A locally minimal, but not globally minimal bridge position of a knot

We give a locally minimal, but not globally minimal bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.Comment: 27 pages, 12 figures, v3: minor corrections throughout the pape

A Procedure of Graph-partitioning for the Mixed Analysis of Electrical Networks

A procedure to get a graph-partitioning for the mixed anaysis of Electrical Networks with the minimum number of equilibrium equations is described. The graph representing the given electrical network is partitioned into subgraphs each of which have a certain specified property, and a partial ordering is given to these subgraphs. The graph-partitioning is then acquired according to the partial ordering. A method to manipulate the diagram showing the partial ordering is given simplifying the partitioning

Solvability of Linear Electrical Networks

The problem of the solvability of a linear active network is discussed on the basis of the two-graph method. It is shown that the topological condition for the solvability is the existence of a common tree of the voltage and current graphs derived from the network. Several conditions for the existence of a common tree are given as well as an algorithm to check whether a common tree exists or not. The algorithm also gives a common tree, if one exists. Then a structure of two-graphs is defined and algorithms to determine the structure are given. The uniqueness and the stability of the structure are discussed. A decomposition of the coefficient matrix of the network equations is derived from the structure. Finally, a classification of the network solvability is given

Some Considerations on the State Equations of Linear Active Networks and the Network Topology

The state equations as a characterization of a linear active network involve two factors, the network topology and the properties of the network elements. In this paper the way these two factors influence the formulation of the state equations is studied. A set of state equations for the networks with certain topological restrictions is derived based on an overnormal tree. Furthermore the order of complexity of a linear active network is investigated in connection with the network topology, and an upper bound on the order of complexity is stated with respect to a particular common tree. Several examples are given to illustrate the investigation

Uniform Plane Graphs

This paper presents two special classes of plane graphs characterized by the sequences M(v) and W(e). M(v) is the circular sequence consisting of the numbers of vertices on the meshes around vertex v ; and W(e) is the sequence consisting of the numbers of vertices on the meshes to the right and the left of edge e and also of the degrees of the head and the tail of e. A graph is called uniform with respect to M(v) or W(e) if its vertices all have the same M(v), or if its edges all have the same W(e), respectively. It is shown that if such a uniform plane graph exists for the given M(v) or W(e), the numbers of its vertices, edges and meshes are uniquely determined. Then, the conditions on M(v) or W(e) for the existence of a graph are investigated. Tables of plane graphs which are uniform with respect to M(v) or W(e) are presented. Besides regular polyhedrons, there are thirteen types of graphs which are uniform with respect to M(v), and only four graphs which are uniform with respect to W(e)

A Computer-Aided Circuit Layout System Based on the Functional Structure and the Physical Structure of Circuits

In this paper, there is reported a computer-aided circuit layout system which is based on a new design philosophy. For a circuit to be designed and laid out, two structures called the functional structure (FS) and the physical structure (PS) are defined. The former is concerned with the behavior of the circuit and is hierarchical, while the latter is dependent on the physical realization of the circuit. The FS has “blocks, ” “components, ” "connections, “terminals” and “external terminals” as its basic elements, whereas the PS has “modules, ” “nets, ” “pins” and “external pins” as its basic elements. A circuit description language to specify the FS is designed and its interpreter is computer programmed. In order to have access to specific basic elements, data reference formulae are defined. The realizations of the basic elements of the PS are displayed on a color CRT. Our circuit layout procedure consists of three steps : (1) inputting the FS by using the circuit description language, (2) assigning components and terminals to modules and pins respectively, and (3) layouting basic elements of the PS on the board by using layout and display commands which take data reference formulae as their operand. The last step is performed conversationally and the layout of the elements can easily be changed

Some Properties of Multicolored-Branch Graphs

A multicolored-branch graph is such a linear graph that the branches of the graph are partitioned into several sets, and a certain color is assigned to the branches belonging to each of the sets. The assignment is called a coloring. The degree of interference of loops or cutsets in such a graph is deffned to be the minimum number of indenpedent loops or cutsets respectively containing all the colors. The maximum of the degree of interference taken over all the possible colorings is studied. Theorems concerning the colorings to give the maximum in a two-colored-branch graph are derived. Moreover, the maximum of the degree of interference is shown to be equal to the topological degree of freedom and to the maximum distance between a pair of trees in the graph. The degree of interference is also related to the rank of a certain submatrix of the fundamental loop or cutset matrix. An upper bound and a lower bound on the degree of interference in a three-colored-branch graph are given

X-Ray Spectrum of a Peculiar Supernova Remnant G359.1-0.5

We present the Suzaku results of a supernova remnant (SNR), G359.1-0.5 in the direction of the Galactic center region. From the SNR, we find prominent K-shell lines of highly ionized Si and S ions, together with unusual structures at 2.5-3.0 and 3.1-3.6 keV. No canonical SNR plasma model, in either ionization equilibrium or under-ionization, can explain the structures. The energies and shapes of the structures are similar to those of the radiative transitions of free electrons to the K-shell of He-like Si and S ions (radiative recombination continuum: RRC). The presence of the strong RRC structures indicates that the plasma is in over-ionization. In fact, the observed spectrum is well fitted with an over-ionized plasma model. The best-fit electron temperature of 0.29 keV is far smaller than the ionization temperature of 0.77 keV, which means that G359.1-0.5 is in extreme condition of over-ionization. We report some cautions on the physical parameters, and comment possible origins for the over-ionized plasma.Comment: 7 pages, 5 figures, accepted for publication in PAS