A note on transitive permutation groups of prime degree

LetG be a nonsolvable transitive permutation group of prime degreep. LetP be a Sylow-p-subgroup ofG and letq be a generator of the subgroup ofN _G(P) fixing one point. Assume that |N _G(P)|=p(p?1) and that there exists an elementj inG such thatj ^?1qj=q^(p+1)/2. We shall prove that a group that satisfies the above condition must be the symmetric group onp points, andp is of the form 4n+1.     

A set intersection theorem and applications

Using Scarf's algorithm for “computing” a fixed point of a continuous mapping, the following is proved: LetM ₁ ? M _n be closed sets inR ⁿ which cover the standard simplexS, so thatM _i coversS _i , the face ofS opposite vertexi. We say a point inS iscompletely labeled if it belongs to everyM _i andk-almost-completely labeled if it belongs to all butM _k . Then there exists a closed setT ofk-almost-completely labeled points which connects vertexk with some completely labeled point. This result is used to prove Browder's theorem (a parametric fixed-point theorem) inR ⁿ . It is also used to generate “algorithms” for the nonlinear complementarity problem which are analogous to the Lemke—Howson algorithm and the Cottle—Dantzig algorithm, respectively, for the linear complementarity problem.     

An existence theorem for the complementarity problem

LetC be a pointed, solid, closed and convex cone in then-dimensional Euclidean spaceE ⁿ ,C* its polar cone,M:C→E ⁿ a map, andq a vector inE ⁿ . The complementarity problem (q|M) overC is that of finding a solution to the system $$(q|M) x \varepsilon C, M(x) + q \varepsilon C{^*} , \left\langle {x, M(x) + q} \right\rangle = 0.$$ It is shown that, ifM is continuous and positively homogeneous of some degree onC, and if (q|M) has a unique solution (namely,x=0) forq=0 and for someq=q ₀ ∈ intC*, then it has a solution for allq ∈E ⁿ .     

Lösung des Dirichlet-Problems bei Jordangebieten mit analytischem Rand durch Interpolation

LetC be an analytic Jordan curve, letG be the interior ofC, and letU (w) be (at least) continuous onC. Here the solution of the Dirichlet problemu(w) which coincides withU(w) onC is approximated by harmonic polynomials. These harmonic polynomialsF _n F(w) are determined by interpolatingU(w) in a given point system. For sufficiently greatn we prove |u(w)?F _n (w)|≤K·logn·E _n in \(\bar G\) , where \(E_n = \mathop {\max }\limits_{w \in C} \left| {U(w) - h_n (w)} \right|\) andh _n (w) is the harmonic polynomial of degreen of best approximation toU(w) onC andK is a constant independent ofn.     

De Rham cohomology of manifolds containing the points of infinite type,and Sobolev estimates for the\bar \partial - Neumann problem

We consider smooth bounded pseudoconvex domains Ω in Cⁿ whose boundary points of infinite type are contained in a smooth submanifoldM (with or without boundary) of the boundary having its (real) tangent space at each point contained in the null space of the Levi form ofbΩ at the point. (In particular, complex submanifolds satisfy this condition.) We consider a certain one-form α onbΩ and show that it represents a De Rham cohomology class on submanifolds of the kind described. We prove that if α represents the trivial cohomology class onM, then the Bergman projection and the \(\bar \partial - Neumann\) operator on Ω are continuous in Sobolev norms. This happens, in particular, ifM has trivial first De Rham cohomology, for instance, ifM is simply connected.     

Regularity of discs attached to a submanifold ofC n

Letp be an analytic disc attached to a generating CR-submanifoldM of C ⁿ . It is proved that some recently introduced conditions onp andM which imply that the family of all smallC ^α holomorphic perturbations ofp alongM is a Banach submanifold of (A^α(D))ⁿ are equivalent. These conditions are given in terms of the partial indices of the discp attached toM and “holomorphic sections” of the conormal bundle ofM along p(∂D). Also, a sufficient geometric conditionon p andM is given so that the family of all smallC ^α holomorphic perturbationsof p alongM, fixed at some boundary point, is a Banach submanifold of (A ^α (D))ⁿ.     

Konstruktion regulärer Hüllen konstanter Breite

LetM be a compact, convex set of diameter 2 inE ^d. There exists a bodyK of constant width 2 containingM such that every symmetry ofM is one ofK and every singular boundary point ofK is a boundary point ofM, for which the set of antipodes inK is the convex hull of the antipodes, which are already inM.

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Mit 1 Abbildung     

Function algebras on which homomorphisms are point evaluations on sequences

In the study of the spectrum of a subalgebraA ofC(X), whereX is a completely regular Hausdorff space, a key question is, whether each homomorphism ?:A→R has the point evaluation property for sequences inA, that is whether, for each sequence (f _n ) inA, there exists a pointa inX such that ?(f _n )=f _n (a) for alln. In this paper it is proved that all algebras, which are closed under composition with functions inC ^∞ (R) and have a certain local property, have the point evaluation property for sequences. Such algebras are, for instance, the spaceC ^m (E) (m=0,1,...,∞) ofC ^m -functions on any real locally convex spaceE. This result yields in a trivial manner that each homomorphism ? onA is a point evaluation, ifX is Lindelöf or ifA contains a sequence which separates points inX. Further, also a well known result as well as some new ones are obtained as a consequence of the main theorem.     

Cohomology of certain moduli spaces of vector bundles

LetX be a smooth irreducible projective curve of genusg over the field of complex numbers. LetM ₀ be the moduli space of semi-stable vector bundles onX of rank two and trivial determinant. A canonical desingularizationN _o ofM _o has been constructed by Seshadri [17]. In this paper we compute the third and fourth cohomology groups ofN _o. In particular we give a different proof of the theorem due to Nitsure [12], that the third cohomology group ofN _o is torsion-free.     

Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues

Suppose thatM ⁿ is a complete, noncompact, Riemannian manifold. If Δ denotes the Laplace operator ofM, one has associated Schrödinger operators ? Δ +V. Conditions onV are formulated, which ensures the essential self-adjointness of ? Δ +V. In particular, ifV ∈ Q_α,loc (M ⁿ), the local Stummel class, andV ≥ ? c outside of a compact set, then ? Δ +V is essentially self-adjoint on C ₀ ^∞ (M ⁿ). In addition, essential self-adjointness is proved for potentials which are strongly singular at a point. The absence of eigenvalues of ?Δ +V is also studied. This relies upon Rellich-type identities. The results on strongly singular potentials make use of a generalization of the classical uncertainty principle, inR ⁿ, to Riemannian manifolds with a pole.     

Continuity of translation invariant linear functionals onC 0(G) for certain locally compact groupsG

LetG be either a non-amenable group or a compact group such that the trivial representation ofG is not weakly contained in the regular representation ofG onL ₂ ⁰ (G). Then every translation invariant linear functional onC ₀(G) or onL _p (G), where 1<p, is continuous.     

A characterization of the Wishart exponential families by an invariance property

E is the space of real symmetric (d, d) matrices, andS and \(\bar S\) are the subsets ofE of positive definite and semipositive-definite matrices. Let there be ap in $$\Lambda = \left\{ {\frac{1}{2},1,\frac{3}{2}, \ldots \frac{{d - 1}}{2}} \right\} \cup \left] {\frac{{d - 1}}{2}, + \infty } \right[$$ The Wishart natural exponential family with parameterp is a set of probability distributions on \(\bar S\) defined by $$F_p = \{ \exp [ - \tfrac{1}{2}Tr(\Gamma x)](det\Gamma )^p \mu _p (dx);\Gamma \in S\} $$ where μ_p is a suitable measure on \(\bar S\) . LetGL(?^d) be the subset ofE of invertible matrices. Fora inGL(?^d), define the automorphismg _a ofE byg _a(x)=^t axa, where ^t a is the transpose ofa. The aim of this paper is to show that a natural exponential familyF onE is invariant byg _a for alla inGL(?^d) if and only if there existsp in Λ such that eitherF=F _p, orF is the image ofF _p byx??x. (Theorem).     

Uniform harmonic approximation with continuous extension to the boundary

Let Ω be an open set in ℝ ⁿ andE be a relatively closed subset of Ω. Further, letC _e(E) be the collection of real-valued continuous functions onE which extend continuously to the closure ofE in ℝ ⁿ . We characterize those pairs (Ω,E) which have the following property: every function inC _e(E) which is harmonic onE ⁰ can be uniformly approximated onE by functions which are harmonic on Ω and whose restrictions toE belong toC _e(E).     

Interpolating sequences in the ball ofC n

LetB be the unit ball ofC ⁿ , I give necessary conditions on sequenceS of points inB to beH ^∞(B) interpolating in term of aC ⁿ valued holomorphic function zero onS (a substitute for the interpolating Blaschke product). These conditions are sufficient to prove that the sequenceS is interpolating for ∩ _p>1 (B) and is also interpolating forH ^p (B) for 1≤p<∞.     

Uniqueness for the harmonic map flow from surfaces to general targets

LetM be a two-dimensional compact Riemannian manifold with smooth (possibly empty) boundary,N an arbitrary compact manifold. Ifu andv are weak solutions of the harmonic map flow inH ¹(Mx[0,T]; N) whose energy is non-increasing in time and having the same initial datau ₀∈H¹(M, N) (and same boundary values if ?M≠Ø) thenu=v. Combined with a result of M. Struwe, this shows any suchu is smooth in the complement of a finite subset ofM×(0,T)c.     

The complex Frobenius Theorem and uniqueness of solutions to the tangential Cauchy-Riemann equations

Suppose M is a C^∞ real k-dimensional CR-submanifold of $\text{C}$ⁿ, n > 1, and suppose that $\text{?}\text{?}\text{t6}{}_{\mathrm{M}}$ is the tangential Cauchy-Riemann operator on M. Let S be a C¹ real (k ? 1)-dimensional submanifold of M which is noncharacteristic for ??t6_M at p?S. Conditions are found so that a C^∞ solution f of $\text{?}\text{?}\text{t6}{}_{\mathrm{M}}\text{f = 0}$ which vanishes on one side of S in M must vanish in a neighborhood of p in M. If M is a real hypersurface, it is known that such unique continuation always exists. If the codimension of M in $\text{C}$ⁿ is greater than 1, and if the excess dimension of the Levi algebra on M is constant, then it is proved that CR-functions on M which vanish on one side of S must vanish in a full neighborhood of p. The assumption on the dimension of the Levi algebra allows us to use the Complex Frobenius Theorem. Other methods to prove such unique continuation results are also developed.     

Complexity Estimates for the Schmüdgen Positivstellensatz

LetKbe a closed basic set inRⁿgiven by the polynomial inequalities φ₁≥ 0, . . . , φ_m≥ 0 and let Σ be the semiring generated by the φ_kand the squares inR[x₁, . . . ,x_n]. Schmüdgen has shown that ifKis compact then any polynomial function strictly positive onKbelongs to Σ. Easy consequences are (1)f≥ 0 onKif and only iff∈R⁺+ Σ (Positivstellensatz) and (2) iff≥ 0 onKbutf∈ Σ then asdtends to 0⁺, in any representation off + das an element of Σ in terms of the φ_k, the squares and semiring operations, the integerN(d) which is the minimum over all representations of the maximum degree of the summands must become arbitrarily large. A one-dimensional example is analyzed to obtain asymptotic lower and upper bounds of the formcd^−1/2≤N(d) ≤Cd^−1/2log (1/d).     

Hyperbolic imbedding and spaces of continuous extensions of holomorphic maps

LetX be a complex subspace of a complex spaceY. We show that hyperbolic imbeddedness ofX inY is characterized by relative compactness in the compact-open topology of certain spaces of continuous extensions of holomorphic maps from the punctured diskD* toX and fromM -A toX whereM is a complex manifold andA is a divisor onM with normal crossings. We apply these characterizations to obtain generalizations and extensions of theorems of Kobayashi, Kiernan, Kwack, Noguchi and Vitali forD and for higher dimensions. Relative compactness ofX inY is not assumed.     

Maximum modulus sets in pseudoconvex boundaries

LedD be a strictly pseudoconvex domain in ? ⁿ withC ^∞ boundary. We denote byA ^∞(D) the set of holomorphic functions inD that have aC ^∞ extension to \(\bar D\) . A closed subsetE of ?D is locally a maximum modulus set forA ^∞(D) if for everyp∈E there exists a neighborhoodU ofp andf∈A ^∞(D∩U) such that |f|=1 onE∩U and |f|<1 on \(\bar D \cap U\backslash E\) . A submanifoldM of ?D is an interpolation manifold ifT _p (M)?T _p ^c (?D) for everyp∈M, whereT _p ^c (?D) is the maximal complex subspace of the tangent spaceT _p (?D). We prove that a local maximum modulus set forA ^∞ (D) is locally contained in totally realn-dimensional submanifolds of ?D that admit a unique foliation by (n?1)-dimensional interpolation submanifolds. LetD =D ₁ x ... xD _r ? ? ⁿ whereD _i is a strictly pseudoconvex domain withC ^∞ boundary in ? ⁿ _i ,i=1,…,r. A submanifoldM of ?D ₁×…×?D _r verifies the cone condition if \(II_p (T_p (M)) \cap \bar C[Jn_1 (p),...,Jn_r (p)] = \{ 0\} \) for everyp∈M, wheren _i (p) is the outer normal toD _i atp, J is the complex structure of ? ⁿ , \(\bar C[Jn_1 (p),...,Jn_r (p)]\) is the closed positive cone of the real spaceV _p generated byJ _{n 1}(p),…,J _{n r}(p), and II _p is the orthogonal projection ofT _p (?D) onV _p . We prove that a closed subsetE of ?D ₁×…×?D _r which is locally a maximum modulus set forA ^∞(D) is locally contained inn-dimensional totally real submanifolds of ?D ₁×…×?D _r that admit a foliation by (n?1)-dimensional submanifolds such that each leaf verifies the cone condition at every point ofE. A characterization of the local peak subsets of ?D ₁×…×?D _r is also given.