On the central limit theorem in R p when p→∞

Summary Let X ₁ , X ₂ , ..., X _n be i.i.d. random vectors in R ^p where p tends to infinity. A theorem is presented showing that the Central Limit Theorem should hold if p ²/n tends to zero. Furthermore, an example is presented with X _i having a mixed multivariate normal distribution (with finite moment generating function) for which a uniform normal approximation to the distribution of the sample mean can not hold if p ²/n does not tend to zero. Research supported in part by National Science Foundation Grants MCS 80-02340, MCS 83-01834, and DMS 85-03785     

On minimizers for the isotropic–nematic interface problem

In this paper, we investigate the structure and stability of the isotropic-nematic interface in 1-D. In the absence of the anisotropic energy, the uniaxial solution is the only global minimizer. In the presence of the anisotropic energy, the uniaxial solution with the homeotropic anchoring is stable for \(L_2<0\) and unstable for \(L_2>0\). We also present many interesting open questions, some of which are related to De Giorgi conjecture.     

On existence of minimizers for the Hardy–Sobolev–Maz’ya inequality

We show existence of minimizers for the Hardy–Sobolev–Maz’ya inequality in when either m > 2, n≥ 1 or m = 1, n≥ 3. The authors expresses their gratitude to the faculties of mathematics departments at Technion - Haifa Institute of Technology, at the University of Crete and at the University of Cyprus for their hospitality. A.T. acknowledges partial support by the RTN European network Fronts–Singularities, HPRN-CT-2002-00274. K.T acknowledges support as a Lady Davis Visiting Professor at Technion and partial support from University of Crete, University of Cyprus and Swedish Research Council.     

Density lower bound estimates for local minimizers of the 2d Mumford–Shah energy

We prove, using direct variational arguments, an explicit energy-treshold criterion for regular points of 2-dimensional Mumford-Shah energy minimizers. From this we infer an explicit constant for the density lower bound of De Giorgi, Carriero and Leaci.     

On the behaviour ast→∞ of solutions of the second mixed problem for a second-order parabolic equation

In this paper is distinguished a geometric characteristic of the unbounded domain , that determines the rate of stabilization fort of the solution in (t>0)× of the second boundary value problem for a second-order parabolic equation, in which the initial function decreases sufficiently rapidly as |x|.     

On the global convergence of a generalized iterative procedure for the minisum location problem with ℓ p distances for p > 2

This paper presents a procedure to solve the classical location median problem where the distances are measured with ? _p -norms with p > 2. In order to do that we consider an approximated problem. The global convergence of the sequence generated by this iterative scheme is proved. Therefore, this paper closes the still open question of giving a modification of the Weiszfeld algorithm that converges to an optimal solution of the median problem with ? _p norms and ${p \in (2, \infty)}$ . The paper ends with a computational analysis of the different provided iterative schemes.     

On a majorant–minorant criterion for the total preservation of global solvability of distributed controlled systems

For distributed controlled systems that can be represented by a functional-operator equation of the Hammerstein type with an additional term on the right-hand side in the form of a linear operator (an extended equation of the Hammerstein type), we prove a criterion for the total (over the entire set of admissible controls) preservation of global solvability. In this connection, we develop an earlier-proved majorant–minorant criterion for the total preservation of global solvability. As an example, we study a controlled semilinear integro-differential equation describing radiation transport with Compton scattering diagram. The choice of this example is explained, in particular, by the fact that the preceding version of the majorant–minorant criterion is not applicable to it.     

On the limit cycles of a quintic planar vector field

This paper concerns the number and distributions of limit cycles in a Z_2-equivariant quintic planar vector field.25 limit cycles are found in this special planar polynomial system and four different configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis of the differential equation.It can be concluded that H(5)≥25=5~2, where H(5)is the Hilbert number for quintic polynomial systems.The results obtained are useful to study the weakened 16th Hilbert problem.     

On an upper bound for the global dimension of Auslander–Dlab–Ringel algebras

Archiv der Mathematik - Lin and Xi introduced Auslander–Dlab–Ringel (ADR) algebras of semilocal modules as a generalization of original ADR algebras and showed that they are...     

Characterization of the Bernoulli–Navier model for a rectangular section beam as the limit of the Kirchhoff–Love model for a plate

In this paper, we compare the Kirchhoff–Love model for a linearly elastic rectangular plate \({\Omega^{t\varepsilon}=(0,L)\times(-t,t)\times(-\varepsilon,\varepsilon)}\) of thickness \({2\varepsilon}\) with the Bernoulli–Navier model for the same solid considered as a linearly elastic beam of length \({L}\) and cross section \({\omega_1^{t\varepsilon}=(-t,t)\times(-\varepsilon,\varepsilon)}\). We assume that the solid is clamped on both ends \({\{0,L\}\times[-t,t]\times[-\varepsilon,\varepsilon]}\). We show that the scaled version of the displacements field \({{\bf{\zeta}}^t}\) in the middle plane, solution of the Kirchhoff–Love model, converges strongly to the unique solution of a one-dimensional problem when the plate width parameter \({t}\) tends to zero. Moreover, after rescaling this limit, we show that, as a matter of fact, it is the solution of the Bernoulli–Navier model for the beam. This means that, under appropriate assumptions on the order of magnitude of the data, the Bernoulli–Navier displacement field is the natural approximation of the Kirchhoff–Love displacement field when the cross section of the plate is rectangular and its width is sufficiently small and homothetic to thickness.