Minimization of the eigenvalues of the dirichlet-laplacian with a diameter constraint

In this paper we look for the domains minimizing the hth eigenvalue of the Dirichlet-Laplacian λh with a constraint on the diameter. Existence of an optimal domain is easily obtained and is attained at a constant width body. In the case of a simple eigenvalue, we provide nonstandard (i.e., nonlocal) optimality conditions. Then we address the question of whether the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane

Phase field approach to optimal packing problems and related Cheeger clusters

In a fixed domain of $\Bbb{R}^N$ we study the asymptotic behaviour of optimal clusters associated to $\alpha$-Cheeger constants and natural energies like the sum or maximum: we prove that, as the parameter $\alpha$ converges to the "critical" value $\Big (\frac{N-1}{N}\Big ) _+$, optimal Cheeger clusters converge to solutions of different packing problems for balls, depending on the energy under consideration. As well, we propose an efficient phase field approach based on a multiphase Gamma convergence result of Modica-Mortola type, in order to compute $\alpha$-Cheeger constants, optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions

Studies on the effect of the number of farrowed sows in heat per box

This scientific paper has as main objective the way how the type of box influences the number of farrowed sow. The number of sows of the Landrace breed that are in heat after weaning the piglets, tidmatter the duration of lactation or the number of boxes they are located in, is considerably larger than that of oestrus sows during lactation. Animal density in the box obviously influenced the appearance of heat, the largest share being in sows located by seven, i.e. 96.25%, and the lowest one, 59.68%, was in the sows located in larger boxes (32 capita). Achieving high performances of production and reproduction in raising swine greatly depends on the way animals are taken care of and exploited, i.e. the shedding system, the exploitation technology, the building material of the sheds, inside equipment, and degree of mechanisation. In sows with a 35-day lactation, the share of post partum heat occurrence had close values (though somewhat smaller), oscillating between 19.52% and 23.33%

Studies on the effect of the number of farrowed sows in heat per box

This scientific paper has as main objective the way how the type of box influences the number of farrowed sow. The number of sows of the Landrace breed that are in heat after weaning the piglets, tidmatter the duration of lactation or the number of boxes they are located in, is considerably larger than that of oestrus sows during lactation. Animal density in the box obviously influenced the appearance of heat, the largest share being in sows located by seven, i.e. 96.25%, and the lowest one, 59.68%, was in the sows located in larger boxes (32 capita). Achieving high performances of production and reproduction in raising swine greatly depends on the way animals are taken care of and exploited, i.e. the shedding system, the exploitation technology, the building material of the sheds, inside equipment, and degree of mechanisation. In sows with a 35-day lactation, the share of post partum heat occurrence had close values (though somewhat smaller), oscillating between 19.52% and 23.33%

Minimization of the eigenvalues of the Dirichlet-LapIacian with a diameter constraint

International audienceIn this paper we look for the domains minimizing the h-th eigenvalue of the Dirichlet-Laplacian λ h with a constraint on the diameter. Existence of an optimal domain is easily obtained, and is attained at a constant width body. In the case of a simple eigenvalue, we provide non standard (i.e., non local) optimality conditions. Then we address the question whether or not the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane

3D positive lattice walks and spherical triangles

International audienceIn this paper we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. As shown in [29], both the exponential growth and the critical exponent admit universal formulas, respectively in terms of the inventory of the step set and of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. We focus on the critical exponent, and our main objective is to relate combinatorial properties of the step set (structure of the so-called group of the walk, existence of a Hadamard factorization, existence of differential equations satisfied by the generating functions) to geometric or analytic properties of the associated spherical triangle (remarkable angles, tiling properties, existence of an exceptional closed-form formula for the principal eigenvalue). As in general the eigenvalues of the Dirichlet problem on a spherical triangle are not known in closed form, we also develop a finite-elements method to compute approximate values, typically with ten digits of precision