Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation
Svetlana
Borovkova;
Robert
Burton;
Herold
Dehling
4261-4318
Abstract: In this paper we develop a general approach for investigating the asymptotic distribution of functionals $X_n=f((Z_{n+k})_{k\in\mathbf{Z}})$of absolutely regular stochastic processes $(Z_n)_{n\in \mathbf{Z}}$. Such functionals occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to $U$-statistics \begin{displaymath}U_n(h) =\frac{1}{{n\choose 2}}\sum_{1\leq i<j\leq n} h(X_i,X_j) \end{displaymath} with symmetric kernel $h:\mathbf{R}\times \mathbf{R}\rightarrow \mathbf{R}$. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for $U$-processes $(U_n(h))_{h}$, indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.
Uniqueness of volume-minimizing submanifolds calibrated by the first Pontryagin form
Daniel
A.
Grossman;
Weiqing
Gu
4319-4332
Abstract: One way to understand the geometry of the real Grassmann manifold $G_k(\mathbf{R}^{k+n})$ parameterizing oriented $k$-dimensional subspaces of $\mathbf{R}^{k+n}$ is to understand the volume-minimizing subvarieties in each homology class. Some of these subvarieties can be determined by using a calibration. In previous work, one of the authors calculated the set of $4$-planes calibrated by the first Pontryagin form $p_1$ on $G_k(\mathbf{R}^{k+n})$for all $k,n\geq 4$, and identified a family of mutually congruent round $4$-spheres which are consequently homologically volume-minimizing. In the present work, we associate to the family of calibrated planes a Pfaffian system on the symmetry group $SO(k+n,\mathbf R)$, an analysis of which yields a uniqueness result; namely, that any connected submanifold of $G_k(\mathbf{R}^{k+n})$ calibrated by $p_1$ is contained in one of these $4$-spheres. A similar result holds for $p_1$-calibrated submanifolds of the quotient Grassmannian $G_k^\natural(\mathbf{R}^{k+n})$ of non-oriented $k$-planes.
Representation theory and ADHM-construction on quaternion symmetric spaces
Yasuyuki
Nagatomo
4333-4355
Abstract: We determine all irreducible homogeneous bundles with anti-self-dual canonical connections on compact quaternion symmetric spaces. To deform the canonical connections, we give a relation between the representation theory and the theory of monads on the twistor space. The moduli spaces are described via the Bott-Borel-Weil Thereom. The Horrocks bundle is also generalized to higher-dimensional projective spaces.
On the semisimplicity conjecture and Galois representations
Lei
Fu
4357-4369
Abstract: The semisimplicity conjecture says that for any smooth projective scheme $X_0$ over a finite field $\mathbf{F}_q$, the Frobenius correspondence acts semisimply on $H^i(X\otimes_{\mathbf{ F}_q} \mathbf{ F}, \overline{\mathbf{ Q}}_l)$, where $\mathbf{ F}$ is an algebraic closure of $\mathbf{ F}_q$. Based on the works of Deligne and Laumon, we reduce this conjecture to a problem about the Galois representations of function fields. This reduction was also achieved by Laumon a few years ago (unpublished).
$S_{\infty }$ representations and combinatorial identities
Amitai
Regev
4371-4404
Abstract: For various probability measures on the space of the infinite standard Young tableaux we study the probability that in a random tableau, the $(i,j)^{th}$ entry equals a given number $n$. Beside the combinatorics of finite standard tableaux, the main tools here are from the Vershik-Kerov character theory of $S_{\infty}$. The analysis of these probabilities leads to many explicit combinatorial identities, some of which are related to hypergeometric series.
A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves
I.
P.
Goulden;
J.
L.
Harer;
D.
M.
Jackson
4405-4427
Abstract: We determine an expression $\xi^s_g(\gamma)$for the virtual Euler characteristics of the moduli spaces of $s$-pointed real $(\gamma=1/2$) and complex ($\gamma=1$) algebraic curves. In particular, for the space of real curves of genus $g$ with a fixed point free involution, we find that the Euler characteristic is $(-2)^{s-1}(1-2^{g-1})(g+s-2)!B_g/g!$ where $B_g$ is the $g$th Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is $(-1)^{s}(g+s-2)!B_{g+1}/(g+1)(g-1)!$ The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to count cells. The approach involves a parameter $\gamma$ that permits specialization of the formula to the real and complex cases. This suggests that $\xi^s_g(\gamma)$ itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.
Gauss sums and Kloosterman sums over residue rings of algebraic integers
Ronald
Evans
4429-4445
Abstract: Let $\mathcal{O}$ denote the ring of integers of an algebraic number field of degree $m$ which is totally and tamely ramified at the prime $p$. Write $\zeta_q= \exp(2\pi i/q)$, where $q=p^r$. We evaluate the twisted Kloosterman sum \begin{displaymath}\sum\limits_{\alpha\in(\mathcal{O}/q \mathcal{O})^*} \chi(N(\alpha)) \zeta_q^{T(\alpha)+z/N(\alpha)},\end{displaymath} where $T$ and $N$ denote trace and norm, and where $\chi$ is a Dirichlet character (mod $q$). This extends results of Salié for $m=1$ and of Yangbo Ye for prime $m$ dividing $p-1.$ Our method is based upon our evaluation of the Gauss sum \begin{displaymath}\sum\limits_{\alpha\in (\mathcal{O}/q\mathcal{O})^*} \chi(N(\alpha)) \zeta_q^{T(\alpha)},\end{displaymath} which extends results of Mauclaire for $m=1$.
Spherical classes and the Lambda algebra
Nguyen
H. V.
Hu'ng
4447-4460
Abstract: Let $\Gamma^{\wedge}= \bigoplus_k \Gamma_k^{\wedge}$ be Singer's invariant-theoretic model of the dual of the lambda algebra with $H_k(\Gamma^{\wedge})\cong Tor_k^{\mathcal{A}}(\mathbb{F} _2, \mathbb{F} _2)$, where $\mathcal{A}$ denotes the mod 2 Steenrod algebra. We prove that the inclusion of the Dickson algebra, $D_k$, into $\Gamma_k^{\wedge}$ is a chain-level representation of the Lannes-Zarati dual homomorphism \begin{displaymath}\varphi_k^*: \mathbb{F} _2\underset{\mathcal{A}}{\otimes} D_k... ..._k(\mathbb{F} _2, \mathbb{F} _2) \cong H_k(\Gamma^{\wedge})\,. \end{displaymath} The Lannes-Zarati homomorphisms themselves, $\varphi_k$, correspond to an associated graded of the Hurewicz map \begin{displaymath}H:\pi_*^s(S^0)\cong \pi_*(Q_0S^0)\to H_*(Q_0S^0)\,. \end{displaymath} Based on this result, we discuss some algebraic versions of the classical conjecture on spherical classes, which states that Only Hopf invariant one and Kervaire invariant one classes are detected by the Hurewicz homomorphism. One of these algebraic conjectures predicts that every Dickson element, i.e. element in $D_k$, of positive degree represents the homology class $0$ in $Tor^{\mathcal{A}}_k(\mathbb{F} _2,\mathbb{F} _2)$ for $k>2$. We also show that $\varphi_k^*$ factors through $\Fd\underset{\mathcal{A}}{\otimes} Ker\partial_k$, where $\partial_k : \Gamma^{\wedge}_k \to \Gamma^{\wedge}_{k-1}$ denotes the differential of $\Gamma^{\wedge}$. Therefore, the problem of determining $\mathbb{F} _2 \underset{\mathcal{A}}{\otimes} Ker\partial_k$ should be of interest.
Embeddings of $\mathrm{DI}_2$ in $\mathrm{F}_4$
Carles
Broto;
Jesper
M.
Møller
4461-4479
Abstract: We show that there is only one embedding of $\mathrm B\mathrm{DI}_2$ in $\mathrm B\mathrm{F}_4$ at the prime $p=3$, up to self-maps of $\mathrm B\mathrm{DI}_2$. We also describe the effect of the group of self-equivalences of $\mathrm B\mathrm{F}_4$ at the prime $p=3$ on this embedding and then show that the Friedlander exceptional isogeny composed with a suitable Adams map is an involution of $\mathrm B\mathrm{F}_4$ whose homotopy fixed point set coincide with $\mathrm B\mathrm{DI}_2$
Homotopy commutativity of $H$-spaces with finitely generated cohomology
Yusuke
Kawamoto;
James
P.
Lin
4481-4496
Abstract: We show that a simply connected homotopy associative and homotopy commutative mod $3$ $H$-space with finitely generated mod $3$ cohomology is homotopy equivalent to a finite product of $K({\mathbb Z},2)$, $Sp(2)$, the three-connected cover $Sp(2)\langle 3\rangle$and the homotopy fiber $Sp(2)\langle 3;3^i\rangle$of the map $[3^i]:Sp(2)\to K({\mathbb Z},3)$ for $i\ge 1$. Our result also shows that a connected $C_p$-space in the sense of Sugawara with finitely generated mod $p$ cohomology has the homotopy type of a finite product of $K({\mathbb Z},1)$, $K({\mathbb Z},2)$ and $K({\mathbb Z}/p^i,1)$ for $i\ge 1$.
Ground states and spectrum of quantum electrodynamics of nonrelativistic particles
Fumio
Hiroshima
4497-4528
Abstract: A system consisting of finitely many nonrelativistic particles bound on an external potential and minimally coupled to a massless quantized radiation field without the dipole approximation is considered. An ultraviolet cut-off is imposed on the quantized radiation field. The Hamiltonian of the system is defined as a self-adjoint operator in a Hilbert space. The existence of the ground states of the Hamiltonian is established. It is shown that there exist asymptotic annihilation and creation operators. Hence the location of the absolutely continuous spectrum of the Hamiltonian is specified.
Periodic solutions of conservation laws constructed through Glimm scheme
Hermano
Frid
4529-4544
Abstract: We present a periodic version of the Glimm scheme applicable to special classes of $2\times 2$ systems for which a simplication first noticed by Nishida (1968) and further extended by Bakhvalov (1970) and DiPerna (1973) is available. For these special classes of $2\times 2$ systems of conservation laws the simplification of the Glimm scheme gives global existence of solutions of the Cauchy problem with large initial data in $L^\infty\cap BV_{loc}(\mathbb{R} )$, for Bakhvalov's class, and in $L^\infty\cap BV(\mathbb{R} )$, in the case of DiPerna's class. It may also happen that the system is in Bakhvalov's class only at a neighboorhood $\mathcal{V}$ of a constant state, as it was proved for the isentropic gas dynamics by DiPerna (1973), in which case the initial data is taken in $L^\infty\cap BV(\mathbb{R} )$ with $\text{TV}\,(U_0)<\text{const.}$, for some constant which is $O((\gamma-1)^{-1})$ for the isentropic gas dynamics systems. For periodic initial data, our periodic formulation establishes that the periodic solutions so constructed, $u(\cdot ,t)$, are uniformly bounded in $L^\infty\cap BV([0,\ell])$, for all $t>0$, where $\ell$ is the period. We then obtain the asymptotic decay of these solutions by applying a theorem of Chen and Frid in (1999) combined with a compactness theorem of DiPerna in (1983). The question about the decay of Nishida's solution was proposed by Glimm and Lax in (1970) and has remained open since then. The classes considered include the $p$-systems with $p(v)=\gamma v^{-\gamma}$, $-1<\gamma<+\infty$, $\gamma\ne0$, which, for $\gamma\ge 1$, model isentropic gas dynamics in Lagrangian coordinates.
Boundedness and differentiability for nonlinear elliptic systems
Jana
Björn
4545-4565
Abstract: We consider the elliptic system $\operatorname{div} (\mathcal{A}^j (x,u,\nabla u)) = \mathcal{B}^j (x,u,\nabla u)$, $j=1,\ldots,N,$and an obstacle problem for a similar system of variational inequalities. The functions $\mathcal{A}^j$ and $\mathcal{B}^j$ satisfy certain ellipticity and boundedness conditions with a $p$-admissible weight $w$ and exponent $1<p\le2$. The growth of $\mathcal{B}^j$ in $\vert\nabla u\vert$ and $\vert u\vert$ is of order $p-1$. We show that weak solutions of the above systems are locally bounded and differentiable almost everywhere in the classical sense.
Unbounded components of the singular set of the distance function in $\mathbb R^n$
Piermarco
Cannarsa;
Roberto
Peirone
4567-4581
Abstract: Given a closed set $F\subseteq \mathbb{R}^{n}$, the set $\Sigma _{F}$ of all points at which the metric projection onto $F$ is multi-valued is nonempty if and only if $F$ is nonconvex. The authors analyze such a set, characterizing the unbounded connected components of $\Sigma _{F}$. For $F$ compact, the existence of an asymptote for any unbounded component of $\Sigma _{F}$ is obtained.
The Bott--Borel--Weil Theorem for direct limit groups
Loki
Natarajan;
Enriqueta
Rodríguez-Carrington;
Joseph
A.
Wolf
4583-4622
Abstract: We show how highest weight representations of certain infinite dimensional Lie groups can be realized on cohomology spaces of holomorphic vector bundles. This extends the classical Bott-Borel-Weil Theorem for finite-dimensional compact and complex Lie groups. Our approach is geometric in nature, in the spirit of Bott's original generalization of the Borel-Weil Theorem. The groups for which we prove this theorem are strict direct limits of compact Lie groups, or their complexifications. We previously proved that such groups have an analytic structure. Our result applies to most of the familiar examples of direct limits of classical groups. We also introduce new examples involving iterated embeddings of the classical groups and see exactly how our results hold in those cases. One of the technical problems here is that, in general, the limit Lie algebras will have root systems but need not have root spaces, so we need to develop machinery to handle this somewhat delicate situation.
Arithmetic rigidity and units in group rings
F.
E. A.
Johnson
4623-4635
Abstract: For any finite group $G$ the group $U(\mathbf{Z}[G])$ of units in the integral group ring $\mathbf{Z}[G]$ is an arithmetic group in a reductive algebraic group, namely the Zariski closure of $\mathbf{SL}_1(\mathbf{Q}[G])$. In particular, the isomorphism type of the $\mathbf{Q}$-algebra $\mathbf{Q}[G]$ determines the commensurability class of $U(\mathbf{Z}[G])$; we show that, to a large extent, the converse is true. In fact, subject to a certain restriction on the $\mathbf{Q}$-representations of $G$ the converse is exactly true.
Isometries of Hilbert $C^*$-modules
Baruch
Solel
4637-4660
Abstract: Let $X$ and $Y$ be right, full, Hilbert $C^*$-modules over the algebras $A$ and $B$ respectively and let $T:X\to Y$ be a linear surjective isometry. Then $T$ can be extended to an isometry of the linking algebras. $T$ then is a sum of two maps: a (bi-)module map (which is completely isometric and preserves the inner product) and a map that reverses the (bi-)module actions. If $A$(or $B$) is a factor von Neumann algebra, then every isometry $T:X\to Y$ is either a (bi-)module map or reverses the (bi-)module actions.
Metric properties of the group of area preserving diffeomorphisms
Michel
Benaim;
Jean-Marc
Gambaudo
4661-4672
Abstract: Area preserving diffeomorphisms of the 2-disk which are identity near the boundary form a group ${\mathcal D}_2$ which can be equipped, using the $L^2$-norm on its Lie algebra, with a right invariant metric. With this metric the diameter of ${\mathcal D}_2$ is infinite. In this paper we show that ${\mathcal D}_2$ contains quasi-isometric embeddings of any finitely generated free group and any finitely generated abelian free group.
Generic Finiteness for Dziobek Configurations
Richard
Moeckel
4673-4686
Abstract: The goal of this paper is to show that for almost all choices of $n$ masses, $m_i$, there are only finitely many central configurations of the Newtonian $n$-body problem for which the bodies span a space of dimension $n-2$ (such a central configuration is called a Dziobek configuration). The result applies in particular to two-dimensional configurations of four bodies and three-dimensional configurations of five bodies.