An inner product space on irreducible and synchronizable probabilistic finite state automata

Probabilistic finite state automata (PFSA) have found their applications in diverse systems. This paper presents the construction of an inner-product space structure on a class of PFSA over the real field via an algebraic approach. The vector space is constructed in a stationary setting, which eliminates the need for an initial state in the specification of PFSA. This algebraic model formulation avoids any reference to the related notion of probability measures induced by a PFSA. A formal language-theoretic and symbolic modeling approach is adopted. Specifically, semantic models are constructed in the symbolic domain in an algebraic setting. Applicability of the theoretical formulation has been demonstrated on experimental data for robot motion recognition in a laboratory environment.     

一类完全非线性S-盒的构造

完全非线性S-盒在对称密码中有着重要的运用。给出有限域上完全非线性S-盒的一种构造方法。与在向量空间上构造的方法比,有限域上置换多项式的代数次数等性质更容易研究。该方法可以构造多类完全非线性S-盒,例如,通过选择幂函数形式的置换α^x,得到Satoh等人构造的S-盒;通过选取指数形式的置换x^d,所得完全非线性S-盒的分量函数的任意非零线性组合的代数次数达到最高。     

Metric concepts and implications in describing compositional changes for world river's water chemistry

The paper is designed to give the reader an outline that is useful for understanding the importance of distance, as a metric concept, and its implications when compositional (geochemical) data are managed from a statistical point of view in a given sample space. Application examples are shown by considering the construction of confidence regions and mixing models. The analyzed data are related to the chemistry of the most important rivers of the world as referring to the GEMS/WATER Global Register of River Inputs when each sample (river) is represented as a composition. A compositional vector of d parts, x=[x₁,x₂,…,x_d], is defined as a vector in which the only relevant information is contained in the ratios between its components. All the components of the vector are assumed positive and are called parts (variables), while the whole compositional vector, with the sum of the parts equal to a constant, represents the composition. In this case data are not represented by variables free to vary from −∞ to +∞ within a Euclidean space but occupy a restricted part of it called the simplex. The d-part simplex, S^d, is a subset of a d-dimensional real space. In this context the metric of the R space, with the definition of basic algebraic operations and of inner product, norm and distance, thus giving an Euclidean vector space structure, cannot be applied since the scale is relative and not absolute.     

命题对象的空间逻辑运算模型

在现有逻辑系统中,各连接词的运算模型都可以归结为某些“代数算子”,其共同特征是仅考虑了命题所描述集合的代数测度大小,而没有考虑它们在几何空间中的位置关系。文章以“空间位置相关性”为中心,提出了“摸天花板问题”,分析了逻辑运算中存在的几何位置相关性。在“命题对象”、“真位向量”、“空间图像”等概念的基础上,提出了命题对象的空间逻辑运算模型,并结合格分维理论给出了在几何图像中的具体应用形式。本文工作拓展了泛逻辑学中广义相关性的含义,为连接词的运算形式提供了一种新的模型。     

Graphic Objects: A Mathematical Abstract Model for Computer Graphics

This paper defines an abstract mathematical model for graphic systems which incorporates modelling operations. The model is based on the “graphic object” concept. A graphic object is an abstraction of a graphic entity. Graphic objects are a generalization of the Fiume graphic object concept, supporting modelling operations, which give the set of graphic objects an algebraic structure of vector space and booelan algebra. Fiume visualization formalism can be applied to graphic objects. Further, product operations of graphic objects are defined, which are a generalization of solid sweep operations. Graphic object products may be used to build common 2D and 3D objects such as triangles, squares, cubes, circles and spheres from lines and arcs.     

Chance-constrained programming on sugeno measure space

Uncertain programming is a theoretical tool to handle optimization problems under uncertain environment, it is mainly established in probability, possibility, or credibility measure spaces. Sugeno measure space is an interesting and important extension of probability measure space. This motivates us to discuss the uncertain programming based on Sugeno measure space. We have constructed the first type of uncertain programming on Sugeno measure space, i.e. the expected value models of uncertain programming on Sugeno measure space. In this paper, the second type of uncertain programming on Sugeno measure space, i.e. chance-constrained programming on Sugeno measure space, is investigated. Firstly, the definition and the characteristic of α-optimistic value and α-pessimistic value as a ranking measure are provided. Secondly, Sugeno chance-constrained programming (SCCP) is introduced. Lastly, in order to construct an approximate solution to the complex SCCP, the ideas of a Sugeno random number generation and a Sugeno simulation are presented along with a hybrid approach.     

基于匹配理论的无线广播编码重传算法

针对成批数据在无线广播过程中发生丢包后的重传策略,利用网络编码技术,研究尽可能多地减少重传次数的方法.首先通过矩阵及向量的运算给出了待重传数据满足编码条件的代数表达形式,再根据关联矩阵构造相应的图,最后通过图论中的最佳匹配理论给出了一种寻找编码机会的优化算法.此算法中不但能够找出最多的两两编码机会,而且还考虑了将尽可能多的数据包编在一起的可能性,从而尽可能地减少了重传次数,实现了编码的优化,有效提高了网络带宽效率和吞吐量.     

广义的以物体为中心的行程编码：一种新的三维物体模型

三维物体的表达和识别是图象理解和场景分析的核心问题,三维模型在三维物体的识别和场景分析中具有十分重要的作用.三维模型应该是以物体为中心的,能够提供该场景的所有有用信息.物体的大小,形状及朝向应均可从该模型中提取得到.本文提出了一种新的三维物体模型——广义的以物体为中心的行程编码.它包括物体的GORC物理数据结构,详细的形状描述和抽象描述.物体的高层次的表达可以通过以GORC编码的物理数据直接提取得到.三维的GORC是二维的以物体为中心的行程编码在三维上的推广,它兼有物体的体积表达和表面表达的优点.三维物体的GORC模型可以很容易地由其深度信息构造得出,基于GORC的投影运算,图象代数运算以及特征提取均可非常有效地实现.     

C-fuzzy numbers and a dual of extension principle

In this paper, we introduce and investigate the concept of c-fuzzy numbers. We extend the algebraic operations on c-fuzzy numbers, and specially study the properties of these operations on LR type c-fuzzy numbers. In addition, a dual of extension principle is introduced. It is shown that the algebraic operations with c-fuzzy numbers have a representation based on the dual of extension principle.     

On efficiency measurement of algebraic data structures encodings

Since the Second Symposium on Symbolic and Algebraic Manipulation (1971) with the development of many algebraic manipulation systems, there has been demand for an «efficiency measurement» for these systems. Some of the frequently used systems, asAltran andCamal, provide for numerical encodings of polynomials and of rational functions. This kind of representation for algebraic structures is particularly useful from the space saving point of view. This paper introduces a methodology for comparing different kinds of encodings for monomials, and hence for polynomials. This new methodological approach to the evaluation of the encodings uses four criteria, which are related both to static and dynamic aspects for algebraic data structures processing. The set of monomials withn variables is considered as a vector space and its topology is discussed. In order to exemplify our approach two particular encodings are considered. According to the proposed criteria it will be shown how the so-called encoding by juxtaposition can be considered very good. A direction for possible development is also presented.     

Algorithms of information authentication for the automatic control systems on the basis of structures in finite vector spaces

New algebraic structures differing from the existing ones in new kinds of operations over finite sets of m-dimensional vectors (m ≥ 2) with discrete components were developed to construct information authentication algorithms for the automatic control systems. The operation of vector multiplication including multiplication and summation of the coordinates of the multiplied vectors modulo simple number p was defined. Depending on the structure of module p, dimensionality m, and value of the “extension” coefficients defined in terms of the rules of multiplication of the base vectors, the vector spaces were shown to include high-order finite groups or be extended fields that can be efficiently used to construct effective algorithms of electronic digital signature.     

Some remarks on Kähler differentials and ordinary differentials in nonlinear control theory

In the algebraic approach to nonlinear control systems two similar notions, namely Kähler differentials and the formal vector space of differential one-forms having the properties of ordinary differentials, are frequently used to study the systems. This technical note explains that the formal vector space of differential one-forms is isomorphic to a quotient space (module) of Kähler differentials. These two modules coincide when they are modules over a ring of linear differential operators over the field of algebraic functions. Some remarks and examples demonstrating when the use of Kähler differentials might not be appropriate are included.     

Text document clustering using global term context vectors

Despite the advantages of the traditional vector space model (VSM) representation, there are known deficiencies concerning the term independence assumption. The high dimensionality and sparsity of the text feature space and phenomena such as polysemy and synonymy can only be handled if a way is provided to measure term similarity. Many approaches have been proposed that map document vectors onto a new feature space where learning algorithms can achieve better solutions. This paper presents the global term context vector-VSM (GTCV-VSM) method for text document representation. It is an extension to VSM that: (i) it captures local contextual information for each term occurrence in the term sequences of documents; (ii) the local contexts for the occurrences of a term are combined to define the global context of that term; (iii) using the global context of all terms a proper semantic matrix is constructed; (iv) this matrix is further used to linearly map traditional VSM (Bag of Words—BOW) document vectors onto a ‘semantically smoothed’ feature space where problems such as text document clustering can be solved more efficiently. We present an experimental study demonstrating the improvement of clustering results when the proposed GTCV-VSM representation is used compared with traditional VSM-based approaches.     

基于模糊相似性的彩色形态学图像处理方法

为了将传统灰度图像数学形态学扩展到彩色图像,提出一种结合矢量空间模糊相似性的彩色形态学图像处理方法。首先,在RGB彩色空间中利用彩色矢量间的距离和角度定义模糊相似性测度,以刻画与人类视觉感知相一致的彩色相似程度;以上述相似性测度为准则定义彩色空间中任意一组彩色的上确界和下确界;利用中心像素及其结构单元内像素的上确界和下确界构建彩色形态学的基本操作,包括膨胀、腐蚀、开、闭等操作;进一步将提出的彩色形态学操作应用于高分辨率遥感图像,通过实验对比验证其对地物目标的形变和平滑能力,说明其实用性和有效性。     

Unconstrained optimal control of regular languages

This paper formulates an unconstrained optimal policy for control of regular languages realized as deterministic finite state automata (DFSA). A signed real measure quantifies the behavior of controlled sublanguages based on a state transition cost matrix and a characteristic vector as reported in an earlier publication. The state-based optimal control policy is obtained by selectively disabling controllable events to maximize the measure of the controlled plant language without any further constraints. Synthesis of the optimal control policy requires at most n iterations, where n is the number of states of the DFSA model. Each iteration solves a set of n simultaneous linear algebraic equations. As such, computational complexity of the control synthesis is polynomial in n.     

An Axiom Foundation for Uncertain Reasonings in Rule-Based Expert Systems: NT-Algebra

This paper identifies an axiom foundation for uncertain reasonings in rule-based expert systems: a near topological algebra (NT-algebra for short), which holds some basic notions hidden behind the uncertain reasoning models in rule-based expert systems. According to the basic means of topological connection in an inference network, an NT-algebraic structure has five basic operators, i.e. AND, OR, NOT, Sequential combination and Parallel combination, which obey some axioms. An NT-algebraic structure is defined on a near-degree space introduced by the authors, which is a special topological space. The continuities of real functions, of fuzzy functions and functions in other senses can be uniformly considered in the framework of a near-degree space. This paper also proves that EMYCIN’s and PROSPECTOR’s uncertain reasoning models correspond to good NT-algebras. Moreover, the existence of any finite NT-algebraic structure is constructively proved. Compared to other related research efforts, the NT-algebra as an axiom foundation has the following characteristics: (1) various cases of assessments for uncertainties of evidence and rules are put into a unified algebraic structure; and (2) major emphasis has been placed on the basic laws of the propagation for them in an inference network, especially the continuity of propagation operations and the relationships between propagation operations.     

Hypermatrix Algebra: Applications in Parallel Image Processing

A unique algebraic methodology is presented for implementing parallel tasks on multistage multidimensional architectures. Such architectures are motivated by the potential of digital optical and optoelectronic computing. Structured tasks can be decomposed from their initial algebraic casting and shown to match known canonical forms representing desirable computing structures such as 2D Omega processor. Algebraic manipulation of two-dimensional data structures—typical in image processing—requires more powerful operations than those afforded by classical matrix algebra. To this end, hypermatrix algebra provides a compact treatment of multidimensional objects and associated operations. Mappings of linear transformations are demonstrated including 2D Hadamard transformations, matrix rotations, and transposition.     

Symbolic and numerical analysis of the restricted five body problem using computer algebra

This paper is devoted to the application of the computer algebra system Mathematica for the investigation of a nonlinear system of algebraic equations with parameters that describes the dynamics of a passively gravitating mass in the space of four bodies forming a central convex deltoid configuration. Domains of existence of solutions in the plane of parameters are found, and the number of solutions depending on the parameters of the system is determined. Bifurcational analysis is performed, and bifurcation curves in the plane of parameters are constructed that separate the domains of parameters with different number of solutions. Corresponding analysis algorithms for the system under examination are developed and implemented in Mathematica.