We consider the rational flow ?R(z)=R(z)(d/dz) where R is given by quotient of two polynomials without common factors on Riemann sphere. The separatrix graph ?R boundary between trajectories with different properties. characterize properties a planar directed to be vector field

In the last lecture, we have proved the harder side of the Cheegar’s Inequality, i.e. the conductance of a graph is upper bounded by second smallest eigenvalue, in formula, ΦG ≤ √ 2λ2. To prove that we took any vector ~ X having the property ~ X ⊥ ~1 and Rayleigh Quotient R(LG, ~ X) = ~ XLG ~ X ~ XT ~ X = λ2 to construct a set so that the conductance of the graph is upper bounded by the inequal...

Tree Pattern Queries (TPQ), Branching Path Queries (BPQ), and Core XPath (CXPath) are subclasses of the XML query language XPath, TPQ ⊂ BPQ ⊂ CXPath ⊂ XPath. Let TPQ = TPQ ⊂ BPQ ⊂ CXPath ⊂ XPath denote the corresponding subclasses, consisting of queries that do not involve the boolean negation operator not in their predicates. Simulation and bisimulation are two different binary relations on gr...

Let be a finite G-symmetric graph whose vertex set admits a nontrivial G-invariant partition B. It was observed that the quotient graph B of relative to B can be (G,2)-arc transitive even if itself is not necessarily (G,2)-arc transitive. In a previous article of Iranmanesh et al., this observation motivated a study of G-symmetric graphs ( ,B) such that B is (G,2)-arc transitive and, for blocks...

Denote by G the set of triples (Γ, X,B), where Γ is a finite X-symmetric graph of valency val(Γ) ≥ 1, B is a nontrivial X-invariant partition of V (Γ) such that ΓB, the quotient graph of Γ with respect to B, is nonempty and Γ is not a multicover of ΓB. In this article, for any given X-symmetric graph Σ, we aim to give a sufficient and necessary condition for the existence of (Γ, X,B) ∈ G, such ...

This extended abstract presents a submission to the Music Information Retrieval Evaluation eXchange (MIREX) in the symbolic Melodic Similarity task. This submission is an implementation of an extension of the edit-distance technique to the polyphonic context. Melodies are represented by quotiented sequences. A quotiented sequence is a sequence graph defined with an additional equivalent relatio...

We consider planar bar-and-joint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetry-forced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2k with odd k, unifying and extending previous work on this subject. We also explo...

Space and time problems still occasionally dog the functional programmer, despite increasingly efficient implementations and the recent spate of useful profiling tools. There is a need for a model of program reduction that relates directly to the user’s code and has a simple graphical representation. Naı̈ve graph reduction provides this. We address the problems of displaying a series of program ...

The C∗-algebra C∗(E) of a directed graph E is generated by partial isometries satisfying relations which reflect the path structure of the graph. In [10], Kumjian and Pask considered the action of a group G on C∗(E) induced by an action of G on E. They proved that if G acts freely and E is locally finite, then the crossed product C∗(E) × G is Morita equivalent to the C∗-algebra of the quotient ...