Hilbert-Speiser number fields for a prime p inside the p-cyclotomic field

Let p be a prime number. We say that a number field F satisfies the condition when for any cyclic extension N/F of degree p, the ring of p-integers of N has a normal integral basis over . It is known that F=Q satisfies for any p. It is also known that when p?19, any subfield F of Q(ζp) satisfies . In this paper, we prove that when p?23, an imaginary subfield F of Q(ζp) satisfies if and only if and p=43, 67 or 163 (under GRH). For a real subfield F of Q(ζp) with F≠Q, we give a corresponding but weaker assertion to the effect that it quite rarely satisfies .     

On the univalence of functions with logharmonic laplacian

In this paper, we prove Radó’s theorem holds for functions of the form is logharmonic. We show that if F is of the form , where is logharmonic, then F is starlike iff ψ(z)=h(z)/g(z) is starlike. In addition, when , where L is logharmonic and H is harmonic, we give the sufficient conditions for F to be locally univalent.     

Strong influence of a small fiber on shear stress in fiber-reinforced composites

In stiff fiber-reinforced composites, it has been known that the shear stress increases at the rate of as the distance ? between adjacent fibers approaches 0. This paper reveals a strong influence of a combination of a triple fiber, as well as the distance between a pair of fibers, on the blow-up so that the stress concentration can be significantly accelerated by adding a small fiber in-between fibers. Specifically, if a fiber F₂ with a small diameter δ is located in-between fibers F₁ and F₃, ?₁=dist(F₁,F₂) and ?₂=dist(F₂,F₃), then the stress blows up at the exact rates of and between F₁ and F₂ and between F₂ and F₃, respectively. This estimate still holds even when a part of F₂ overlaps with F₃. The magnification factor yields the enormous increase in the stress that greatly surpasses the expectancy by previous methods.     

The third homology of the special linear group of a field

We prove that for any infinite field F, the map is an isomorphism for all n≥3. When n=2 the cokernel of this map is naturally isomorphic to , where is the nth Milnor K-group of F. We deduce that the natural homomorphism from to the indecomposable K₃ of F, , is surjective for any infinite field F.     

Removable singularities for a Sobolev space

Let Ω⊂RN be an open set and F a relatively closed subset of Ω. We show that if the (N−1)-dimensional Hausdorff measure of F is finite, then the spaces and coincide, that is, F is a removable singularity for . Here is the closure of in H¹(Ω) and H¹(Ω) denotes the first order Sobolev space. We also give a relative capacity criterium for this removability. The space is important for defining realizations of the Laplacian with Neumann and with Robin boundary conditions. For example, if the boundary of Ω has finite (N−1)-dimensional Hausdorff measure, then our results show that we may replace Ω by the better set (which is regular in topology), i.e., Neumann boundary conditions (respectively Robin boundary conditions) on Ω and on coincide.     

Some classes of non-analytic Markov semigroups

We deal with Markov semigroups T_t corresponding to second order elliptic operators Au=Δu+〈Du,F〉, where F is an unbounded locally Lipschitz vector field on . We obtain new conditions on F under which T_t is not analytic in . In particular, we prove that the one-dimensional operator Au=u″−x³u′, with domain , , is not sectorial in . Under suitable hypotheses on the growth of F, we introduce a class of non-analytic Markov semigroups in , where μ is an invariant measure for T_t.     

Projective Schur groups of Henselian fields

One of the open questions that has emerged in the study of the projective Schur group of a field F is whether or not is an algebraic relative Brauer group over F, i.e. does there exist an algebraic extension L/F such that ? We show that the same question for the Schur group of a number field has a negative answer. For the projective Schur group, no counterexample is known. In this paper we prove that is an algebraic relative Brauer group for all Henselian valued fields F of equal characteristic whose residue field is a local or global field. For this, we first show how is determined by for an equicharacteristic Henselian field with arbitrary residue field k.     

A general regularity theorem for functionals with φ-growth

We prove C1,α-regularity for local minimizers of functionals

     

Limits in compact Abelian groups

For X a compact Abelian group and B an infinite subset of its dual , let CB be the set of all x∈X such that converges to 1. If F is a free filter on , let . The sets CB and DF are subgroups of X. CB always has Haar measure 0, while the measure of DF depends on F. We show that there is a filter F such that DF has measure 0 but is not contained in any CB. This generalizes previous results for the special case where X is the circle group.     

Polynomial approximation, local polynomial convexity, and degenerate CR singularities

We begin with the following question: given a closed disc and a complex-valued function , is the uniform algebra on generated by z and F equal to ? When F∈C¹(D), this question is complicated by the presence of points in the surface that have complex tangents. Such points are called CR singularities. Let p∈S be a CR singularity at which the order of contact of the tangent plane with S is greater than 2; i.e. a degenerate CR singularity. We provide sufficient conditions for S to be locally polynomially convex at the degenerate singularity p. This is useful because it is essential to know whether S is locally polynomially convex at a CR singularity in order to answer the initial question. To this end, we also present a general theorem on the uniform algebra generated by z and F, which we use in our investigations. This result may be of independent interest because it is applicable even to non-smooth, complex-valued F.     

Euler systems and special units

Let F be a number field. We investigate the group of Rubin's special units, S_F defined over F. The group of special units is a subgroup of the group of global units containing the group of Sinnott's cyclotomic units, C_F of F. It plays an important role in studying the ideal class group of F. Let be a sequence of decreasing subgroups (defined in Section 2) of the group of global units of any real abelian field K which lie between Rubin's special units and the circular units of K. Motivated by a question of whether the group of special units equals the group of cyclotomic units, which is stated by Rubin (Invent. Math. 89 (1987) 511), we propose the following question which relates the group structure of the ideal class group with the group structure of units modulo special units. Are and isomorphic as Z[Gal(F/Q)] modules? Let Ξ be the set of p-adic valued Dirichlet characters of Gal(F/Q). Let and be the χ-eigenspaces of and Cl_F⊗Z_p respectively. Using Euler system methods and Thaine's results we obtain that the Z/pZ-rank of is less than or equal to the Z/pZ-rank of with some inequalities on the cardinalities of both sides. This gives us the following corollary. If , then for all χ∈Ξ, we have is a cyclic group.     

Category and compact leaves

We prove that if F is a C¹-foliation of a compact manifold M with finite transverse saturated LS category, , then F has a compact leaf. In contrast, we show that if F is expansive on some non-trivial minimal set of F, then . Examples of foliations are given to illustrate the main results of the paper.     

On real quadratic function fields

Let q be a power of an odd prime number p and K:=Fq(T) be the rational function field with a fixed indeterminate T. For P a prime of K, let be the maximal real subfield of the Pth-cyclotomic function field and its ring of integers. We prove that there exists infinitely many primes P of even degree such that the cardinal of the ideal class group is divisible by q. We prove also an analogous result for imaginary extensions.     

Covering numbers of different points in Dvoretzky covering

Consider the Dvoretzky random covering on the circle T with a decreasing length sequence {?n}_n?1 such that . We study, for a given β?0, the set Fβ of points which are asymptotically covered by a number βLn of the first n randomly placed intervals where . Three typical situations arise, delimited by two “phase transitions”, according to is zero, positive-finite or infinite, where . More precisely, if ?n tends to zero rapidly enough so that then, with probability one, dimHFβ=1 for all β?0; if ?n is moderate so that then, with probability one, we have for and Fβ=∅ for where and is the interval consisting of β's such that ; eventually, if ?n is so slow that then, with probability one, F₁=T. This solves a problem raised by L. Carleson in a rather satisfactory fashion.Analogous results are obtained for the Poisson covering of the line, which is studied as a tool.     

ε-Regularity theorem and its application to the blow-up solutions of Keller-Segel systems in higher dimensions

Let us consider m(KS) below for all N?2 and general exponents m and q. In particular, the 2-D semi-linear case such as N=2, m=1 and q=2 is included. We establish an ε-regularity theorem for weak solutions. As an application, we give an extension criterion in which coincides with a scaling invariant class of weak solutions associated with m(KS). In addition, the Hausdorff dimension of its singular set is zero if and .     

Bounds for the bias of the empirical CTE

The Conditional Tail Expectation (CTE) is gaining an increasing level of attention as a measure of risk. It is known that nonparametric unbiased estimators of the CTE do not exist, and that , the empirical α-level CTE (the average of the n(1−α) largest order statistics in a random sample of size n), is negatively biased. In this article, we show that increasing convex order among distributions is preserved by . From this result it is possible to identify the specific distributions, within some large classes of distributions, that maximize the bias of . This in turn leads to best possible bounds on the bias under various sets of conditions on the sampling distribution F. In particular, we show that when the α-level quantile is an isolated point in the support of a non-degenerate distribution (for example, a lattice distribution) then the bias is either of the order or vanishes exponentially fast. This is intriguing as the bias of vanishes at the in-between rate of 1/n when F possesses a positive derivative at the αth quantile.     

Homology rings of homotopy associative H-spaces

Let X be a homotopy associative mod p H-space for p an odd prime. The homology H_*(X;Fp) is an associative ring, but not necessarily commutative. We study conditions when for elements of H_*(X;Fp). Under certain conditions imply for l=p−2 or p−1. These methods can be used to prove results about homology commutators that were previously obtained using the adjoint action [H. Hamanaka, S. Hara, A. Kono, Adjoint action of Lie groups on the loop spaces and cohomology of exceptional Lie groups, Transform. Group Theory (1996) 44-50, Korea Adv. Inst. Sci. Tech.; A. Kono, K. Kozima, The adjoint action of a Lie group on the space of loops, J. Math. Soc. Japan 45 (3) (1993) 495-509; A. Kono, J. Lin, O. Nishimura, Characterization of the mod 3 cohomology of E₇, Proc. Amer. Math. Soc. 131 (10) (2003) 3289-3295]. We also generalize results of Kane [R. Kane, Torsion in homotopy associative H-spaces, Illinois J. Math. 20 (1976) 476-485] to nonfinite mod p homotopy associative H-spaces.