$L^1$--framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations
Gui-Qiang
Chen;
Kenneth
H.
Karlsen
937-963
Abstract: We develop a general $L^1$-framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach. We apply our $L^1$-framework to establish an explicit estimate for continuous dependence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method, without imposition of bounded variation of the approximate solutions. Finally, as an example of a direct application of this framework to numerical methods, we focus on a linear convection-diffusion model equation and derive an $L^1$ error estimate for an upwind-central finite difference scheme.
Generalized interpolation in $H^\infty$ with a complexity constraint
Christopher
I.
Byrnes;
Tryphon
T.
Georgiou;
Anders
Lindquist;
Alexander
Megretski
965-987
Abstract: In a seminal paper, Sarason generalized some classical interpolation problems for $H^\infty$ functions on the unit disc to problems concerning lifting onto $H^2$ of an operator $T$ that is defined on $\EuScript{K} =H^2\ominus\phi H^2$ ($\phi$ is an inner function) and commutes with the (compressed) shift $S$. In particular, he showed that interpolants (i.e., $f\in H^\infty$ such that $f(S)=T$) having norm equal to $\Vert T\Vert$ exist, and that in certain cases such an $f$ is unique and can be expressed as a fraction $f=b/a$ with $a,b\in\EuScript{K}$. In this paper, we study interpolants that are such fractions of $\EuScript{K}$ functions and are bounded in norm by $1$ (assuming that $\Vert T\Vert<1$, in which case they always exist). We parameterize the collection of all such pairs $(a,b)\in\EuScript{K}\times\EuScript{K}$ and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where $\phi$ is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.
Invariants, Boolean algebras and ACA$_{0}^{+}$
Richard
A.
Shore
989-1014
Abstract: The sentences asserting the existence of invariants for mathematical structures are usually third order ones. We develop a general approach to analyzing the strength of such statements in second order arithmetic in the spirit of reverse mathematics. We discuss a number of simple examples that are equivalent to ACA$_{0}$. Our major results are that the existence of elementary equivalence invariants for Boolean algebras and isomorphism invariants for dense Boolean algebras are both of the same strength as ACA$_{0}^{+}$. This system corresponds to the assertion that $X^{(\omega)}$(the arithmetic jump of $X$) exists for every set $X$. These are essentially the first theorems known to be of this proof theoretic strength. The proof begins with an analogous result about these invariants on recursive (dense) Boolean algebras coding $0^{(\omega)}$.
Horrocks theory and the Bernstein-Gel'fand-Gel'fand correspondence
I.
Coanda;
G.
Trautmann
1015-1031
Abstract: We construct an explicit equivalence between a category of complexes over the exterior algebra, which we call HT-complexes, and the stable category of vector bundles on the corresponding projective space, essentially translating into more fancy terms the results of Trautmann (1978) which, in turn, were influenced by ideas of Horrocks (1964), (1980). However, the result expressed by Theorem 5.1 and its corollary, which establishes a relation between the Tate resolutions over the exterior algebra (described in a paper by Eisenbud, Fløystad, and Schreyer) and HT-complexes, might be new, although, perhaps, not a surprise to experts.
The integral cohomology of the Bianchi groups
Ethan
Berkove
1033-1049
Abstract: We calculate the integral cohomology ring structure for various members of the Bianchi group family. The main tools we use are the Bockstein spectral sequence and a long exact sequence derived from Bass-Serre theory.
Distance between toroidal surgeries on hyperbolic knots in the $3$-sphere
Masakazu
Teragaito
1051-1075
Abstract: For a hyperbolic knot in the $3$-sphere, at most finitely many Dehn surgeries yield non-hyperbolic $3$-manifolds. As a typical case of such an exceptional surgery, a toroidal surgery is one that yields a closed $3$-manifold containing an incompressible torus. The slope corresponding to a toroidal surgery, called a toroidal slope, is known to be integral or half-integral. We show that the distance between two integral toroidal slopes for a hyperbolic knot, except the figure-eight knot, is at most four.
Mayer brackets and solvability of PDEs -- II
Boris
Kruglikov;
Valentin
Lychagin
1077-1103
Abstract: For the Spencer $\delta$-cohomologies of a symbolic system we construct a spectral sequence associated with a subspace. We calculate the sequence for the systems of Cohen-Macaulay type and obtain a reduction theorem, which facilitates computation of $\delta$-cohomologies by reducing dimension of the system. Using this algebraic result we prove an efficient compatibility criterion for a system of two scalar non-linear PDEs on a manifold of any dimension in terms of (generalized) Mayer brackets.
Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces
Gil
Solanes
1105-1115
Abstract: We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.
Duality for Hopf orders
Robert
G.
Underwood;
Lindsay
N.
Childs
1117-1163
Abstract: In this paper we use duality to construct new classes of Hopf orders in the group algebra $KC_{p^3}$, where $K$ is a finite extension of $\mathbb{Q} _p$ and $C_{p^3}$ denotes the cyclic group of order $p^3$. Included in this collection is a subcollection of Hopf orders which are realizable as Galois groups.
Damped wave equation with a critical nonlinearity
Nakao
Hayashi;
Elena
I.
Kaikina;
Pavel
I.
Naumkin
1165-1185
Abstract: We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity $\begin{displaymath}\left\{ \begin{array}{c} \partial _{t}^{2}u+\partial _{t}u-\... ...u_{1}\left( x\right) ,x\in \mathbf{R}^{n}, \end{array}\right. \end{displaymath}$ where $\varepsilon >0,$ and space dimensions $n=1,2,3$. Assume that the initial data \begin{displaymath}u_{0}\in \mathbf{H}^{\delta ,0}\cap \mathbf{H}^{0,\delta },\t... ..._{1}\in \mathbf{H}^{\delta -1,0}\cap \mathbf{H}^{-1,\delta }, \end{displaymath} where $\delta >\frac{n}{2},$ weighted Sobolev spaces are $\begin{displaymath}\mathbf{H}^{l,m}=\left\{ \phi \in \mathbf{L}^{2};\left\Vert \... ...left( x\right) \right\Vert _{\mathbf{L}^{2}}<\infty \right\} , \end{displaymath}$ $\left\langle x\right\rangle =\sqrt{1+x^{2}}.$ Also we suppose that $\begin{displaymath}\lambda \theta ^{\frac{2}{n}}>0,\int u_{0}\left( x\right) dx>0, \end{displaymath}$ where $\begin{displaymath}\text{ }\theta =\int \left( u_{0}\left( x\right) +u_{1}\left( x\right) \right) dx\text{.} \end{displaymath}$ Then we prove that there exists a positive $\varepsilon _{0}$ such that the Cauchy problem above has a unique global solution $u\in \mathbf{C}\left( \left[ 0,\infty \right) ;\mathbf{H}^{\delta ,0}\right)$ satisfying the time decay property \begin{displaymath}\left\Vert u\left( t\right) -\varepsilon \theta G\left( t,x\r... ...gle t\right\rangle ^{-\frac{n}{2}\left( 1-\frac{1}{p}\right) } \end{displaymath} for all $t>0,$ $1\leq p\leq \infty ,$ where $\varepsilon \in \left( 0,\varepsilon _{0}\right] .$
Automorphisms of fiber surfaces of genus $2$, inducing the identity in cohomology
Jin-Xing
Cai
1187-1201
Abstract: Let $S$ be a complex non-singular projective surface of general type with a genus $2$ fibration and $\chi (\mathcal O_S)\geq 5$. Let $G \subset\operatorname{Aut}S$ be a non-trivial subgroup of automorphisms of $S$, inducing trivial actions on $H^i(S,\mathbb{Q})$ for all $i$. Then $\vert G\vert=2$, $K_S^2=4\chi (\mathcal O_S)$ and $q(S)=1$. Examples of such surfaces are given.
Constructive recognition of $\mathrm{PSL}(2, q)$
M.
D. E.
Conder;
C.
R.
Leedham-Green;
E.
A.
O'Brien
1203-1221
Abstract: Existing black box and other algorithms for explicitly recognising groups of Lie type over $\mathrm{GF}(q)$ have asymptotic running times which are polynomial in $q$, whereas the input size involves only $\log q$. This has represented a serious obstruction to the efficient recognition of such groups. Recently, Brooksbank and Kantor devised new explicit recognition algorithms for classical groups; these run in time that is polynomial in the size of the input, given an oracle that recognises $\mathrm{PSL}(2,q)$ explicitly. The present paper, in conjunction with an earlier paper by the first two authors, provides such an oracle. The earlier paper produced an algorithm for explicitly recognising $\mathrm{SL}(2,q)$ in its natural representation in polynomial time, given a discrete logarithm oracle for $\mathrm{GF}(q)$. The algorithm presented here takes as input a generating set for a subgroup $G$ of $\mathrm{GL}(d,F)$ that is isomorphic modulo scalars to $\mathrm{PSL}(2,q)$, where $F$ is a finite field of the same characteristic as $\mathrm{GF}(q)$; it returns the natural representation of $G$ modulo scalars. Since a faithful projective representation of $\mathrm{PSL}(2,q)$ in cross characteristic, or a faithful permutation representation of this group, is necessarily of size that is polynomial in $q$ rather than in $\log q$, elementary algorithms will recognise $\mathrm{PSL} (2,q)$ explicitly in polynomial time in these cases. Given a discrete logarithm oracle for $\mathrm{GF}(q)$, our algorithm thus provides the required polynomial time oracle for recognising $\mathrm{PSL}(2,q)$ explicitly in the remaining case, namely for representations in the natural characteristic. This leads to a partial solution of a question posed by Babai and Shalev: if $G$ is a matrix group in characteristic $p$, determine in polynomial time whether or not $O_p(G)$ is trivial.
Brownian intersection local times: Exponential moments and law of large masses
Wolfgang
König;
Peter
Mörters
1223-1255
Abstract: Consider $p$ independent Brownian motions in $\mathbb{R} ^d$, each running up to its first exit time from an open domain $B$, and their intersection local time $\ell$ as a measure on $B$. We give a sharp criterion for the finiteness of exponential moments, \begin{displaymath}\mathbb{E}\Big[\exp\Big(\sum_{i=1}^n \langle\varphi_i, \ell \rangle^{1/p}\Big) \Big],\end{displaymath} where $\varphi_1, \dots, \varphi_n$ are nonnegative, bounded functions with compact support in $B$. We also derive a law of large numbers for intersection local time conditioned to have large total mass.
Gorenstein projective dimension for complexes
Oana
Veliche
1257-1283
Abstract: We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.
Complex symmetric operators and applications
Stephan
Ramon
Garcia;
Mihai
Putinar
1285-1315
Abstract: We study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. The existence of this additional symmetry has notable implications and, in particular, it explains from a unifying point of view some classical results. We explore applications of this symmetry to Jordan canonical models, self-adjoint extensions of symmetric operators, rank-one unitary perturbations of the compressed shift, Darlington synthesis and matrix-valued inner functions, and free bounded analytic interpolation in the disk.
Invariant pre-foliations for non-resonant non-uniformly hyperbolic systems
Ernest
Fontich;
Rafael
de la Llave;
Pau
Martín
1317-1345
Abstract: Given an orbit whose linearization has invariant subspaces satisfying some non-resonance conditions in the exponential rates of growth, we prove existence of invariant manifolds tangent to these subspaces. The exponential rates of growth can be understood either in the sense of Lyapunov exponents or in the sense of exponential dichotomies. These manifolds can correspond to ``slow manifolds'', which characterize the asymptotic convergence. Let $\{x_i\}_{i \in \mathbb{N} }$ be a regular orbit of a $C^2$ dynamical system $f$. Let $S$ be a subset of its Lyapunov exponents. Assume that all the Lyapunov exponents in $S$ are negative and that the sums of Lyapunov exponents in $S$ do not agree with any Lyapunov exponent in the complement of $S.$ Denote by $E^S_{x_i}$ the linear spaces spanned by the spaces associated to the Lyapunov exponents in $S.$ We show that there are smooth manifolds $W^S_{x_i}$ such that $f(W^S_{x_i}) \subset W^S_{x_{i+1}}$ and $T_{x_i} W^S_{x_i} = E^S_{x_i}$. We establish the same results for orbits satisfying dichotomies and whose rates of growth satisfy similar non-resonance conditions. These systems of invariant manifolds are not, in general, a foliation.
Quivers with relations arising from clusters $(A_n$ case)
P.
Caldero;
F.
Chapoton;
R.
Schiffler
1347-1364
Abstract: Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let $U$ be a cluster algebra of type $A_n$. We associate to each cluster $C$ of $U$ an abelian category $\mathcal{C}_C$ such that the indecomposable objects of $\mathcal{C}_C$ are in natural correspondence with the cluster variables of $U$ which are not in $C$. We give an algebraic realization and a geometric realization of $\mathcal{C}_C$. Then, we generalize the ``denominator theorem'' of Fomin and Zelevinsky to any cluster.
Contact reduction and groupoid actions
Marco
Zambon;
Chenchang
Zhu
1365-1401
Abstract: We introduce a new method to perform reduction of contact manifolds that extends Willett's and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map $J:M \rightarrow \Gamma_0$ from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of $\Gamma_0$ on $M$, and we show that the quotients of fibers $J^{-1}(x)$ by suitable Lie subgroups $\Gamma_x$ are either contact or locally conformal symplectic manifolds with structures induced by the one on $M$. We show that Willett's reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett's reduction can be performed only at distinguished points. As an application we obtain Kostant's prequantizations of coadjoint orbits. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.