Construction and asymptotic stability of structurally stable internal layer solutions
Xiao-Biao
Lin
2983-3043
Abstract: We introduce a geometric/asymptotic method to treat structurally stable internal layer solutions. We consider asymptotic expansions of the internal layer solutions and the critical eigenvalues that determine their stability. Proofs of the existence of exact solutions and eigenvalue-eigenfunctions are outlined. Multi-layered solutions are constructed by a new shooting method through a sequence of pseudo Poincaré mappings that do not require the transversality of the flow to cross sections. The critical eigenvalues are determined by a coupling matrix that generates the SLEP matrix. The transversality of the shooting method is related to the nonzeroness of the critical eigenvalues. An equivalent approach is given to mono-layer solutions. They can be determined by the intersection of a fast jump surface and a slow switching curve, which reduces Fenichel's transversality condition to the slow manifold. The critical eigenvalue is determined by the angle of the intersection. We present three examples. The first treats the critical eigenvalues of the system studied by Angenent, Mallet-Paret & Peletier. The second shows that a key lemma in the SLEP method may not hold. The third is a perturbed activator-inhibitor system that can have any number of mono-layer solutions. Some of the solutions can only be found with the new shooting method.
Compactifying the relative Jacobian over families of reduced curves
Eduardo
Esteves
3045-3095
Abstract: We construct natural relative compactifications for the relative Jacobian over a family $X/S$ of reduced curves. In contrast with all the available compactifications so far, ours admit a Poincaré sheaf after an étale base change. Our method consists of studying the étale sheaf $F$ of simple, torsion-free, rank-1 sheaves on $X/S$, and showing that certain open subsheaves of $F$ have the completeness property. Strictly speaking, the functor $F$ is only representable by an algebraic space, but we show that $F$ is representable by a scheme after an étale base change. Finally, we use theta functions originating from vector bundles to compare our new compactifications with the available ones.
Cohomology of projective space seen by residual complex
I-Chiau
Huang
3097-3114
Abstract: A subcomplex of a residual complex on projective space is constructed for computing the cohomology modules of locally free sheaves. A constructive new proof of the Bott formula is given by explicitly exhibiting bases for the cohomology modules.
Computing the homology of Koszul complexes
Bernhard
Köck
3115-3147
Abstract: Let $R$ be a commutative ring and $I$ an ideal in $R$ which is locally generated by a regular sequence of length $d$. Then, each f. g. projective $R/I$-module $V$ has an $R$-projective resolution $P.$ of length $d$. In this paper, we compute the homology of the $n$-th Koszul complex associated with the homomorphism $P_1 \rightarrow P_0$ for all $n \ge 1$, if $d=1$. This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if $d=2$, we compute the homology of the complex $N\, \operatorname{Sym}^2 \, \Gamma(P.)$ where $\Gamma$ and $N$ denote the functors occurring in the Dold-Kan correspondence.
On the commutation of the test ideal with localization and completion
Gennady
Lyubeznik;
Karen
E.
Smith
3149-3180
Abstract: Let $R$ be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of $R$ commutes with localization and, if $R$ is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull $E$ of the residue field of every local ring of $R$ is equal to the finitistic tight closure of zero in $E$. It is conjectured that this latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities. In order to prove the result on the commutation of the test ideal with localization and completion, a ring of Frobenius operators associated to each $R$-module is introduced and studied. This theory gives rise to an ideal of $R$ which defines the non-strongly F-regular locus, and which commutes with localization and completion. This ideal is conjectured to be the test ideal of $R$ in general, and shown to equal the test ideal under the hypothesis that $0_E^*=0_E^{fg*}$in every local ring of $R$.
Classifying subcategories of modules
Mark
Hovey
3181-3191
Abstract: Let $R$ be the quotient of a regular coherent commutative ring by a finitely generated ideal. In this paper, we classify all abelian subcategories of finitely presented $R$-modules that are closed under extensions. We also classify abelian subcategories of arbitrary $R$-modules that are closed under extensions and coproducts, when $R$ is commutative and Noetherian. The method relies on comparison with the derived category of $R$.
Igusa's local zeta functions of semiquasihomogeneous polynomials
W.
A.
Zúñiga-Galindo
3193-3207
Abstract: In this paper, we prove the rationality of Igusa's local zeta functions of semiquasihomogeneous polynomials with coefficients in a non-archimedean local field $K$. The proof of this result is based on Igusa's stationary phase formula and some ideas on Néron $\pi$-desingularization.
Sums of $k$ unit fractions
Christian
Elsholtz
3209-3227
Abstract: Erdos and Straus conjectured that for any positive integer $n\geq 2$ the equation $\frac{4}{n}= \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$has a solution in positive integers $x,\, y$, and $z$. Let $m > k \geq 3$ and \begin{displaymath}E_{m,k}(N)= \, \mid \{ n \leq N \mid \frac{m}{n} = \frac{1}... ...t_k} \text{ has no solution with }t_i \in \mathbb{N}\} \mid . \end{displaymath} We show that parametric solutions can be used to find upper bounds on $E_{m,k}(N)$where the number of parameters increases exponentially with $k$. This enables us to prove \begin{displaymath}E_{m,k}(N) \ll N \exp \left( -c_{m,k} (\log N)^{1-\frac{1}{2^{k-1}-1}} \right) \text{ with } c_{m,k}>0. \end{displaymath} This improves upon earlier work by Viola (1973) and Shen (1986), and is an ``exponential generalization'' of the work of Vaughan (1970), who considered the case $k=3$.
Link complements and the Bianchi modular groups
Mark
D.
Baker
3229-3246
Abstract: We determine the values of $m$ for which the Bianchi modular group $\operatorname{PSL}_2(\mathcal{O}_m)$ contains a link group.
$L^2$-determinant class and approximation of $L^2$-Betti numbers
Thomas
Schick
3247-3265
Abstract: A standing conjecture in $L^2$-cohomology says that every finite $CW$-complex $X$ is of $L^2$-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class $\mathcal G$ of groups containing, e.g., all extensions of residually finite groups with amenable quotients, all residually amenable groups, and free products of these. If, in addition, $X$ is $L^2$-acyclic, we also show that the $L^2$-determinant is a homotopy invariant -- giving a short and easy proof independent of and encompassing all known cases. Under suitable conditions we give new approximation formulas for $L^2$-Betti numbers.
The Siegel modular variety of degree two and level three
J.
William
Hoffman;
Steven
H.
Weintraub
3267-3305
Abstract: Let $\mathcal{A}_{2}(n)$ denote the quotient of the Siegel upper half space of degree two by $\Gamma_{2}(n)$, the principal congruence subgroup of level $n$in $\mathbf{Sp}(4,\mathbf{Z})$. $\mathcal{A}_{2}(n)$ is the moduli space of principally polarized abelian varieties of dimension two with a level $n$structure, and has a compactification $\mathcal{A}_{2}(n)^{\ast}$ first constructed by Igusa. When $n\ge 3$ this is a smooth projective algebraic variety of dimension three. In this work we analyze the topology of $\mathcal{A}_{2}(3)^{\ast}$ and the open subset $\mathcal{A}_{2}(3)$. In this way we obtain the rational cohomology ring of $\Gamma_{2}(3)$. The key is that one has an explicit description of $\mathcal{A}_{2}(3)^{\ast}$: it is the resolution of the 45 nodes on a projective quartic threefold whose equation was first written down about 100 years ago by H. Burkhardt. We are able to compute the zeta function of this variety reduced modulo certain primes.
Bilinear estimates and applications to 2d NLS
J.
E.
Colliander;
J.-M.
Delort;
C.
E.
Kenig;
G.
Staffilani
3307-3325
Abstract: The three bilinearities $u v, \overline{uv},\overline{u}v$ for functions $u, v : \mathbb{R} ^2 \times [0,T] \longmapsto \mathbb{C}$ are sharply estimated in function spaces $X_{s,b}$ associated to the Schrödinger operator $i \partial_t + \Delta$. These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cubic nonlinear Schrödinger equation (and certain generalizations) are also obtained.
Theta lifting of holomorphic discrete series: The case of $ U(n, n) \times U(p, q) $
Kyo
Nishiyama;
Chen-bo
Zhu
3327-3345
Abstract: Let $K_{\mathbb{C} }$-module structure of the regular function rings on the closure of the associated nilpotent $K_{\mathbb{C} }$-orbits in $\mathfrak{s}$, where $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{s}$ is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free.
Wandering vector multipliers for unitary groups
Deguang
Han;
D.
Larson
3347-3370
Abstract: A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system into itself. A special case of unitary system is a discrete unitary group. We prove that for many (and perhaps all) discrete unitary groups, the set of wandering vector multipliers is itself a group. We completely characterize the wandering vector multipliers for abelian and ICC unitary groups. Some characterizations of special wandering vector multipliers are obtained for other cases. In particular, there are simple characterizations for diagonal and permutation wandering vector multipliers. Similar results remain valid for irrational rotation unitary systems. We also obtain some results concerning the wandering vector multipliers for those unitary systems which are the ordered products of two unitary groups. There are applications to wavelet systems.
Projective metrics and mixing properties on towers
Véronique
Maume-Deschamps
3371-3389
Abstract: We study the decay of correlations for towers. Using Birkhoff's projective metrics, we obtain a rate of mixing of the form: \begin{displaymath}c_n (f,g) \leq \text{\rm Ct} \alpha(n) \Vert f \Vert \, \Vert g \Vert_1\end{displaymath} where $\alpha(n)$ goes to zero in a way related to the asymptotic mass of upper floors, $\Vert f\Vert$ is some Lipschitz norm and $\Vert g \Vert_1$ is some $L^1$ norm. The fact that the dependence on $g$ is given by an $L^1$ norm is useful to study asymptotic laws of successive entrance times.
Vector fields with topological stability
Kazumine
Moriyasu;
Kazuhiro
Sakai;
Naoya
Sumi
3391-3408
Abstract: In this paper, we give a characterization of the structurally stable vector fields by making use of the notion of topological stability. More precisely, it is proved that the $C^1$ interior of the set of all topologically stable $C^1$ vector fields coincides with the set of all vector fields satisfying Axiom A and the strong transversality condition.