Search

This paper dwells upon two aspects of affine supergroup theory, investigating the links among them. First, the "splitting" properties of affine supergroups are discussed, i.e., special kinds of factorizations they may admit -- either globally, or pointwise. Almost everything should be more or less known, but seems to be not as clear in the literature (to the author's knowledge) as it ought to. Second, a new contribution to the study of affine supergroups by means of super Harish-Chandra pairs is presented (a method already introduced by Koszul, and later extended by other authors). Namely, a new functorial construction ψ is provided which, with each super Harish-Chandra pair, associates an affine supergroup that is always globally strongly split (in short, gs-split) -- thus setting a link with the first part of the paper. One knows that there exists a natural functor Φ from affine supergroups to super Harish-Chandra pairs. Then we show that the new functor ψ -- which goes the other way round -- is indeed a quasi-inverse to Φ, provided we restrict our attention to the subcategory of affine supergroups that are gs-split. Therefore, (the restrictions of) Φ and ψ are equivalences between the categories of gs-split affine supergroups and of super Harish-Chandra pairs. Such a result was known in other contexts, such as the smooth differential or the complex analytic one, via different approaches. Nevertheless, the novelty in the present paper lies in the construction of a different functor ψ and thus extends the result to a much larger setup, with a totally different, more geometrical method. In fact, this method (very concrete, indeed) is universal and characteristic-free and is presented here for the algebro-geometric setting, but actually it can be easily adapted to the frameworks of differential or complex analytic supergeometry. The case of linear supergroups is treated also as an intermediate, inspiring step. Some examples, applications and further generalizations are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper we investigate nilpotenct and probabilistically nilpotent Hopf algebras. We define nilpotency via a descending chain of commutators and give a criterion for nilpotency via a family of central invertible elements. These elements can be obtained from a commutator matrix A which depends only on the Grothendieck ring of H. When H is almost cocommutative we introduce a probabilistic method. We prove that every semisimple quasitriangular Hopf algebra is probabilistically nilpotent. In a sense we thereby answer the title of our paper Are we counting or measuring anything? by Yes, we are. [ABSTRACT FROM AUTHOR]

Let A be a commutative Banach algebra with a BAI (=bounded approximate identity). We equip A** with the (first) Arens multiplication. To each idempotent element u of A** we associate the closed ideal Iu = {a ∊ A : au = 0} in A. In this paper we present a characterization of the closed ideals of A with BAI's in terms of idempotent elements of A**. The main results are: a) A closed ideal I of A has a BAI iff there is an idempotent u ∊ A** such that I = Iu and the subalgebra Au is norm closed in A**. b) For any closed ideal I of A with a BAI, the quotient algebra A/I is isomorphic to a subalgebra of A**. We also show that a weak* closed invariant subspace X of the group von Neumann algebra VN(G) of an amenable group G is naturally complemented in VN(G) iff the spectrum of X belongs to the closed coset ring Rc(Gd) of Gd, the discrete version of G. This paper contains several applications of these results. [ABSTRACT FROM AUTHOR]

We investigate the Dirichlet problem for the parablic equation [ u_t = \Delta u^m, m > 0, \] in a non-smooth domain $\Omega \subset \mathbb{R}^{N+1}, N \geq 2$. In a recent paper [{\em U.G. Abdulla, J. Math. Anal. Appl., 260, 2 (2001), 384-403}] existence and boundary regularity results were established. In this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial-boundary data. In particular, we prove $L_1$-contraction estimation in general non-smooth domains. [ABSTRACT FROM AUTHOR]

The Weyl algebra over a field k of characteristic 0 is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all Z-graded simple rings of GK-dimension 2 and show that they are graded Morita equivalent to generalized Weyl algebras as defined by Bavula. More generally, we study Z-graded simple rings A of any dimension which have a graded quotient ring of the form K[t, t-1; σ] for a field K. Under some further hypotheses, we classify all such A in terms of a new construction of simple rings which we introduce in this paper. In the important special case that GKdimA = tr. deg(K/k) + 1, we show that K and σ must be of a very special form. The new simple rings we define should warrant further study from the perspective of noncommutative geometry. [ABSTRACT FROM AUTHOR]

In this paper we study the following question related to Diophantine approximations and geometric measure theory: for a given set Ω find α such that α - θ has bad Diophantine properties simultaneously for all θ ∊ Ω. How do the arising Diophantine inequalities depend on the geometry of the set Ω? We provide several methods which yield different answers in terms of the metric entropy of Ω and consider various examples. Furthermore, we apply these results to explore the asymptotic behavior of the directional discrepancy, i.e., the discrepancy with respect to rectangles rotated in certain sets of directions. It is well known that the extremal cases of this problem (fixed direction vs. all possible rotations) yield completely different bounds. We use rotated lattices to obtain directional discrepancy estimates for general rotation sets and investigate the sharpness of these methods. [ABSTRACT FROM AUTHOR]

In this paper, we show that the chordal Loewner differential equation with C° driving function generates a C°1/2 slit for 1/2 < ° ≼ 2, except when ° = 3/2 the slit is only proved to be weakly C1,1. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit combinatorial realization of the crystal B(λ) for an irreducible highest weight Uq(q(n))-module V (λ) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms for decomposing the tensor product of q(n)-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients. [ABSTRACT FROM AUTHOR]

The main result of this paper gives criteria for extendibility of mappings defined on symmetric subsets of $ *$ [ABSTRACT FROM AUTHOR]

For an $n$-component link, Milnor's isotopy invariants are defined for each multi-index $I=i_1i_2...i_m~(i_jin {1,...,n})$. Here $m$ is called the length. Let $r(I)$ denote the maximum number of times that any index appears in $I$. It is known that Milnor invariants with $r=1$, i.e., Milnor invariants for all multi-indices $I$ with $r(I)=1$, are link-homotopy invariant. N. Habegger and X. S. Lin showed that two string links are link-homotopic if and only if their Milnor invariants with $r=1$ coincide. This gives us that a link in $S^3$ is link-homotopic to a trivial link if and only if all Milnor invariants of the link with $r=1$ vanish. Although Milnor invariants with $r=2$ are not link-homotopy invariants, T. Fleming and the author showed that Milnor invariants with $rleq 2$ are self $Delta $-equivalence invariants. In this paper, we give a self $Delta $-equivalence classification of the set of $n$-component links in $S^3$ whose Milnor invariants with length $leq 2n-1$ and $rleq 2$ vanish. As a corollary, we have that a link is self $Delta $-equivalent to a trivial link if and only if all Milnor invariants of the link with $rleq 2$ vanish. This is a geometric characterization for links whose Milnor invariants with $rleq 2$ vanish. The chief ingredient in our proof is Habiro's clasper theory. We also give an alternate proof of a link-homotopy classification of string links by using clasper theory. [ABSTRACT FROM AUTHOR]

Let $G$ be a finite group, $S in mathit {Syl}_p(G)$, and $mathcal S$ be the set subgroups containing $S$. For $M in mathcal S$ and $V = Omega _1Z(O_p(M))$, the paper discusses the action of $M$ on $V$. Apart from other results, it is shown that for groups of parabolic characteristic $p$ either $S$ is contained in a unique maximal $p$-local subgroup, or there exists a maximal $p$-local subgroup in $M in mathcal S$ such that $V$ is a nearly quadratic 2F-module for $M$. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Murre's conjectures on the structure of rational Chow groups and exhibit explicit Chow--Künneth projectors for some examples. More precisely, the examples we study are the varieties which have a nef tangent bundle. For surfaces and threefolds which have a nef tangent bundle, explicit Chow--Künneth projectors are obtained which satisfy Murre's conjectures, and the motivic Hard Lefschetz theorem is verified. [ABSTRACT FROM AUTHOR]

In this paper we let $G$ be a simple algebraic group and $r$ be a natural number, and consider the codimension in $G$ of the variety of elements $g\in G$ satisfying $g^r=1$. We shall obtain a lower bound for this codimension which is independent of characteristic, and show that it is attained if $G$ is of adjoint type. [ABSTRACT FROM AUTHOR]

We consider subgroups of $C^{\infty}$-diffeomorphisms of the circle $\mathbb S^1$ which act transitively on $4$-tuples of points. We show, in particular, that these subgroups are dense in the group of homeomorphisms of $\mathbb S^1$. A stronger result concerning $C^{\infty}$-approximations is obtained as well. The techniques employed in this paper rely on Lie algebra ideas and they also provide partial generalizations to the differentiable case of some results previously established in the analytic category. [ABSTRACT FROM AUTHOR]

This paper investigates varieties of tori and Cartan subalgebras of a finite-dimensional restricted Lie algebra $(\mathfrak{g},[p])$, defined over an algebraically closed field $k$ of positive characteristic $p$. We begin by showing that schemes of tori may be used as a tool to retrieve results by A. Premet on regular Cartan subalgebras. Moreover, they give rise to principal fibre bundles, whose structure groups coincide with the Weyl groups in case $\mathfrak{g}= \operatorname{Lie}(\mathcal{G})$ is the Lie algebra of a smooth group $\mathcal{G}$. For solvable Lie algebras, varieties of tori are full affine spaces, while simple Lie algebras of classical or Cartan type cannot have varieties of this type. In the final sections the quasi-projective variety of Cartan subalgebras of minimal dimension ${\rm rk}(\mathfrak{g})$ is shown to be irreducible of dimension $\dim_k\mathfrak{g}-{\rm rk}(\mathfrak{g})$, with Premet's regular Cartan subalgebras belonging to the regular locus. [ABSTRACT FROM AUTHOR]

In this paper we construct an example of a complete minimal disk which is properly immersed in a ball of $\mathbb{R}^3$. [ABSTRACT FROM AUTHOR]

A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope $G(A)$ of a finite dimensional superalgebra $A$. In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field. The importance of such algebras is readily proved: $A$ is a minimal superalgebra if and only if the ideal of identities of $G(A)$ is a product of verbally prime T-ideals. Also, such superalgebras allow us to classify all minimal varieties of a given exponent i.e., varieties $\mathcal{V}$ such that $\exp({\mathcal{V}})=d\ge 2$ and $\exp(\mathcal{U})

The main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds. Precisely, given a smooth projective threefold $X$, a rank-two vector bundle $\mathcal{E}$ on $X$, and integers $k\geq 0$, $\delta >0 $, denote by ${\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ the subscheme of ${\mathbb{P}}(H^0({\mathcal{E}}(k)))$ parametrizing global sections of ${\mathcal{E}}(k)$ whose zero-loci are irreducible $\delta$-nodal curves on $X$. We present a new cohomological description of the tangent space $T_{[s]}({\mathcal{V}}_{\delta} ({\mathcal{E}} (k)))$ at a point $[s]\in {\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$. This description enables us to determine effective and uniform upper bounds for $\delta$, which are linear polynomials in $k$, such that the family ${\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ is smooth and of the expected dimension ({\em regular}, for short). The almost sharpness of our bounds is shown by some interesting examples. Furthermore, when $X$ is assumed to be a Fano or a Calabi-Yau threefold, we study in detail the regularity property of a point $[s] \in {\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ related to the postulation of the nodes of its zero-locus $C = V(s) \subset X$. Roughly speaking, when the nodes of $C$ are assumed to be in general position either on $X$, or on an irreducible divisor of $X$ having at worst log-terminal singularities or to lie on a l.c.i. and subcanonical curve in $X$, we find upper bounds on $\delta$ which are, respectively, cubic, quadratic and linear polynomials in $k$ ensuring the regularity of ${\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ at $[s]$. Finally, when $X= \mathbb{P}^3$, we also discuss some interesting geometric properties of the curves given by sections parametrized by ${\mathcal{V}}_{\delta} ({\mathcal{E}} \otimes \mathcal{O}_X(k))$. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit formula for the first two terms of the Taylor expansion of a classical Eisenstein series of weight $2k+1$ for $\Gamma_{0}(q)$. Both the first term and the second term have interesting arithmetic interpretations. We apply the result to compute the central derivative of some Hecke $L$-functions. [ABSTRACT FROM AUTHOR]

In one of the main results of this paper, we find the degrees of the generators of the canonical ring of a regular algebraic surface $X$ of general type defined over a field of characteristic $0$, under the hypothesis that the canonical divisor of $X$ determines a morphism $\varphi $ from $X$ to a surface of minimal degree $Y$. As a corollary of our results and results of Ciliberto and Green, we obtain a necessary and sufficient condition for the canonical ring of $X$ to be generated in degree less than or equal to $2$. We construct new examples of surfaces satisfying the hypothesis of our theorem and prove results which show that many a priori plausible examples cannot exist. Our methods are to exploit the $\mathcal{O}_{Y}$-algebra structure on $\varphi_{*}\mathcal{O}_{X}$. These methods have other applications, including those on Calabi-Yau threefolds. We prove new results on homogeneous rings associated to a polarized Calabi-Yau threefold and also prove some existence theorems for Calabi-Yau covers of threefolds of minimal degree. These have consequences towards constructing new examples of Calabi-Yau threefolds. [ABSTRACT FROM AUTHOR]

In this article, we consider a measure in Wiener space, induced by the sum of measures associated with an uncountable set of positive real numbers, and investigate the basic properties of this measure. We apply this measure to the various theories related to Wiener space. In particular, we can obtain a partial answer to Johnson and Skoug's open problems, raised in their 1979 paper. Moreover, we can improve and clarify some theories related to Wiener space. [ABSTRACT FROM AUTHOR]

This paper concerns the closed points, closed subspaces, open subspaces, weakly closed and weakly open subspaces, and effective divisors, on a non-commutative space. [ABSTRACT FROM AUTHOR]

Let $T(x)=\sum_{j=0}^{n-1}\pm e^{ijx}$ where $\pm$ stands for a random choice of sign with equal probability. The first author recently showed that for any $\epsilon>0$ and most choices of sign, $\min_{x\in[0,2\pi)}|T(x)|0$ and large $n$ most choices of sign satisfy $\min_{x\in[0,2\pi)}|T(x)|> \eps n^{-1/2}$. Furthermore, we study the case of more general random coefficients and applications of our methods to complex zeros of random polynomials. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and nonexistence of global positive solutions to $u_t=\Delta\ln(1+u)$ in $\Omega \times (0, +\infty)$, \[\frac{\partial\ln(1+u)}{\partial n}=\sqrt{1+u}(\ln(1+u))^{\alpha} \quad\text{on} \partial \Omega \times (0, +\infty),] and $u(x, 0)=u_0(x)$ in $\Omega$. Here $\alpha\geq 0$ is a parameter, $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain. After pointing out the mistakes in {\em Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary}, SIAM J. Math. Anal. {\bf 24} (1993), 317--326, by N. Wolanski, which claims that, for $\Omega=B_R$ the ball of $\mathbb{R}^N$, the positive solution exists globally if and only if $\alpha\leq 1$, we reconsider the same problem in general bounded domain $\Omega$ and obtain that every positive solution exists globally if and only if $\alpha\leq {1/2}$. [ABSTRACT FROM AUTHOR]

Let $\Omega$ be a bounded smooth domain in $R^{n}$, and $u(x)$ a $C^{1}$ function with compact support in $\Omega$. Moser's inequality states that there is a constant $c_{o}$, depending only on the dimension $n$, such that \begin{equation*} \frac{1}{|\Omega|} \int_{\Omega} e^{n \omega_{n-1}^{\frac{1}{n-1}} u^{\frac{n}{n-1}}} dx \leq c_{o} , \end{equation*} where $|\Omega|$ is the Lebesgue measure of $\Omega$, and $\omega_{n-1}$ the surface area of the unit ball in $R^{n}$. We prove in this paper that there are extremal functions for this inequality. In other words, we show that the \begin{equation*} \sup \{\frac{1}{|\Omega|} \int_{\Omega} e^{n \omega_{n-1}^{\frac{1}{n-1}} u^{\frac{n}{n-1}}} dx: u \in W_{o}^{1,n}, \|\nabla u\|_{n} \leq 1 } \end{equation*} is attained. Earlier results include Carleson-Chang (1986, $\Omega$ is a ball in any dimension) and Flucher (1992, $\Omega$ is any domain in 2-dimensions). [ABSTRACT FROM AUTHOR]

In this paper we generalize the $ \mathcal{E}$ and $ \mathcal{N}$-type resolutions used by Martin-Deschamps and Perrin for curves in $ \mathbb{P}^{3}$ to subschemes of pure codimension in projective space, and shows that these resolutions are interchanged by the mapping cone procedure under a simple linkage. Via these resolutions, Rao's correspondence is extended to give a bijection between even linkage classes of subschemes of pure codimension two and stable equivalence classes of reflexive sheaves $ \mathcal{E}$ satisfying $H^{1}_{*}( \mathcal{E})=0$ and $ \mathop{\mathcal{E}xt}^{1}( \mathcal{E}^{\vee }, \mathcal{O})=0$. Further, these resolutions are used to extend the work of Martin-Deschamps and Perrin for Cohen-Macaulay curves in $ \mathbb{P}^{3}$ to subschemes of pure codimension two in $ \mathbb{P}^{n}$. In particular, even linkage classes of such subschemes satisfy the Lazarsfeld-Rao property and any minimal subscheme for an even linkage class links directly to a minimal subscheme for the dual class. [ABSTRACT FROM AUTHOR]

Let $E/F$ be a quadratic extension of number fields. Suppose that every real place of $F$ splits in $E$ and let $H$ be the unitary group in 3 variables. Suppose that $\Pi$ is an automorphic cuspidal representation of $GL(3,E_{\mathbb{A}})$. We prove that there is a form $\phi$ in the space of $\Pi$ such that the integral of $\phi$ over $H(F)\setminus H(F_{\mathbb{A}})$ is non zero. Our proof is based on earlier results and the notion, discussed in this paper, of Shalika germs for certain Kloosterman integrals. [ABSTRACT FROM AUTHOR]