252 research outputs found
A note on the complete rotational invariance of biradial solutions to semilinear elliptic equations
We investigate symmetry properties of solutions to equations of the form in R^N for , with at most
critical nonlinearities. By using geometric arguments, we prove that solutions
with low Morse index (namely 0 or 1) and which are biradial (i.e. are invariant
under the action of a toric group of rotations), are in fact completely radial.
A similar result holds for the semilinear Laplace-Beltrami equations on the
sphere. Furthermore, we show that the condition on the Morse index is sharp.
Finally we apply the result in order to estimate best constants of Sobolev type
inequalities with different symmetry constraints
On the regularization of the collision solutions of the one-center problem with weak forces
We study the possible regularization of collision solutions for one centre problems with a weak singularity. In the case of logarithmic singularities, we consider the method of regularization via smoothing of the potential. With this technique, we prove that the extended ow, where collision solutions are replaced with transmission trajectories, is continuous, though not differentiable, with respect to the initial data
Regularity of the optimal sets for some spectral functionals
In this paper we study the regularity of the optimal sets for the shape optimization problem min{λ1(Ω)+⋯+λk(Ω) : Ω⊂Rd open, |Ω|=1}, where λ1(·) , … , λk(·) denote the eigenvalues of the Dirichlet Laplacian and | · | the d-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer Ωk∗ is composed of a relatively open regular part which is locally a graph of a C∞ function and a closed singular part, which is empty if d< d∗, contains at most a finite number of isolated points if d= d∗ and has Hausdorff dimension smaller than (d- d∗) if d> d∗, where the natural number d∗∈ [ 5 , 7 ] is the smallest dimension at which minimizing one-phase free boundaries admit singularities. To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case
The nonlinear Schrödinger equation ground states on product spaces
We study the nature of the nonlinear Schrödinger equation ground states on the product spaces Rn x Mk ,
where Mk is a compact Riemannian manifold. We prove that for small L2 masses the ground states
coincide with the corresponding Rn ground states. We also prove that above a critical mass the ground
states have nontrivial Mk dependence. Finally, we address the Cauchy problem issue, which transforms
the variational analysis into dynamical stability results
Action minimizing orbits in the n-body problem with simple choreography constraint
In 1999 Chenciner and Montgomery found a remarkably simple choreographic
motion for the planar 3-body problem (see \cite{CM}). In this solution 3 equal
masses travel on a eight shaped planar curve; this orbit is obtained minimizing
the action integral on the set of simple planar choreographies with some
special symmetry constraints. In this work our aim is to study the problem of
masses moving in \RR^d under an attractive force generated by a potential
of the kind , , with the only constraint to be a simple
choreography: if are the orbits then we impose the
existence of x \in H^1_{2 \pi}(\RR,\RR^d) such that q_i(t)=x(t+(i-1) \tau),
i=1,...,n, t \in \RR, where . In this setting, we first
prove that for every d,n \in \NN and , the lagrangian action
attains its absolute minimum on the planar circle. Next we deal with the
problem in a rotating frame and we show a reacher phenomenology: indeed while
for some values of the angular velocity minimizers are still circles, for
others the minima of the action are not anymore rigid motions.Comment: 24 pages; 4 figures; submitted to Nonlinearit
Symbolic dynamics for the -centre problem at negative energies
We consider the planar -centre problem, with homogeneous potentials of
degree -\a<0, \a \in [1,2). We prove the existence of infinitely many
collisions-free periodic solutions with negative and small energy, for any
distribution of the centres inside a compact set. The proof is based upon
topological, variational and geometric arguments. The existence result allows
to characterize the associated dynamical system with a symbolic dynamics, where
the symbols are the partitions of the centres in two non-empty sets
Lines on projective varieties and applications
The first part of this note contains a review of basic properties of the
variety of lines contained in an embedded projective variety and passing
through a general point. In particular we provide a detailed proof that for
varieties defined by quadratic equations the base locus of the projective
second fundamental form at a general point coincides, as a scheme, with the
variety of lines. The second part concerns the problem of extending embedded
projective manifolds, using the geometry of the variety of lines. Some
applications to the case of homogeneous manifolds are included.Comment: 15 pages. One example removed; one remark and some references added;
typos correcte
Methodological issues in estimating survival in patients with multiple primary cancers: an application to women with breast cancer as a first tumour
<p>Abstract</p> <p>Background</p> <p>Comparing survival of patients with a single tumour and patients with multiple primaries poses different methodological problems. In population based studies, where we cannot rely on detailed clinical information, the issue is disentangling the share of survival probability from the first and second cancer, and their compounded effect. We examined three hypotheses: A) the survival probability since the first tumour does not change with the occurrence of a second tumour; B) the probability of surviving a tumour does not change with the presence of a previous primary; C) the probabilities of surviving two subsequent primary tumours are independent (additivity hypothesis on mortality rates).</p> <p>Methods</p> <p>We studied the survival probabilities modelling mortality rates according to hypotheses A), B) and C). Mortality rates were calculated using Aalen-Johansen estimators which allowed to discount for the lag-time survival before developing a second tumour. We applied this approach to a cohort of 436 women with breast cancer (BC) and a subsequent tumour in the resident population of Turin, Italy, between 1985 and 2002.</p> <p>Results</p> <p>We presented our results in term of a Standardised Mortality Ratio calculated (<it>SMR</it><sub><it>AJ</it></sub>) after 10 years of follow-up. For hypothesis A we observed a significant excess mortality of 2.21 (95% C.I. 1.94 – 2.45). Concerning hypothesis B we found a not significant <it>SMR</it><sub><it>AJ </it></sub>of 0.98 (95% C.I. 0.87 – 1.10). The additivity hypothesis (C) was not confirmed as it overestimated the risk of death, in fact <it>SMRs</it><sub><it>AJ </it></sub>were all below 1: 0.75 (95% C.I. 0.66 – 0.84) for BC and all subsequent cancers, 0.72 (95% C.I. 0.55 – 0.94) for BC and colon-rectum cancer, 0.76 (95% C.I. 0.48 – 1.14) for BC and corpus uteri cancer (not significant).</p> <p>Conclusion</p> <p>This method proved to be useful in disentangling the effect of different subsequent cancers on mortality. In our application it shows a worse long-term mortality for women with two cancers than that with BC only. However, the increase in mortality was lower than expected under the additivity assumption.</p
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