Higher homotopy commutativity of $H$-spaces and the permuto-associahedra
Yutaka
Hemmi;
Yusuke
Kawamoto
3823-3839
Abstract: In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an $A_n$-space. To give the definition, we use polyhedra called the permuto-associahedra which are constructed by Kapranov. We also show that if a connected $A_p$-space has the finitely generated mod $p$ cohomology for a prime $p$ and the multiplication of it is homotopy commutative of the $p$-th order, then it has the mod $p$ homotopy type of a finite product of Eilenberg-Mac Lane spaces $K(\mathbb{Z},1)$s, $K(\mathbb{Z},2)$s and $K(\mathbb{Z}/p^i,1)$s for $i\ge 1$.
Latroids and their representation by codes over modules
Dirk
Vertigan
3841-3868
Abstract: It has been known for some time that there is a connection between linear codes over fields and matroids represented over fields. In fact a generator matrix for a linear code over a field is also a representation of a matroid over that field. There are intimately related operations of deletion, contraction, minors and duality on both the code and the matroid. The weight enumerator of the code is an evaluation of the Tutte polynomial of the matroid, and a standard identity relating the Tutte polynomials of dual matroids gives rise to a MacWilliams identity relating the weight enumerators of dual codes. More recently, codes over rings and modules have been considered, and MacWilliams type identities have been found in certain cases. In this paper we consider codes over rings and modules with code duality based on a Morita duality of categories of modules. To these we associate latroids, defined here. We generalize notions of deletion, contraction, minors and duality, on both codes and latroids, and examine all natural relations among these. We define generating functions associated with codes and latroids, and prove identities relating them, generalizing above-mentioned generating functions and identities.
Quantum deformations of fundamental groups of oriented $3$-manifolds
Uwe
Kaiser
3869-3880
Abstract: We compute two-term skein modules of framed oriented links in oriented $3$-manifolds. They contain the self-writhe and total linking number invariants of framed oriented links in a universal way. The relations in a natural presentation of the skein module are interpreted as monodromies in the space of immersions of circles into the $3$-manifold.
The Deligne complex for the four-strand braid group
Ruth
Charney
3881-3897
Abstract: This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on $\mathbb C^n$. A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend those results to another class of Coxeter groups. The key technical result is that the spherical Deligne complex for the 4-strand braid group is CAT(1).
Surface superconductivity in $3$ dimensions
Xing-Bin
Pan
3899-3937
Abstract: We study the Ginzburg-Landau system for a superconductor occupying a $3$-dimensional bounded domain, and improve the estimate of the upper critical field $H_{C_{3}}$ obtained by K. Lu and X. Pan in J. Diff. Eqns., 168 (2000), 386-452. We also analyze the behavior of the order parameters. We show that, under an applied magnetic field lying below and not far from $H_{C_{3}}$, order parameters concentrate in a vicinity of a sheath of the surface that is tangential to the applied field, and exponentially decay both in the normal and tangential directions away from the sheath in the $L^{2}$sense. As the applied field decreases further but keeps in between and away from $H_{C_{2}}$ and $H_{C_{3}}$, the superconducting sheath expands but does not cover the entire surface, and superconductivity at the surface portion orthogonal to the applied field is always very weak. This phenomenon is significantly different to the surface superconductivity on a cylinder of infinite height studied by X. Pan in Comm. Math. Phys., 228 (2002), 327-370, where under an axial applied field lying in-between $H_{C_{2}}$ and $H_{C_{3}}$ the entire surface is in the superconducting state.
Identities of graded algebras and codimension growth
Yu.
A.
Bahturin;
M.
V.
Zaicev
3939-3950
Abstract: Let $A=\oplus_{g\in G}A_g$ be a $G$-graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component $A_e$ to that of the whole of $A$, in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where $A$ is finite dimensional and $A_e$ has polynomial growth.
When does the subadditivity theorem for multiplier ideals hold?
Shunsuke
Takagi;
Kei-ichi
Watanabe
3951-3961
Abstract: Demailly, Ein and Lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals. We prove that, in the two-dimensional case, the subadditivity theorem holds on log terminal singularities. However, in the higher dimensional case, we have several counterexamples. We consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three-dimensional toric ring.
A modified Brauer algebra as centralizer algebra of the unitary group
Alberto
Elduque
3963-3983
Abstract: The centralizer algebra of the action of $U(n)$ on the real tensor powers $\otimes_\mathbb{R}^r V$ of its natural module, $V=\mathbb{C}^n$, is described by means of a modification in the multiplication of the signed Brauer algebras. The relationships of this algebra with the invariants for $U(n)$ and with the decomposition of $\otimes_\mathbb{R}^r V$ into irreducible submodules is considered.
On the asymptotic behavior of a complete bounded minimal surface in $\mathbb{R}^3$
Francisco
Martín;
Santiago
Morales
3985-3994
Abstract: In this paper we construct an example of a complete minimal disk which is properly immersed in a ball of $\mathbb{R} ^3$.
Flat holomorphic connections on principal bundles over a projective manifold
Indranil
Biswas;
S.
Subramanian
3995-4018
Abstract: Let $G$ be a connected complex linear algebraic group and $R_u(G)$ its unipotent radical. A principal $G$-bundle $E_G$ over a projective manifold $M$ will be called polystable if the associated principal $G/R_u(G)$-bundle is so. A $G$-bundle $E_G$ over $M$ is polystable with vanishing characteristic classes of degrees one and two if and only if $E_G$ admits a flat holomorphic connection with the property that the image in $G/R_u(G)$ of the monodromy of the connection is contained in a maximal compact subgroup of $G/R_u(G)$.
Quadratic forms and Pfister neighbors in characteristic 2
Detlev
W.
Hoffmann;
Ahmed
Laghribi
4019-4053
Abstract: We study Pfister neighbors and their characterization over fields of characteristic $2$, where we include the case of singular forms. We give a somewhat simplified proof of a theorem of Fitzgerald which provides a criterion for when a nonsingular quadratic form $q$ is similar to a Pfister form in terms of the hyperbolicity of this form over the function field of a form $\varphi$ which is dominated by $q$. From this, we derive an analogue in characteristic $2$ of a result by Knebusch saying that, in characteristic $\neq 2$, a form is a Pfister neighbor if its anisotropic part over its own function field is defined over the base field. Our result includes certain cases of singular forms, but we also give examples which show that Knebusch's result generally fails in characteristic $2$ for singular forms. As an application, we characterize certain forms of height $1$ in the sense of Knebusch whose quasi-linear parts are of small dimension. We also develop some of the basics of a theory of totally singular quadratic forms. This is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper.
Conservation laws for a class of third order evolutionary differential systems
Sung Ho
Wang
4055-4073
Abstract: Conservation laws of third order quasi-linear scalar evolution equations are studied via exterior differential system and characteristic cohomology. We find a subspace of 2-forms in the infinite prolongation space in which every conservation law has a unique representative. Analysis of the structure of this subspace based upon the symbol of the differential equation leads to a universal integrability condition for an evolution equation to admit any higher order (weight) conservation laws. As an example, we give a complete classification of a class of evolution equations which admit conservation laws of the first three consecutive weights $-1$, $1$, $3$. The differential system describing the flow of a curve in the plane by the derivative of its curvature with respect to the arc length is also shown to exhibit the KdV property, i.e., an infinite sequence of conservation laws of distinct weights.
Homotopy groups of $K$-contact toric manifolds
Eugene
Lerman
4075-4083
Abstract: Contact toric manifolds of Reeb type are a subclass of contact toric manifolds which have the property that they are classified by the images of the associated moment maps. We compute their first and second homotopy group terms of the images of the moment map. We also explain why they are $K$-contact.
Green's functions for elliptic and parabolic equations with random coefficients II
Joseph
G.
Conlon
4085-4142
Abstract: This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green's functions for the equations are then random variables. Regularity properties for expectation values of Green's functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green's function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.
On the classification of full factors of type III
Dimitri
Shlyakhtenko
4143-4159
Abstract: We introduce a new invariant $\mathscr{S}(M)$ for type III factors $M$ with no almost-periodic weights. We compute this invariant for certain free Araki-Woods factors. We show that Connes' invariant $\tau$cannot distinguish all isomorphism classes of free Araki-Woods factors. We show that there exists a continuum of mutually non-isomorphic free Araki-Woods factors, each without almost-periodic weights.
Dual Radon transforms on affine Grassmann manifolds
Fulton
B.
Gonzalez;
Tomoyuki
Kakehi
4161-4180
Abstract: Fix $0 \leq p < q \leq n-1$, and let $G(p,n)$ and $G(q,n)$denote the affine Grassmann manifolds of $p$- and $q$-planes in $\mathbb{R} ^n$. We investigate the Radon transform $\mathcal{R}^{(q,p)} : C^{\infty} (G(q,n)) \to C^{\infty} (G(p,n))$associated with the inclusion incidence relation. For the generic case $\dim G(q,n) < \dim G(p,n)$ and $p+q > n$, we will show that the range of this transform is given by smooth functions on $G(p,n)$ annihilated by a system of Pfaffian type differential operators. We also study aspects of the exceptional case $p+q =n$.
Varieties of tori and Cartan subalgebras of restricted Lie algebras
Rolf
Farnsteiner
4181-4236
Abstract: This paper investigates varieties of tori and Cartan subalgebras of a finite-dimensional restricted Lie algebra $(\mathfrak{g},[p])$, defined over an algebraically closed field $k$ of positive characteristic $p$. We begin by showing that schemes of tori may be used as a tool to retrieve results by A. Premet on regular Cartan subalgebras. Moreover, they give rise to principal fibre bundles, whose structure groups coincide with the Weyl groups in case $\mathfrak{g}= \operatorname{Lie}(\mathcal{G})$ is the Lie algebra of a smooth group $\mathcal{G}$. For solvable Lie algebras, varieties of tori are full affine spaces, while simple Lie algebras of classical or Cartan type cannot have varieties of this type. In the final sections the quasi-projective variety of Cartan subalgebras of minimal dimension ${\rm rk}(\mathfrak{g})$ is shown to be irreducible of dimension $\dim_k\mathfrak{g}-{\rm rk}(\mathfrak{g})$, with Premet's regular Cartan subalgebras belonging to the regular locus.
Elliptic Apostol sums and their reciprocity laws
Shinji
Fukuhara;
Noriko
Yui
4237-4254
Abstract: We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter $\tau$ having positive imaginary part. When $\tau\to i\infty$, these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable $\tau$. We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).