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The following problem, suggested by Laguerre’s Theorem (1884),
remains open: Characterize all real sequences {μk} k=0...∞
which have the zero-diminishing property; that is, if k=0...n, p(x) = ∑(ak x^k) is any P real polynomial, then
k=0...n, p(x) = ∑(μk ak x^k) has no more real zeros than p(x).
In this paper this problem is solved under the additional assumption of a weak
growth condition on the sequence {μk} k=0...∞, namely lim n→∞ | μn |^(1/n) < ∞.
More precisely, it is established that...
Unilateral deflection problem of a clamped plate above a rigid inner obstacle is considered. The variable thickness of the plate is to be optimized to reach minimal weight under some constraints for maximal stresses. Since the constraints are expressed in terms of the bending moments only, Herrmann-Hellan finite element scheme is employed. The existence of an optimal thickness is proved and some convergence analysis for approximate penalized optimal design problem is presented.
Extending the results of the previous paper [1], the authors consider elastic bodies with two design variables, i.e. "curved trapezoids" with two curved variable sides. The left side is loaded by a hydrostatic pressure. Approximations of the boundary are defined by cubic Hermite splines and piecewise linear finite elements are used for the displacements. Both existence and some convergence analysis is presented for approximate penalized optimal design problems.
Shape optimization of a two-dimensional elastic body is considered, provided the material is weakly supporting tension. The problem generalizes that of a masonry dam subjected to its own weight and to the hydrostatic presure. Existence of an optimal shape is proved. Using a penalty method and finite element technique, approximate solutions are proposed and their convergence is analyzed.
The problem to find an optimal thickness of the plate in a set of bounded Lipschitz continuous functions is considered. Mean values of the intensity of shear stresses must not exceed a given value. Using a penalty method and finite element spaces with interpolation to overcome the “locking” effect, an approximate optimization problem is proposed. We prove its solvability and present some convergence analysis.
In this paper, a weighted regularization method for the time-harmonic Maxwell equations
with perfect conducting or impedance boundary condition in composite materials is presented.
The computational domain Ω is the union
of polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularities
near geometric singularities, which are the interior and exterior edges and corners. The variational formulation of the
weighted regularized problem...
We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in with . The constant appearing in Céa’s lemma and coming from its standard proof can be very large when the coefficients of an elliptic operator attain considerably different values. We restrict ourselves to regular families of uniform partitions and linear simplicial elements. Using a lower bound of the interpolation error and the supercloseness between the finite element solution and...
We give conditions such that the least degree solution of a Bézout identity is nonnegative on the interval [-1,1].
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