Order,degree, and complexity

The order of a system not in state-space form has been defined elsewhere. Degree has been defined for rational transfer function matrices, and by extension to systems of least order : this definition is generalized for systems not having least order. A new concept of ‘ complexity ’ is introduced, and these ideas are used to resolve a confusion which exists in the literature.     

Graphics, Bifurcation, Order and Chaos

Chaos theory involves the study of how complicated behaviour can arise in systems which are based on simple rules, and how minute changes in the input of a system can lead to large differences in the output. In this paper, bifurcation maps of the education X_t+1=αλX_t [1+X_t] ^-β, where α= 1 or α=e^-Xi, are presented, and they reveal a visually striking and intricate class of patterns ranging from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations. The computer-based system presented is special in its primary focus on the fast characterization of simple "chacs equation" data using an interactive graphics system with a variety of controlling parameters.     

Order,stepsize and stiffness switching

The question discussed in this paper is “When should a code switch between stiff and non-stiff options, when should it switch between one order and another and how should it adjust its stepsize from one step to the next.” Criteria are proposed for switching between the options that become available as the integration progresses and these are presented in algorithmic form.     

Combining Non-stably Infinite, Non-first Order Theories

A crucial step in the assertional verification of concurrent programs is deciding whether some sets of literals are satisfiable or not. In this context, the Nelson-Oppen combination scheme is often used. This scheme combines decision procedures for two disjoint theories into a decision procedure for the union of these theories. However, the standard version of the Nelson-Oppen technique tackles only one-sorted, stably infinite first-order theories. The scheme has previously been adapted to a many-sorted framework [C. Tinelli and C. G. Zarba. Combining decision procedures for theories in sorted logics. Technical Report 04-01, Department of Computer Science, The University of Iowa, Feb. 2004], and to handle non-stably infinite theories [C. Tinelli and C. G. Zarba. Combining non-stably infinite theories, in: I. Dahn and L. Vigneron, editors, Proceedings of the 4th International Workshop on First Order Theorem Proving, FTP'03 (Valencia, Spain), Electronic Notes in Theoretical Computer Science 86.1 (2003), Elsevier Science Publishers]. Those two enhancements were presented independently. We propose a unifying version in the continuity of both previous ones, which further relaxes the stably infinite requirement. Notably, some non-stably infinite theories can now be combined with the theory of arrays. Also, the combination scheme is presented here using a semantic notion of theory, allowing to handle non-first order theories.     

The Hidden Order of Hadrian’s Villa, and the Order of Modern Architecture

The casual violence with which the various parts of the second-century Villa of Hadrian crash into each other seems to belie any possibility of a rational plan, although an overall master plan seems a necessity. Amazingly, in the twentieth century, Le Corbusier extolled Hadrian’s Villa as the archetype of great planning. This paper will describe the clear, mathematically ordered process by which the Villa was planned, and the compositional principles of its design will be shown to be exactly those which underlie the theories of Corbusier, 1800 years later.     

Decoupled,Unconditionally Stable,Higher Order Discretizations for MHD Flow Simulation

We propose, analyze, and test a new MHD discretization which decouples the system into two Oseen problems at each timestep yet maintains unconditional stability with respect to the time step size, is optimally accurate in space, and behaves like second order in time in practice. The proposed method chooses a parameter \(\theta \in [0,1]\), dependent on the viscosity \(\nu \) and magnetic diffusivity \(\nu _m\), so that the explicit treatment of certain viscous terms does not cause instabilities, and gives temporal accuracy \(O(\Delta t^2 + (1-\theta )|\nu -\nu _m|\Delta t)\). In practice, \(\nu \) and \(\nu _m\) are small, and so the method behaves like second order. When \(\theta =1\), the method reduces to a linearized BDF2 method, but it has been proven by Li and Trenchea that such a method is stable only in the uncommon case of \(\frac{1}{2}< \frac{\nu }{\nu _m} < 2\). For the proposed method, stability and convergence are rigorously proven for appropriately chosen \(\theta \), and several numerical tests are provided that confirm the theory and show the method provides excellent accuracy in cases where usual BDF2 is unstable.     

A Practical,Accurate, Information Criterion for Nth Order Markov Processes

The recent increase in the breath of computational methodologies has been matched with a corresponding increase in the difficulty of comparing the relative explanatory power of models from different methodological lineages. In order to help address this problem a Markovian information criterion (MIC) is developed that is analogous to the Akaike information criterion (AIC) in its theoretical derivation and yet can be applied to any model able to generate simulated or predicted data, regardless of its methodology. Both the AIC and proposed MIC rely on the Kullback–Leibler (KL) distance between model predictions and real data as a measure of prediction accuracy. Instead of using the maximum likelihood approach like the AIC, the proposed MIC relies instead on the literal interpretation of the KL distance as the inefficiency of compressing real data using modelled probabilities, and therefore uses the output of a universal compression algorithm to obtain an estimate of the KL distance. Several Monte Carlo tests are carried out in order to (a) confirm the performance of the algorithm and (b) evaluate the ability of the MIC to identify the true data-generating process from a set of alternative models.     

Set systems: Order types, continuous nondeterministic deformations, and quasi-orders

By reformulating a learning process of a set system L as a game between Teacher and Learner, we define the order type of L to be the order type of the game tree, if the tree is well-founded. The features of the order type of L (dimL in symbol) are (1) we can represent any well-quasi-order (wqo for short) by the set system L of the upper-closed sets of the wqo such that the maximal order type of the wqo is equal to dimL; (2) dimL is an upper bound of the mind-change complexity of L. dimL is defined iff L has a finite elasticity (fe for short), where, according to computational learning theory, if an indexed family of recursive languages has fe then it is learnable by an algorithm from positive data. Regarding set systems as subspaces of Cantor spaces, we prove that fe of set systems is preserved by any continuous function which is monotone with respect to the set-inclusion. By it, we prove that finite elasticity is preserved by various (nondeterministic) language operators (Kleene-closure, shuffle-closure, union, product, intersection, …). The monotone continuous functions represent nondeterministic computations. If a monotone continuous function has a computation tree with each node followed by at most n immediate successors and the order type of a set system L is α, then the direct image of L is a set system of order type at most n-adic diagonal Ramsey number of α. Furthermore, we provide an order-type-preserving contravariant embedding from the category of quasi-orders and finitely branching simulations between them, into the complete category of subspaces of Cantor spaces and monotone continuous functions having Girard’s linearity between them.     

Interpolatory Model Order Reduction Method for Second Order Systems

In this paper, we propose a structure‐preserving model reduction method for second‐order systems based on H₂ optimal interpolation. In the iterative process of the proposed method, an algorithm is presented for selecting interpolation points in order to control the dimension of the reduced system. Result about error analysis of the interpolation points selection algorithm is obtained and the property of the new model reduction method is also given. Finally, three numerical examples are performed to illustrate the effectiveness of the new method.     

Transitive Matrices, Strict Preference Order and Ordinal Evaluation Operators

Let X ={x ₁, x ₂, ..., x _n } be a set of alternatives and a _ij a positive number expressing how much the alternative x _i is preferred to the alternative x _j . Under suitable hypothesis of no indifference and transitivity over the pairwise comparison matrix A=(a _ij ), the actual qualitative ranking on the set X is achievable. Then a coherent priority vector is a vector giving a weighted ranking agreeing with the actual ranking and an ordinal evaluation operator is a functional F that, acting on the row vectors , translates A in a coherent priority vector. In this paper we focus our attention on the matrix A, looking for conditions ensuring the existence of coherent priority vectors. Then, given a type of matrices, we look for ordinal evaluation operators, including OWA operators, associated to it.     

Geodesic Order Types

The geodesic between two points a and b in the interior of a simple polygon P is the shortest polygonal path inside P that connects a to b. It is thus the natural generalization of straight line segments on unconstrained point sets to polygonal environments. In this paper we use this extension to generalize the concept of the order type of a set of points in the Euclidean plane to geodesic order types. In particular, we show that, for any set S of points and an ordered subset \(\mathcal {B} \subseteq S\) of at least four points, one can always construct a polygon P such that the points of \(\mathcal {B} \) define the geodesic hull of S w.r.t. P, in the specified order. Moreover, we show that an abstract order type derived from the dual of the Pappus arrangement can be realized as a geodesic order type.     

IDL-PMCFG,a Grammar Formalism for Describing Free Word Order Languages

Journal of Logic, Language and Information - We introduce Interleave-Disjunction-Lock parallel multiple context-free grammars (IDL-PMCFG), a novel grammar formalism designed to describe the syntax...     

Discrete Rotational Symmetry,Moment Isotropy,and Higher Order Lattice Boltzmann Models

Conventional lattice Boltzmann models only satisfy moment isotropy up to fourth order. In order to accurately describe important physical effects beyond the isothermal Navier-Stokes fluid regime, higher-order isotropy is required. In this paper, we present some basic results on moment isotropy and its relationship to the rotational symmetry of a generating discrete vector set. The analysis provides a geometric understanding for popular lattice Boltzmann models, while offering a systematic procedure to construct higher-order models.     

基于高阶统计特性的非高斯AR模型的阶次辨识

提出一种基于输入输出信号的三阶累积量 (third ordercumulants)对非高斯AR模型阶次p进行辨识的新算法 .该算法依模型阶次递推 ,通过求解代价函数J(^p) 的首次最小值 ,确定AR模型的阶次估计值 ^p .理论分析和仿真结果均显示 ,该算法具有良好的收敛性和精确性 .     

Variable Order Panel Clustering

We present a new version of the panel clustering method for a sparse representation of boundary integral equations. Instead of applying the algorithm separately for each matrix row (as in the classical version of the algorithm) we employ more general block partitionings. Furthermore, a variable order of approximation is used depending on the size of blocks. We apply this algorithm to a second kind Fredholm integral equation and show that the complexity of the method only depends linearly on the number, say n, of unknowns. The complexity of the classical matrix oriented approach is O(n ²) while, for the classical panel clustering algorithm, it is O(nlog⁷ n). Received July 28, 1999; revised September 21, 1999     

分数阶控制系统

随着基于分数阶次的数学理论日益完善,分数阶控制系统也越来越广泛地被研究和讨论。为了完善分数阶控制系统的理论体系,给出了分数阶系统的总体综述。介绍了分数微积分的定义,给出了分数阶线性定常系统的传递函数和状态空间描述,简要介绍了分数阶控制系统的复频域分析和分数阶控制器及控制器的设计方法,并分析了几种设计方法的优缺点。分数系统的研究必然能找到合适的切入点广泛地进入现代控制领域,为真正实现工业控制自动化提供强有力的理论依据。     

Higher Order Ray Marching

Rendering participating media is still a challenging and time consuming task. In such media light interacts at every differential point of its path. Several rendering algorithms are based on ray marching: dividing the path of light into segments and calculating interactions at each of them. In this work, we revisit and analyze ray marching both as a quadrature integrator and as an initial value problem solver, and apply higher order adaptive solvers that ensure several interesting properties, such as faster convergence, adaptiveness to the mathematical definition of light transport and robustness to singularities. We compare several numerical methods, including standard ray marching and Monte Carlo integration, and illustrate the benefits of different solvers for a variety of scenes. Any participating media rendering algorithm that is based on ray marching may benefit from the application of our approach by reducing the number of needed samples (and therefore, rendering time) and increasing accuracy.