On the Fredholm Alternative for the -Laplacian

Consider

where and and let be the principal eigenvalue of the problem with . For , we discuss for which values of and the Fredholm alternative holds.

     

Bifurcation problems for the

In this paper we consider the bifurcation problem

in with . We show that a continuum of positive solutions bifurcates out from the principal eigenvalue of the problem

Here both functions and may change sign.

     

On polarized surfaces

Let be a smooth projective surface over and an ample Cartier divisor on . If the Kodaira dimension or , the author proved , where . If , then the author studied with . In this paper, we study the polarized surface with , , and .

     

On the Diophantine equation

If and are positive integers with and , then the equation of the title possesses at most one solution in positive integers and , with the possible exceptions of satisfying , and . The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometric functions, the theory of linear forms in logarithms and recent computational methods related to lattice-basis reduction. Additionally, we compare and contrast a number of these last mentioned techniques.

     

The

In this paper we analyze the localization of , the fiber of the double suspension map , with respect to . If four cells at the bottom of , the th extended power spectrum of the Moore spectrum, are collapsed to a point, then one obtains a spectrum . Let be the James-Hopf map followed by the collapse map. Then we show that the secondary suspension map has a lifting to the fiber of and this lifting is shown to be a -periodic equivalence, hence an -equivalence.

     

On the diophantine equation

One of the purposes of this note is to correct the proof of a recent result of Y. Guo & M. Le on the equation . Moreover, we prove that the diophantine equation , , , , , gcd, , has only finitely many solutions, all of which satisfying .

     

Bounds for eigenvalues and condition numbers in the -version of the finite element method

In this paper, we present a theory for bounding the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices arising from the -version of finite element analysis. Bounds are derived for the eigenvalues and the condition numbers, which are valid for stiffness matrices based on a set of general basis functions that can be used in the -version. For a set of hierarchical basis functions satisfying the usual local support condition that has been popularly used in the -version, explicit bounds are derived for the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices. We prove that the condition numbers of the stiffness matrices grow like , where is the number of dimensions. Our results disprove a conjecture of Olsen and Douglas in which the authors assert that ``regardless of the choice of basis, the condition numbers grow like or faster". Numerical results are also presented which verify that our theoretical bounds are correct.

     

On the suspension order of

It is shown that the suspension order of the -fold cartesian product of real projective -space is less than or equal to the suspension order of the -fold symmetric product of and greater than or equal to , where and satisfy and . In particular has suspension order , and for fixed the suspension orders of the spaces are unbounded while their stable suspension orders are constant and equal to .

     

Calibrated thin

Let be a compact metric space, and let be a calibrated thin -ideal. Then is . This solves an open problem, which was posed by Kechris, Louveau and Woodin. Using our result we obtain a new proof of Kaufman's theorem concerning -sets and -sets.

     

Gaps in

For a partial order , let denote the statement that for every -increasing -sequence there is a -decreasing -sequence on top of such that is an -gap in . The main result of this paper is that . It is also shown, as a corollary, that but .

     

without sharps

We show that the supremum of the lengths of prewellorderings of the reals can be , with inaccessible to reals, assuming only the consistency of an inaccessible.

     

Steiner systems

It is proved that there are precisely 4204 pairwise non-isomorphic Steiner systems invariant under the group and which can be constructed using only short orbits.

It is further proved that there are precisely 38717 pairwise non-isomorphic Steiner systems invariant under the group and which can be constructed using only short orbits.

     

Decomposition of

For any locally compact group , let and be the Fourier and the Fourier-Stieltjes algebras of , respectively. is decomposed as a direct sum of and , where is a subspace of consisting of all elements that satisfy the property: for any and any compact subset , there is an with and such that is characterized by the following: an element is in if and only if, for any there is a compact subset such that for all with and . Note that we do not assume the amenability of . Consequently, we have for all if is noncompact. We will apply this characterization of to investigate the general properties of and we will see that is not a subalgebra of even for abelian locally compact groups. If is an amenable locally compact group, then is the subspace of consisting of all elements with the property that for any compact subset , .

     

On the weak uniform convexity of

We will discuss the geometry of the unit sphere in the Banach space of integrable holomorphic quadratic differentials on a Riemann surface and answer some questions posed by L.R. Goldberg (Proc. Amer. Math. Soc. 118 (1993), 1179--1185).

     

Some characterizations of

We show that a function on the unit disk extends continuously to , the maximal ideal space of iff it is uniformly continuous (in the hyperbolic metric) and close to constant on the complementary components of some Carleson contour.

     

The irrationality of

We shall show that the numbers and
are linearly independent over for any natural number . The key is to construct explicit Padé-type approximations using Legendre-type polynomials.

     

Normal subgroups of

We classify the normal subgroups of of index less than 960; they are all congruence subgroups.