Lower and upper Loeb-integrals
D.
Landers;
L.
Rogge
3263-3283
Abstract: We introduce the concepts of lower and upper Loeb-integrals for an internal integration structure. These are concepts which are similarly useful for Loebs internal integration theory as the concepts of inner and outer Loeb-measures for Loebs measure theory.
Prime geodesic theorem for higher-dimensional hyperbolic manifold
Maki
Nakasuji
3285-3303
Abstract: For a $(d+1)$-dimensional hyperbolic manifold $\mathcal{M}$, we consider an estimate of the error term of the prime geodesic theorem. Put the fundamental group $\Gamma$ of $\mathcal{M}$ to be a discrete subgroup of $SO_e(d+1, 1)$ with cofinite volume. When the contribution of the discrete spectrum of the Laplace-Beltrami operator is larger than that of the continuous spectrum in Weyl's law, we obtained a lower estimate $\Omega_{\pm}(\tfrac{x^{d/2}(\log\log x)^{1/(d+1)}}{\log x})$ as $x$ goes to $\infty$.
On decompositions in homotopy theory
Brayton
Gray
3305-3328
Abstract: We first describe Krull-Schmidt theorems decomposing $H$ spaces and simply-connected co-$H$ spaces into atomic factors in the category of pointed nilpotent $p$-complete spaces of finite type. We use this to construct a 1-1 correspondence between homotopy types of atomic $H$ spaces and homotopy types of atomic co-$H$ spaces, and construct a split fibration which connects them and illuminates the decomposition. Various properties of these constructions are analyzed.
Kirwan-Novikov inequalities on a manifold with boundary
Maxim
Braverman;
Valentin
Silantyev
3329-3361
Abstract: We extend the Novikov Morse-type inequalities for closed 1-forms in 2 directions. First, we consider manifolds with boundary. Second, we allow a very degenerate structure of the critical set of the form, assuming only that the form is non-degenerated in the sense of Kirwan. In particular, we obtain a generalization of a result of Floer about the usual Morse inequalities on a manifold with boundary. We also obtain an equivariant version of our inequalities. Our proof is based on an application of the Witten deformation technique. The main novelty here is that we consider the neighborhood of the critical set as a manifold with a cylindrical end. This leads to a considerable simplification of the local analysis. In particular, we obtain a new analytic proof of the Morse-Bott inequalities on a closed manifold.
Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients
Günther
Hörmann;
Michael
Oberguggenberger;
Stevan
Pilipovic
3363-3383
Abstract: We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest.
Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces
J.
William
Hoffman;
Hao
Hao
Wang
3385-3398
Abstract: Let $I = \langle f_1 , f_2 , f_3\rangle$ be a bigraded ideal in the bigraded polynomial ring $k[s, u; t, v]$. Assume that $I$ has codimension 2. Then $Z = \mathbb{V}(I) \subset \mathbf{P}^{1} \times \mathbf{P}^{1}$ is a finite set of points. We prove that if $Z$ is a local complete intersection, then any syzygy of the $f_i$ vanishing at $Z$, and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).
Signature invariants of covering links
Jae
Choon
Cha;
Ki
Hyoung
Ko
3399-3412
Abstract: We apply the theory of signature invariants of links in rational homology spheres to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, we derive an explicit formula to compute signature invariants of their covering links. Using the formula, we produce fused boundary links that are positive mutants of ribbon links but are not concordant to boundary links. We also show that for any finite collection of patterns, there are homology boundary links that are not concordant to any homology boundary links admitting a pattern in the collection.
A general theory of almost convex functions
S.
J.
Dilworth;
Ralph
Howard;
James
W.
Roberts
3413-3445
Abstract: Let $\Delta_m=\{(t_0,\dots, t_m)\in \mathbf{R}^{m+1}: t_i\ge 0, \sum_{i=0}^mt_i=1\}$ be the standard $m$-dimensional simplex and let $\varnothing\ne S\subset \bigcup_{m=1}^\infty\Delta_m$. Then a function $h\colon C\to \mathbf{R}$ with domain a convex set in a real vector space is $S$-almost convex iff for all $(t_0,\dots, t_m)\in S$ and $x_0,\dots, x_m\in C$ the inequality $\displaystyle h(t_0x_0+\dots+t_mx_m)\le 1+ t_0h(x_0)+\cdots+t_mh(x_m)$ holds. A detailed study of the properties of $S$-almost convex functions is made. If $S$ contains at least one point that is not a vertex, then an extremal $S$-almost convex function $E_S\colon \Delta_n\to \mathbf{R}$ is constructed with the properties that it vanishes on the vertices of $\Delta_n$ and if $h\colon \Delta_n\to \mathbf{R}$ is any bounded $S$-almost convex function with $h(e_k)\le 0$ on the vertices of $\Delta_n$, then $h(x)\le E_S(x)$ for all $x\in \Delta_n$. In the special case $S=\{(1/(m+1),\dotsc, 1/(m+1))\}$, the barycenter of $\Delta_m$, very explicit formulas are given for $E_S$ and $\kappa_S(n)=\sup_{x\in\Delta_n}E_S(x)$. These are of interest, as $E_S$ and $\kappa_S(n)$ are extremal in various geometric and analytic inequalities and theorems.
Explicit bounds for the finite jet determination problem
Bernhard
Lamel
3447-3457
Abstract: We introduce biholomorphic invariants for (germs of) rigid holomorphically nondegenerate real hypersurfaces in complex space and show how they can be used to compute explicit bounds on the order of jets for which biholomorphisms of the hypersurface are determined uniquely by their jets. The main result which allows us to derive these bounds is a theorem which shows that solutions of certain singular analytic equations are uniquely determined by their $1$-jet.
A geometric characterization of interpolation in $\hat{\mathcal{E}}^\prime(\mathbb{R})$
Xavier
Massaneda;
Joaquim
Ortega-Cerdà;
Myriam
Ounaïes
3459-3472
Abstract: We give a geometric description of the interpolating varieties for the algebra of Fourier transforms of distributions (or Beurling ultradistributions) with compact support on the real line.
Dynamical forcing of circular groups
Danny
Calegari
3473-3491
Abstract: In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties. As an application, we show that the set $X \subset \mathbb{R} /\mathbb{Z}$ consisting of rotation numbers $\theta$ which can be forced by finitely presented groups is an infinitely generated $\mathbb{Q}$-module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number $\theta$ is forced by a pair $(G_\theta,\alpha)$, where $G_\theta$ is a finitely presented group $G_\theta$ and $\alpha \in G_\theta$ is some element, if the set of rotation numbers of $\rho(\alpha)$ as $\rho$varies over $\rho \in \operatorname{Hom}(G_\theta,\operatorname{Homeo}^+(S^1))$ is precisely the set $\lbrace 0, \pm \theta \rbrace$. We show that the set of subsets of $\mathbb{R} /\mathbb{Z}$ which are of the form \begin{displaymath}\operatorname{rot}(X(G,\alpha)) = \lbrace r \in \mathbb{R} /... ... \in \operatorname{Hom}(G,\operatorname{Homeo}^+(S^1)) \rbrace,\end{displaymath} where $G$ varies over countable groups, are exactly the set of closed subsets which contain $0$ and are invariant under $x \to -x$. Moreover, we show that every such subset can be approximated from above by $\operatorname{rot}(X(G_i,\alpha_i))$ for finitely presented $G_i$. As another application, we construct a finitely generated group $\Gamma$ which acts faithfully on the circle, but which does not admit any faithful $C^1$action, thus answering in the negative a question of John Franks.
Exponents for $B$-stable ideals
Eric
Sommers;
Julianna
Tymoczko
3493-3509
Abstract: Let $G$ be a simple algebraic group over the complex numbers containing a Borel subgroup $B$. Given a $B$-stable ideal $I$ in the nilradical of the Lie algebra of $B$, we define natural numbers $m_1, m_2, \dots, m_k$ which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types $A_n, B_n, C_n$ and some other types. When $I = 0$, we recover the usual exponents of $G$ by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.
Bilinear restriction estimates for surfaces with curvatures of different signs
Sanghyuk
Lee
3511-3533
Abstract: Recently, the sharp $L^2$-bilinear (adjoint) restriction estimates for the cone and the paraboloid were established by Wolff and Tao, respectively. Their results rely on the fact that for the cone and the paraboloid, the nonzero principal curvatures have the same sign. We generalize those bilinear restriction estimates to surfaces with curvatures of different signs.
Nonnegative solvability of linear equations in certain ordered rings
Philip
Scowcroft
3535-3570
Abstract: In the integers and in certain densely ordered rings that are not fields, projections of the solution set of finitely many homogeneous weak linear inequalities may be defined by finitely many congruence inequalities, where a congruence inequality combines a weak inequality with a system of congruences. These results extend well-known facts about systems of weak linear inequalities over ordered fields and imply corresponding analogues of Farkas' Lemma on nonnegative solvability of systems of linear equations.
Extended degree functions and monomial modules
Uwe
Nagel;
Tim
Römer
3571-3589
Abstract: The arithmetic degree, the smallest extended degree, and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not Cohen-Macaulay. We compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule. This leads to various bounds and to counterexamples to a conjecture of Gunston and Vasconcelos, respectively. Particular attention is given to the class of sequentially Cohen-Macaulay modules. The results in this case lead to an algorithm that computes the smallest extended degree.
Frobenius morphisms and representations of algebras
Bangming
Deng;
Jie
Du
3591-3622
Abstract: By introducing Frobenius morphisms $F$ on algebras $A$ and their modules over the algebraic closure ${\overline {\mathbb{F}}}_q$ of the finite field ${\mathbb{F}}_q$ of $q$ elements, we establish a relation between the representation theory of $A$ over $\overline {\mathbb{F}}_q$ and that of the $F$-fixed point algebra $A^F$ over ${\mathbb{F}}_q$. More precisely, we prove that the category mod-$A^F$ of finite-dimensional $A^F$-modules is equivalent to the subcategory of finite-dimensional $F$-stable $A$-modules, and, when $A$ is finite dimensional, we establish a bijection between the isoclasses of indecomposable $A^F$-modules and the $F$-orbits of the isoclasses of indecomposable $A$-modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over ${\mathbb{F}}_q$ can be interpreted as $F$-stable representations of the corresponding quiver over $\overline {\mathbb{F}}_q$. We further prove that every finite-dimensional hereditary algebra over ${\mathbb{F}}_q$ is Morita equivalent to some $A^F$, where $A$ is the path algebra of a quiver $Q$ over $\overline {\mathbb{F}}_q$ and $F$ is induced from a certain automorphism of $Q$. A close relation between the Auslander-Reiten theories for $A$ and $A^F$ is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of $A^F$ is obtained by ``folding" the Auslander-Reiten quiver of $A$. Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver over ${\mathbb{F}}_q$ with a given dimension vector and to generalize Kac's theorem for all modulated quivers and their associated Kac-Moody algebras defined by symmetrizable generalized Cartan matrices.
On Cartan matrices and lower defect groups for covering groups of symmetric groups
Christine
Bessenrodt;
Jørn
B.
Olsson
3623-3635
Abstract: We determine the elementary divisors of the Cartan matrices of spin $p$-blocks of the covering groups of the symmetric groups when $p$ is an odd prime. As a consequence, we also compute the determinants of these Cartan matrices, and in particular we confirm a conjecture by Brundan and Kleshchev that these determinants depend only on the weight but not on the sign of the block.
Unital bimodules over the simple Jordan superalgebra $D(t)$
Consuelo
Martínez;
Efim
Zelmanov
3637-3649
Abstract: We classify indecomposable finite dimensional bimodules over Jordan superalgebras $D(t)$, $t \neq -1,0,1$.
Geometric structures as deformed infinitesimal symmetries
Anthony
D.
Blaom
3651-3671
Abstract: A general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries. Specifically, this model consists of a Lie algebroid, equipped with an affine connection compatible with the Lie algebroid structure. The curvature of this connection vanishes precisely when the structure is locally symmetric. This model generalizes Cartan geometries, a substantial class, to the intransitive case. Simple examples are surveyed and corresponding local obstructions to symmetry are identified. These examples include foliations, Riemannian structures, infinitesimal $G$-structures, symplectic and Poisson structures.
On the Eshelby-Kostrov property for the wave equation in the plane
M.
A.
Herrero;
G.
E.
Oleaga;
J.
J. L.
Velázquez
3673-3695
Abstract: This work deals with the linear wave equation considered in the whole plane $\mathbb{R}^{2}$ except for a rectilinear moving slit, represented by a curve $\Gamma\left( t\right) =\left\{ \left( x_{1},0\right) :-\infty<x_{1}<\lambda\left( t\right) \right\}$ with $t\geq0.$ Along $\Gamma\left( t\right) ,$ either homogeneous Dirichlet or Neumann boundary conditions are imposed. We discuss existence and uniqueness for these problems, and derive explicit representation formulae for solutions. The latter have a simple geometrical interpretation, and in particular allow us to derive precise asymptotic expansions for solutions near the tip of the curve. In the Neumann case, we thus recover a classical result in fracture dynamics, namely the form of the stress intensity factor in crack propagation under antiplane shear conditions.
On linear transformations preserving the Pólya frequency property
Petter
Brändén
3697-3716
Abstract: We prove that certain linear operators preserve the Pólya frequency property and real-rootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by Bóna, Brenti and Reiner-Welker.
Length, multiplicity, and multiplier ideals
Tommaso
de Fernex
3717-3731
Abstract: Let $(R,\mathfrak{m})$ be an $n$-dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an $\mathfrak{m}$-primary ideal $\mathfrak{a}$ of $R$, the relationship between the singularities of the scheme defined by $\mathfrak{a}$ and those defined by the multiplier ideals $\mathcal{J}(\mathfrak{a}^c)$, with $c$ varying in $\mathbb{Q}_+$, are quantified in this paper by showing that the Samuel multiplicity of $\mathfrak{a}$ satisfies $e(\mathfrak{a}) \ge (n+k)^n/c^n$ whenever $\mathcal{J}(\mathfrak{a}^c) \subseteq \mathfrak{m}^{k+1}$. This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustata and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.
On higher syzygies of ruled surfaces
Euisung
Park
3733-3749
Abstract: We study higher syzygies of a ruled surface $X$ over a curve of genus $g$ with the numerical invariant $e$. Let $L \in$ Pic$X$ be a line bundle in the numerical class of $aC_0 +bf$. We prove that for $0 \leq e \leq g-3$, $L$ satisfies property $N_p$ if $a \geq p+2$ and $b-ae \geq 3g-1-e+p$, and for $e \geq g-2$, $L$ satisfies property $N_p$ if $a \geq p+2$ and $b-ae\geq 2g+1+p$. By using these facts, we obtain Mukai-type results. For ample line bundles $A_i$, we show that $K_X + A_1 + \cdots + A_q$ satisfies property $N_p$ when $0 \leq e < \frac{g-3}{2}$ and $q \geq g-2e+1 +p$ or when $e \geq \frac{g-3}{2}$ and $q \geq p+4$. Therefore we prove Mukai's conjecture for ruled surface with $e \geq \frac{g-3}{2}$. We also prove that when $X$ is an elliptic ruled surface with $e \geq 0$, $L$ satisfies property $N_p$ if and only if $a \geq 1$ and $b-ae\geq 3+p$.