Milnor classes of local complete intersections

Let be a compact local complete intersection defined as the zero set of a section of a holomorphic vector bundle over the ambient space. For each connected component of the singular set of , we define the Milnor class in the homology of . The difference between the Schwartz-MacPherson class and the Fulton-Johnson class of is shown to be equal to the sum of over the connected components of . This is done by proving Poincaré-Hopf type theorems for these classes with respect to suitable tangent frames. The -degree component coincides with the Milnor numbers already defined by various authors in particular situations. We also give an explicit formula for when is a non-singular component and satisfies the Whitney condition along .

     

Compressions of resolvents and maximal radius of regularity

Suppose that is left invertible in for all , where is an open subset of the complex plane. Then an operator-valued function is a left resolvent of in if and only if has an extension , the resolvent of which is a dilation of of a particular form. Generalized resolvents exist on every open set , with included in the regular domain of . This implies a formula for the maximal radius of regularity of in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by
J. Zemánek is obtained.

     

Singular solutions of parabolic -Laplacian with absorption

We consider, for and , the -Laplacian evolution equation with absorption

We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in , and satisfy for all . We prove the following:

(i)

When , there does not exist any such singular solution.

(ii)

When , there exists, for every , a unique singular solution that satisfies as .

Also, as , where is a singular solution that satisfies as .

Furthermore, any singular solution is either or for some finite positive .

     

Existence of curves with prescribed topological singularities

Throughout this paper we study the existence of irreducible curves on smooth projective surfaces with singular points of prescribed topological types . There are necessary conditions for the existence of the type for some fixed divisor on and suitable coefficients , and , and the main sufficient condition that we find is of the same type, saying it is asymptotically proper. Ten years ago general results of this quality were not known even for the case . An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up of the form , deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in , and K3-surfaces.

     

Good measures on Cantor space

While there is, up to homeomorphism, only one Cantor space, i.e. one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor space which are not topologically equivalent. The clopen values set for a full, nonatomic measure is the countable dense subset is clopen of the unit interval. It is a topological invariant for the measure. For the class of good measures it is a complete invariant. A full, nonatomic measure is good if whenever are clopen sets with , there exists a clopen subset of such that . These measures have interesting dynamical properties. They are exactly the measures which arise from uniquely ergodic minimal systems on Cantor space. For some of them there is a unique generic measure-preserving homeomorphism. That is, within the Polish group of such homeomorphisms there is a dense, conjugacy class.

     

Asymptotic distribution of the largest off-diagonal entry of correlation matrices

Suppose that we have observations from a -dimensional population. We are interested in testing that the variates of the population are independent under the situation where goes to infinity as . A test statistic is chosen to be , where is the sample correlation coefficient between the -th coordinate and the -th coordinate of the population. Under an independent hypothesis, we prove that the asymptotic distribution of is an extreme distribution of type , by using the Chen-Stein Poisson approximation method and the moderate deviations for sample correlation coefficients. As a statistically more relevant result, a limit distribution for , where is Spearman's rank correlation coefficient between the -th coordinate and the -th coordinate of the population, is derived.

     

Vertices for characters of -solvable groups

Suppose that is a finite -solvable group. We associate to every irreducible complex character of a canonical pair , where is a -subgroup of and , uniquely determined by up to -conjugacy. This pair behaves as a Green vertex and partitions into ``families" of characters. Using the pair , we give a canonical choice of a certain -radical subgroup of and a character associated to which was predicted by some conjecture of G. R. Robinson.

     

On the unique representation of families of sets

Let and be uncountable Polish spaces. represents a family of sets provided each set in occurs as an -section of . We say that uniquely represents provided each set in occurs exactly once as an -section of . is universal for if every -section of is in . is uniquely universal for if it is universal and uniquely represents . We show that there is a Borel set in which uniquely represents the translates of if and only if there is a Vitali set. Assuming there is a Borel set with all sections sets and all non-empty sets are uniquely represented by . Assuming there is a Borel set with all sections which uniquely represents the countable subsets of . There is an analytic set in with all sections which represents all the subsets of , but no Borel set can uniquely represent the sets. This last theorem is generalized to higher Borel classes.

     

Solutions of fully nonlinear elliptic equations with patches of zero gradient: Existence, regularity and convexity of level curves

In this paper, we first construct ``viscosity' solutions (in the Crandall-Lions sense) of fully nonlinear elliptic equations of the form

In fact, viscosity solutions are surprisingly weak. Since candidates for solutions are just continuous, we only require that the ``test' polynomials (those tangent from above or below to the graph of at a point ) satisfy the correct inequality only if . That is, we simply disregard those test polynomials for which .

Nevertheless, this is enough, by an appropriate use of the Alexandroff-Bakelman technique, to prove existence, regularity and, in two dimensions, for , (0$">) and constant boundary conditions on a convex domain, to prove that there is only one convex patch.

     

Block representation type of reduced enveloping algebras

Let be an algebraically closed field of characteristic , a connected, reductive -group, , and the reduced enveloping algebra of associated with . Assume that is simply-connected, is good for and has a non-degenerate -invariant bilinear form. All blocks of having finite and tame representation type are determined.

     

Tight closure, plus closure and Frobenius closure in cubical cones

We consider tight closure, plus closure and Frobenius closure in the rings , where is a field of characteristic and . We use a -grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring . We show that Frobenius closure is the same as tight closure in certain classes of ideals when . Since , we conclude that for these ideals. Using injective modules over the ring , the union of all th roots of elements of , we reduce the question of whether for -graded ideals to the case of -graded irreducible modules. We classify the irreducible -primary -graded ideals. We then show that for most irreducible -primary -graded ideals in , where is a field of characteristic and . Hence for these ideals.

     

An application of the Littlewood restriction formula to the Kostant-Rallis Theorem

Consider a symmetric pair of linear algebraic groups with , where and are defined as the +1 and -1 eigenspaces of the involution defining . We view the ring of polynomial functions on as a representation of . Moreover, set , where is the space of homogeneous polynomial functions on of degree . This decomposition provides a graded -module structure on . A decomposition of is provided for some classical families when is within a certain stable range.

The stable range is defined so that the spaces are within the hypothesis of the classical Littlewood restriction formula. The Littlewood restriction formula provides a branching rule from the general linear group to the standard embedding of the symplectic or orthogonal subgroup. Inside the stable range the decomposition of is interpreted as a -analog of the Kostant-Rallis theorem.

     

Surfaces with

We classify minimal complex surfaces of general type with . More precisely, we show that such a surface is either the symmetric product of a curve of genus or a free quotient of the product of a curve of genus and a curve of genus . Our main tools are the generic vanishing theorems of Green and Lazarsfeld and the characterization of theta divisors given by Hacon in Corollary 3.4 of Fourier transforms, generic vanishing theorems and polarizations of abelian varieties.

     

Finely -harmonic functions of a Markov process

Let be a Borel right process and a fixed excessive measure. Given a finely open nearly Borel set we define an operator which we regard as an extension of the restriction to of the generator of . It maps functions on to (locally) signed measures on not charging -semipolars. Given a locally smooth signed measure we define to be (finely) -harmonic on provided on and denote the class of such by . Under mild conditions on we show that is equivalent to a local ``Poisson' representation of . We characterize by an analog of the mean value property under secondary assumptions. We obtain global Poisson type representations and study the Dirichlet problem for elements of under suitable finiteness hypotheses. The results take their nicest form when specialized to Hunt processes.

     

Analyticity on translates of a Jordan curve

Let be a domain in which is symmetric with respect to the real axis and whose boundary is a real analytic simple closed curve. Translate vertically to get where is such that . We prove that if is a continuous function on such that for each , the function has a continuous extension to which is holomorphic on , then is holomorphic on .

     

A Pieri-type formula for isotropic flag manifolds

We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from the Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type (respectively, type ) Schubert polynomial by the Schur -polynomial (respectively, the Schur -polynomial ). Geometric constructions and intermediate results allow us to ultimately deduce this formula from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the infinite Coxeter group , identities of the structure constants for the Schubert basis of cohomology, and intersections of Schubert varieties. We show that most of these identities follow from the Pieri-type formula, and our analysis leads to a new partial order on the Coxeter group and formulas for many of these structure constants.

     

Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let be the blow-up of a projective scheme along the ideal sheaf of . It is known that there are embeddings for , where denotes the maximal generating degree of , and that there exists a Cohen-Macaulay ring of the form (which gives an arithmetic Macaulayfication of ) if and only if , for , and is equidimensional and Cohen-Macaulay. We show that under these conditions, there are well-determined invariants and such that is Cohen-Macaulay for all d(I)e + \varepsilon$"> and e_0$">, and that these bounds are the best possible. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form . If has negative -invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if , for 0$">, and is equidimensional and Cohen-Macaulay. Moreover, these conditions imply the Cohen-Macaulayness of for all d(I)e + \varepsilon$"> and e_0$">.

     

Summing inclusion maps between symmetric sequence spaces

In 1973/74 Bennett and (independently) Carl proved that for the identity map id: is absolutely -summing, i.e., for every unconditionally summable sequence in the scalar sequence is contained in , which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a -concave symmetric Banach sequence space the identity map is absolutely -summing, i.e., for every unconditionally summable sequence in the scalar sequence is contained in . Various applications are given, e.g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator on with values in a -concave symmetric Banach sequence space is a multiplier from into . Furthermore, we prove an asymptotic formula for the -th approximation number of the identity map , where denotes the linear span of the first standard unit vectors in , and apply it to Lorentz and Orlicz sequence spaces.

     

An analogue of minimal surface theory in

We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semi-simple Lie groups (e.g. ), which contains minimal surfaces in and constant mean curvature surfaces in . A Weierstrass type representation formula and a Chern-Osserman type inequality for such surfaces are given.

     

Coloring

If and , then define the graph to be the graph whose vertex set is with two vertices being adjacent iff there are distinct such that . For various and and various , typically or , the graph can be properly colored with colors. It is shown that in some cases such a coloring can also have the additional property that if is an isometric embedding, then the restriction of to is a bijection onto .