Coordinates in two variables over a $\mathbb{Q}$-algebra
Arno
van den Essen;
Peter
van Rossum
1691-1703
Abstract: This paper studies coordinates in two variables over a $\mathbb{Q}$-algebra. It gives several ways to characterize such coordinates. Also, various results about coordinates in two variables that were previously only known for fields, are extended to arbitrary $\mathbb{Q}$-algebras.
Root vectors for geometrically simple two-parameter eigenvalues
Paul
Binding;
Tomaz
Kosir
1705-1726
Abstract: A class of two-parameter eigenvalue problems involving generally nonselfadjoint and unbounded operators is studied. A basis for the root subspace at a geometrically simple eigenvalue of Fredholm type is computed in terms of the underlying two-parameter system. Comparison with Faierman's work on two-parameter boundary value problems of Sturm-Liouville type is given as an application.
Commuting Toeplitz operators on the polydisk
Boo
Rim
Choe;
Hyungwoon
Koo;
Young
Joo
Lee
1727-1749
Abstract: We obtain characterizations of (essentially) commuting Toeplitz operators with pluriharmonic symbols on the Bergman space of the polydisk. We show that commuting and essential commuting properties are the same for dimensions bigger than 2, while they are not for dimensions less than or equal to 2. Also, the corresponding results for semi-commutators are obtained.
Newton polyhedra, unstable faces and the poles of Igusa's local zeta function
Kathleen
Hoornaert
1751-1779
Abstract: In this paper we examine when the order of a pole of Igusa's local zeta function associated to a polynomial $f$ is smaller than ``expected''. We carry out this study in the case that $f$ is sufficiently non-degenerate with respect to its Newton polyhedron $\Gamma(f)$, and the main result of this paper is a proof of one of the conjectures of Denef and Sargos. Our technique consists in reducing our question about the polynomial $f$ to the same question about polynomials $f_\mu$, where $\mu$ are faces of $\Gamma(f)$ depending on the examined pole and $f_\mu$ is obtained from $f$ by throwing away all monomials of $f$ whose exponents do not belong to $\mu$. Secondly, we obtain a formula for Igusa's local zeta function associated to a polynomial $f_\mu$, with $\mu$ unstable, which shows that, in this case, the upperbound for the order of the examined pole is obviously smaller than ``expected''.
A nonstandard Riemann existence theorem
Rahim
Moosa
1781-1797
Abstract: We study elementary extensions of compact complex spaces and deduce that every complete type of dimension $1$ is internal to projective space. This amounts to a nonstandard version of the Riemann Existence Theorem, and answers a question posed by Anand Pillay.
Algebraic $\mathbb{Z}^d$-actions of entropy rank one
Manfred
Einsiedler;
Douglas
Lind
1799-1831
Abstract: We investigate algebraic $\mathbb Z^d$-actions of entropy rank one, namely those for which each element has finite entropy. Such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery. This leads to numerous alternative characterizations of entropy rank one, both geometric and algebraic. We then compute the measure entropy of a class of skew products, where the fiber maps are elements from an algebraic $\mathbb Z^d$-action of entropy rank one. This leads, via the relative variational principle, to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures. We use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms.
Poincaré series of resolutions of surface singularities
Steven
Dale
Cutkosky;
Jürgen
Herzog;
Ana
Reguera
1833-1874
Abstract: Let $X\rightarrow\mathrm{spec}(R)$ be a resolution of singularities of a normal surface singularity $\mathrm{spec}(R)$, with integral exceptional divisors $E_1,\dotsc,E_r$. We consider the Poincaré series \begin{displaymath}g= \sum_{\underline{n}\in\mathbf{N}^r} h(\underline{n})t^{\underline{n}}, \end{displaymath} where \begin{displaymath}h(\underline{n})=\ell(R/\Gamma(X,\mathcal{O}_X(-n_1E-1-\cdots-n_rE_r)). \end{displaymath} We show that if $R/m$ has characteristic zero and $\mathrm{Pic}^0(X)$ is a semi-abelian variety, then the Poincaré series $g$ is rational. However, we give examples to show that this series can be irrational if either of these conditions fails.
Uniqueness of varieties of minimal degree containing a given scheme
M.
Casanellas
1875-1888
Abstract: We prove that if $X \subset \mathbb{P} ^N$ has dimension $k$ and it is $r$-Buchsbaum with $r>\max{(\operatorname{codim}{X}-k,0)}$, then $X$ is contained in at most one variety of minimal degree and dimension $k+1$.
Asymptotic behavior of the solutions of linear and quasilinear elliptic equations on $\mathbb{R}^{N}$
Patrick
J.
Rabier
1889-1907
Abstract: We investigate the relationship between the decay at infinity of the right-hand side $f$ and solutions $u$ of an equation $Lu=f$ when $L$ is a second order elliptic operator on $\mathbb{R} ^{N}.$ It is shown that when $L$is Fredholm, $u$ inherits the type of decay of $f$ (for instance, exponential, or power-like). In particular, the generalized eigenfunctions associated with all the Fredholm eigenvalues of $L,$ isolated or not, decay exponentially. No use is made of spectral theory. The result is next extended when $L$ is replaced by a Fredholm quasilinear operator. Various generalizations to other unbounded domains, higher order operators or elliptic systems are possible and briefly alluded to, but not discussed in detail.
Automorphic forms and differentiability properties
Fernando
Chamizo
1909-1935
Abstract: We consider Fourier series given by a type of fractional integral of automorphic forms, and we study their local and global properties, especially differentiability and fractal dimension of the graph of their real and imaginary parts. In this way we can construct fractal objects and continuous non-differentiable functions associated with elliptic curves and theta functions.
Simple Bratteli diagrams with a Gödel-incomplete C*-equivalence problem
Daniele
Mundici
1937-1955
Abstract: An abstract simplicial complex is a finite family of subsets of a finite set, closed under subsets. Every abstract simplicial complex $\mathcal{C}$ naturally determines a Bratteli diagram and a stable AF-algebra $A(\mathcal{C})$. Consider the following problem: INPUT: a pair of abstract simplicial complexes $\mathcal{C}$ and $A(\mathcal{C})$ isomorphic to
Semilattices of finitely generated ideals of exchange rings with finite stable rank
F.
Wehrung
1957-1970
Abstract: We find a distributive \ensuremath{(\vee,0,1)}-semilattice $S_{\omega_1}$ of size $\aleph_1$ that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular: -- There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with finite stable rank whose semilattice of finitely generated, idempotent-generated two-sided ideals is isomorphic to $S_{\omega_1}$. -- There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to $S_{\omega_1}$. These results are established by constructing an infinitary statement, denoted here by $\mathrm{URP_{sr}}$, that holds in the maximal semilattice quotient of every Riesz monoid endowed with an order-unit of finite stable rank, but not in the semilattice $S_{\omega_1}$.
On restrictions of modular spin representations of symmetric and alternating groups
Alexander
S.
Kleshchev;
Pham
Huu
Tiep
1971-1999
Abstract: Let $\mathbb F$ be an algebraically closed field of characteristic $p$ and $H$ be an almost simple group or a central extension of an almost simple group. An important problem in representation theory is to classify the subgroups $G$ of $H$ and $\mathbb F H$-modules $V$ such that the restriction $V{\downarrow}_G$ is irreducible. For example, this problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where $H$ is the Schur's double cover $\hat A_n$ or $\hat S_n$.
Varying the time-frequency lattice of Gabor frames
Hans
G.
Feichtinger;
Norbert
Kaiblinger
2001-2023
Abstract: A Gabor or Weyl-Heisenberg frame for $L^2(\mathbb{R}^d)$is generated by time-frequency shifts of a square-integrable function, the Gabor atom, along a time-frequency lattice. The dual frame is again a Gabor frame, generated by the dual atom. In general, Gabor frames are not stable under a perturbation of the lattice constants; that is, even for arbitrarily small changes of the parameters the frame property can be lost. In contrast, as a main result we show that this kind of stability does hold for Gabor frames generated by a Gabor atom from the modulation space $M^1(\mathbb{R}^d)$, which is a dense subspace of $L^2(\mathbb{R}^d)$. Moreover, in this case the dual atom depends continuously on the lattice constants. In fact, we prove these results for more general weighted modulation spaces. As a consequence, we obtain for Gabor atoms from the Schwartz class that the continuous dependence of the dual atom holds even in the Schwartz topology. Also, we complement these main results by corresponding statements for Gabor Riesz sequences and their biorthogonal system.
The periodic isoperimetric problem
Laurent
Hauswirth;
Joaquín
Pérez;
Pascal
Romon;
Antonio
Ros
2025-2047
Abstract: Given a discrete group $G$ of isometries of $\mathbb{R} ^3$, we study the $G$-isoperimetric problem, which consists of minimizing area (modulo $G$) among surfaces in $\mathbb{R} ^3$ which enclose a $G$-invariant region with a prescribed volume fraction. If $G$ is a line group, we prove that solutions are either families of round spheres or right cylinders. In the doubly periodic case we prove that for most rank two lattices, solutions must be spheres, cylinders or planes. For the remaining rank two lattices we show, among other results, an isoperimetric inequality in terms of the topology of the isoperimetric surfaces. Finally, we study the case where $G=Pm\overline{3}m$ (the group of symmetries of the integer rank three lattice $\mathbb{Z} ^3$) and other crystallographic groups of cubic type. We prove that isoperimetric solutions must be spheres if the prescribed volume fraction is less than $1/6$, and we give an isoperimetric inequality for $G$-invariant regions that, for instance, implies that the area (modulo $\mathbb{Z} ^3$) of a surface dividing the three space in two $G$-invariant regions with equal volume fractions, is at least $2.19$ (the conjectured solution is the classical $P$ Schwarz triply periodic minimal surface whose area is $\sim 2.34$). Another consequence of this isoperimetric inequality is that $Pm\overline{3}m$-symmetric surfaces (other than families of spheres) cannot be isoperimetric for the lattice group $\mathbb{Z} ^3$.
Spinors as automorphisms of the tangent bundle
Alexandru
Scorpan
2049-2066
Abstract: We show that, on a $4$-manifold $M$ endowed with a $\operatorname{spin}^{\mathbb{C} }$-structure induced by an almost-complex structure, a self-dual (positive) spinor field $\phi\in\Gamma(W^+)$ is the same as a bundle morphism $\phi:T_M\to T_M$ acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of $\phi$ on tangent vectors, and that the squaring map $\sigma:\mathcal{W}^+\to\Lambda^+$ acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kähler and symplectic structures.
Gromov translation algebras over discrete trees are exchange rings
P.
Ara;
K.
C.
O'Meara;
F.
Perera
2067-2079
Abstract: It is shown that the Gromov translation ring of a discrete tree over a von Neumann regular ring is an exchange ring. This provides a new source of exchange rings, including, for example, the algebras $G(0)$ of $\omega\times\omega$ matrices (over a field) of constant bandwidth. An extension of these ideas shows that for all real numbers $r$ in the unit interval $[0,1]$, the growth algebras $G(r)$(introduced by Hannah and O'Meara in 1993) are exchange rings. Consequently, over a countable field, countable-dimensional exchange algebras can take any prescribed bandwidth dimension $r$in $[0,1]$.
Positive laws in fixed points
Pavel
Shumyatsky
2081-2091
Abstract: Let $A$ be an elementary abelian group of order at least $q^3$ acting on a finite $q'$-group $G$in such a manner that $C_G(a)$ satisfies a positive law of degree $n$ for any $a\in A^\char93$. It is proved that the entire group $G$ satisfies a positive law of degree bounded by a function of $q$ and $n$ only.
Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems
Donatella
Donatelli;
Pierangelo
Marcati
2093-2121
Abstract: In this paper we investigate the diffusive zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form: \begin{displaymath}W_{t}(x,t) + \frac{1}{\varepsilon}A(x,D)W(x,t)= \frac{1}{\varepsilon ^2} B(x,W(x,t))+\frac{1}{\varepsilon} D(W(x,t))+E(W(x,t)).\end{displaymath} We analyze the singular convergence, as $\varepsilon \downarrow 0$, in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps: (i) We single out algebraic ``structure conditions'' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories. (ii) We deduce ``energy estimates '', uniformly in $\varepsilon$, by assuming the existence of a symmetrizer having the so-called block structure and by assuming ``dissipativity conditions'' on $B$. (iii) We assume a Kawashima type condition and perform the convergence analysis by using generalizations of compensated compactness due to Tartar and Gérard. Finally, we include examples that show how to use our theory to approximate any quasilinear parabolic systems, satisfying the Petrowski parabolicity condition, or general reaction diffusion systems, including Chemotaxis and Brusselator type systems.
The steepest point of the boundary layers of singularly perturbed semilinear elliptic problems
T.
Shibata
2123-2135
Abstract: We consider the nonlinear singularly perturbed problem \begin{displaymath}-\epsilon^2\Delta u = f(u), \enskip u > 0 \quad \mbox{in} \enskip \Omega, u = 0 \quad \mbox{on} \enskip \partial\Omega, \end{displaymath} where $\Omega \subset {\mathbf{R}}^N$ ($N \ge 2$) is an appropriately smooth bounded domain and $\epsilon > 0$ is a small parameter. It is known that under some conditions on $f$, the solution $u_\epsilon$ corresponding to $\epsilon$ develops boundary layers when $\epsilon \to 0$. We determine the steepest point of the boundary layers on the boundary by establishing an asymptotic formula for the slope of the boundary layers with exact second term.
Erratum to ``The central limit problem for convex bodies''
Milla
Anttila;
Keith
Ball;
Irini
Perissinaki
2137-2137