Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In recent years, quite a lot of work has been done on the extremal problem for these two indices, i.e., the problem of determining the graphs within certain prescribed classes that maximize or minimize the index value. This survey collects and classifies these results, and also provides some useful auxiliary results, tools and techniques that are frequently used in the study of this type of problem.     

Gauss Sum of Index Four

Let p be a prime number, q=p^f and m|q-1. For a multiplicative character x of Fq with order m, the Gauss sum G(x) over Fq is defined by     

“Asymptotically Unbiased” Estimators of the Tail Index Based on External Estimation of the Second Order Parameter

In this paper we shall deal with the asymptotic and finite sample properties of asymptotically unbiased estimators of the tail index , based on external adequate estimators of the second order parameter . The behavior of the -estimator considered has indeed a high impact on the distributional properties of the final estimator of , and must be carefully chosen. As a by-product of the final study we present also the finite sample properties of a few -estimators available in the literature.     

Nonexistence and Existence of Multiple Positive Solutions for Superlinear Three-point Boundary Value Problems via Index Theory

This paper discusses both the nonexistence of positive solutions for second-order three-point boundary value problems when the nonlinear term f（t, x, y） is superlinear in y at y = 0 and the existence of multiple positive solutions for second-order three-point boundary value problems when the nonlinear term f（t, x,y） is superlinear in x at ＋∞.     

Asymptotic Formulae for Brillouin Index on Riemannian Manifolds

We establish several asymptotic formulae for Brillouin index on fiat tori. As an application of these formulae it is proved that the topological entropy of a geodesic flow on a fiat torus is zero.     

Multiple Solutions of the Singular Impulsive Boundary Value Problems on the Half-line

This paper presents some results on the existence of double solutions for singular impulsive bound-ary value problems on the half line.     

The Tensor Products of Nilrings with Bounded Index of Nilpotence

R is a ring, for any nilpotent element r∈R if there exists a fixed integer nsuch that r~n=0, then R is said to be with bounded index of nilpotence, theleast of such integer n is called the index of R, denoted by i(R). If R is a nil ring with bounded index i(R)=n, R′ is a commutative ring,A·A·Klein〔1〕 has discussed the property of bounded index of nilpotence of RR′. In this paper we shall discuss the properties of bounhed index of nil-potence of RR′ when R′ is not commutative.     

Good Index Behaviour of θ-Representations, I

Let Q be an algebraic group with ${\mathfrak q}={\mathrm{Lie\,}} Q$ and V a Q-module. The index of V is the minimal codimension of the Q-orbits in the dual space V ^*. There is a general inequality, due to Vinberg, relating the index of V and the index of a Q _v -module $V/{\mathfrak q}{\cdot}v$ for v?∈?V. A pair (Q, V) is said to have GIB if Vinberg’s inequality turns into an equality for all v?∈?V. In this article, we are interested in the GIB property of θ-representations, where θ is a finite order automorphism of a simple Lie algebra ${\mathfrak g}$ . An automorphism of order m defines a ${{\mathbb Z}}/m{{\mathbb Z}}$ -grading ${\mathfrak g} =\bigoplus{\mathfrak g}_i$ . If G ₀ is the identity component of G ^θ , then it acts on ${\mathfrak g}_1$ and this action is called a θ-representation. We classify inner automorphisms of ${\mathfrak gl}_n$ and all finite order autmorphisms of the exceptional Lie algebras such that $(G_0,{\mathfrak g}_1)$ has GIB and ${\mathfrak g}_1$ contains a non-zero semisimple element.     

Values in preservice mathematics teachers’ discussions of the Body Mass Index - A critical perspective

This article explores the values that come to the fore when preservice mathematics teachers (PTs) ¹ engage in critical discussions about the role of mathematical models in society. The specific model that was discussed was the Body Mass Index (BMI) ². From the analysis of the PTs’ discussions of the BMI from a mathematical and societal point of view several mathematical and mathematics educational values were identified such as openness, rationalism, progress, reasoning, evaluating, and problematizing the instrumental understanding of mathematics. In addition, critical thinking about mathematics in society as emphasized in curricula in the three countries involved in the study, was identified with four categories of complementary pairs. Knowing the mathematical and mathematics educational values underpinning PTs’ discussions and their connection to critical thinking is important for successfully engaging with the role of mathematics in society.     

Exponential Synchronization of the Linearly Coupled Dynamical Networks with Delays

In this paper,the authors investigate the synchronization of an array of linearly coupled identical dynamical systems with a delayed coupling.Here the coupling matrix can be asymmetric and reducible.Some criteria ensuring delay-independent and delay- dependent global synchronization are derived respectively.It is shown that if the coupling delay is less than a positive threshold,then the coupled network will be synchronized.On the other hand,with the increase of coupling delay,the synchronization stability of the network will be restrained,even eventually de-synchronized.     

Iteration of closed geodesics in stationary Lorentzian manifolds

Following the lines of Bott in (Commun Pure Appl Math 9:171–206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic γ, we prove the existence of a locally constant integer valued map Λ_γ on the unit circle with the property that the Morse index of the iterated γ ^N is equal, up to a correction term ε_γ∈{0,1}, to the sum of the values of Λ_γ at the N-th roots of unity. The discontinuities of Λ_γ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ. We discuss some applications of the theory.     

Maslov P-Index Theory for a Symplectic Path with Applications

Abstract The Maslov P-index theory for a symplectic path is defined. Various properties of this index theory such as homotopy invariant, symplectic additivity and the relations with other Morse indices are studied. As an application, the non-periodic problem for some asymptotically linear Hamiltonian systems is considered. *Project supported by the National Natural Science Foundation of China (No.10531050) and FANEDD.     

Spectral Flow,Maslov Index and Bifurcation of Semi-Riemannian Geodesics

We give a functional analytical proof of the equalitybetween the Maslov index of a semi-Riemannian geodesicand the spectral flow of the path of self-adjointFredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals imply that each nondegenerate and nonnull conjugate (or P-focal)point along a semi-Riemannian geodesic is a bifurcation point.In particular, the semi-Riemannian exponential map is notinjective in any neighborhood of a nondegenerate conjugate point,extending a classical Riemannian result originally due to Morse and Littauer.     

Perturbed closed geodesics are periodic orbits: Index and transversality

We study the classical action functional ${\cal S}_V$ on the free loop space of a closed, finite dimensional Riemannian manifold M and the symplectic action on the free loop space of its cotangent bundle. The critical points of both functionals can be identified with the set of perturbed closed geodesics in M. The potential $V\in C^\infty(M\times S^1,\mathbb{R})$ serves as perturbation and we show that both functionals are Morse for generic V. In this case we prove that the Morse index of a critical point x of equals minus its Conley-Zehnder index when viewed as a critical point of and if is trivial. Otherwise a correction term +1 appears. Received: 21 May 2001; in final form: 10 October 2001 / Published online: 4 April 2002     

Index parity of closed geodesics and rigidity of Hopf fibrations

The diameter rigidity theorem of Gromoll and Grove [1987] states that a Riemannian manifold with sectional curvature ≥ 1 and diameter ≥ π/2 is either homeomorphic to a sphere, locally isometric to a rank one symmetric space, or it has the cohomology ring of the Cayley plane Caℙ. The reason that they were only able to recognize the cohomology ring of Caℙ is due to an exceptional case in another theorem [Gromoll and Grove, 1988]: A Riemannian submersion σ:?^m→B ^b with connected fibers that is defined on the Euclidean sphere ?^m is metrically congruent to a Hopf fibration unless possibly (m,b)=(15,8). We will rule out the exceptional cases in both theorems. Our argument relies on a rather unusal application of Morse theory. For that purpose we give a general criterion which allows to decide whether the Morse index of a closed geodesic is even or odd. Oblatum 7-II-2000 & 11-X-2000?Published online: 29 January 2001     

MASLOV-TYPE INDEX THEORY FOR SYMPLECTIC PATHS AND SPECTRAL FLOW （II）

1.IntroductionandMainResultsStartingfromthepioneeringworks[5,61ofH.AmannandE.Zehnderin1980,C.ConleyandE.Zehnderestablishedanindextheoryin1984intheircelebratedwork[if]fornondegeneratepathsinSp(Zn)startedfromtheidentitymatrixwithn22.Thisindextheorywasextendedtothenondegeneratecaseofn~1byE.Zehnderandthefirstauthorin[28]of1990.ThisindextheoryforthedegenerateHamiltoniansystemswasestablishedbythefirstauthorin[21]of1990andC.Viterboin[34]of1990viadifferentmethods,andthenextendedtoalldegenerates…     

A Morse Index Theorem for Elliptic Operators on Bounded Domains

Given a selfadjoint, elliptic operator L, one would like to know how the spectrum changes as the spatial domain Ω ? ? ⁿ is deformed. For a family of domains {Ω _t } _{t∈[a, b]} we prove that the Morse index of L on Ω _a differs from the Morse index of L on Ω _b by the Maslov index of a path of Lagrangian subspaces on the boundary of Ω. This is particularly useful when Ω _a is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the difference of Morse indices for the “original” problem (on Ω _b ) and the “simplified” problem (on Ω _a ). This generalizes previous multi-dimensional Morse index theorems that were only available on star-shaped domains or for Dirichlet boundary conditions. We also discuss how one can compute the Maslov index using crossing forms, and present some applications to the spectral theory of Dirichlet and Neumann boundary value problems.     

Hofer'sL ∞-geometry: energy and stability of Hamiltonian flows,part I

Consider the group Ham ^c (M) of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold (M, ) with the HoferL -norm. A path in Ham ^c (M) will be called a geodesic if all sufficiently short pieces of it are local minima for the Hofer length functional . In this paper, we give a necessary condition for a path to be a geodesic. We also develop a necessary condition for a geodesic to be stable, that is, a local minimum for . This condition is related to the existence of periodic orbits for the linearization of the path, and so extends Ustilovsky's work on the second variation formula. Using it, we construct a symplectomorphism ofS ² which cannot be reached from the identity by a shortest path. In later papers in this series, we will use holomorphic methods to prove the sufficiency of the condition given here for the characterisation of geodesics as well as the sufficiency of the condition for the stability of geodesics. We will also investigate conditions under which geodesics are absolutely length-minimizing.Oblatum 13-X-1994 & 8-V-1995Partially supported by NSERC grant OGP 0092913 and FCAR grant ER-1199Partially supported by NSF grant DMS 9103033 and NSF Visiting Professorship for Women GER 9350075     

A Note on the Game Chromatic Index of Graphs

In this note we prove that the game chromatic index χ _g ^′(G) of a graph G of arboricity k is at most Δ + 3k − 1. This improves a bound obtained by Cai and Zhu [J. Graph Theory 36 (2001), 144–155] for k-degenerate graphs. Tomasz Bartnicki: Research of the first author is supported by a PhD grant from Polish Ministry of Science and Higher Education N201 2128 33. Received: November 1, 2006. Final version received: December 22, 2007.     

Some bendings of a long cylinder

Elementary tools are applied to describe piecewise-linear isometric embeddings of cylindrical surfaces in ℝ³. Let T² be a flat torus, let γ⊂T² be the shortest closed geodesic of length l_o, and let k be a fixed positive integer. We assume that if l is the length of any closed geodesic on T² which is homotopic neither to γ nor to any power of γ, then l>kl₀. It is shown how to embed T² in ℝ³ if k is sufficiently large. The same problem is solved for a flat skew torus T². It is also shown that if a knot of arbitrary type in ℝ³ is fixed and k is sufficiently large, then T² can be isometrically embedded in ℝ³ as a tube knotted according to the type of fixed knot. Bibliography; 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 66–83. Translated by S. Yu. Pilyugin.