Nested ε-decompositions and clustering of complex systems

A graph-theoretic method of extreme simplicity is proposed for decomposition and clustering of complex systems into interconnected subsystems with prescribed interaction strength. The method is suitable for stability analysis via vector Lyapunov functions and decentralized control, and estimation of large-scale systems. The new decomposition scheme can be used in clustering (aggregation) of models in economics, queuing networks and computer systems, as well as decompositions in block iterative methods for solutions of linear algebraic equations.     

Design and Performance Evaluation of Residual Generators for the FDI of an Aircraft

In this work, several procedures for the fault detection and isolation （FDI） on general aviation aircraft sensors are presented. In order to provide a comprehensive wide-pectrum treatment, both linear and nonlinear, model-based and data-driven methodologies are considered. The main contributions of the paper are related to the development of both FDI polynomial method （PM） and FDI scheme based on the nonLinear geometric approach （NLGA）. As to the PM, the obtained results highlight a good trade-off between solution complexity and resulting performances. Moreover, the proposed PM is especially useful when robust solutions are required for minimising the effects of modelling errors and noise, while maximising fault sensitivity. As to the NLGA, the proposed work is the first development and robust application of the NLGA to an aircraft model in flight conditions characterised by tight-oupled longitudinal and lateral dynamics. In order to verify the robustness of the residual generators related to the previous FDI techniques, the simulation results adopt a typical aircraft reference trajectory embedding several steady-tate flight conditions, such as straight flight phases and coordinated turns. Moreover, the simulations are performed in the presence of both measurement and modelling errors. Finally, extensive simulations are used for assessing the overall capabilities of the developed FDI schemes and a comparison with neural networks （NN） and unknown input Kalman filter （UIKF） diagnosis methods is performed.     

Analogical Learning and Automated Rule Constructions

This paper describes some experiments of analogical learning and automated rule construction.The present investigation focuses on knowledge acquisition,learning by analyogy,and knowledge retention.The developed system initially learns from scratch,gradually acquires knowledge from its environment through trial-and-error interaction,incrementally augments its knowledge base,and analogically solves new tasks in a more efficient and direct manner.     

The design of absorbing Bayesian pursuit algorithms and the formal analyses of their ε-optimality

The fundamental phenomenon that has been used to enhance the convergence speed of learning automata (LA) is that of incorporating the running maximum likelihood (ML) estimates of the action reward probabilities into the probability updating rules for selecting the actions. The frontiers of this field have been recently expanded by replacing the ML estimates with their corresponding Bayesian counterparts that incorporate the properties of the conjugate priors. These constitute the Bayesian pursuit algorithm (BPA), and the discretized Bayesian pursuit algorithm. Although these algorithms have been designed and efficiently implemented, and are, arguably, the fastest and most accurate LA reported in the literature, the proofs of their \(\epsilon\)-optimal convergence has been unsolved. This is precisely the intent of this paper. In this paper, we present a single unifying analysis by which the proofs of both the continuous and discretized schemes are proven. We emphasize that unlike the ML-based pursuit schemes, the Bayesian schemes have to not only consider the estimates themselves but also the distributional forms of their conjugate posteriors and their higher order moments—all of which render the proofs to be particularly challenging. As far as we know, apart from the results themselves, the methodologies of this proof have been unreported in the literature—they are both pioneering and novel.     

T-Efficient Calculation of the ε-Solutions to Problems of Calculus and Applied Mathematics. II

An approach to the construction of algorithms that are efficient for complexity and that calculate -solutions to computation and applied mathematics problems is described in the first part of the present paper. This approach is applied for creation of T-efficient algorithms used to solve some classes of nonlinear integral equations, ordinary differential equations, and global optimization.     

Enhancement of Heat Transfer in a Tube with Swirl Generators

Influence of strip self-rotating plastic spiral elements inserted in a tube on heat transfer enhancement is studied experimentally. The strip consists of 100 self-rotating spiral elements made of plastic polycarbonate inserted in the inner tube of a concentric tube heat exchanger with a view to generating swirl flow that helps to increase the heat transfer rate of the tube. Cold water flows in the annulus whereas the hot water flows in the inner tube. The obtained experimental data are compared with the data obtained from plain tube .Experimental results confirmed that the use of self-rotating plastic spiral elements leads to higher overall heat transfer coefficient than the plain tube. This technology is not only useful for heat transfer enhancement but also can be used for self cleaning deposition fouling in heat exchanger when the flow velocity is higher than 0.2 m/sec.     

One more Global Property of ε-Subdifferentials

The paper presents one more global interrelation between the values of a convex function and its -subdifferential. A direct consequence of this interrelation is the proof of the fact that functions with identical -subdifferentials have an identical limit behavior.     

Range Estimation Is NP-Hard for ε2 Accuracy and Feasible for ε2−δ

The basic problem of interval computations is: given a function f(x ₁,..., x _n) and n intervals [x _i, x _i], find the (interval) range yof the given function on the given intervals. It is known that even for quadratic polynomials f(x ₁,..., x _n), this problem is NP-hard. In this paper, following the advice of A. Neumaier, we analyze the complexity of asymptotic range estimation, when the bound on the width of the input intervals tends to 0. We show that for small c > 0, if we want to compute the range with an accuracy c ₂, then the problem is still NP-hard; on the other hand, for every > 0, there exists a feasible algorithm which asymptotically, estimates the range with an accuracy c ^2–.     

Semantics of Constructions（I）—The Traditional Approach

It is well known that impredicative type systems do not have set theoretical semantics.This paper takes a look at semantics of inductive types in impredicative type systems.A generalized inductive type is interpreted as an omega set generated by effectivizing a certain rule set .The result provedes a semantic justification of inductive types in the calculus of constructions.     

Semantics of Constructions（Ⅱ） —The Initial Algebraic Approach

Inductive types can be formulated by incorporating the idea of initial T-algebra.The interpretation of an inductive type of this kind boils down to finding out the initial T-algebra defined by the inductive type.In this paper the issue in the semantic domain of omega sets is examined.Based on the semantic results a new class of inductive types,that of local inductive types,in propsed.     

Semantics of Constructions (Ⅱ)─ The Initial Algebraic Approach

1 IntroductionInductive types play an important role in type theory. In Martin-L5f type theory[1],all types are inductive types. The idea of inductive types is very much set theoretical.Recursion rules for inductive types, for example, are type theoretical version of set recursionin classical set theory. As a matter of fact, inductive types in Martin--L6f type theory haveset theoretical semantics in classical set theory. The situation is different in the calculusof con.t..ctions[2]. As the …     

Geometric properties of satisfying assignments of random ε-1-in-k SAT

We study the geometric structure of the set of solutions of a random ε-1-in-k SAT problem [D. Achlioptas, A. Chtcherba, G. Istrate, and C. Moore, The phase transition in 1-in-K SAT and NAE3SAT, in Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms, 2001, pp. 721–722; A. Percus, G. Istrate, and C. Moore (eds.), Computational Complexity and Statistical Physics, Oxford University Press, Oxford, UK, 2006]. For l≥1, two satisfying assignments A and B are l-connected if there exists a sequence of satisfying assignments connecting them by changing at most l bits at a time.

We first identify a subregion of the satisfiable phase where the set of solutions provably forms one cluster. Next, we provide a range of parameters (c, ε) such that w.h.p. (with high probability) two assignments of a random ε-1-in-k SAT instance with n variables and cn clauses are O(log n)-connected, conditional on being satisfying assignments. Also, for random instances of 1-in-k SAT in the satisfiable phase we show that there exists ν _k ∈(0, 1/(k?2)] such that w.h.p. no two satisfying assignments at distance at least ν _k ·n form a ‘hole’. We believe that this is true for all ν _k >0, and in fact solutions of a random 1-in-k SAT instance in the satisfiable phase form one cluster.     

On the Formalization of the Modal μ-Calculus in the Calculus of Inductive Constructions

We present a natural deduction proof system for the propositional modal μ-calculus and its formalization in the calculus of inductive constructions. We address several problematic issues, such as the use of higher-order abstract syntax in inductive sets in the presence of recursive constructors, and the formalization of modal (sequent-style) rules and of context sensitive grammars. The formalization can be used in the system Coq, providing an experimental computer-aided proof environment for the interactive development of error-free proofs in the modal μ-calculus. The techniques we adopt can be readily ported to other languages and proof systems featuring similar problematic issues.     

Fractional Wishart processes and ε-fractional Wishart processes with applications

In this paper, we introduce two new matrix stochastic processes: fractional Wishart processes and $\epsilon$-fractional Wishart processes with integer indices which are based on the fractional Brownian motions and then extend $\epsilon$-fractional Wishart processes to the case with non-integer indices. Both processes include classic Wishart processes (if the Hurst index $H$ equals $\frac{1}{2}$) and present serial correlation of stochastic processes. Applying $\epsilon$-fractional Wishart processes to financial volatility theory, the financial models account for the stochastic volatilities of the assets and for the stochastic correlations not only between the underlying assets’ returns but also between their volatilities and for stochastic serial correlation of the relevant assets.     

A formal proof of the ε-optimality of absorbing continuous pursuit algorithms using the theory of regular functions

The most difficult part in the design and analysis of Learning Automata (LA) consists of the formal proofs of their convergence accuracies. The mathematical techniques used for the different families (Fixed Structure, Variable Structure, Discretized etc.) are quite distinct. Among the families of LA, Estimator Algorithms (EAs) are certainly the fastest, and within this family, the set of Pursuit algorithms have been considered to be the pioneering schemes. Informally, if the environment is stationary, their ε-optimality is defined as their ability to converge to the optimal action with an arbitrarily large probability, if the learning parameter is sufficiently small/large. The existing proofs of all the reported EAs follow the same fundamental principles, and to clarify this, in the interest of simplicity, we shall concentrate on the family of Pursuit algorithms. Recently, it has been reported Ryan and Omkar (J Appl Probab 49(3):795–805, 2012) that the previous proofs for ε-optimality of all the reported EAs have a common flaw. The flaw lies in the condition which apparently supports the so-called “monotonicity” property of the probability of selecting the optimal action, which states that after some time instant t ₀, the reward probability estimates will be ordered correctly forever. The authors of the various proofs have rather offered a proof for the fact that the reward probability estimates are ordered correctly at a single point of time after t ₀, which, in turn, does not guarantee the ordering forever, rendering the previous proofs incorrect. While in Ryan and Omkar (J Appl Probab 49(3):795–805, 2012), a rectified proof was presented to prove the ε-optimality of the Continuous Pursuit Algorithm (CPA), which was the pioneering EA, in this paper, a new proof is provided for the Absorbing CPA (ACPA), i.e., an algorithm which follows the CPA paradigm but which artificially has absorbing states whenever any action probability is arbitrarily close to unity. Unlike the previous flawed proofs, instead of examining the monotonicity property of the action probabilities, it rather examines their submartingale property, and then, unlike the traditional approach, invokes the theory of Regular functions to prove that the probability of converging to the optimal action can be made arbitrarily close to unity. We believe that the proof is both unique and pioneering, and adds insights into the convergence of different EAs. It can also form the basis for formally demonstrating the ε-optimality of other Estimator algorithms which are artificially rendered absorbing.     

Differential evolution based on ε-domination and orthogonal design method for power environmentally-friendly dispatch

This paper proposes a differential evolution algorithm based on ε-domination and orthogonal design method (ε-ODEMO) to solve power dispatch problem considering environment protection and saving energy. Besides the operation costs of thermal power plant, contaminative gas emission is also optimized as an objective. In the proposed algorithm, ε-dominance is adopted to make genetic algorithm obtain a good distribution of Pareto-optimal solutions in a small computational time, and the orthogonal design method can generate an initial population of points that are scattered uniformly over the feasible solution space, these modify the differential evolution algorithm (DE) to make it suit for multi-objective optimization (MOO) problems and improve its performance. A test hydrothermal system is used to verify the feasibility and effectiveness of the proposed method. Compared with other methods, the results obtained demonstrate the effectiveness of the proposed algorithm for solving the power environmentally-friendly dispatch problem.     

Kolmogorov ε-Entropy in the Problems on Global Attractors for Evolution Equations of Mathematical Physics

We study the Kolmogorov -entropy and the fractal dimension of global attractors for autonomous and nonautonomous equations of mathematical physics. We prove upper estimates for the -entropy and fractal dimension of the global attractors of nonlinear dissipative wave equations.