Functions -orthogonal with respect to their own zeros

In 1939, G. H. Hardy proved that, under certain conditions, the only functions satisfying

where the are the zeros of , are the Bessel functions. We replace the above integral by the Jackson -integral and give the -analogue of Hardy's result.

     

Integral representation for a class of vector valued operators

Let be a compact space and let , be a (real, for simplicity) Banach space. We consider the space of all continuous -valued functions on , with the supremum norm .

We prove in this paper a Bochner integral representation theorem for bounded linear operators

which satisfy the following condition:

where is the conjugate space of . In the particular case where , this condition is obviously satisfied by every bounded linear operator

and the result reduces to the classical Riesz representation theorem.

If the dimension of is greater than , we show by a simple example that not every bounded linear admits an integral representation of the type above, proving that the situation is different from the one dimensional case.

Finally we compare our result to another representation theorem where the integration process is performed with respect to an operator valued measure.

     

Arbitrarily large solutions of the conformal scalar curvature problem at an isolated singularity

We study the conformal scalar curvature problem

where is a continuous function. We show that a necessary and sufficient condition on for this problem to have positive solutions which are arbitrarily large at is that be less than 1 on a sequence of points in which tends to .

     

Unbounded solutions and periodic solutions for second order differential equations with asymmetric nonlinearity

In this paper we will prove the coexistence of unbounded solutions and periodic solutions for the asymmetric oscillator

where and are positive constants satisfying the nonresonant condition

and is periodic in the first variable and bounded.

     

A Cauchy-Schwarz type inequality for bilinear integrals on positive measures

If is Borel measurable, define for -finite positive Borel measures on the bilinear integral expression

We give conditions on such that there is a constant , independent of and , with

Our results apply to a much larger class of functions than known before.

     

Cauchy-Schwarz and means inequalities for elementary operators into norm ideals

The Cauchy-Schwarz norm inequality for normal elementary operators

implies a means inequality for generalized normal derivations

for all , as well as an inequality for normal contractions and

for all in and for all unitarily invariant norms

     

On a subspace perturbation problem

We discuss the problem of perturbation of spectral subspaces for linear self-adjoint operators on a separable Hilbert space. Let and be bounded self-adjoint operators. Assume that the spectrum of consists of two disjoint parts and such that 0$">. We show that the norm of the difference of the spectral projections

for and is less than one whenever either (i) or (ii) and certain assumptions on the mutual disposition of the sets and are satisfied.

     

Global Hölder regularity for discontinuous elliptic equations in the plane

-regularity up to the boundary is proved for solutions of boundary value problems for elliptic equations with discontinuous coefficients in the plane.

In particular, we deal with the Dirichlet boundary condition

where , 2$">, or with the following normal derivative boundary conditions:

where , 2$">, 0$"> and is the unit outward normal to the boundary .

     

The -primary classifying space of the fiber of the double suspension

Gray showed that the homotopy fiber of the double suspension has an integral classifying space , which fits in a homotopy fibration . In addition, after localizing at an odd prime , is an -space and if , then is homotopy associative and homotopy commutative, and is an -map. We positively resolve a conjecture of Gray's that the same multiplicative properties hold for as well. We go on to give some exponent consequences.

     

Knot signature functions are independent

A Seifert matrix is a square integral matrix satisfying

To such a matrix and unit complex number there corresponds a signature,

Let denote the set of unit complex numbers with positive imaginary part. We show that is linearly independent, viewed as a set of functions on the set of all Seifert matrices.

If is metabolic, then unless is a root of the Alexander polynomial, . Let denote the set of all unit roots of all Alexander polynomials with positive imaginary part. We show that is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices.

To each knot one can associate a Seifert matrix , and induces a knot invariant. Topological applications of our results include a proof that the set of functions is linearly independent on the set of all knots and that the set of two-sided averaged signature functions, , forms a linearly independent set of homomorphisms on the knot concordance group. Also, if is the root of some Alexander polynomial, then there is a slice knot whose signature function is nontrivial only at and . We demonstrate that the results extend to the higher-dimensional setting.

     

Cardinal spline interpolation from

Let be the discrete Hardy space, consisting of those sequences , such that , where , , is the discrete Hilbert transform of . For a sequence , let be the unique cardinal spline of degree interpolating to at the integers. The norm of this operator, , is called a Lebesgue constant from to , and it was proved that .

It is proved in this paper that

     

Weierstrass functions in Zygmund's class

Consider the function

where 1$"> and is an almost periodic function. It is well known that the function lives in the so-called Zygmund class. We prove that is generically nowhere differentiable. This is the case in particular if the elementary condition is satisfied. We also give a sufficient condition on the Fourier coefficients of which ensures that is nowhere differentiable.

     

Extremal properties of the derivatives of the Newman polynomials

Let be a set of distinct positive numbers. The span of

over will be denoted by

Our main result of this note is the following.

Theorem. Suppose . Let be a non-negative integer. Then there are constants 0$"> and 0$"> depending only on , , and such that

where the lower bound holds for all and for all , while the upper bound holds when and and when , , and .

     

An expression of spectral radius via Aluthge transformation

For an operator , the Aluthge transformation of is defined by . And also for a natural number , the -th Aluthge transformation of is defined by and . In this paper, we shall show

where is the spectral radius.

     

Ample divisors on the blow up of at points

Fix integers with and ; if assume . Let be general points of the complex projective space and let be the blow up of at with exceptional divisors , . Set . Here we prove that the divisor is ample if and only if , i.e. if and only if .

     

Strong comparison principle for solutions of quasilinear equations

Let , , be a bounded smooth connected open set and be a map satisfying the hypotheses (H1)-(H4) below. Let with , in and with be two weak solutions of

Suppose that in . Then we show that u_1$"> in under the following assumptions: either u_1$"> on , or on and in . We also show a measure-theoretic version of the Strong Comparison Principle.

     

On the Betti numbers of sign conditions

Let be a real closed field and let and be finite subsets of such that the set has elements, the algebraic set defined by has dimension and the elements of and have degree at most . For each we denote the sum of the -th Betti numbers over the realizations of all sign conditions of on by . We prove that

This generalizes to all the higher Betti numbers the bound on . We also prove, using similar methods, that the sum of the Betti numbers of the intersection of with a closed semi-algebraic set, defined by a quantifier-free Boolean formula without negations with atoms of the form or for , is bounded by

making the bound more precise.

     

Twisted higher moments of Kloosterman sums

Let be a nontrivial Dirichlet character modulo an odd prime . Write

We shall prove

and, for complex ,

0, \end{displaymath}">

where is a constant depending only on .

     

Convergence theorem for zeros of generalized Lipschitz generalized phi-quasi-accretive operators

Let be a uniformly smooth real Banach space and let be a mapping with . Suppose is a generalized Lipschitz generalized -quasi-accretive mapping. Let and be real sequences in [0,1] satisfying the following conditions: (i) ; (ii) ; (iii) ; (iv) Let be generated iteratively from arbitrary by

where is defined by and is an arbitrary bounded sequence in . Then, there exists such that if the sequence converges strongly to the unique solution of the equation . A related result deals with approximation of the unique fixed point of a generalized Lipschitz and generalized -hemi-contractive mapping.

     

Super solutions of the dynamical Yang-Baxter equation

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical matrices. A super dynamical matrix satisfies the zero weight condition if

    for all 

In this paper we classify super dynamical matrices with zero weight.