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A connected graph G is distance hereditary if every induced path in G is a shortest path. The Interval function I G (u , v) of a connected graph G is defined as the set of all vertices that lie on some shortest u , v -path in G. In this paper, we consider certain types of first-order betweenness axioms framed on an arbitrary function known as transit function and used to characterize the interval function I G of a distance hereditary graph. As a byproduct, we give a new characterization of distance hereditary graphs using these betweenness axioms. [ABSTRACT FROM AUTHOR]

AbstractInterval function of a graph is a well-known notion in metric graph theory and the axiomatic characterization using a set of first order axioms of different graph classes is an interesting problem in this area. Partial cubes and partial Hamming graphs form one of the central graph class that can be embedded into hypercubes and Hamming graphs respectively. In this paper we present an axiomatic characterization of the interval function of partial cubes and partial Hamming graphs, using different first order axioms. Further, we give an axiomatic characterization of the interval function of these graphs using axioms on an arbitrary function R on V.

Transit functions are introduced to study betweenness, intervals and convexity in an axiomatic setup on graphs and other discrete structures. Prime example of a transit function on graphs is the well studied interval function of a connected graph. In this paper, we study the Cycle transit function C(u; v) on graphs which is a transit function derived from the interval function. We study the betweenness properties and also characterize graphs in which the cycle transit function coincides with the interval function. We also characterize graphs where ... as an analogue of median graphs. [ABSTRACT FROM AUTHOR]

The paper presents the Finite element method (FEM) that calculates the internal force and the displacement of beams on elastic foundation in the case of the uncertainty input parameters described in terms of the number intervals. Using the interval function optimization algorithm combined with the FEM to determine the internal force values of the span reinforced concrete beam structure. This study applies the hybrid differential evolutionary optimization algorithm combined with FEM interval functions to determine the required internal force results of reinforced concrete beams placed on an elastic foundation. The calculation process is programmed using Maple.17 software to determine the displacement output and the resulting internal force in the beam. To check the correctness of the program calculated on Maple.17, the input declared dataset corresponds to the central values of the interval numbers and is recalculated by SAP2000v21 software, then evaluates the results error of internal force on the beam under consideration.

The interval function and the induced path function are two well studied class of set functions of a connected graph having interesting properties and applications to convexity, metric graph theory. Both these functions can be framed as special instances of a general set function termed as a transit function defined on the Cartesian product of a non-empty set \(V\) to the power set of \(V\) satisfying the expansive, symmetric and idempotent axioms. In this paper, we propose a set of independent first order betweenness axioms on an arbitrary transit function and provide characterization of the interval function of Ptolemaic graphs and the induced path function of chordal graphs in terms of an arbitrary transit function. This in turn gives new characterizations of the Ptolemaic and chordal graphs.

The interval function and the induced path function are two well studied class of set functions of a connected graph having interesting properties and applications to convexity, metric graph theory. Both these functions can be framed as special instances of a general set function termed as a transit function defined on the Cartesian product of a non-empty set V to the power set of V satisfying the expansive, symmetric and idempotent axioms. In this paper, we propose a set of independent first order betweenness axioms on an arbitrary transit function and provide characterization of the interval function of Ptolemaic graphs and the induced path function of chordal graphs in terms of an arbitrary transit function. This in turn gives new characterizations of the Ptolemaic and chordal graphs. [ABSTRACT FROM AUTHOR]

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Betweenness is a universal notion present in several disciplines of mathematics. The notion of betweenness has a profound history and many pioneers like Euclid, Pasch, Hilbert have studied betweenness axiomatically. In discrete mathematics too, betweenness is present and several authors have worked on this concept from an axiomatic view. In graph theory, betweenness is developed mainly as metric betweenness, studied using the shortest path metric in a connected graph, thus resulting in the notion of the interval function. Many interesting results are available in graph theory using the interval function. The interval function is generalized to induced path function by replacing shortest paths by induced paths. The induced path betweenness also captured attention among graph theorists with several interesting results to date. From an axiomatic point of view, two pertinent questions can be framed on these functions. Is it possible to axiomatically characterize the interval function of some special graphs using some set of first order axioms defined on an arbitrary transit function? Is it possible to characterize the graphs with the help of their interval functions? In this paper, we survey the results as answers to these questions available from the research papers.

In this note we extend the Mulder-Nebeský characterization of the interval function of a connected graph to the disconnected case. One axiom needs to be adapted, but also a new axiom is needed in addition.

We study the global stability of interval optimization problems and focus on set-type solutions that are defined by means of the Kulisch–Miranker order between intervals. In order to carry out this, we employ a variational convergence notion for vector functions taken from the literature and provide geometric and metric characterizations of this variational convergence for interval functions. We describe the global behaviour of the solution, level and colevel sets under variations of the data and show that the coercivity and coercive existence conditions for these problems are preserved locally within certain classes of functions. Then, we compare the variational convergence with other convergence notions taken from the literature. Finally, we study the behaviour of the operations with interval functions under variations of the data. [ABSTRACT FROM AUTHOR]

The axiomatic study on the interval function, induced path function and all-paths function of a connected graph is a well-known area in metric graph theory and related areas. In this paper, we introduce the following new axiom: (cp) v ∈ R (u , w) and v ∈ R (u , x) ⇒ w ∈ R (v , x) or x ∈ R (v , w) , for all distinct u , v , w , x ∈ V. We present characterizations of (claw, paw)-free graphs using axiom (cp) on the standard path transit functions on graphs, namely the interval function, the induced path function, and the all-paths function. We study the underlying graphs of the transit functions which are (claw, paw)-free and Hamiltonian. We present an axiomatic characterization of the interval function on (claw, paw)-free graphs. Furthermore, we obtain an axiomatic characterization of the induced path function on a subclass of (claw, paw)-free graphs. [ABSTRACT FROM AUTHOR]

Copyright of Acta Sting is the property of Akademie STING, o.p.s. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Intervals and convexity play crucial roles in the applications of graph theory such as town planning and design of graphics. In this article, the concept of geodetic interval in graphs is extended to fuzzy graphs. Intervals are useful in the study of properties of fuzzy graphs which depend on the geodetic distance between vertices. The axiomatic definition of intervals in fuzzy graphs are used to define intervals in different fuzzy graph structures like fuzzy trees and complete fuzzy graphs. Finally a set theoretic operations of intervals like union, intersection are also discussed and some results are obtained.

The consistency of the Aalen—Johansen-derived estimator of state occupation probabilities in non-Markov multi-state settings is studied and established via a new route. This new route is based on interval functions and relies on a close connection between additive and multiplicative transforms of interval functions, which is established. Under certain assumptions, the consistency follows from explicit expressions of the additive and multiplicative transforms related to the transition probabilities as interval functions, which are obtained, in combination with certain censoring and positivity assumptions [ABSTRACT FROM AUTHOR]

This paper proposes a geometric method for solving the inverse kinematics (IK) of redundant manipulators while satisfying the collision avoidance criterion. The proposed approach focuses on creating an algorithm for solving the redundant inverse kinematics (RIK) of manipulators in a known environment. Central to the method is the creation of an explicit expression of C-free space o f the manipulators by using two new notions, named "inverse Denavit-Hartenberg (D-H) variables" and "interval function". Thus, a single optimized solution far from the obstacles is obtained automatically by exploring in the C-free space, following the principle of selecting the median value of the interval function preferentially. The proposed method is demonstrated using a snake robot, the EAST Articulated Maintenance Arm (EAMA), which is utilized in the EAST (Experimental Advanced Superconducting Tokamak) for maintenance tasks in the vacuum vessel. C-free space of EAMA in EAST is created based on the Obstacle Topology Partition Projection (OTPP) approach and formulated by the interval functions. With the expression of C-free space, a single optimum solution is obtained during the exploration that starts at the end position or pose of the extremity point Pend. Eventually, several tip positions of EAMA are sampled to test the accuracy and correctness of the algorithm. [ABSTRACT FROM AUTHOR]

Copyright of Radio Electronics, Computer Science, Control is the property of Zaporizhzhia National Technical University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Radio Electronics, Computer Science, Control is the property of Zaporizhzhia National Technical University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The discrete Fourier transform on pitch-class sets, proposed by David Lewin and advanced by Ian Quinn, may provide a new lease on life for Allen Forte's idea of a general theory of harmony for the twentieth century based on the intervallic content of pitch-class collections. This article proposes the use of phase spaces and Quinn's harmonic qualities in analysis of a wide variety of twentieth-century styles. The main focus is on how these ideas relate to scale-theoretic concepts and the repertoires to which they are applied, such as the music of Debussy, Satie, Stravinsky, Ravel, and Shostakovich. Diatonicity, one of the harmonic qualities, is a basic concern for all of these composers. Phase spaces and harmonic qualities also help to explain the "scale-network wormhole" phenomenon in Debussy and Ravel and better pinpoint the role of octatonicism in Stravinsky's and Ravel's music. [ABSTRACT FROM AUTHOR]