On minimal non-potentially closed subsets of the plane

We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions. We show the existence of a perfect antichain made of minimal sets among non-potentially closed sets. We apply this result to graphs, quasi-orders and partial orders. We also give a non-potentially closed set minimum for another notion of comparison. Finally, we show that we cannot have injectivity in the Kechris-Solecki-Todor?evi? dichotomy about analytic graphs.     

A Cantor set in the plane and its monotone subsets

Given c >  0 a planar Cantor set X with a dim _H (X) < 2 is constructed such that each c-monotone subspace of X has a smaller Hausdorff dimension than X.     

Topologically free subsets of universal algebras

We prove that every infinite Abelian algebra and every countable field contain infinite topologically free subsets. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 651–661, May, 1997.     

On the Neumann problem for the Helmholtz equation in a plane angle

We prove that the solution of the Neumann problem for the Helmholtz equation in a plane angle Ω with boundary conditions from the space H_−1/2(Γ), where Γ is the boundary of Ω, which is provided by the well‐known Sommerfeld integral, belongs to the Sobolev space H₁(Ω) and depends continuously on the boundary values. To this end, we use another representation of the solution given by the inverse two‐dimensional Fourier transform of an analytic function depending on the Cauchy data of the solution. Copyright © 2000 John Wiley & Sons, Ltd.     

On free fermions and plane partitions

We use free fermion methods to re-derive a result of Okounkov and Reshetikhin relating charged fermions to random plane partitions, and to extend it to relate neutral fermions to strict plane partitions.     

Maximal subsets free of arithmetic progressions in arbitrary sets

The problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length k in a given set of size n is considered. It is proved that it is sufficient, in a certain sense, to consider the interval [1,..., n]. The study continues the work of Komlós, Sulyok, and Szemerédi.     

Deformation of free boundaries in plane viscous creep

Summary and Final Comments The deformation of a cylindrical free boundary lying in a thick infinite viscous body loaded by uniform stresses at infinity has been investigated. It is shown that when the free boundary can be initially conformally mapped onto a circle by a polynomial of degreen, it deforms so that its configuration remains within the family of curves described by a polynomial mapping of exactly degreen. A method of obtaining the exact motion of boundaries in this class is formulated and an example illustrating the calculations is offered; the example shows that a small irregularity on a free boundary may grow when compression is applied at infinity.The fact that polynomial free boundaries remain polynomial curves during deformation is one aspect of the kinematic reversibility (or memory) of linear viscous deformation, which greatly facilitates analysis of their motion. The deformation of completely arbitrary free boundaries is more difficult to obtain. In plane elasticity one often seeks approximate solutions to a problem by using a polynomia approximation to the exact mapping of a boundary.Muskhelishvili [3] has shown that if one chooses a sequence {_n} of mappings which converges onto the true mapping of the boundary , the sequences of stress functions {_n} and {_n} generated by {_n} converge onto the stress functions and associated with . In the viscous case Muskhelishvili's theorem always applies for computation of incipient motion; but, (21) and (26) show that when finite motion is considered it is not always adequate to replace the true mapping of the surface with a polynomial which is initially a close approximation. The general conditions under which one may expect the magnitude of higher order terms in the mapping of the free surface to diminish during deformation, so that a polynomial approximation to a mapping improves with time, have not been worked out in detail; however, it seems sufficient at this writing to require that the hydrostatic component of stress at infinity be tensile (P ₁+P₂>0). In the case that the higher order coefficients diminish, Equations (14), (15) and (18) indicate that the free boundary will tend to become an ellipse.

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Zusammenfassung Es wird die Verformung einer zylindrischen freien Oberfläche in einem unendlichen viskosen Körper untersucht, der sich unter dem Einfluss eines gleichförmigen Spannungszustandes im Unendlichen in einem ebenen Fliesszustand befindet. Es wird gezeigt, dass dann, wenn die freie Oberfläche ursprünglich durch ein Polynomn-ten Grades konform auf einen Kreis abgebildet werden kann, das gleiche auch für die deformierte Oberfläche gilt. Für diese Problemklasse wird eine Methode zur Ermittlung der exakten Oberflächenbewegung angegeben und an einem Beispiel illustriert. Dabei zeigt sich, dass eine kleine Unregelmässigkeit in der freien Oberfläche unter Druck im Unendlichen anwachsen kann.

     

Inverse monoids and rational Schreier subsets of the free group

An inverse monoidM is an idempotent-pure image of the free inverse monoid on a setX if and only ifM has a presentation of the formM=Inv<X:e_o=f_i, i∈I>, wheree _i ,f _i are idempotents of the free inverse monoid: every inverse monoid is an idempotent-separating image of one of this type. IfR is anR-class of such an inverse monoid, thenR may be regarded as a Schreier subset of the free group onX. This paper is concerned with an examination of which Schreier subsets arise in this way. In particular, ifI is finite, thenR is a rational Schreier subset of the free group. Not every rational Schreier set arises in this way, but every positively labeled rational Schreier set does. Research supported by National Science Foundation grant #DMS8702019.     

Extension of collineations defined on subsets of a translation plane

The T-closure of a setA in a projective translation plane is defined as the image ofA under the group generated by translations such that there exist proper points A, B with(A) = B. The sets, called quasi-anchors, make up the concepts of anchors introduced in [11]. It is proved that any collineation defined on a quasi-anchor can be extended to a collineation of the T-closure ofA. An application to the problem of uniqueness of the nomography is provided in a special case.     

A free boundary problem for p-Laplacian in the plane

We consider the following free boundary problem in an unbounded domain Ω in two dimensions: Δpu=0 in Ω, on J₀, on J₁, where ∂Ω=J₀∪J₁. We prove that if 0<u<1 in Ω, Ji is the graph of a function in and gi is a constant for each i=0,1, then the free boundary ∂Ω must be two parallel straight lines and the solution u must be a linear function. The proof is based on maximum principle.     

Some subsets of points in the plane associated to truncated Reed-Muller codes with good parameters

We gives some examples of subsets of points in the projective plane associated to truncated generalized projective Reed-Muller codes with good parameters, of dimensions 6 and 10 over GF(7), GF(8) and GF(9).     

Closed free hyperelastic curves in the hyperbolic plane and Chen-Willmore rotational hypersurfaces

We show that closed Chen-Willmore rotational hypersurfaces of nonnegative curved real space forms are shaped on closed hyperelastic curves of the hyperbolic plane. Then, we study the variational problem associated to this class of curves, proving that there exist a rationally dependent family of closed solutions. They give rise to the first non-trivial examples of Chen-Willmore hypersurfaces in real space forms. Research partially supported by a MCYT-FEDER grant No. BFM2001-2871-C04 and by a UPV grant 9/UPV00127.310-13574/01.     

On the best estimate for perimeters of plane sets with the angle property

We investigate the problem of finding the maximum length of perimeters of plane sets with fixed diameter d, such that every point of the boundary of the set is a vertex of an open angle of opening which does not intersect the set. First we consider plane curves which satisfy such angle property in a finite number of directions, and among them we find the one of maximum length. Then we prove that the perimeter of any plane set with the angle property is less than or equal to d(sin /2)^-2; this is the best estimate when /2.     

The capillary contact angle,II: The inclined plane

In an earlier paper an hypothesis on the nature of the forces resisting motion of a liquid drop on a support surface II was introduced and tested in the (rotationally symmetric) case for which ll is a horizontal plane. In the present work II is chosen as an inclined plane, so that rotational symmetry fails. The hypothesis leads to a formal series development; in a restricted case the result is verified by a perturbation analysis, which yields the identical series. The series predictions are compared with results of computer calculations. A consequence of the analysis is an estimate for a Bohd number B₀, such that a further increase in B would result in the fluid penetrating the support surface; thus an absolute upper bound B₀, for stability of the envisaged configuration is obtained.     

A note on plane nonlinear creep of free boundaries

Résumé On montre que la déformation plane et quasi-statique d'un corps incompressible qui est non-linéairement visqueux du second ordre, et dont les frontières sont soumises aux tractions données, ne peut être distinguée de celle d'un corps fluide, ayant les mêmes frontières et les mêmes tractions prescrites, qui est linéairement visqueux et incompressible. Ainsi, les solutions données préalablement (Berg, 1964) pour le fluage plan des frontières libres de tractions s'appliquent également au fluage plan et nonlinéaire du second ordre des frontières libres de tractions. Cela signifie en particulier que dans le fluage plan les effets visqueux du second ordre ne peuvent pas donner lieu à l'irréversibilité cinématique de la déformation d'une surface libre. Les résultats des effets du second ordre sont quelque peu étendus pour montrer que dans le cas le plus général d'un fluide non-linéaire et incompressible les problèmes de fluage plan peuvent se simplifier si l'on incorpore dans le champ de pression une partie des contraintes non-linéairement visqueuses.