Monophonic Distance in Graphs

For any two vertices u and v in a connected graph G, a u − v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) is the length of a longest u − v monophonic path in G. For any vertex v in G, the monophonic eccentricity of v is em(v) = max {dm(u, v) : u ∈ V}. The subgraph induced by the vertices of G having minimum monophonic eccentricity is the monophonic center of G, and it is proved that every graph is the monophonic center of some graph. Also it is proved that the monophonic center of every connected graph G lies in some block of G. With regard to convexity, this monophonic distance is the basis of some detour monophonic parameters such as detour monophonic number, upper detour monophonic number, forcing detour monophonic number, etc. The concept of detour monophonic sets and detour monophonic numbers by fixing a vertex of a graph would be introduced and discussed. Various interesting results based on these parameters are also discussed in this chapter

The Outer Connected Detour Monophonic Number of a Graph

For a connected graph ???? = (????, ????) of order  a set is called a monophonic set of ????if every vertex of ????is contained in a monophonic path joining some pair of vertices in ????. The monophonic number (????) of is the minimum cardinality of its monophonic sets. If  or the subgraph  is connected, then a detour monophonic set  of a connected graph is said to be an outer connected detour monophonic setof .The outer connecteddetourmonophonic number of , indicated by the symbol , is the minimum cardinality of an outer connected detour monophonic set of . The outer connected detour monophonic number of some standard graphs are determined. It is shown that for positive integers , and ???? ≥ 2 with ,there exists a connected graph ????with???????????????????? = , ????????????m???????? = and  = ????. Also, it is shown that for every pair of integers ????and b with 2 ≤ ???? ≤ ????, there exists a connected graph with and

RESTRAINED DOUBLE MONOPHONIC NUMBER OF A GRAPH

For a connected graph $G$ of order at least two, a double monophonic set $S$ of a graph $G$ is a restrained double monophonic set if  either $S=V$ or the subgraph induced by $V-S$ has no isolated vertices. The minimum cardinality of a restrained double  monophonic set of $G$ is the restrained double monophonic number of $G$ and is denoted by $dm_{r}(G)$. The restrained double monophonic number of certain classes graphs are determined. It is shown that for any integers $a,\, b,\, c$ with $3 \leq a \leq b \leq c$, there is a connected graph $G$ with $m(G) = a$, $m_r(G) = b$ and $dm_{r}(G) = c$, where $m(G)$ is the monophonic number and $m_r(G)$ is the restrained monophonic number of a graph $G$

Restrained Double Monophonic Number of a Graph

For a connected graph G of order at least two, a double monophonic set S of a graph G is a restrained double monophonic set if either S=V or the subgraph induced by V−S has no isolated vertices. The minimum cardinality of a restrained double monophonic set of G is the restrained double monophonic number of G and is denoted by dmr(G). The restrained double monophonic number of certain classes graphs are determined. It is shown that for any integers a,b,c with 3≤a≤b≤c, there is a connected graph G with m(G)=a, mr(G)=b and dmr(G)=c, where m(G) is the monophonic number and mr(G) is the restrained monophonic number of a graph G.The second author research work was supported by National Board for Higher Mathematics, INDIA (Project No. NBHM/R.P.29/2015/Fresh/157).The authors are thankful to the reviewers for their useful comments for the improvement of this paper

Developing a Mathematically Informed Approach to Musical Narrative through the Analysis of Three Twentieth-Century Monophonic Woodwind Works

This project applies mathematically informed narrative to monophonic music in the twentieth century, with a focus on three works for solo woodwinds: Debussy’s Syrinx (flute), Stravinsky’s Three Pieces for Clarinet, and Britten’s “Bacchus” from Six Metamorphoses after Ovid, Op. 49 (oboe). This music poses difficulties for traditional analytical methods due to a lack of explicit harmonies and unusual pitch language that is neither functionally tonal nor serially atonal. Additionally, these pieces present a variety of challenges due to differences in length, number of movements, and presence or absence of programmatic elements. Therefore, nontraditional methods could be beneficial for understanding these idiosyncratic pieces. Mathematical and transformational approaches have shown that such descriptions can elegantly illustrate pitch language in a wide variety of tonal and atonal styles. Visual transformational and geometric approaches, such as oriented networks and graphic representations, can assist in illustrating important changes that take place during a piece. Narrative theory approaches analysis from another viewpoint. While not all music can be considered narrative, a narrative paradigm is applicable to a wide range of musical styles. Because narrative theories focus on large-scale topical and gestural changes for building interpretations, it complements the locally focused nature of transformational theory. Together, a mathematically informed narrative method can reveal connections that are not immediately obvious in these works, and help a listener or performer create an informed interpretation

Order-Theoretic Combination Techniques and the Electronic Schrödinger Equation

Most standard constructions of the combination technique [M. Griebel et al., Iterative Methods in Linear Algebra, Elsevier, North Holland, p. 263] manipulate families of functions organised by downward-closed subsets of Nd. We introduce instead an alternative formulation, with functions indexed from a more general kind of partially ordered set (poset). The combinatorial and order-theoretic machinery of Möbius inversion helps us to construct combination sums of functions organised by order ideals of a poset grid. An adaptive algorithm is given for the quasi-optimal assembly of such an order ideal. This order-theoretic combination technique (OTCT) formalism is applied in the quantum-chemical setting of the high-dimensional electronic Schrödinger equation. Here, the OTCT allows us to connect, understand, and improve on a number of existing approaches. We consider first a selection of existing extrapolative composite methods. Extending on the idea of the CQML approach [P. Zaspel et al., J. Chem. Theory Comput., 15(3), 2018], an application of basically just the standard version of the combination technique leads to a generalised composite method (GCM). This approach is systematically improvable and appears comparable, if not yet truly competitive with standard composite methods from the perspectives of both accuracy and of cost. We turn then to energy-based fragmentation methods, which are often founded upon a truncated many-body expansion (MBE). It is well-known that Möbius inversion can provide non-recursive expressions for the individual MBE terms, and so the OTCT delivers by construction a framework for the adaptive truncation of MBE-like formulae. The same also functions for a class of related graph-based decompositions described in the existing literature. We term these in our context as SUPANOVA (SUbgraph Poset ANOVA) decompositions, and motivate them as extensions to the BOSSANOVA decomposition [M. Griebel et al., Extraction of Quantifiable Information from Complex Systems, Springer, Cham, 2014, p. 211; F. Heber, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, 2014]. We identify a subtle technical issue that afflicts BOSSANOVA in certain cases, and apply instead an adaptive SUPANOVA decomposition defined for convex subgraphs. Finally, we combine the GCM and SUPANOVA ideas to obtain a poset grid that recovers many existing multilevel fragmentation methods. We extend the ML-BOSSANOVA method [S. R. Chinnamsetty et al., Multiscale Model. Simul., 16(2), 2018], exploring now also a hierarchy of ab initio theories. Although an initial assessment is inconclusive, this ML-SUPANOVA formulation appears well-founded and paves the way to a number of interesting possible applications in the future

Brian Ferneyhough : the logic of the figure

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