On a Condition for the Nonexistence of W-Solutions of Nonlinear High-Order Equations with L¹ -Data

En un conjunto abierto y acotado de consideramos el problema de Dirichlet para ecuaciones no lineales de orden en la forma divergente con lados L¹ -right-hand. Se supone que , y los coeficientes de las ecuaciones admiten el radio de crecimiento con respecto a las derivadas de orden m de la función desconocida. Establecemos que bajo la condición para algn data el problema de Dirichlet correspondiente no tiene W-soluciones.In a bounded open set of we consider the Dirichlet problem for nonlinear order equations in divergence form with right-hand sides. It is supposed that , and the coefficients of the equations admit the growth of rate with respect to the derivatives of order m of unknown function. We establish that under the condition for some L¹ -data the corresponding Dirichlet problem does not have W-solutions

CONVERGENCE OF SOLUTIONS OF BILATERAL PROBLEMS IN VARIABLE DOMAINS AND RELATED QUESTIONS

We discuss some results on the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral constraints in variable domains. We consider the case of regular constraints, i.e., constraints lying in the corresponding Sobolev space, and the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative function. The first case concerns a larger class of integrands and requires the positivity almost everywhere of the difference between the upper and lower constraints. In the second case, this requirement is absent. Moreover, in the latter case, the exhaustion condition of an n-dimensional domain by a sequence of n-dimensional domains plays an important role. We give a series of results involving this condition. In particular, using the exhaustion condition, we prove a certain convergence of sets of functions defined by bilateral (generally irregular) constraints in variable domains

ON THE SETS OF BOUNDEDNESS OF SOLUTIONS TO DEGENERATE FOURTH-ORDER EQUATIONS WITH STRENGTHENINGLY MONOTONE PRINCIPAL PARTS, ABSORPTION AND L1-DATA

We consider the Dirichlet problem for a class of degenerate nonlinear elliptic fourth-order equations with strengtheningly monotone principal parts, absorbing lower-order terms and L1-right-hand sides. We establish existence of solutions of the given problem bounded on the sets where the behaviour of the data of the problem is regular enough.We consider the Dirichlet problem for a class of degenerate nonlinearelliptic fourth-order equations with strengtheningly monotone principalparts, absorbing lower-order terms and L1-right-hand sides. We establish existence of solutions of the given problem bounded on the sets where the behaviour of the data of the problem is regular enough

Dimension on Discrete Spaces

In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by means of axioms, and the axioms are based on an obvious geometrical background. This work presents some discrete models of n-dimensional Euclidean spaces, n-dimensional spheres, a torus and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities It also analyzes the topology of (3+1)-spacetime. We are also discussing the question by R. Sorkin [19] about how to derive the system of simplicial complexes from a system of open covering of a topological space S.Comment: 16 pages, 8 figures, Latex. Figures are not included, available from the author upon request. Preprint SU-GP-93/1-1. To appear in "International Journal of Theoretical Physics

Novel complex MAD phasing and RNase H structural insights using selenium oligonucleotides

The crystal structures of protein–nucleic acid complexes are commonly determined using selenium-derivatized proteins via MAD or SAD phasing. Here, the first protein–nucleic acid complex structure determined using selenium-derivatized nucleic acids is reported. The RNase H–RNA/DNA complex is used as an example to demonstrate the proof of principle. The high-resolution crystal structure indicates that this selenium replacement results in a local subtle unwinding of the RNA/DNA substrate duplex, thereby shifting the RNA scissile phosphate closer to the transition state of the enzyme-catalyzed reaction. It was also observed that the scissile phosphate forms a hydrogen bond to the water nucleophile and helps to position the water molecule in the structure. Consistently, it was discovered that the substitution of a single O atom by a Se atom in a guide DNA sequence can largely accelerate RNase H catalysis. These structural and catalytic studies shed new light on the guide-dependent RNA cleavage

Correlated errors in Hipparcos parallaxes towards the Pleiades and the Hyades

We show that the errors in the Hipparcos parallaxes towards the Pleiades and the Hyades open clusters are spatially correlated over angular scales of 2 to 3 deg, with an amplitude of up to 2 mas. This correlation is stronger than expected based on the analysis of the Hipparcos catalog. We predict the parallaxes of individual cluster members, pi_pm, from their Hipparcos proper motions, assuming that all cluster members have the same space velocity. We compare pi_pm with their Hipparcos parallaxes, pi_Hip, and find that there are significant spatial correlations in pi_Hip. We derive a distance modulus to the Pleiades of 5.58 +- 0.18 mag using the radial-velocity gradient method. This value, agrees very well with the distance modulus of 5.60 +- 0.04 mag determined using the main-sequence fitting technique, compared with the value of 5.33 +- 0.06 inferred from the average of the Hipparcos parallaxes of the Pleiades members. We show that the difference between the main-sequence fitting distance and the Hipparcos parallax distance can arise from spatially correlated errors in the Hipparcos parallaxes of individual Pleiades members. Although the Hipparcos parallax errors towards the Hyades are spatially correlated in a manner similar to those of the Pleiades, the center of the Hyades is located on a node of this spatial structure. Therefore, the parallax errors cancel out when the average distance is estimated, leading to a mean Hyades distance modulus that agrees with the pre-Hipparcos value. We speculate that these spatial correlations are also responsible for the discrepant distances that are inferred using the mean Hipparcos parallaxes to some open clusters. Finally, we note that our conclusions are based on a purely geometric method and do not rely on any models of stellar isochrones.Comment: 33 pages including 10 Figures, revised version accepted for publication in Ap

<title language="eng">On a Condition for the Nonexistence of W-Solutions of Nonlinear High-Order Equations with L¹ -Data

En un conjunto abierto y acotado de consideramos el problema de Dirichlet para ecuaciones no lineales de orden en la forma divergente con lados L¹ -right-hand. Se supone que , y los coeficientes de las ecuaciones admiten el radio de crecimiento con respecto a las derivadas de orden m de la función desconocida. Establecemos que bajo la condición para algn data el problema de Dirichlet correspondiente no tiene W-soluciones.In a bounded open set of we consider the Dirichlet problem for nonlinear order equations in divergence form with right-hand sides. It is supposed that , and the coefficients of the equations admit the growth of rate with respect to the derivatives of order m of unknown function. We establish that under the condition for some L¹ -data the corresponding Dirichlet problem does not have W-solutions

Solvability of degenerate anisotropic elliptic second-order equations with L^1-data

In this article, we study the Dirichlet problem for degenerate anisotropic elliptic second-order equations with $L^1$-right-hand sides on a bounded open set of $mathbb{R}^n$ ($ngeqslant 2$). These equations are described with a set of exponents and of a set of weighted functions. The exponents characterize the rates of growth of the coefficients of the equations with respect to the corresponding derivatives of the unknown function, and the weighted functions characterize degeneration or singularity of the coefficients of the equations with respect to the spatial variable. We prove theorems on the existence of entropy solutions, T-solutions, W-solutions, and weighted weak solutions of the problem under consideration