Semilattices of left reversible semigroups

Let S be a cancellative semigroup which is a semilattice of left reversible semigroups S, . This article studies the relationship between the group of quotients G of S and the groups of quotients G of S, . It is shown that G is the maximum group homomorphic image of an inverse semigroup which is a semilattice of groups G (up to isomorphism).The technique used here which involves the use of Ore's quotients also applies to the study of the maximum group homomorphic image of a semigroup which is a semilattice of inverse semigroups.     

On a Darboux problem for a third-order hyperbolic equation with multiple characteristics

A Darboux type problem for a model hyperbolic equation of the third order with multiple characteristics is considered in the case of two independent variables. The Banach space, 0, is introduced where the problem under consideration is investigated. The real number ₀ is found such that for > ₀ the problem is solved uniquely and for < ₀ it is normally solvable in Hausdorff's sense. In the class of uniqueness an estimate of the solution of the problem is obtained which ensures stability of the solution.     

Semi-Classical Analysis of a Dirac Equation Without Adiabatic Decoupling

We study adiabatic decoupling for Dirac equation with some scaling which yields that the mass appears with a coefficient where is the semi-classical parameter and > 0. Therefore, the system presents an avoided crossing. The scale = 1/2 is critical: adiabatic decoupling holds for (0,1/2) while for 1/2, there is energy transfer at leading order between the two modes. We describe this transfer in terms of two-scale Wigner measures by means of Landau-Zener formula which takes into account the change of polarization of the measures after the crossing.     

Uniqueness of kernel functions of the heat equation

We consider the heat equation on ={(x,t) R ²;t<0, ¦x¦<(–t)} and give the uniqueness of kernel functions at the infinity (see Theorem 5). For the proof, we examine the continuity of the density of the parabolic measure onD ={(x,t);t>x}, closely related to . By this theorem, we can decide the Martin boundary of (<1) with respect to the heat equation.     

Random reals and the relation 171-1171-1171-1

It is consistent that ₁(₁,(:n))² holds in any random extension for n finite and countable.     

On cyclic biquadratic fields related to a problem of Hasse

In this note we shall prove that there exist infinitely many cyclic biquadratic fieldsK whose integral bases are neither {1, , ², } nor {1, , , ³) for any numbers , inK. Next, we shall construct infinitely many cyclic biquadratic fieldsK which have the index 1, but still have not the integral basis {1, , ², ³) for every inK. Finally we shall give a class of biquadratic fields for a problem of Hasse concerning an integral basis.     

A contribution to shape theory for the ising model at low temperature

Summary We consider low temperature limits of Gibbs states of the ferromagnetic nearest-neighbour Ising Hamiltonian in the positive orthant of the lattice ^d ,d=1, 2,..., under a negative boundary condition and a small positive external fieldh that decreases linearly with the temperatureT. It is shown that positive and negative spins are separated by a staircase-shaped random boundary. Its explicit distribution is computed in the case that the ratio =h/T exceeds some positive ₀. For < ₀, our results do not rule out infinite negative areas.     

Galois theory for the family of partial differential equationsΔ α ψ=0

Summary The partial differential fields most suited for the purpose of construction of Galois theory for the family (1) are endowed with the symmetric bilinear form (2iv) and are called -differential fields. In Section 1 are defined certain algebraic notions related to the symmetric bilinear form (2iv) and which are necessary for the construction of any Galois theory. Necessary and sufficient condition for the extension of the domain of the operator (this operator is not a derivation although it commutes with the partial derivations of the -differential field) from an -differential fieldK to a finitely generated -differential extension field is given in Theorem 1.Section 2 defines the notion of -differential mapping as linear mappings which preserve the symmetric bilinear form and commute with the partial derivations. The group properties of the set of -differential mappings are discussed and the Galois correspondence theorems set up for -differential fields.Section 3 sets up the notion of -Liouvillian extensions of -differential fields and briefly discusses the Galois groups associated with these -Liouvillian extension fields.Section 4 points to the procedure for the algebraic characterization of -simple--differential field extensions by elementary solutions of the partial differential equation _m =0.     

Comparaison des semi-groupes et des résolvantes d'ordre α associés à des opérateurs différentiels de type divergence

We prove that the densities of the semi-groups of order , 0<<1 associated with differential operators of second order and of divergence type, and the density of Riesz semi-groups of order are comparables.We give a necessary and sufficient condition such that the semi-group of order and its resolvent family and their perturbated with a nonnegative and regular Radon measure are comparables.When =1, we prove that the semi-group of brownian motion and its perturbated with a radial and nonnegative measure are comparables if and only if the measure generates a bounded potential, but the result is not true if the measure is not radial.

Ce travail est soutenu par la Fondation Nationale pour la Recherche Scientifique.     

On the Density of Elliptic Curves

We show that 17.9% of all elliptic curves over Q, ordered by their exponential height, are semistable, and that there is a positive density subset of elliptic curves for which the root numbers are uniformly distributed. Moreover, for any > 1/6 (resp. > 1/12) the set of Frey curves (resp. all elliptic curves) for which the generalized Szpiro Conjecture |(E)| N _E ¹² is false has density zero. This implies that the ABC Conjecture holds for almost all Frey triples. These results remain true if we use the logarithmic or the Faltings height. The proofs make use of the fibering argument in the square-free sieve of Gouvêa and Mazur. We also obtain conditional as well as unconditional lower bounds for the number of curves with Mordell–Weil rank 0 and 2, respectively.     

On the dispersion spectrum

A measure for the denseness of sequences (an) mod 1, irrational, is the dispersion constantD() introduced byH. Niederreiter. In this paper the smallest accumulation point ₁ of the set of theD() is determined and all those are explicitely given for whichD () < ₁ holds.     

α-Sober spaces via the orthogonal closure operator

Each ordinal equipped with the upper topology is a T ₀-space. It is well known that for =2 the reflective hull of in Top₀ is the subcategory of sober spaces. Here, we define -sober space for each 2 in such a way that the reflective hull of in Top₀ is the subcategory of -sober spaces. Moreover, we obtain an order-preserving bijective correspondence between a proper class of ordinals and the corresponding (epi)reflective hulls. Our main tool is the concept of orthogonal closure operator, first introduced in [12].The author acknowledges financial support from Instituto Politécnico de Viseu and from Centro de Matemática da Universidade de Coimbra.     

Perturbation of the Laplacian by the Coulomb potential and a point interaction in L p (ℝ)3

Perturbations of -+/|x| (with >0) by a point interaction centered at zero are defined in L ^p(³). This is done for 3/20 (³{0}), such that the extension is the negative generator of an analytic semigroup on L ^p(³).     

K1 of Twisted Rings of Polynomials

We prove that for an arbitrary endomorphism of a ring R the group K1(R[t]) splits into the direct sum of K1(R) and Ñil (r;). Moreover, for any such R and Ñil (R; ) is isomorphic to Ñil (R ; ) for some ring R with : R R – an isomorphism.     

The capillary problem for an infinite trough

Consider an infinite trough (or wedge) with dihedral angle 2, 0 < < and a quantity of fluid inside contacting the edge. In equilibrium the free interface of the fluid will be a surface of constant mean curvature meeting the planar walls at a constant angle determined from physical considerations. One obvious configuration is for the free surface to be a section of a round circular cylinder parallel to the axis of the wedge whose position is determined by the angles and . For + > /2 the cylinder configuration is unstable and bifurcation occurs. We exhibit the full family of bifurcating solutions starting with the round cylinder solution and proceeding through a beading up process into a series of spherical sections suitably positioned. Furthermore, if the edge of the wedge is a re-entrant corner ( > /2) then there are further bifurcating families. One is a secondary bifurcation from the family initially constructed while the other is a primary bifurcation from the cylinder which are less symmetric than the initial families.This paper was completed while the author was on a subbatical from the University of Toledo and was a visitor at Stanford University and also with SFB 256, University of Bonn.     

Characterization of Dirichlet Forms by their Excessive and Co–Excessive Elements

It is proved that if S, T are two elliptic Dirichlet operators on an ordered Hilbert space such that the excessive (resp. coexcessive) elements with respect to S and T are the same then there exists > 0 with T = S. Particularly if , are two elliptic Dirichlet forms on L² ( ) having the same domain of definition and the same -excessive (resp. -coexcessive) elements for any > 0 then = .     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

A class of non-Moufang Bol loops isomorphic to all their loop isotopes

For any two primes, , such that< and divides–1, it is shown that there exists a non-Moufang Bol loop of order ² which is isomorphic to each of its loop isotopes.     

On the integrability of “finite energy” solutions for <Emphasis Type="Italic">p</Emphasis>-harmonic equations

We prove a higher integrability result for the gradient of solutions to some degenerate elliptic PDEs, whose model arises in the study of mappings with finite distortion.The nonnegative function which measures the degree of degeneracy of ellipticity bounds lies in the exponential class, i.e. is integrable for some > 0.Our result states that if is sufficiently large, then the gradient of a finite energy solution actually belongs to the Zygmund space L^plog^{L, 1}.     

A relationship between the Mahler measure and the discriminant of algebraic numbers

In this note we show that in the well-known Dobrowolski estimate lnM() (ln lnd/ lnd)3,d , where is a nonzero algebraic number of degreed that is not a root of unity andM() is its Mahler measure, the parameterd can be replaced by the quantity=d/() ^1/d, where () is the modulus of the discriminant of. To this end, must satisfy the condition deg ^p=deg for any primep.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 415–420, March, 1996.