Dimension formula for graded Lie algebras and its applications
Seok-Jin
Kang;
Myung-Hwan
Kim
4281-4336
Abstract: In this paper, we investigate the structure of infinite dimensional Lie algebras $L=\bigoplus _{\alpha \in \Gamma } L_{\alpha }$ graded by a countable abelian semigroup $\Gamma$ satisfying a certain finiteness condition. The Euler-Poincaré principle yields the denominator identities for the $\Gamma$-graded Lie algebras, from which we derive a dimension formula for the homogeneous subspaces $L_{\alpha }$ $(\alpha \in \Gamma )$. Our dimension formula enables us to study the structure of the $\Gamma$-graded Lie algebras in a unified way. We will discuss some interesting applications of our dimension formula to the various classes of graded Lie algebras such as free Lie algebras, Kac-Moody algebras, and generalized Kac-Moody algebras. We will also discuss the relation of graded Lie algebras and the product identities for formal power series.
Limit theorems for the convex hull of random points in higher dimensions
Irene
Hueter
4337-4363
Abstract: We give a central limit theorem for the number $N_n$ of vertices of the convex hull of $n$ independent and identically distributed random vectors, being sampled from a certain class of spherically symmetric distributions in $\mathbb{R}^d \; (d> 1),$ that includes the normal family. Furthermore, we prove that, among these distributions, the variance of $N_n$ exhibits the same order of magnitude as the expectation as $n \rightarrow \infty.$ The main tools are Poisson approximation of the point process of vertices of the convex hull and (sub/super)-martingales.
The $\overline{\partial}$ problem on domains with piecewise smooth boundaries with applications
Joachim
Michel;
Mei-Chi
Shaw
4365-4380
Abstract: Let $\Omega$ be a bounded domain in $\mathbb C^n$ such that $\Omega$ has piecewise smooth boudnary. We discuss the solvability of the Cauchy-Riemann equation \begin{equation*}\overline{\partial}u=\alpha\quad \text{in}\quad \Omega\tag{0.1} \end{equation*} where $\alpha$ is a smooth $\overline{\partial}$-closed $(p,q)$ form with coefficients $C^\infty$ up to the bundary of $\Omega$, $0\le p\le n$ and $1\le q\le n$. In particular, Equation (0.1) is solvable with $u$ smooth up to the boundary (for appropriate degree $q)$ if $\Omega$ satisfies one of the following conditions: i) $\Omega$ is the transversal intersection of bounded smooth pseudoconvex domains. ii) $\Omega=\Omega _1\setminus\overline\Omega _2$ where $\Omega _2$ is the union of bounded smooth pseudoconvex domains and $\Omega _1$ is a pseudoconvex convex domain with a piecewise smooth boundary. iii) $\Omega=\Omega _1\setminus\overline{\Omega}_2$ where $\Omega _2$ is the intersection of bounded smooth pseudoconvex domains and $\Omega _1$ is a pseudoconvex domain with a piecewise smooth boundary. The solvability of Equation (0.1) with solutions smooth up to the boundary can be used to obtain the local solvability for $\overline{\partial}_b$ on domains with piecewise smooth boundaries in a pseudoconvex manifold.
Arithmetically Buchsbaum divisors on varieties of minimal degree
Uwe
Nagel
4381-4409
Abstract: In this paper we consider integral arithmetically Buchsbaum subschemes of projective space. First we show that arithmetical Buchsbaum varieties of sufficiently large degree have maximal Castelnuovo-Mumford regularity if and only if they are divisors on a variety of minimal degree. Second we determine all varieties of minimal degree and their divisor classes which contain an integral arithmetically Buchsbaum subscheme. Third we investigate these varieties. In particular, we compute their Hilbert function, cohomology modules and (often) their graded Betti numbers and obtain an existence result for smooth arithmetically Buchsbaum varieties.
Some interesting nonspherical tempered representations of graded Hecke algebras
C.
Kriloff
4411-4428
Abstract: Lusztig's presentation of the graded Hecke algebra in terms of generators and relations allows for the definition of algebras associated to noncrystallographic root systems. The representation theory of general graded Hecke algebras is investigated, the expected number of tempered representations for $\mathbb{H}(H_3)$ are accounted for, and it is shown that one of these representations has the unexpected property of being nonspherical despite being the only tempered representation appearing at its infinitesimal character. Additional nonspherical tempered representations of $\mathbb{H}(H_4)$ are also included.
A Littlewood-Richardson rule for factorial Schur functions
Alexander
I.
Molev;
Bruce
E.
Sagan
4429-4443
Abstract: We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.
Class numbers of cyclotomic function fields
Li
Guo;
Linghsueh
Shu
4445-4467
Abstract: Let $q$ be a prime power and let ${\mathbb F}_q$ be the finite field with $q$ elements. For each polynomial $Q(T)$ in ${\mathbb F}_q [T]$, one could use the Carlitz module to construct an abelian extension of ${\mathbb F}_q (T)$, called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of ${\mathbb F}_q(T)$, similar to the role played by cyclotomic number fields for abelian extensions of ${\mathbb Q}$. We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in ${\mathbb F}_q [T]$. Two types of properties are obtained for the $l$-parts of the class numbers of the fields in this tower, for a fixed prime number $l$. One gives congruence relations between the $l$-parts of these class numbers. The other gives lower bound for the $l$-parts of these class numbers.
On 2-Generator Subgroups of SO(3)
Charles
Radin;
Lorenzo
Sadun
4469-4480
Abstract: We classify all subgroups of $SO(3)$ that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of $\pi$. In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple consequences of standard facts about rotations by $\pi$ and $\pi/2$. Embedded in the subgroups are explicit free groups on 2 generators, as used in the Banach-Tarski paradox.
Hecke algebras and cohomotopical Mackey functors
Norihiko
Minami
4481-4513
Abstract: In this paper, we define the concept of the cohomotopical Mackey functor, which is more general than the usual cohomological Mackey functor, and show that Hecke algebra techniques are applicable to cohomotopical Mackey functors. Our theory is valid for any (possibly infinite) discrete group. Some applications to topology are also given.
Densities of idempotent measures and large deviations
Marianne
Akian
4515-4543
Abstract: Considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by Maslov. Then, idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure. In this paper, we give conditions under which an idempotent measure has a density and show by many examples that they are often satisfied. These conditions depend on the lattice structure of the semiring and on the Boolean algebra in which the measure is defined. As an application, we obtain a necessary and sufficient condition for a family of probabilities to satisfy the large deviation principle.
Quadratic differential equations in $\mathbb{Z}_2$-graded algebras
Nora
C.
Hopkins;
Michael
K.
Kinyon
4545-4559
Abstract: Quadratic differential equations whose associated algebra has an automorphism of order two are studied. Under hypotheses that naturally generalize the cases where the even or odd part of the algebra is one dimensional, the following are examined: structure theory of the associated algebra (ideal structure, simplicity, solvability, and nilpotence), derivations and first integrals, trajectories given by derivations, and Floquet decompositions.
The support of the equilibrium measure in the presence of a monomial external field on $[-1,1]$
S.
B.
Damelin;
A.
B. J.
Kuijlaars
4561-4584
Abstract: The support of the equilibrium measure associated with an external field of the form $Q(x) = - cx^{2m+1}$, $x \in [-1,1]$, with $c > 0$ and $m$ a positive integer is investigated. It is shown that the support consists of at most two intervals. This resolves a question of Deift, Kriecherbauer and McLaughlin.
A sharp version of Zhang's theorem on truncating sequences of gradients
Stefan
Müller
4585-4597
Abstract: Let $K \subset \mathbf{R}^{mn}$ be a compact and convex set of $m \times n$ matrices and let $\{u_j\}$ be a sequence in $W_{\operatorname{loc}} ^{1,1}(\mathbf{R}^n;\mathbf{R}^m)$ that converges to $K$ in the mean, i.e. $\int _{\mathbf{R}^n} {\operatorname{dist}} (Du_j, K) \to 0$. I show that there exists a sequence $v_j$ of Lipschitz functions such that $\parallel {\operatorname{dist}} (Dv_j, K)\parallel _\infty \to 0$ and $\mathcal{L}^n (\{u_j \not= v_j\}) \to 0$. This refines a result of Kewei Zhang (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 313-326), who showed that one may assume $\parallelDv_j \parallel _\infty \le C$. Applications to gradient Young measures and to a question of Kinderlehrer and Pedregal (Arch. Rational Mech. Anal. 115 (1991), 329-365) regarding the approximation of $\mathbf{R} \cup \{+\infty\}$ valued quasiconvex functions by finite ones are indicated. A challenging open problem is whether convexity of $K$ can be replaced by quasiconvexity.
A classification of Baire-1 functions
P.
Kiriakouli
4599-4609
Abstract: In this paper we give some topological characterizations of bounded Baire-1 functions using some ranks. Kechris and Louveau classified the Baire-1 functions to the subclasses $\mathbb{B}^\xi _1(K)$ for every $\xi<\omega _1$ (where $K$ is a compact metric space). The first basic result of this paper is that for $\xi<\omega$, $f\in \mathbb{B}^{\xi+1}_1(K)$ iff there exists a sequence $(f_n)$ of differences of bounded semicontinuous functions on $K$ with $f_n\to f$ pointwise and $\gamma((f_n))\le \omega^\xi$ (where ``$\gamma$'' denotes the convergence rank). This extends the work of Kechris and Louveau who obtained this result for $\xi=1$. We also show that the result fails for $\xi\ge \omega$. The second basic result of the paper involves the introduction of a new ordinal-rank on sequences $(f_n)$, called the $\delta$-rank, which is smaller than the convergence rank $\gamma$. This result yields the following characterization of $\mathbb{B}^\xi _1(K): f\in \mathbb{B}^\xi _1(K)$ iff there exists a sequence $(f_n)$ of continuous functions with $f_n\to f$ pointwise and $\delta((f_n))\le \omega^{\xi-1}$ if $1\le \xi<\omega$, resp. $\delta((f_n))\le \omega^\xi$ if $\xi\ge \omega$.
An $L^p$ a priori estimate for the Tricomi equation in the upper half space
Jong
Uhn
Kim
4611-4628
Abstract: We establish an $L^{p}$ a priori estimate for the Tricomi equation. Our main tool is Mihlin's multiplier theorem combined with well-known estimates of the Newtonian potential.
Admissibility of Weights on Non-normed $*$-Algebras
S.
J.
Bhatt;
A.
Inoue;
H.
Ogi
4629-4656
Abstract: The notion of weights on (topological) $*$-algebras is defined and studied. The primary purpose is to define the notions of admissibility and approximate admissibility of weights, and to investigate when a weight is admissible or approximately admissible. The results obtained are applied to vector weights and tracial weight on unbounded operator algebras, as well as to weights on smooth subalgebras of a C$^*$-algebra.
Quasitriangular $+$ small compact $=$ strongly irreducible
You
Qing
Ji
4657-4673
Abstract: Let $T$ be a bounded linear operator acting on a separable infinite dimensional Hilbert space. Let $\epsilon$ be a positive number. In this article, we prove that the perturbation of $T$ by a compact operator $K$ with $\Vert K\Vert <\epsilon$ can be strongly irreducible if $T$ is a quasitriangular operator with the spectrum $\sigma (T)$ connected. The Main Theorem of this article nearly answers the question below posed by D. A. Herrero. Suppose that $T$ is a bounded linear operator acting on a separable infinite dimensional Hilbert space with $\sigma (T)$ connected. Let $\epsilon >0$ be given. Is there a compact operator $K$ with $\Vert K\Vert <\epsilon$ such that $T+K$ is strongly irreducible?
Decomposition of $B(G)$
Tianxuan
Miao
4675-4692
Abstract: For any locally compact group $G$, let $A(G)$ and $B(G)$ be the Fourier and the Fourier-Stieltjes algebras of $G$, respectively. $B(G)$ is decomposed as a direct sum of $A(G)$ and $B^{s}(G)$, where $B^{s}(G)$ is a subspace of $B(G)$ consisting of all elements $b\in B(G)$ that satisfy the property: for any $\epsilon > 0$ and any compact subset $K\subset G$, there is an $f\in L^{1}(G)$ with $\Vert f\Vert _{C^{*}(G)} \le 1$ and $supp(f) \subset K^{c}$ such that $\vert \langle f, b \rangle \vert > \Vert b\Vert - \epsilon .$ $A(G)$ is characterized by the following: an element $b\in B(G)$ is in $A(G)$ if and only if, for any $\epsilon > 0,$ there is a compact subset $K\subset G$ such that $\vert \langle f, b \rangle \vert < \epsilon$ for all $f\in L^{1}(G)$ with $\Vert f\Vert _{C^{*}(G)} \le 1$ and $supp(f) \subset K^{c}$. Note that we do not assume the amenability of $G$. Consequently, we have $\Vert 1 + a\Vert = 1 + \Vert a\Vert$ for all $a\in A(G)$ if $G$ is noncompact. We will apply this characterization of $B^{s}(G)$ to investigate the general properties of $B^{s}(G)$ and we will see that $B^{s}(G)$ is not a subalgebra of $B(G)$ even for abelian locally compact groups. If $G$ is an amenable locally compact group, then $B^{s}(G)$ is the subspace of $B(G)$ consisting of all elements $b\in B(G)$ with the property that for any compact subset $K\subseteq G$, $\Vert b\Vert = \sup \, \{ \, \Vert a b\Vert : \, a\in A(G), \; supp(a) \subseteq K^{c} \; \text{ and } \; \Vert a\Vert \le 1 \, \}$.