Application of the generalized Weierstrass preparation theorem to the study of homogeneous ideals
Mutsumi
Amasaki
1-43
Abstract: The system of Weierstrass polynomials, defined originally for ideals in convergent power series rings, together with its sequence of degrees allows us to analyze a homogeneous ideal directly. Making use of it, we study local cohomology modules, syzygies, and then graded Buchsbaum rings. Our results give a formula which to some extent clarifies the connection among the matrices appearing in the free resolution starting from a system of Weierstrass polynomials, a rough classification of graded Buchsbaum rings in the general case and a complete classification of graded Buchsbaum integral domains of codimension two.
Invariant tori for the billiard ball map
Valery
Kovachev;
Georgi
Popov
45-81
Abstract: For an $ n$-dimensional domain $\Omega (n \geq 3)$ with a smooth boundary which is strictly convex in a neighborhood of an elliptic closed geodesic $ \mathcal{O}$, the existence of a family of invariant tori for the billiard ball map with a positive measure is proved under the assumptions of nondegeneracy and $N$-elementarity, $N \geq 5$, of the corresponding to $\mathcal{O}$ Poincaré map. Moreover, the conjugating diffeomorphism constructed is symplectic. An analogous result is obtained in the case $n=2$. It is shown that the lengths of the periodic geodesics determine uniquely the invariant curves near the boundary and the billiard ball map on them up to a symplectic diffeomorphism.
Replacing homotopy actions by topological actions. II
Larry
Smith
83-90
Abstract: A homotopy action of a group $G$ on a space $X$ is a homomorphism from $G$ to the group ${\operatorname{HAUT}}(X)$ of homotopy classes of homotopy equivalences of $X$. George Cooke developed an obstruction theory to determine if a homotopy action is equivalent up to homotopy to a topological action. The question studied in this paper is: Given a diagram of spaces with homotopy actions of $G$ and maps between them that are equivariant up to homotopy, when can the diagram be replaced by a homotopy equivalent diagram of $G$-spaces and $G$-equivariant maps? We find that the obstruction theory of Cooke has a natural extension to this context.
Representing sets of ordinals as countable unions of sets in the core model
Menachem
Magidor
91-126
Abstract: We prove the following theorems. Theorem 1 $(\neg {0^\char93 })$. Every set of ordinals which is closed under primitive recursive set functions is a countable union of sets in $L$. Theorem 2. (No inner model with an Erdàs cardinal, i.e. $ \kappa \to {({\omega _1})^{ < \omega }}$.) For every ordinal $ \beta$, there is in $K$ an algebra on $\beta$ with countably many operations such that every subset of $\beta$ closed under the operations of the algebra is a countable union of sets in $K$.
Laws of trigonometry on ${\rm SU}(3)$
Helmer
Aslaksen
127-142
Abstract: The orbit space of congruence classes of triangles in $SU(3)$ has dimension $8$. Each corner is given by a pair of tangent vectors $(X,Y)$, and we consider the $8$ functions $ {\text{tr}}{X^2},i{\text{tr}}{X^3},{\text{tr}}{Y^2},i{\text{tr}}{Y^3},{\text{tr}}XY,i{\text{tr}}{X^2}Y,i{\text{tr}}X{Y^2}$ and ${\text{tr}}{X^2}{Y^2}$ which are invariant under the full isometry group of $SU(3)$. We show that these $8$ corner invariants determine the isometry class of the triangle. We give relations (laws of trigonometry) between the invariants at the different corners, enabling us to determine the invariants at the remaining corners, including the values of the remaining side and angles, if we know one set of corner invariants. The invariants that only depend on one tangent vector we will call side invariants, while those that depend on two tangent vectors will be called angular invariants. For each triangle we then have $ 6$ side invariants and $ 12$ angular invariants. Hence we need $18 - 8 = 10$ laws of trigonometry. If we restrict to $SU(2)$, we get the cosine laws of spherical trigonometry. The basic tool for deriving these laws is a formula expressing $ {\text{tr}}({\operatorname{exp}}X{\operatorname{exp}}Y)$ in terms of the corner invariants.
Topological equivalence of foliations of homogeneous spaces
Dave
Witte
143-166
Abstract: For $i = 1,2$, let $ {\Gamma _i}$ be a lattice in a connected Lie group ${G_i}$, and let ${X_i}$ be a connected Lie subgroup of ${G_i}$. The double cosets ${\Gamma _i}g{X_i}$ provide a foliation ${\mathcal{F}_i}$ of the homogeneous space ${\Gamma _i}\backslash {G_i}$. Assume that ${X_1}$ and ${X_2}$ are unimodular and that ${\mathcal{F}_1}$ has a dense leaf. If $ {G_1}$ and ${G_2}$ are semisimple groups to which the Mostow Rigidity Theorem applies, or are simply connected nilpotent groups (or are certain more general solvable groups), we use an idea of D. Benardete to show that any topological equivalence of ${\mathcal{F}_1}$ and ${\mathcal{F}_2}$ must be the composition of two very elementary maps: an affine map and a map that takes each leaf to itself.
Rapidly decreasing functions in reduced $C\sp *$-algebras of groups
Paul
Jolissaint
167-196
Abstract: Let $\Gamma$ be a group. We associate to any length-function $L$ on $\Gamma$ the space $H_L^\infty (\Gamma )$ of rapidly decreasing functions on $\Gamma$ (with respect to $L$), which coincides with the space of smooth functions on the $k$-dimensional torus when $\Gamma = {{\bf {Z}}^k}$. We say that $\Gamma$ has property (RD) if there exists a length-function $L$ on $\Gamma$ such that $H_L^\infty (\Gamma )$ is contained in the reduced $ {C^*}$-algebra $C_r^*(\Gamma )$ of $\Gamma$. We study the stability of property (RD) with respect to some constructions of groups such as subgroups, over-groups of finite index, semidirect and amalgamated products. Finally, we show that the following groups have property (RD): (1) Finitely generated groups of polynomial growth; (2) Discrete cocompact subgroups of the group of all isometries of any hyperbolic space.
The number of solutions of norm form equations
Wolfgang M.
Schmidt
197-227
Abstract: A norm form is a form $ F({X_1}, \ldots ,{X_n})$ with rational coefficients which factors into linear forms over $ {\mathbf{C}}$ but is irreducible or a power of an irreducible form over ${\mathbf{Q}}$. It is known that a nondegenerate norm form equation $F({x_1}, \ldots ,{x_n}) = m$ has only finitely many solutions $({x_1}, \ldots ,{x_n}) \in {{\mathbf{Z}}^n}$. We derive explicit bounds for the number of solutions. When $ F$ has coefficients in ${\mathbf{Z}}$, these bounds depend only on $ n$, $m$ and the degree of $F$, but are independent of the size of the coefficients of $F$.
Almost periodic operators in ${\rm VN}(G)$
Ching
Chou
229-253
Abstract: Let $G$ be a locally compact group, $ A(G)$ the Fourier algebra of $G$, $B(G)$ the Fourier-Stieltjes algebra of $ G$ and ${\text{VN}}(G)$ the von Neumann algebra generated by the left regular representation $ \lambda$ of $ G$. Then $A(G)$ is the predual of ${\text{VN}}(G)$; $ {\text{VN}}(G)$ is a $ B(G)$-module and $ A(G)$ is a closed ideal of $ B(G)$. Let ${\text{AP}}(\hat G) = \{ T \in {\text{VN}}(G):u \mapsto u \cdot T$ is a compact operator from $A(G)$ into ${\text{VN}}(G)\}$, the space of almost periodic operators in $ {\text{VN}}(G)$. Let $C_\delta ^*(G)$ be the ${C^*}$-algebra generated by $\{ \lambda (x):x \in G\} $. Then $ C_\delta ^*(G) \subset {\text{AP}}(\hat G)$. For a compact $G$, let $E$ be the rank one operator on ${L^2}(G)$ that sends $h \in {L^2}(G)$ to the constant function $\int {h(x)dx}$. We have the following results: (1) There exists a compact group $G$ such that $E \in$ AP$(\hat G)\backslash C_\delta ^*(G)$. (2) For a compact Lie group $G$, $E \in {\text{AP(}}\hat G{\text{)}} \Leftrightarrow E \in C_\delta ^*(G) \Leftrightarrow {L^\infty }(G)$ has a unique left invariant mean $\Leftrightarrow G$ is semisimple. (3) If $ G$ is an extension of a locally compact abelian group by an amenable discrete group then ${\text{AP}}(\hat G) = C_\delta ^*(G)$. (4) Let $ G = {{\mathbf{F}}_r}$, the free group with $r$ generators, $ 1 < r < \infty$. If $ T \in {\text{VN}}(G)$ and $u \mapsto u \cdot T$ is a compact operator from $B(G)$ into $ {\text{VN}}(G)$ then $T \in C_\delta ^*(G)$.
Complex interpolation of normed and quasinormed spaces in several dimensions. II. Properties of harmonic interpolation
Zbigniew
Slodkowski
255-285
Abstract: This paper is a continuation of the study of harmonic interpolation families of normed or quasinormed spaces parametrized by points of a domain in $ {{\mathbf{C}}^k}$. It is shown, among other things, that each of the following properties holds for all the intermediate quasinormed spaces, if it holds for all given boundary spaces: (1) being a normed space; (2) being a Hilbert space; (3) satisfying the triangle inequality by the $ r$th power of the quasinorm; (4) being uniformly convex; and (5) being uniformly smooth. As a principal tool, the notion of a harmonic set valued function (a generalization of analytic multifunction) is introduced and studied.
Extensions of valuation rings in central simple algebras
H.-H.
Brungs;
J.
Gräter
287-302
Abstract: Certain subrings $ R$ of simple algebras $ Q$, finite dimensional over their center $K$, are studied. These rings are called $ Q$-valuation rings since they share many properties with commutative valuation rings. Let $V$ be a valuation ring of $K$, the center of $Q$, and let $ \mathcal{R}$ be the set of $ Q$-valuation rings $ R$ in $Q$ with $ R \cap K = V$, then $ \left\vert \mathcal{R} \right\vert \geq 1$. This extension theorem, which does not hold if one considers only total valuation rings, was proved by N. I. Dubrovin. Here, first a somewhat different proof of this result is given and then information about the set $ \mathcal{R}$ is obtained. Theorem. The elements in $\mathcal{R}$ are conjugate if $ V$ has finite rank. Theorem. The elements in $\mathcal{R}$ are total valuation rings if $\mathcal{R}$ contains one total valuation ring. In this case $Q$ is a division ring. Theorem. $\mathcal{R}$ if $ \mathcal{R}$ contains an invariant total valuation ring.
A fixed point theorem for weakly chainable plane continua
Piotr
Minc
303-312
Abstract: In this paper the fixed point theorem is proven for every plane acyclic continuum $X$ with the property that every indecomposable continuum in the boundary of $X$ is contained in a weakly chainable subcontinuum of $X$.
Zero integrals on circles and characterizations of harmonic and analytic functions
Josip
Globevnik
313-330
Abstract: We determine the kernels of two circular Radon transforms of continuous functions on an annulus and use this to obtain a characterization of harmonic functions in the open unit disc which involves Poisson averages over circles computed at only one point of the disc and to obtain a version of Morera's theorem which involves only the circles which surround the origin.
Eventual finite order generation for the kernel of the dimension group representation
J. B.
Wagoner
331-350
Abstract: The finite order generation problem (FOG) in symbolic dynamics asks whether every element in the kernel of the dimension group representation of a subshift of finite type $({X_A},{\sigma _A})$ is a product of elements of finite order in the group $ {\operatorname{Aut}}({\sigma _A})$ of homeomorphisms of ${X_A}$ commuting with $ {\sigma _A}$. We study the space of strong shift equivalences over the nonnegative integers, and the first application is to prove Eventual FOG which says that every inert symmetry of ${\sigma _A}$ is a product of finite order homeomorphisms of ${X _A}$ commuting with sufficiently high powers of ${\sigma _A}$. Then we discuss the relation of FOG to Williams' lifting problem (LIFT) for symmetries of fixed points. In particular, either FOG or LIFT is false. Finally, we also discuss $p$-adic convergence and other implications of Eventual FOG for gyration numbers.
Extending $H\sp p$ functions from subvarieties to real ellipsoids
Kenzō
Adachi
351-359
Abstract: Let $\Omega$ be a domain in ${C^n}$ which is a somewhat generalized type of the real ellipsoid. Let $V$ be a subvariety in $\Omega$ which intersects $\partial \Omega$ transversally. Then there exists an operator $E:{H^p}(V) \to {H^p}(\Omega )$ satisfying $Ef{\vert _\nu } = f$.
Centers of generic Hecke algebras
Lenny K.
Jones
361-392
Abstract: Let $W$ be a Weyl group and let $W'$ be a parabolic subgroup of $ W$. Define $A$ as follows: $\displaystyle A = R{ \otimes _{{\mathbf{Q}}[u]}}\mathcal{A}(W)$ where $\mathcal{A}(W)$ is the generic algebra of type $ {A_n}$ over ${\mathbf{Q}}[u]$ an indeterminate, associated with the group $W$, and $R$ is a $ {\mathbf{Q}}[u]$-algebra, possibly of infinite rank, in which $u$ is invertible. Similarly, we define $A'$ associated with $W'$. Let $M$ be an $A - A$ bimodule, and let $b \in M$. Define the relative norm [14] $ b \in {Z_M}(A') = \{ m \in M\vert ma' = a'm\quad \forall a' \in A'\}$, then $\alpha = ({k_1},{k_2}, \ldots ,{k_z})$ be a partition of $n$ and let ${S_\alpha } = \Pi _{i = 1}^Z{S_{{k_i}}}$ be a "left-justified" parabolic subgroup of ${S_n}$ of shape $\alpha$. Define $\displaystyle {b_\alpha } = {N_{{S_n},{S_\alpha }}}({\mathcal{N}_\alpha })$ , where $\displaystyle {\mathcal{N}_\alpha } = \prod\limits_{i = 1}^z {{N_{{S_{{k_i} - 1}},{S_1}}}({a_{{w_i}}})}$ with $ {w_i}$ a ${k_i}$-cycle of length ${k_i} - 1$ in $ {S_{{k_i}}}$. Then the main result of this paper is Theorem. The set $\{ {b_\alpha }\vert\alpha \vdash n\}$ is a basis for $ {Z_{A({S_n})}}(A({S_n}))$ over ${\mathbf{Q}}[u,{u^{ - 1}}]$. Remark. The norms $ {b_\alpha }$ in $ {Z_{A({S_n})}}(A({S_n}))$ are analogs of conjugacy class sums in the center of $ {\mathbf{Q}}{S_n}$ and, in fact, specialization of these norms at $ u = 1$ gives the standard conjugacy class sum basis of the center of ${\mathbf{Q}}{S_n}$ up to coefficients from ${\mathbf{Q}}$.
Topological spaces whose Baire measure admits a regular Borel extension
Haruto
Ohta;
Ken-ichi
Tamano
393-415
Abstract: A completely regular, Hausdorff space $X$ is called a Măík space if every Baire measure on $X$ admits an extension to a regular Borel measure. We answer the questions about Măík spaces asked by Wheeler [29] and study their topological properties. In particular, we give examples of the following spaces: A locally compact, measure compact space which is not weakly Bairedominated; i.e., it has a sequence $ {F_n} \downarrow \emptyset$ of regular closed sets such that $ { \cap _{n \in \omega }}{B_n} \ne \emptyset$ whenever ${B_n}$'s are Baire sets with ${F_n} \subset {B_n}$; a countably paracompact, non-Măík space; a locally compact, non-Măík space $X$ such that the absolute $E(X)$ is a Măík space; and a locally compact, Măík space $ X$ for which $ E(X)$ is not. It is also proved that Michael's product space is not weakly Baire-dominated.