Lamé operators with finite monodromy—a combinatorial approach

We are describing Lamé differential operators with a full set of algebraic solutions. For each finite group G, we are describing the possible values of the degree parameter n such that the Lamé operator L_n has the projective monodromy group G. The main technical tool is the combinatorics associated to Belyi functions, ideas that we already used in (Rend. Sem. Mat. Univ. Padova 107 (2002) 191-208) for describing the case n=1. We also supply proofs to some finiteness properties conjectured by Baldassarri and by Dwork, and we work out an explicit formula for the number of essentially different Lamé equations when n=2. This approach can be generalized for arbitrary degree n (see (Counting Integral Lamé Equations by Means of Dessins d'Enfants, arXiv:math.CA/0311510) for n integer).     

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

In this paper, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form x^′=y, y^′=Q₁(x)+εyQ₂(x) with Q₁ and Q₂ polynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation and Poincaré bifurcation, which emerges at most three limit cycles in the plane for sufficiently small positive ε. And the limit cycles can encompass only an equilibrium inside, i.e. the configuration (3,0) of limit cycles can appear for some values of parameters, where (3,0) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.     

Algebraic solutions of the Lamé equation, revisited

A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out (see Baldassarri, J. Differential Equations 41 (1) (1981) 44). It is shown that if the group is the octahedral group S₄, then the degree parameter of the equation may differ by ±1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lamé equation (see Churchill, J. Symbolic Comput. 28 (4-5) (1999) 521). The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.     

Analytic continuation of eigenvalues of the Lamé operator

Eigenvalues of the Lamé operator are studied as complex-analytic functions in period τ of an elliptic function. We investigate the branching of eigenvalues numerically and clarify the relationship between the branching of eigenvalues and the convergent radius of a perturbation series.     

On the spectral radius of bipartite graphs with given diameter

Let GB(n,d) be the set of bipartite graphs with order n and diameter d. This paper characterizes the extremal graph with the maximal spectral radius in GB(n,d). Furthermore, the maximal spectral radius is a decreasing function on d. At last, bipartite graphs with the second largest spectral radius are determined.     

Navier-Stokes equations in 3D thin domains with Navier friction boundary condition

In this article we study the 3D Navier-Stokes equations with Navier friction boundary condition in thin domains. We prove the global existence of strong solutions to the 3D Navier-Stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small. We generalize the techniques developed to study the 3D Navier-Stokes equations in thin domains, see [G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains I: Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993) 503-568; G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains II: Global regularity of spatially periodic conditions, in: Nonlinear Partial Differential Equations and Their Application, College de France Seminar, vol. XI, Longman, Harlow, 1994, pp. 205-247; R. Temam, M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations 1 (1996) 499-546; R. Temam, M. Ziane, Navier-Stokes equations in thin spherical shells, in: Optimization Methods in Partial Differential Equations, in: Contemp. Math., vol. 209, Amer. Math. Soc., Providence, RI, 1996, pp. 281-314], to the Navier friction boundary condition by introducing a new average operator Mε in the thin direction according to the spectral decomposition of the Stokes operator Aε. Our analysis hinges on the refined investigation of the eigenvalue problem corresponding to the Stokes operator Aε with Navier friction boundary condition.     

Morphisms between complete Riemannian pseudogroups

We introduce the concept of morphism of pseudogroups generalizing the étalé morphisms of Haefliger. With our definition, any continuous foliated map induces a morphism between the corresponding holonomy pseudogroups. The main theorem states that any morphism between complete Riemannian pseudogroups is complete, has a closure and its maps are C^∞ along the orbit closures. Here, completeness and closure are versions for morphisms of concepts introduced by Haefliger for pseudogroups. This result is applied to approximate foliated maps by smooth ones in the case of transversely complete Riemannian foliations, yielding the foliated homotopy invariance of their spectral sequence. This generalizes the topological invariance of their basic cohomology, shown by El Kacimi-Alaoui-Nicolau. A different proof of the spectral sequence invariance was also given by the second author.     

On the roots of independence polynomials of almost all very well-covered graphs

If s_k denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then is the independence polynomial of G [I. Gutman, F. Harary, Generalizations of the matching polynomial, Utilitas Math. 24 (1983) 97-106]. A graph G is very well-covered [O. Favaron, Very well-covered graphs, Discrete Math. 42 (1982) 177-187] if it has no isolated vertices, its order equals 2α(G) and it is well-covered, i.e., all its maximal independent sets are of the same size [M.D. Plummer, Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91-98]. For instance, appending a single pendant edge to each vertex of G yields a very well-covered graph, which we denote by G^*. Under certain conditions, any well-covered graph equals G^* for some G [A. Finbow, B. Hartnell, R.J. Nowakowski, A characterization of well-covered graphs of girth 5 or greater, J. Combin. Theory Ser B 57 (1993) 44-68].The root of the smallest modulus of the independence polynomial of any graph is real [J.I. Brown, K. Dilcher, R.J. Nowakowski, Roots of independence polynomials of well-covered graphs, J. Algebraic Combin. 11 (2000) 197-210]. The location of the roots of the independence polynomial in the complex plane, and the multiplicity of the root of the smallest modulus are investigated in a number of articles.In this paper we establish formulae connecting the coefficients of I(G;x) and I(G^*;x), which allow us to show that the number of roots of I(G;x) is equal to the number of roots of I(G^*;x) different from -1, which appears as a root of multiplicity α(G^*)-α(G) for I(G^*;x). We also prove that the real roots of I(G^*;x) are in [-1,-1/2α(G^*)), while for a general graph of order n we show that its roots lie in |z|>1/(2n-1).Hoede and Li [Clique polynomials and independent set polynomials of graphs, Discrete Math. 125 (1994) 219-228] posed the problem of finding graphs that can be uniquely defined by their clique polynomials (clique-unique graphs). Stevanovic [Clique polynomials of threshold graphs, Univ. Beograd Publ. Elektrotehn. Fac., Ser. Mat. 8 (1997) 84-87] proved that threshold graphs are clique-unique. Here, we demonstrate that the independence polynomial distinguishes well-covered spiders among well-covered trees.     

The smallest signless Laplacian spectral radius of graphs with a given clique number

The signless Laplacian spectral radius of a graph G is the largest eigenvalue of its signless Laplacian matrix. In this paper, the first four smallest values of the signless Laplacian spectral radius among all connected graphs with maximum clique of size greater than or equal to 2 are obtained.     

Strong solutions to a nonlinear fluid structure interaction system

In this paper, we prove the existence of local-in-time smooth solutions to the nonlinear fluid structure interaction model first introduced in [J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969] and considered in [V. Barbu, Z. Gruji?, I. Lasiecka, A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in: Fluids and Waves, in: Contemp. Math., vol. 440, Amer. Math. Soc., Providence, RI, 2007, pp. 55-82; V. Barbu, Z. Gruji?, I. Lasiecka, A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J. 57 (3) (2008) 1173-1207]. In particular, the strong solutions here are obtained given initial datum for the Navier-Stokes equation in the space H¹, and initial data for the wave equation w₀ and w₁ in the spaces H²(Ωe) and H¹(Ωe) respectively.     

A compactness result for Landau state in thin-film micromagnetics

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters ε and η and defined over vector fields m:Ω⊂R²→S² that are tangent at the boundary ∂Ω. We are interested in the behavior of minimizers as ε,η→0. They tend to be in-plane away from a region of length scale ε (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that S¹-transition layers of length scale η (Néel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of Néel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields {mε,η}_ε,η↓0 of energies close to the Landau state in the regime where a vortex is energetically more expensive than a Néel wall. Our method uses techniques developed for the Ginzburg-Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of S²-vector fields by S¹-vector fields away from the vortex balls.     

Mixed sums of squares and triangular numbers (III)

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if Tm=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p²=x²+8(y²+z²) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2Tm(m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4.     

A Gel’fand-type spectral radius formula and stability of linear constrained switching systems

Using ergodic theory, in this paper we present a Gel’fand-type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup S⁺ restricted to a subset that need not carry the algebraic structure of S⁺ This generalizes the Berger–Wang formula. Using it as a tool, we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one-sided Markov shift associated to S.     

The Laplacian spectral radii of trees with degree sequences

In this paper, we characterize all extremal trees with the largest Laplacian spectral radius in the set of all trees with a given degree sequence. Consequently, we also obtain all extremal trees with the largest Laplacian spectral radius in the sets of all trees of order n with the largest degree, the leaves number and the matching number, respectively.     

A sharp upper bound on the maximal entry in the principal eigenvector of symmetric nonnegative matrix

An upper bound on the maximal entry in the principal eigenvector of a symmetric nonnegative matrix with zero diagonal entries is investigated in [S. Zhao, Y. Hong, On the bounds of maximal entries in the principal eigenvector of symmetric nonnegative matrix, Linear Algebra Appl. 340 (2002) 245-252]. We obtain a sharp upper bound on the maximal entry y^max_p in the principal eigenvector of symmetric nonnegative matrix in terms of order, the spectral radius, the largest and the smallest diagonal entries of that matrix. Our bound is applicable for any symmetric nonnegative matrix and the upper bound of Zhao and Hong (2002) for the maximal entry y^max_p follows as a special case. Moreover, we find an upper bound on maximal entry in the principal eigenvector for the signless Laplacian matrix of a graph.     

Intersection theorems under dimension constraints

In Ahlswede et al. [Discrete Math. 273(1-3) (2003) 9-21] we posed a series of extremal (set system) problems under dimension constraints. In the present paper, we study one of them: the intersection problem. The geometrical formulation of our problem is as follows. Given integers 0?t, k?n determine or estimate the maximum number of (0,1)-vectors in a k-dimensional subspace of the Euclidean n-space Rⁿ, such that the inner product (“intersection”) of any two is at least t. Also we are interested in the restricted (or the uniform) case of the problem; namely, the problem considered for the (0,1)-vectors of the same weight ω.The paper consists of two parts, which concern similar questions but are essentially independent with respect to the methods used.In Part I, we consider the unrestricted case of the problem. Surprisingly, in this case the problem can be reduced to a weighted version of the intersection problem for systems of finite sets. A general conjecture for this problem is proved for the cases mentioned in Ahlswede et al. [Discrete Math. 273(1-3) (2003) 9-21]. We also consider a diametric problem under dimension constraint.In Part II, we study the restricted case and solve the problem for t=1 and k<2ω, and also for any fixed 1?t?ω and k large.     

Regularity estimates for solutions of the equations of linear elasticity in convex plane polygonal domains

The Dirichlet problem for the plane elasticity problem on a convex polygonal domain is considered and it is proved that for data in L ² the H ² regularity estimate holds with constants independent of the Lamé coefficients.     

The spectral radius of graphs without paths and cycles of specified length

Let G be a graph with n vertices and μ(G) be the largest eigenvalue of the adjacency matrix of G. We study how large μ(G) can be when G does not contain cycles and paths of specified order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give tight bounds on the spectral radius of graphs without given even cycles. We also raise a number of open problems.     

The signless Laplacian spectral radius of tricyclic graphs and trees with k pendant vertices

In this paper, we consider the following problem: of all tricyclic graphs or trees of order n with k pendant vertices (n,k fixed), which achieves the maximal signless Laplacian spectral radius?We determine the graph with the largest signless Laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices. Then we show that the maximal signless Laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at T_n,k, where T_n,k is a tree obtained from a star K_1,k and k paths of almost equal lengths by joining each pendant vertex to one end-vertex of one path. We also discuss the signless Laplacian spectral radius of T_n,k and give some results.