Lyapunov exponents and control sets near singular points

We consider control affine systems of the form

     

The effect of oscillations in the dynamics of differential equations with delay

The purpose of this paper is to study the dynamic behavior of delay differential equations of the form

     

Ground-state solutions for the electrostatic nonlinear Klein-Gordon-Maxwell system

In this paper, we study the nonlinear Klein-Gordon equation coupled with the Maxwell equation in the electrostatic case:

(P)     

On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes

For a finitely triangulated closed surface M ², let α_x be the sum of angles at a vertex x. By the well-known combinatorial version of the 2- dimensional Gauss-Bonnet Theorem, it holds Σ_x(2π - α_x) = 2πχ(M ²), where χ denotes the Euler characteristic of M ², α_x denotes the sum of angles at the vertex x, and the sum is over all vertices of the triangulation. We give here an elementary proof of a straightforward higher-dimensional generalization to Euclidean simplicial complexes K without assuming any combinatorial manifold condition. First, we recall some facts on simplicial complexes, the Euler characteristics and its local version at a vertex. Then we define δ(τ) as the normed dihedral angle defect around a simplex τ. Our main result is Σ_τ (-1)^dim(τ)δ(τ) = χ(K), where the sum is over all simplices τ of the triangulation. Then we give a definition of curvature κ(x) at a vertex and we prove the vertex-version Σ _x∈K0 κ(x) = χ(K) of this result. It also possible to prove Morse-type inequalities. Moreover, we can apply this result to combinatorial (n + 1)-manifolds W with boundary B, where we prove that the difference of Euler characteristics is given by the sum of curvatures over the interior of W plus a contribution from the normal curvature along the boundary B:

$$\chi \left( W \right) - \frac{1}{2}\chi \left( B \right) = \sum {_{\tau \in W - B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}} + \sum {_{\tau \in B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}}\rho \left( \tau \right)$$

.

     

Existence of solutions for a nonlinear elliptic problem on a Riemannian manifold

Let (M,g) be a noncompact, connected, orientable smooth N-dimensional Riemannian manifold without boundary. We consider the existence of solutions of problem

(P)     

Existence and multiple solutions for a variable exponent system

In this paper, we study the following variable exponent system

     

Uniform stabilization of the wave equation on compact manifolds and locally distributed damping - a sharp result

Let (M,g) be an n-dimensional (n?2) compact Riemannian manifold with or without boundary where g denotes a Riemannian metric of class C^∞. This paper is concerned with the study of the wave equation on (M,g) with locally distributed damping, described by

     

Multipeak solutions for asymptotically critical elliptic equations on Riemannian manifolds

Let (M,g) be a smooth compact Riemannian manifold of dimension n≥3. We are concerned with the following asymptotically critical elliptic problem

(0.1)     

Existence of Nonnegative Solutions for a Class of Systems Involving Fractional (p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional (p, q)-Laplacian operators:

$$\left\{ {\begin{array}{*{20}{c}} {\left( { - \Delta } \right)_p^su = \lambda a\left( x \right){{\left| u \right|}^{p - 2}}u + \lambda b\left( x \right){{\left| u \right|}^{\alpha - 2}}{{\left| v \right|}^\beta }u + \frac{{\mu \left( x \right)}}{{\alpha \delta }}{{\left| u \right|}^{\gamma - 2}}{{\left| v \right|}^\delta }uin\Omega ,} \\ {\left( { - \Delta } \right)_q^sv = \lambda c\left( x \right){{\left| v \right|}^{q - 2}}v + \lambda b\left( x \right){{\left| u \right|}^\alpha }{{\left| v \right|}^{\beta - 2}}v + \frac{{\mu \left( x \right)}}{{\beta \gamma }}{{\left| u \right|}^\gamma }{{\left| v \right|}^{\delta - 2}}vin\Omega ,} \\ {u = v = 0on{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$

where λ > 0 is a real parameter, Ω is a bounded domain in R ^N , with boundary ?Ω Lipschitz continuous, s ∈ (0, 1), 1 < p ≤ q < ∞, sq < N, while (?Δ) _p ^s u is the fractional p-Laplacian operator of u and, similarly, (?Δ) _q ^s v is the fractional q-Laplacian operator of v. Since possibly p ≠ q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalue λ₁ for a related system, they prove that there exists a positive solution for the problem when λ < λ₁ by the modified definitions. Moreover, the authors obtain the bifurcation property when λ → λ₁^-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ ≥ λ₁.

     

The solution manifold and C-smoothness for differential equations with state-dependent delay

Let h>0, open, and continuously differentiable. If f satisfies two mild additional smoothness conditions then the set

     

Sasakian structures on CR-manifolds

A contact manifold M can be defined as a quotient of a symplectic manifold X by a proper, free action of \(\mathbb{R}\), with the symplectic form homogeneous of degree 2. If X is also Kähler, and its metric is homogeneous of degree 2, M is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kähler potential on a complex manifold, hence it is equipped with a pseudoconvex CR-structure. We show that any Sasakian manifold M is CR-diffeomorphic to an S ¹-bundle of unit vectors in a positive line bundle on a projective Kähler orbifold. This induces an embedding of M into an algebraic cone C. We show that this embedding is uniquely defined by the CR-structure. Additionally, we classify the Sasakian metrics on an odd-dimensional sphere equipped with a standard CR-structure.     

Classification of Bott towers by matrix

A criterion for the classification of Bott towers is presented, i.e., two Bott towers B _*(A) and B _*(A′) are isomorphic if and only if the matrices A and A′ are equivalent. The equivalence relation is defined by two operations on matrices. And it is based on the observation that any Bott tower B _*(A) is uniquely determined by its structure matrix A, which is a strictly upper triangular integer matrix. The classification of Bott towers is closely related to the cohomological rigidity problem for both Bott towers and Bott manifolds.     

Swing Analysis and Electoral Forecasting

This paper examines the procedure by which votes are converted into seats at U.K. General Elections. In particular it seeks to answer these questions.

1. i)
   What is the meaning of swing when more than two parties fight an election?
    
2. ii)
   How can the distribution of seats at an election be determined from a prediction of national swing?
    
3. iii)
   How can swing analysis be extended to help determine an electoral strategy for political parties?
    
4. iv)
   What explanation can be provided for the swing in Scotland?
    
5. v)
   What explanation can be provided for variations in the English swing?
    

     

The inside outside intersection problem for hexagon triple systems

We give a complete solution of the following two problems:

1. (1)
   For which (n, x) does there exist a pair of hexagon triple systems of order n having x inside triples in common?
    
2. (2)
   For which (n, x) does there exist a pair of hexagon triple systems having x outside triples in common?