We prove the vanishing of the Koszul homology group$H$_{μ}(Kos($M$)_{μ}), where μ is the minimal number of generators of M. We give a counterexample that the Koszul complex of a module is not always acyclic and show its relationship with the homology of commutative rings.
Ilia Binder Department of Mathematics, University of Toronto Mark Braverman Department of Computer Science, University of Toronto Michael Yampolsky Department of Mathematics, University of Toronto
In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of the boundary is given either as a list of pixels, or by a Turing machine.
The topological group $\mathcal{D}^k(\mathbb{S})$ of diffeomorphisms of the unit circle $\mathbb{S}$ of Sobolev class$H$^{$k$}, for$k$large enough, is a Banach manifold modeled on the Hilbert space $H^k(\mathbb{S})$ . In this paper we show that the$H$^{1}right-invariant metric obtained by right-translation of the$H$^{1}inner product on $T_{\rm id}\mathcal{D}^k(\mathbb{S})\simeq H^k(\mathbb{S})$ defines a smooth Riemannian metric on $\mathcal{D}^k(\mathbb{S})$ , and we explicitly construct a compatible smooth affine connection. Once this framework has been established results from the general theory of affine connections on Banach manifolds can be applied to study the exponential map, geodesic flow, parallel translation, curvature etc. The diffeomorphism group of the circle provides the natural geometric setting for the Camassa–Holm equation – a nonlinear wave equation that has attracted much attention in recent years – and in this context it has been remarked in various papers how to construct a smooth Riemannian structure compatible with the$H$^{1}right-invariant metric. We give a self-contained presentation that can serve as a detailed mathematical foundation for the future study of geometric aspects of the Camassa–Holm equation.
We obtain integral boundary decay estimates for solutions of fourth-order elliptic equations on a bounded domain with regular boundary. We apply these estimates to obtain stability bounds for the corresponding eigenvalues under small perturbations of the boundary.
[10] Elliptic CR-manifolds and shear invariant ordinary differential equations with additional symmetries
Vladimir Ezhov School of Mathematics and Statistics, University of South Australia Gerd Schmalz School of Mathematics, Statistics and Computer Science, University of New England
We classify the ordinary differential equations that correspond to elliptic CR-manifolds with maximal isotropy. It follows that the dimension of the isotropy group of an elliptic CR-manifold can only be 10 (for the quadric), 4 (for the listed examples) or less. This is in contrast with the situation of hyperbolic CR-manifolds, where the dimension can be 10 (for the quadric), 6 or 5 (for semi-quadrics) or less than 4. We also prove that, for all elliptic CR-manifolds with non-linearizable isotropy group, except for two special manifolds, the points with non-linearizable isotropy form exactly some complex curve on the manifold.
Stefanie Petermichl Department of Mathematics, University of Texas at Austin Brett D. Wick Department of Mathematics, Vanderbilt University
We show that the Luzin area integral or the square function on the unit ball of ℂ^{$n$}, regarded as an operator in the weighted space$L$^{2}($w$) has a linear bound in terms of the invariant$A$_{2}characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated with the weighted$L$^{2}($w$) norm of the square function. We prove the equivalence of the classical and the invariant$A$_{2}classes.
We prove that weak solutions to the obstacle problem for the porous medium equation are locally Hölder continuous, provided that the obstacle is Hölder continuous.
Universal extensions arise naturally in the Auslander bijections. For an abelian category having Auslander-Reiten duality, we exploit a bijection triangle, which involves the Auslander bijections, universal extensions and the Auslander-Reiten duality. Some consequences are given, in particular, a conjecture by Ringel is verified.
We prove that the cover ideals of all unimodular hypergraphs have the nonincreasing depth function property. Furthermore, we show that the index of depth stability of these ideals is bounded by the number of variables.
Moduli of path families are widely used to mark curves which may be neglected for some applications. We introduce ordinary and approximation modulus with respect to Banach function spaces. While these moduli lead to the same result in reflexive spaces, we show that there are important path families (like paths tangent to a given set) which can be labeled as negligible by the approximation modulus with respect to the Lorentz Lp,1-space for an appropriate p, in particular, to the ordinary L1-space if p=1, but not by the ordinary modulus with respect to the same space.
We study singular Hermitian metrics on vector bundles. There are two main results in this paper. The first one is on the coherence of the higher rank analogue of multiplier ideals for singular Hermitian metrics defined by global sections. As an application, we show the coherence of the multiplier ideal of some positively curved singular Hermitian metrics whose standard approximations are not Nakano semipositive. The aim of the second main result is to determine all negatively curved singular Hermitian metrics on certain type of vector bundles, for example, certain rank 2 bundles on elliptic curves.
We show that if X is a cocompact G-CW-complex such that each isotropy subgroup Gσ is L(2)-good over an arbitrary commutative ring k, then X satisfies some fixed-point formula which is an L(2)-analogue of Brown’s formula in 1982. Using this result we present a fixed point formula for a cocompact proper G-CW-complex which relates the equivariant L(2)-Euler characteristic of a fixed point CW-complex Xs and the Euler characteristic of X/G. As corollaries, we prove Atiyah’s theorem in 1976, Akita’s formula in 1999 and a result of Chatterji-Mislin in 2009. We also show that if X is a free G-CW-complex such that C∗(X) is chain homotopy equivalent to a chain complex of finitely generated projective Zπ1(X)-modules of finite length and X satisfies some fixed-point formula over Q or C which is an L(2)-analogue of Brown’s formula, then χ(X/G)=χ(2)(X). As an application, we prove that the weak Bass conjecture holds for any finitely presented group G satisfying the following condition: for any finitely dominated CW-complex Y with π1(Y )=G, Y satisfies some fixed-point formula over Q or C which is an L(2)-analogue of Brown’s formula.
We show that if X is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of X as we wish. Furthermore, we show that, under the condensation open set condition, the lower dimension of an inhomogeneous self-similar set EC coincides with the lower dimension of the condensation set C, while the Assouad dimension of EC is the maximum of the Assouad dimensions of the corresponding self-similar set E and the condensation set C. If the Assouad dimension of C is strictly smaller than the Assouad dimension of E, then the upper regularity dimension of any measure supported on EC is strictly larger than the Assouad dimension of EC. Surprisingly, the corresponding statement for the lower regularity dimension fails. 1
What is the shape of a uniformly massive object that generates a gravitational potential equivalent to that of two equal point-masses? If the weight of each point-mass is sufficiently small compared to the distance between the points then the answer is a pair of balls of equal radius, one centered at each of the two points, but otherwise it is a certain domain of revolution about the axis passing through the two points. The existence and uniqueness of such a domain is known, but an explicit parameterization is known only in the plane where the region is referred to as a Neumann oval. We construct a four-dimensional “Neumann ovaloid”, solving explicitly this inverse potential problem.
We derive an integral representation for Herglotz-Nevanlinna functions in two variables, which provides a complete characterization of this class in terms of a real number, two non-negative numbers and a positive measure satisfying certain conditions. Further properties of the class of representing measures are also discussed.
Let F :RN→RN be a locally Lipschitz continuous function. We prove that F is a global homeomorphism or only injective, under suitable assumptions on the subdifferential ∂F(x). We use variational methods, nonsmooth inverse function theorem and extensions of the Hadamard-Levy Theorem. We also address questions on the Markus-Yamabe conjecture.
We extend a recent result of Avelin, Hed, and Persson about approximation of functions f that are plurisubharmonic on a domain Ω and continuous on ˙Ω, with functions that are plurisubharmonic on (shrinking) neighborhoods of ˙Ω. We show that such approximation is possible if the boundary of Ω is C0 outside a countable exceptional set E⊂∂Ω. In particular, approximation is possible on the Hartogs triangle. For Hölder continuous u, approximation is possible under less restrictive conditions on E. We next give examples of domains where this kind of approximation is not possible, even when approximation in the Hölder continuous case is possible.
This paper is devoted to a functional analytic approach to the subelliptic oblique derivative problem for the usual Laplacian with a complex parameter λ. We solve the long-standing open problem of the asymptotic eigenvalue distribution for the homogeneous oblique derivative problem when |λ| tends to ∞. We prove the spectral properties of the closed realization of the Laplacian similar to the elliptic (non-degenerate) case. In the proof we make use of Boutet de Monvel calculus in order to study the resolvents and their adjoints in the framework of L2 Sobolev spaces.