency-Division Multiple Access with amplify-and-forward （AF） relaying systems in the presence of frequency offsets is evaluated. Unlike previous work, per-subcarrier adaptive power allocation is performed on each relay to optimize the system ergodic information rate. For a given frequency offset and total number of relays M, the AF ergodic information rate is proven to be a monotonically increasing function of α （the ratio of the power allocated to the source node and the total transmit power）, implying that the maximum ergodic information rate can be obtained at α=1 （i.e., there is no cooperative relay）. Furthermore, the proof of ＂cooperative relays cannot improve the AF ergodic information rate in a quasi-static wireless channel＂ is also provided in this letter.

ROBUST WEAK ERGODICITY AND STABLE ERGODICITY

In this paper, we define robust weak ergodicity and study the relation between robust weak ergodicity and stable ergodicity for conservative partially hyperbolic systems. We prove that a C^r（r 〉 1） conservative partially hyperbolic diffeomorphism is stably ergodic if it is robustly weakly ergodic and has positive （or negative） central exponents on a positive measure set. Furthermore, if the condition of robust weak ergodicity is replaced by weak ergodicity, then the diffeomophism is an almost stably ergodic system. Additionally, we show in dimension three, a C^r（r 〉 1） conservative partially hyperbolic diffeomorphism can be approximated by stably ergodic systems if it is robustly weakly ergodic and robustly has non-zero central exponents.