An existence theorem for solutions tof(x) = 0

InR ¹, if a continuous function has opposite signs at the endpoints of an interval, then the function has a zero in the interval. If the function has a nonvanishing derivative at a zero, then there is an interval such that the function has opposite signs at the endpoints. In this paper each of these results is extended toR ⁿ .The research of this author has been partially supported by ONR Contract N000-14-67-A-0285-0019 (NR #047-095) and by NSF Contract SOC-7402516.     

Schrödinger operators on the half line: Resolvent expansions and the Fermi golden rule at thresholds

We consider Schrodinger operatorsH = - d² /dr ² +V onL ²([0, ∞)) with the Dirichlet boundary condition. The potentialV may be local or non-local, with polynomial decay at infinity. The point zero in the spectrum ofH is classified, and asymptotic expansions of the resolvent around zero are obtained, with explicit expressions for the leading coefficients. These results are applied to the perturbation of an eigenvalue embedded at zero, and the corresponding modified form of the Fermi golden rule. Dedicated to K B Sinha on the occasion of his sixtieth birthday     

Strongly meager sets can be quite big

AssumeCH. There exists a strongly meager setX⊆2^ω and a continuous functionF: 2^ω → 2^ω such thatF″ (X)=2^ω. The analogous statement for the strong measure zero, the notion dual to strongly meager, is false. The first author was partially supported by NSF grant DMS 9971282 and the Alexander von Humboldt Foundation. The second author was partially supported by grant BW 5100-5-0231-2.     

Infinite retrogression in the Eaves—Saigal algorithm

For certain functionsf fromR ⁿ toR ⁿ , the Eaves—Saigal algorithm computes a path inR ⁿ × (0, 1] which converges to a zero off. In this case, it is shown that even whenf is of classC and has a unique zero, the converging path may retrogress infinitely many times.Army Research Office, Durham, Contract No. DAAG-29-78-G-0026; National Science Foundation Grant No. MCS-77-05623.     

On separating a fixed point from zero by invariants

Assume a fixed point v∈V^G can be separated from zero by a homogeneous invariant f∈𝕜[V]^G of degree p^rd, where p>0 is the characteristic of the ground field 𝕜 and p,d are coprime. We show that then v can also be separated from zero by an invariant of degree p^r, which we obtain explicitly from f. It follows that the minimal degree of a homogeneous invariant separating v from zero is a p-power.     

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S ¹-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S ⁴, ℂP ², ²,S ²×S ²and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁴,ℂP ²,S ²×S ²,ℂP ²#  ² or ℂP ²# ℂP ². Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁵,S ³×S ², then on trivial S ³-bundle over S ² or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.     

Orlicz capacities and Hausdorff measures on metric spaces

In the setting of doubling metric measure spaces with a 1-Poincaré inequality, we show that sets of Orlicz Φ-capacity zero have generalized Hausdorff h-measure zero provided thatwhere Θ⁻¹ is the inverse of the function Θ(t)=Φ(t)/t, and s is the “upper dimension” of the metric measure space. This condition is a generalization of a well known condition in Rⁿ. For spaces satisfying the weaker q-Poincaré inequality, we obtain a similar but slightly more restrictive condition. Several examples are also provided.     

Polynomial cycles in finitely generated domains

LetR be a finitely generated domain of zero characteristics, and letN be a given integer. We show that the lengths of all cycles of polynomial mappingsR ^NR ^N defined overR are bounded by a number depending only onR andN.The work of the second author has been supported by the KBN-grant 2-1037-91-01.     

On the central limit theorem in R p when p→∞

Summary Let X ₁ , X ₂ , ..., X _n be i.i.d. random vectors in R ^p where p tends to infinity. A theorem is presented showing that the Central Limit Theorem should hold if p ²/n tends to zero. Furthermore, an example is presented with X _i having a mixed multivariate normal distribution (with finite moment generating function) for which a uniform normal approximation to the distribution of the sample mean can not hold if p ²/n does not tend to zero. Research supported in part by National Science Foundation Grants MCS 80-02340, MCS 83-01834, and DMS 85-03785     

Twisted traces of singular values

Let p be a prime for which the congruence group Γ₀(p)^* is of genus zero, and j _p ^* be the corresponding Hauptmodul. We investigate the twisted traces of singular values of j _p ^* and construct infinite products related to them. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-331-C00006).     

Simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds

Ideas of a simplicial variable dimension restart algorithm to approximate zero points onR ⁿ developed by the authors and of a linear complementarity problem pivoting algorithm are combined to an algorithm for solving the nonlinear complementarity problem with lower and upper bounds. The algorithm can be considered as a modification of the2n-ray zero point finding algorithm onR ⁿ . It appears that for the new algorithm the number of linear programming pivot steps is typically less than for the2n-ray algorithm applied to an equivalent zero point problem. This is caused by the fact that the algorithm utilizes the complementarity conditions on the variables. This work is part of the VF-program “Equilibrium and Disequilibrium in Demand and Supply,” which has been approved by the Netherlands Ministry of Education and Sciences.     

On Banach spaces whose dual balls are not weak∗ sequentially compact

Theorem 1. LetX be a Banach space. (a) IfX ^∗ has a closed subspace in which no normalized sequence converges weak^∗ to zero, thenl ₁ is isomorphic to a subspace ofX. (b) IfX ^∗ contains a bounded sequence which has no weak^∗ convergent subsequence, thenX contains a separable subspace whose dual is not separable. The second-named author was supported in part by NSF-MPS 72-04634-A03.     

On the cotangent cohomology of rational surface singularities with almost reduced fundamental cycle

We prove dimension formulas for the cotangent spaces T ¹ and T ² for a class of rational surface singularities by calculating a correction term in the general dimension formulas. We get that it is zero if the dual graph of the rational surface singularity X does not contain a particular type of configurations, and this generalizes a result of Theo de Jong stating that the correction term c (X ) is zero for rational determinantal surface singularities. In particular our result implies that c (X ) is zero for Riemenschneiders quasi‐determinantal rational surface singularities, and this also generalizes results for quotient singularities. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zero sets of holomorphic functions in the bidisk

We characterize in geometric terms the zero sets of holomorphic functionsf in the bidisk such that log |f|∈L ^p (D ²) for 1<p<∞. Partially supported by the DGCYT grant PB95-0956-C02-02 and grant 1996-SGR-26.     

On the Index of Solvability for Variational Inequalities in Banach Spaces

We define the index of solvability, a topological characteristic, whose difference from zero provides the existence of a solution for variational inequalities of Stampacchia’s type with S ⁺-type and pseudo-monotone multimaps on reflexive separable Banach spaces. Some applications to a minimization problem and to a problem of economical dynamics are presented. The work is supported by the Russian FBR Grants 05-01-00100 and 07-01-00137 and by the NATO Grant ICS.NR.CLG 981757.     

On the solution of nonlinear equations in interval arithmetic

We consider the problem of finding a simple zero of a continuously differentiable functionf:R ⁿ R ⁿ . There is given an intervalvectorX ₀ ^I containing one zero off, and we will construct a contracting sequence of intervalvectors enclosing this zero. This can be done by Newton's method, which gives quadratic convergence, but requires inversion of an intervalmatrix at each step of the iteration. Alefeld and Herzberger, [1], give a modification of Newton's method, without the necessity of inversion, the convergence being superlinear. We give a slight modification of the latter method, with the property that the sequence of interval widths is dominated by a quadratically convergent sequence.     

On directional convexity

Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties of D-convexity. A function f: R ^d → R is called D-convex, where D is a set of vectors in R ^d, if its restriction to each line parallel to a nonzero v ∈ D is convex. The D-convex hull of a compact set A ⊂ R ^d, denoted by co^D(A), is the intersection of the zero sets of all nonnegative D-convex functions that are zero on A. It also equals the zero set of the D-convex envelope of the distance function of A. We give an example of an n-point set A ⊂ R ² where the D-convex envelope of the distance function is exponentially close to zero at points lying relatively far from co ^D(A), showing that the definition of the D-convex hull can be very nonrobust. For separate convexity in R ³ (where D is the orthonormal basis of R ³), we construct arbitrarily large finite sets A with co ^D(A) ≠ A whose proper subsets are all equal to their D-convex hull. This implies the existence of analogous sets for rank-one convexity and for quasiconvexity on 3 × 3 (or larger) matrices. This research was supported by Charles University Grants No. 158/99 and 159/99.     

On the class number of p-th cyclotomic field

Let h^[-(p)h^-(p) be the relative class number of the p-th cyclotomic field. We show that logh^-(p) = [(p+3)/4] logp - [(p)/2] log2p+ log(1-b) + O(log₂² p)\log h^-(p) = {{p+3} \over {4}} \log p - {{p} \over {2}} \log 2\pi + \log (1-\beta ) + O(\log _2^2 p), where b\beta denotes a Siegel zero, if such a zero exists and p o -1 mod 4p\equiv -1\pmod {4}. Otherwise this term does not appear.     

On Interpolating Blaschke Products and?Blaschke-Oscillatory Equations

This research is partially a continuation of a 2007 paper by the author. Growth estimates for generalized logarithmic derivatives of Blaschke products are provided under the assumption that the zero sequences are either uniformly separated or exponential. Such Blaschke products are known as interpolating Blaschke products. The growth estimates are then proven to be sharp in a rather strong sense. The sharpness discussion yields a solution to an open problem posed by E. Fricain and J. Mashreghi in 2008. Finally, several aspects are pointed out to illustrate that interpolating Blaschke products appear naturally in studying the oscillation of solutions of a differential equation f″+A(z)f=0, where A(z) is analytic in the unit disc. In particular, a unit disc analogue of a 1988 result due to S. Bank on prescribed zero sequences for entire solutions is obtained, and a more careful analysis of a 1955 example due to B. Schwarz on the case A(z)=\frac1+4g²(1-z²)²A(z)=\frac{1+4\gamma^{2}}{(1-z^{2})^{2}} reveals that an infinite zero sequence is always a union of two exponential sequences.