We show that robustly transitive endomorphisms of a closed manifold must have a non-trivial dominated splitting or be a local diffeomorphism. This allows to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result, we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation.

We establish some existence results for a class of critical $N$-Laplacian problems in a bounded domain in $\mathbb {R}^{N}$. In the absence of a suitable direct sum decomposition of the underlying Sobolev space to which the classical linking theorem can be applied, we use an abstract linking theorem based on the $\mathbb {Z}_2$-cohomological index to obtain a non-trivial critical point.

Let $p:\mathbb{C}\rightarrow \mathbb{C}$ be a polynomial. The Gauss–Lucas theorem states that its critical points, $p^{\prime }(z)=0$, are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose that $p$ has $n+m$ roots, where $n$ are inside the unit disk,

$$\begin{eqnarray}\max _{1\leq i\leq n}|a_{i}|\leq 1~\text{and}~m~\text{are outside}~\min _{n+1\leq i\leq n+m}|a_{i}|\geq d>1+\frac{2m}{n};\end{eqnarray}$$

$p^{\prime }$

$n-1$

$m$

$(dn-m)/(n+m)>1$

$n$

$m$

${\sim}n^{-1}$

In this paper quasilinear elliptic boundary value equations without an Ambrosetti and Rabinowitz growth condition are considered. Existence of a non-trivial solution result is established. For this, we show the existence of a Cerami sequence by using a variant of the mountain-pass theorem due to Schechter. The novelty here is that we may consider nonlinearities that satisfy a local p-superlinear condition and may change sign.

In this paper, we deal with the multiplicity of solutions for a fourth-order impulsive differential equation with a parameter. Using variational methods and a ‘three critical points’ theorem, we give some new criteria to guarantee that the impulsive problem has at least three classical solutions. An example is also given in order to illustrate the main results.

Let f be a polynomial of degree n≥2 with f(0)=0 and f′(0)=1. We prove that there is a critical point ζ of f with ∣f(ζ)/ζ∣≤1/2 provided that the critical points of f lie in the sector {reiθ:r>0,∣θ∣≤π/6}, and ∣f(ζ)/ζ∣<2/3 if they lie in the union of the two rays {1+re±iθ:r≥0}, where 0<θ≤π/2.

We study the following problem: establish existence and classification of closed curves which are critical points for the total curvature functional, defined on spaces of curves in a Riemannian manifold. This problem is completely solved in a real space form. Next, we give examples of critical points for this functional in a class of metrics with constant scalar curvature on the three sphere. Also, we obtain a rational one-parameter family of closed helices which are critical points for that functional in ℂℙ2 (4) when it is endowed with its usual Kaehlerian structure. Finally, we use the principle of symmetric criticality to get equivariant submanifolds, constructed on the above curves, which are critical points for the total mean curvature functional.

Critical points of a variant of the Ambrosio-Tortorelli functional,for which non-zero Dirichlet boundary conditions replace thefidelity term, are investigated. They are shown to converge toparticular critical points of the corresponding variant of theMumford-Shah functional; those exhibit many symmetries. ThatDirichlet variant is the natural functional when addressing aproblem of brittle fracture in an elastic material.

We show that the norm of a Bethe vector in the $sl_{r+1}$ Gaudin model is equal to the Hessian of the corresponding master function at the corresponding critical point. In particular the Bethe vectors corresponding to non-degenerate critical points are non-zero vectors. This result is a byproduct of functorial properties of Bethe vectors studied in this paper. As another byproduct of functoriality we show that the Bethe vectors form a basis in the tensor product of several copies of first and last fundamental $sl_{r+1}$-modules.

In this paper we prove the existence of at least three classical solutions for the problem

$$ \left\{ \begin{aligned} \amp-(|u'|^{p-2}u')'=\lambda f(t,u)h(u'), \\ \ampu(a)=u(b)=0, \end{aligned} \right. $$

when $\lambda$ lies in an explicitly determined open interval.

Our main tool is a very recent three-critical-points theorem stated in a paper by D. Averna and G. Bonanno (Topolog. Meth. Nonlin. Analysis22 (2003), 93–103).

AMS 2000 Mathematics subject classification: Primary 34B15

In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature KM [ges ] 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than π/2 and contains a geodesic loop of length 2π. We also prove that if M is an n([ges ]3)-dimensional complete connected Riemannian manifold with KM [ges ] 1 and radius bigger than π/2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.

We study finite and infinite entangled graphsin the bond percolation process in three dimensionswith density $p$.After a discussion of the relevant definitions,the entanglement critical probabilities are defined.The size of the maximal entangled graph at the origin is studied for small $p$, and it is shown that this graph has radius whose tail decays at least as fast as $\exp(-\alpha n/\log n)$; indeed, the logarithm may be replaced by any iterate of logarithmfor an appropriate positive constant $\alpha$. We explore the question of almost sure uniqueness of the infinite maximal open entangled graph when $p$ is large, and we establish two relevant theorems. We make several conjectures concerning the properties of entangled graphs in percolation. http://www.statslab.cam.ac.uk/$\sim$grg/ 1991 Mathematics Subject Classification: primary 60K35; secondary 05C10, 57M25, 82B41, 82B43, 82D60.

Let $M$ be a compact, smooth, $2$-connected, $2m$-dimensional manifold with $\partial M$ simply connected. If $M$ has the homotopytype of an $m$-dimensional $CW$-complex, then it supports a smooth, self-indexed function, maximal, constant and regular on $\partial M$ with at most $\mbox{cat}(M)+2$ critical points, all of which are of a certain ‘reasonable’ type. To such a critical point there corresponds, homotopically, the attachment of a cone. Conversely, to a cone attachment we may associate, under certain dimensionality and connectivity conditions, a ‘reasonable’ critical point.

1991 Mathematics Subject Classification: primary 55M30, 57R70; secondary 58E05, 55P50, 55P62.

In Pickard (1976) limit theorems were obtained for the classical Ising model at non-critical points. These determined the asymptotic distribution of the sample nearest-neighbour correlation, thereby providing a basis for statistical inference by confidence intervals. In this paper, these limit theorems are extended to the statistically significant case of different vertical and horizontal interactions. Results at critical points are also obtained. Critical points clearly have the potential to seriously distort statistical inferences, especially in their immediate neighbourhoods. For our Ising model it turns out that such distortion is relatively minor. Surprisingly, in the two-parameter case the correlation between the sufficient statistics exhibits peculiar asymptotic behaviour resulting in a singular covariance matrix at critical points in the central limit theorem. Finally, at critical points, unusual norming constants are required for the central limit theorem, and our results are much more sensitive to the relative rate at which m, n tend to infinity.