Search results for: Synchronous dataflow graph.
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 487

Search results for: Synchronous dataflow graph.

487 Analysis of a Singular Perturbed Synchronous Generator with a Bond Graph Approach

Authors: Gilberto Gonzalez-A, Noe Barrera-G

Abstract:

An analysis of a synchronous generator in a bond graph approach is proposed. This bond graph allows to determine the simplified models of the system by using singular perturbations. Firstly, the nonlinear bond graph of the generator is linearized. Then, the slow and fast state equations by applying singular perturbations are obtained. Also, a bond graph to get the quasi-steady state of the slow dynamic is proposed. In order to verify the effectiveness of the singularly perturbed models, simulation results of the complete system and reduced models are shown.

Keywords: Bond graph modelling, synchronous generator, singular perturbations

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486 Phasor Analysis of a Synchronous Generator: A Bond Graph Approach

Authors: Israel Núñez-Hernández, Peter C. Breedveld, Paul B. T. Weustink, Gilberto Gonzalez-A

Abstract:

This paper presents the use of phasor bond graphs to obtain the steady-state behavior of a synchronous generator. The phasor bond graph elements are built using 2D multibonds, which represent the real and imaginary part of the phasor. The dynamic bond graph model of a salient-pole synchronous generator is showed, and verified viz. a sudden short-circuit test. The reduction of the dynamic model into a phasor representation is described. The previous test is executed on the phasor bond graph model, and its steady-state values are compared with the dynamic response. Besides, the widely used power (torque)-angle curves are obtained by means of the phasor bond graph model, to test the usefulness of this model.

Keywords: Bond graphs, complex power, phasors, synchronous generator, short-circuit, open-circuit, power-angle curve.

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485 The Vulnerability Analysis of Java Bytecode Based on Points-to Dataflow

Authors: Tang Hong, Zhang Lufeng, Chen Hua, Zhang Jianbo

Abstract:

Today many developers use the Java components collected from the Internet as external LIBs to design and develop their own software. However, some unknown security bugs may exist in these components, such as SQL injection bug may comes from the components which have no specific check for the input string by users. To check these bugs out is very difficult without source code. So a novel method to check the bugs in Java bytecode based on points-to dataflow analysis is in need, which is different to the common analysis techniques base on the vulnerability pattern check. It can be used as an assistant tool for security analysis of Java bytecode from unknown softwares which will be used as extern LIBs.

Keywords: Java bytecode, points-to dataflow, vulnerability analysis

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484 Research on IBR-Driven Distributed Collaborative Visualization System

Authors: Yin Runmin, Song Changfeng

Abstract:

Image-based Rendering(IBR) techniques recently reached in broad fields which leads to a critical challenge to build up IBR-Driven visualization platform where meets requirement of high performance, large bounds of distributed visualization resource aggregation and concentration, multiple operators deploying and CSCW design employing. This paper presents an unique IBR-based visualization dataflow model refer to specific characters of IBR techniques and then discusses prominent feature of IBR-Driven distributed collaborative visualization (DCV) system before finally proposing an novel prototype. The prototype provides a well-defined three level modules especially work as Central Visualization Server, Local Proxy Server and Visualization Aid Environment, by which data and control for collaboration move through them followed the previous dataflow model. With aid of this triple hierarchy architecture of that, IBR oriented application construction turns to be easy. The employed augmented collaboration strategy not only achieve convenient multiple users synchronous control and stable processing management, but also is extendable and scalable.

Keywords: Image-Based Rendering, Distributed CollaborativeVisualization, Computer Supported Cooperative Work, Model andSimulation, Modular Visualization Environment.

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483 Design of an Efficient Retimed CIC Compensation Filter

Authors: Vishal Awasthi, Krishna Raj

Abstract:

Unwanted side effects because of spectral aliasing and spectral imaging during signal processing would be the major concern over the sampling rate alteration. Multirate-multistage implementation of digital filter could come about a large computational saving than single rate filter suitable for sample rate conversion. This implementation can further improve through high-level architectural transformation in circuit level. Reallocating registers and  relocating flip-flops across logic gates through retiming certainly a prominent sequential transformation technology, that optimize hardware circuits to achieve faster clocking speed without affecting the functionality. In this paper, we proposed an efficient compensated cascade Integrator comb (CIC) decimation filter structure that analyze the consequence of filter order variation which has a retimed FIR filter being compensator while using the cutset retiming technique and achieved an improvement in the passband droop by 14% to 39%, in computation time by 38.04%, 25.78%, 12.21%, 6.69% and 4.44% and reduction in path delay by 62.27%, 72%, 86.63%, 91.56% and 94.42% of 3, 6, 8, 12 and 24 order filter respectively than the non-retimed CIC compensation filter.

Keywords: Multirate Filtering, CIC decimation filter, Compensation theory, Retiming, Retiming algorithm, Filter order, Synchronous dataflow graph.

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482 A New Self-stabilizing Algorithm for Maximal 2-packing

Authors: Zhengnan Shi

Abstract:

In the self-stabilizing algorithmic paradigm, each node has a local view of the system, in a finite amount of time the system converges to a global state with desired property. In a graph G = (V, E), a subset S C V is a 2-packing if Vi c V: IN[i] n SI <1. In this paper, an ID-based, constant space, self-stabilizing algorithm that stabilizes to a maximal 2-packing in an arbitrary graph is proposed. It is shown that the algorithm stabilizes in 0(n3) moves under any scheduler (daemon). Specifically, it is shown that the algorithm stabilizes in linear time-steps under a synchronous daemon where every privileged node moves at each time-step.

Keywords: self-stabilization, 2-packing, distributed computing, fault tolerance, graph algorithms

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481 Efficient Filtering of Graph Based Data Using Graph Partitioning

Authors: Nileshkumar Vaishnav, Aditya Tatu

Abstract:

An algebraic framework for processing graph signals axiomatically designates the graph adjacency matrix as the shift operator. In this setup, we often encounter a problem wherein we know the filtered output and the filter coefficients, and need to find out the input graph signal. Solution to this problem using direct approach requires O(N3) operations, where N is the number of vertices in graph. In this paper, we adapt the spectral graph partitioning method for partitioning of graphs and use it to reduce the computational cost of the filtering problem. We use the example of denoising of the temperature data to illustrate the efficacy of the approach.

Keywords: Graph signal processing, graph partitioning, inverse filtering on graphs, algebraic signal processing.

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480 Using Spectral Vectors and M-Tree for Graph Clustering and Searching in Graph Databases of Protein Structures

Authors: Do Phuc, Nguyen Thi Kim Phung

Abstract:

In this paper, we represent protein structure by using graph. A protein structure database will become a graph database. Each graph is represented by a spectral vector. We use Jacobi rotation algorithm to calculate the eigenvalues of the normalized Laplacian representation of adjacency matrix of graph. To measure the similarity between two graphs, we calculate the Euclidean distance between two graph spectral vectors. To cluster the graphs, we use M-tree with the Euclidean distance to cluster spectral vectors. Besides, M-tree can be used for graph searching in graph database. Our proposal method was tested with graph database of 100 graphs representing 100 protein structures downloaded from Protein Data Bank (PDB) and we compare the result with the SCOP hierarchical structure.

Keywords: Eigenvalues, m-tree, graph database, protein structure, spectra graph theory.

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479 A Neighborhood Condition for Fractional k-deleted Graphs

Authors: Sizhong Zhou, Hongxia Liu

Abstract:

Abstract–Let k ≥ 3 be an integer, and let G be a graph of order n with n ≥ 9k +3- 42(k - 1)2 + 2. Then a spanning subgraph F of G is called a k-factor if dF (x) = k for each x ∈ V (G). A fractional k-factor is a way of assigning weights to the edges of a graph G (with all weights between 0 and 1) such that for each vertex the sum of the weights of the edges incident with that vertex is k. A graph G is a fractional k-deleted graph if there exists a fractional k-factor after deleting any edge of G. In this paper, it is proved that G is a fractional k-deleted graph if G satisfies δ(G) ≥ k + 1 and |NG(x) ∪ NG(y)| ≥ 1 2 (n + k - 2) for each pair of nonadjacent vertices x, y of G.

Keywords: Graph, minimum degree, neighborhood union, fractional k-factor, fractional k-deleted graph.

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478 The Extremal Graph with the Largest Merrifield-Simmons Index of (n, n + 2)-graphs

Authors: M. S. Haghighat, A. Dolati, M. Tabari, E. Mohseni

Abstract:

The Merrifield-Simmons index of a graph G is defined as the total number of its independent sets. A (n, n + 2)-graph is a connected simple graph with n vertices and n + 2 edges. In this paper we characterize the (n, n+2)-graph with the largest Merrifield- Simmons index. We show that its Merrifield-Simmons index i.e. the upper bound of the Merrifield-Simmons index of the (n, n+2)-graphs is 9 × 2n-5 +1 for n ≥ 5.

Keywords: Merrifield-Simmons index, (n, n+2)-graph.

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477 N-Sun Decomposition of Complete, Complete Bipartite and Some Harary Graphs

Authors: R. Anitha, R. S. Lekshmi

Abstract:

Graph decompositions are vital in the study of combinatorial design theory. A decomposition of a graph G is a partition of its edge set. An n-sun graph is a cycle Cn with an edge terminating in a vertex of degree one attached to each vertex. In this paper, we define n-sun decomposition of some even order graphs with a perfect matching. We have proved that the complete graph K2n, complete bipartite graph K2n, 2n and the Harary graph H4, 2n have n-sun decompositions. A labeling scheme is used to construct the n-suns.

Keywords: Decomposition, Hamilton cycle, n-sun graph, perfect matching, spanning tree.

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476 The Diameter of an Interval Graph is Twice of its Radius

Authors: Tarasankar Pramanik, Sukumar Mondal, Madhumangal Pal

Abstract:

In an interval graph G = (V,E) the distance between two vertices u, v is de£ned as the smallest number of edges in a path joining u and v. The eccentricity of a vertex v is the maximum among distances from all other vertices of V . The diameter (δ) and radius (ρ) of the graph G is respectively the maximum and minimum among all the eccentricities of G. The center of the graph G is the set C(G) of vertices with eccentricity ρ. In this context our aim is to establish the relation ρ = δ 2  for an interval graph and to determine the center of it.

Keywords: Interval graph, interval tree, radius, center.

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475 Completion Number of a Graph

Authors: Sudhakar G

Abstract:

In this paper a new concept of partial complement of a graph G is introduced and using the same a new graph parameter, called completion number of a graph G, denoted by c(G) is defined. Some basic properties of graph parameter, completion number, are studied and upperbounds for completion number of classes of graphs are obtained , the paper includes the characterization also.

Keywords: Completion Number, Maximum Independent subset, Partial complements, Partial self complementary

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474 On Fractional (k,m)-Deleted Graphs with Constrains Conditions

Authors: Sizhong Zhou, Hongxia Liu

Abstract:

Let G be a graph of order n, and let k  2 and m  0 be two integers. Let h : E(G)  [0, 1] be a function. If e∋x h(e) = k holds for each x  V (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh = {e  E(G) : h(e) > 0}. A graph G is called a fractional (k,m)-deleted graph if there exists a fractional k-factor G[Fh] of G with indicator function h such that h(e) = 0 for any e  E(H), where H is any subgraph of G with m edges. In this paper, it is proved that G is a fractional (k,m)-deleted graph if (G)  k + m + m k+1 , n  4k2 + 2k − 6 + (4k 2 +6k−2)m−2 k−1 and max{dG(x), dG(y)}  n 2 for any vertices x and y of G with dG(x, y) = 2. Furthermore, it is shown that the result in this paper is best possible in some sense.

Keywords: Graph, degree condition, fractional k-factor, fractional (k, m)-deleted graph.

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473 Metric Dimension on Line Graph of Honeycomb Networks

Authors: M. Hussain, Aqsa Farooq

Abstract:

Let G = (V,E) be a connected graph and distance between any two vertices a and b in G is a−b geodesic and is denoted by d(a, b). A set of vertices W resolves a graph G if each vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G. In this paper line graph of honeycomb network has been derived and then we calculated the metric dimension on line graph of honeycomb network.

Keywords: Resolving set, metric dimension, honeycomb network, line graph.

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472 Comparison of Full Graph Methods of Switched Circuits Solution

Authors: Zdeňka Dostálová, David Matoušek, Bohumil Brtnik

Abstract:

As there are also graph methods of circuit analysis in addition to algebraic methods, it is, in theory, clearly possible to carry out an analysis of a whole switched circuit in two-phase switching exclusively by the graph method as well. This article deals with two methods of full-graph solving of switched circuits: by transformation graphs and by two-graphs. It deals with the circuit switched capacitors and the switched current, too. All methods are presented in an equally detailed steps to be able to compare.

Keywords: Switched capacitors of two phases, switched currents of two phases, transformation graph, two-graph, Mason's formula, voltage transfer, summary graph.

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471 Speedup Breadth-First Search by Graph Ordering

Authors: Qiuyi Lyu, Bin Gong

Abstract:

Breadth-First Search (BFS) is a core graph algorithm that is widely used for graph analysis. As it is frequently used in many graph applications, improving the BFS performance is essential. In this paper, we present a graph ordering method that could reorder the graph nodes to achieve better data locality, thus, improving the BFS performance. Our method is based on an observation that the sibling relationships will dominate the cache access pattern during the BFS traversal. Therefore, we propose a frequency-based model to construct the graph order. First, we optimize the graph order according to the nodes’ visit frequency. Nodes with high visit frequency will be processed in priority. Second, we try to maximize the child nodes’ overlap layer by layer. As it is proved to be NP-hard, we propose a heuristic method that could greatly reduce the preprocessing overheads.We conduct extensive experiments on 16 real-world datasets. The result shows that our method could achieve comparable performance with the state-of-the-art methods while the graph ordering overheads are only about 1/15.

Keywords: Breadth-first search, BFS, graph ordering, graph algorithm.

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470 Investigating the Regulation System of the Synchronous Motor Excitation Mode Serving as a Reactive Power Source

Authors: Baghdasaryan Marinka, Ulikyan Azatuhi

Abstract:

The efficient usage of the compensation abilities of the electrical drive synchronous motors used in production processes can essentially improve the technical and economic indices of the process.  Reducing the flows of the reactive electrical energy due to the compensation of reactive power allows to significantly reduce the load losses of power in the electrical networks. As a result of analyzing the scientific works devoted to the issues of regulating the excitation of the synchronous motors, the need for comprehensive investigation and estimation of the excitation mode has been substantiated. By means of the obtained transmission functions, in the Simulink environment of the software package MATLAB, the transition processes of the excitation mode have been studied. As a result of obtaining and estimating the graph of the Nyquist plot and the transient process, the necessity of developing the Proportional-Integral-Derivative (PID) regulator has been justified. The transient processes of the system of the PID regulator have been investigated, and the amplitude–phase characteristics of the system have been estimated. The analysis of the obtained results has shown that the regulation indices of the developed system have been improved. The developed system can be successfully applied for regulating the excitation voltage of different-power synchronous motors, operating with a changing load, ensuring a value of the power coefficient close to 1.

Keywords: Transient process, synchronous motor, excitation mode, regulator, reactive power.

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469 On Detour Spectra of Some Graphs

Authors: S.K.Ayyaswamy, S.Balachandran

Abstract:

The Detour matrix (DD) of a graph has for its ( i , j) entry the length of the longest path between vertices i and j. The DD-eigenvalues of a connected graph G are the eigenvalues for its detour matrix, and they form the DD-spectrum of G. The DD-energy EDD of the graph G is the sum of the absolute values of its DDeigenvalues. Two connected graphs are said to be DD- equienergetic if they have equal DD-energies. In this paper, the DD- spectra of a variety of graphs and their DD-energies are calculated.

Keywords: Detour eigenvalue (of a graph), detour spectrum(of a graph), detour energy(of a graph), detour - equienergetic graphs.

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468 Comparison of an Interior Mounted Permanent Magnet Synchronous Generator with a Synchronous Reluctance Generator for a Wind Application

Authors: Poopak Roshanfekr, Torbjörn Thiringer, Sonja Lundmark, Mikael Alatalo

Abstract:

This article presents a performance comparison of an interior mounted permanent magnet synchronous generator (IPMSG) with a synchronous reluctance generator (SynRG) with the same size for a wind application. It is found that using the same geometrical dimensions, a SynRG can convert 74 % of the power that an IPMSG can convert, while it has 80% of the IPMSG weight. Moreover it is found that the efficieny for the IMPSG is 99% at rated power compared to 98.7% for the SynRG.

Keywords: Interior mounted permanent magnet synchronous generator (IPMSG), synchronous reluctance generator (SynRG), wind energy, annual energy efficiency.

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467 An Algorithm for Preventing the Irregular Operation Modes of the Drive Synchronous Motor Providing the Ore Grinding

Authors: Baghdasaryan Marinka

Abstract:

The current scientific and engineering interest concerning the problems of preventing the emergency manifestations of drive synchronous motors, ensuring the ore grinding technological process has been justified. The analysis of the known works devoted to the abnormal operation modes of synchronous motors and possibilities of protection against them, has shown that their application is inexpedient for preventing the impermissible displays arising in the electrical drive synchronous motors ensuring the ore-grinding process. The main energy and technological factors affecting the technical condition of synchronous motors are evaluated. An algorithm for preventing the irregular operation modes of the electrical drive synchronous motor applied in the ore-grinding technological process has been developed and proposed for further application which gives an opportunity to provide smart solutions, ensuring the safe operation of the drive synchronous motor by a comprehensive consideration of the energy and technological factors.

Keywords: Synchronous motor, abnormal operating mode, electric drive, algorithm, energy factor, technological factor.

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466 Developing a Regulator for Improving the Operation Modes of the Electrical Drive Motor

Authors: Baghdasaryan Marinka

Abstract:

The operation modes of the synchronous motors used in the production processes are greatly conditioned by the accidentally changing technological and power indices.  As a result, the electrical drive synchronous motor may appear in irregular operation regimes. Although there are numerous works devoted to the development of the regulator for the synchronous motor operation modes, their application for the motors working in the irregular modes is not expedient. In this work, to estimate the issues concerning the stability of the synchronous electrical drive system, the transfer functions of the electrical drive synchronous motors operating in the synchronous and induction modes have been obtained.  For that purpose, a model for investigating the frequency characteristics has been developed in the LabView environment. Frequency characteristics for assessing the transient process of the electrical drive system, operating in the synchronous and induction modes have been obtained, and based on their assessment, a regulator for improving the operation modes of the motor has been proposed. The proposed regulator can be successfully used to prevent the irregular modes of the electrical drive synchronous motor, as well as to estimate the operation state of the drive motor of the mechanism with a changing load.

Keywords: Electrical drive system, synchronous motor, regulator, stability, transition process.

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465 Analysis of Electrical Networks Using Phasors: A Bond Graph Approach

Authors: Israel Núñez-Hernández, Peter C. Breedveld, Paul B. T. Weustink, Gilberto Gonzalez-A

Abstract:

This paper proposes a phasor representation of electrical networks by using bond graph methodology. A so-called phasor bond graph is built up by means of two-dimensional bonds, which represent the complex plane. Impedances or admittances are used instead of the standard bond graph elements. A procedure to obtain the steady-state values from a phasor bond graph model is presented. Besides the presentation of a phasor bond graph library in SIDOPS code, also an application example is discussed.

Keywords: Bond graphs, phasor theory, steady-state, complex power, electrical networks.

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464 Topological Queries on Graph-structured XML Data: Models and Implementations

Authors: Hongzhi Wang, Jianzhong Li, Jizhou Luo

Abstract:

In many applications, data is in graph structure, which can be naturally represented as graph-structured XML. Existing queries defined on tree-structured and graph-structured XML data mainly focus on subgraph matching, which can not cover all the requirements of querying on graph. In this paper, a new kind of queries, topological query on graph-structured XML is presented. This kind of queries consider not only the structure of subgraph but also the topological relationship between subgraphs. With existing subgraph query processing algorithms, efficient algorithms for topological query processing are designed. Experimental results show the efficiency of implementation algorithms.

Keywords: XML, Graph Structure, Topological query.

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463 An Efficient Graph Query Algorithm Based on Important Vertices and Decision Features

Authors: Xiantong Li, Jianzhong Li

Abstract:

Graph has become increasingly important in modeling complicated structures and schemaless data such as proteins, chemical compounds, and XML documents. Given a graph query, it is desirable to retrieve graphs quickly from a large database via graph-based indices. Different from the existing methods, our approach, called VFM (Vertex to Frequent Feature Mapping), makes use of vertices and decision features as the basic indexing feature. VFM constructs two mappings between vertices and frequent features to answer graph queries. The VFM approach not only provides an elegant solution to the graph indexing problem, but also demonstrates how database indexing and query processing can benefit from data mining, especially frequent pattern mining. The results show that the proposed method not only avoids the enumeration method of getting subgraphs of query graph, but also effectively reduces the subgraph isomorphism tests between the query graph and graphs in candidate answer set in verification stage.

Keywords: Decision Feature, Frequent Feature, Graph Dataset, Graph Query

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462 Notes on Fractional k-Covered Graphs

Authors: Sizhong Zhou, Yang Xu

Abstract:

A graph G is fractional k-covered if for each edge e of G, there exists a fractional k-factor h, such that h(e) = 1. If k = 2, then a fractional k-covered graph is called a fractional 2-covered graph. The binding number bind(G) is defined as follows, bind(G) = min{|NG(X)| |X| : ├ÿ = X Ôèå V (G),NG(X) = V (G)}. In this paper, it is proved that G is fractional 2-covered if δ(G) ≥ 4 and bind(G) > 5 3 .

Keywords: graph, binding number, fractional k-factor, fractional k-covered graph.

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461 Syntactic Recognition of Distorted Patterns

Authors: Marek Skomorowski

Abstract:

In syntactic pattern recognition a pattern can be represented by a graph. Given an unknown pattern represented by a graph g, the problem of recognition is to determine if the graph g belongs to a language L(G) generated by a graph grammar G. The so-called IE graphs have been defined in [1] for a description of patterns. The IE graphs are generated by so-called ETPL(k) graph grammars defined in [1]. An efficient, parsing algorithm for ETPL(k) graph grammars for syntactic recognition of patterns represented by IE graphs has been presented in [1]. In practice, structural descriptions may contain pattern distortions, so that the assignment of a graph g, representing an unknown pattern, to a graph language L(G) generated by an ETPL(k) graph grammar G is rejected by the ETPL(k) type parsing. Therefore, there is a need for constructing effective parsing algorithms for recognition of distorted patterns. The purpose of this paper is to present a new approach to syntactic recognition of distorted patterns. To take into account all variations of a distorted pattern under study, a probabilistic description of the pattern is needed. A random IE graph approach is proposed here for such a description ([2]).

Keywords: Syntactic pattern recognition, Distorted patterns, Random graphs, Graph grammars.

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460 Analysis of Synchronous Machine Excitation Systems: Comparative Study

Authors: Shewit Tsegaye, Kinde A. Fante

Abstract:

This paper presents the comparison and performance evaluation of synchronous machine excitation models. The two models, DC1A and AC4A, are among the IEEE standardized model structures for representing the wide variety of synchronous machine excitation systems. The performance evaluation of these models is done using SIMULINK simulation software. The simulation results obtained using transient analysis show that the DC1A excitation system is more reliable and stable than AC4A excitation system.

Keywords: Excitation system, synchronous machines, AC and DC regulators.

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459 Automatic Fingerprint Classification Using Graph Theory

Authors: Mana Tarjoman, Shaghayegh Zarei

Abstract:

Using efficient classification methods is necessary for automatic fingerprint recognition system. This paper introduces a new structural approach to fingerprint classification by using the directional image of fingerprints to increase the number of subclasses. In this method, the directional image of fingerprints is segmented into regions consisting of pixels with the same direction. Afterwards the relational graph to the segmented image is constructed and according to it, the super graph including prominent information of this graph is formed. Ultimately we apply a matching technique to compare obtained graph with the model graphs in order to classify fingerprints by using cost function. Increasing the number of subclasses with acceptable accuracy in classification and faster processing in fingerprints recognition, makes this system superior.

Keywords: Classification, Directional image, Fingerprint, Graph, Super graph.

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458 Graphs with Metric Dimension Two-A Characterization

Authors: Sudhakara G, Hemanth Kumar A.R

Abstract:

In this paper, we define distance partition of vertex set of a graph G with reference to a vertex in it and with the help of the same, a graph with metric dimension two (i.e. β (G) = 2 ) is characterized. In the process, we develop a polynomial time algorithm that verifies if the metric dimension of a given graph G is two. The same algorithm explores all metric bases of graph G whenever β (G) = 2 . We also find a bound for cardinality of any distance partite set with reference to a given vertex, when ever β (G) = 2 . Also, in a graph G with β (G) = 2 , a bound for cardinality of any distance partite set as well as a bound for number of vertices in any sub graph H of G is obtained in terms of diam H .

Keywords: Metric basis, Distance partition, Metric dimension.

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