On the variety generated by all nilpotent lattice-ordered groups
V.
V.
Bludov;
A.
M. W.
Glass
5179-5192
Abstract: In 1974, J. Martinez introduced the variety ${\mathcal W}$ of weakly Abelian lattice-ordered groups; it is defined by the identity $\displaystyle x^{-1}(y\vee 1)x\vee (y\vee 1)^2=(y\vee 1)^2.$
Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds
John
Douglas
Moore
5193-5256
Abstract: The purpose of this article is to study conformal harmonic maps $f:\Sigma \rightarrow M$, where $\Sigma$ is a closed Riemann surface and $M$ is a compact Riemannian manifold of dimension at least four. Such maps define parametrized minimal surfaces, possibly with branch points. We show that when the ambient manifold $M$ is given a generic metric, all prime closed parametrized minimal surfaces are free of branch points, and are as Morse nondegenerate as allowed by the group of automorphisms of $\Sigma$. They are Morse nondegenerate in the usual sense if $\Sigma$ has genus at least two, lie on two-dimensional nondegenerate critical submanifolds if $\Sigma$ has genus one, and on six-dimensional nondegenerate critical submanifolds if $\Sigma$ has genus zero.
The 3-manifold recognition problem
Robert
J.
Daverman;
Thomas
L.
Thickstun
5257-5270
Abstract: We introduce a natural Relative Simplicial Approximation Property for maps from a 2-cell to a generalized 3-manifold and prove that, modulo the Poincaré Conjecture, 3-manifolds are precisely the generalized 3-manifolds satisfying this approximation property. The central technical result establishes that every generalized 3-manifold with this Relative Simplicial Approximation Property is the cell-like image of some generalized 3-manifold having just a 0-dimensional set of nonmanifold singularities.
Low-pass filters and representations of the Baumslag Solitar group
Dorin
Ervin
Dutkay
5271-5291
Abstract: We analyze representations of the Baumslag Solitar group $\displaystyle BS(1,N)=\langle u,t\,\vert\,utu^{-1}=t^N\rangle$ that admit wavelets and show how such representations can be constructed from a given low-pass filter. We describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations. As another application, we construct a Fourier transform for some Hausdorff measures.
Sign-changing critical points from linking type theorems
M.
Schechter;
W.
Zou
5293-5318
Abstract: In this paper, the relationships between sign-changing critical point theorems and the linking type theorems of M. Schechter and the saddle point theorems of P. Rabinowitz are established. The abstract results are applied to the study of the existence of sign-changing solutions for the nonlinear Schrödinger equation $-\Delta u +V(x)u = f(x, u), u \in H^1({\mathbf{R}}^N),$ where $f(x, u)$ is a Carathéodory function. Problems of jumping or oscillating nonlinearities and of double resonance are considered.
Scattering theory for the elastic wave equation in perturbed half-spaces
Mishio
Kawashita;
Wakako
Kawashita;
Hideo
Soga
5319-5350
Abstract: In this paper we consider the linear elastic wave equation with the free boundary condition (the Neumann condition), and formulate a scattering theory of the Lax and Phillips type and a representation of the scattering kernel. We are interested in surface waves (the Rayleigh wave, etc.) connected closely with situations of boundaries, and make the formulations intending to extract this connection. The half-space is selected as the free space, and making dents on the boundary is considered as a perturbation from the flat one. Since the lacuna property for the solutions in the outgoing and incoming spaces does not hold because of the existence of the surface waves, instead of it, certain decay estimates for the free space solutions and a weak version of the Morawetz arguments are used to formulate the scattering theory. We construct the representation of the scattering kernel with outgoing scattered plane waves. In this step, again because of the existence of the surface waves, we need to introduce new outgoing and incoming conditions for the time dependent solutions to ensure uniqueness of the solutions. This introduction is essential to show the representation by reasoning similar to the case of the reduced wave equation.
Boundary relations and their Weyl families
Vladimir
Derkach;
Seppo
Hassi;
Mark
Malamud;
Henk
de Snoo
5351-5400
Abstract: The concepts of boundary relations and the corresponding Weyl families are introduced. Let $S$ be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space $\mathfrak{H}$, let $\mathcal{H}$ be an auxiliary Hilbert space, let $\displaystyle J_\mathfrak{H}=\begin{pmatrix}0&-iI_\mathfrak{H} iI_\mathfrak{H} & 0\end{pmatrix},$ and let $J_\mathcal{H}$ be defined analogously. A unitary relation $\Gamma$ from the Krein space $(\mathfrak{H}^2,J_\mathfrak{H})$ to the Krein space $(\mathcal{H}^2,J_\mathcal{H})$ is called a boundary relation for the adjoint $S^*$ if $\ker \Gamma=S$. The corresponding Weyl family $M(\lambda)$ is defined as the family of images of the defect subspaces $\widehat{\mathfrak{N}}_\lambda$, $\lambda\in \mathbb{C}\setminus\mathbb{R}$, under $\Gamma$. Here $\Gamma$ need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space $\mathcal{H}$ and the class of unitary relations $\Gamma:(\mathfrak{H}^2,J_\mathfrak{H})\to(\mathcal{H}^2,J_\mathcal{H})$, it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every $\mathcal{H}$-valued maximal dissipative (for $\lambda\in\mathbb{C}_+$) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.
Hilbert functions of points on Schubert varieties in the symplectic Grassmannian
Sudhir
R.
Ghorpade;
K.
N.
Raghavan
5401-5423
Abstract: We give an explicit combinatorial description of the multiplicity as well as the Hilbert function of the tangent cone at any point on a Schubert variety in the symplectic Grassmannian.
The braid index is not additive for the connected sum of 2-knots
Seiichi
Kamada;
Shin
Satoh;
Manabu
Takabayashi
5425-5439
Abstract: Any $2$-dimensional knot $K$ can be presented in a braid form, and its braid index, ${Braid}(K)$, is defined. For the connected sum $K_1\char93 K_2$ of $2$-knots $K_1$ and $K_2$, it is easily seen that ${Braid}(K_1\char93 K_2)\leq {B}(K_1) + {B}(K_2) -1$ holds. Birman and Menasco proved that the braid index (minus one) is additive for the connected sum of $1$-dimensional knots; the equality holds for $1$-knots. We prove that the equality does not hold for $2$-knots unless $K_1$ or $K_2$ is a trivial $2$-knot. We also prove that the $2$-knot obtained from a granny knot by Artin's spinning is of braid index $4$, and there are infinitely many $2$-knots of braid index $4$.
Deformation theory of abelian categories
Wendy
Lowen;
Michel
Van den Bergh
5441-5483
Abstract: In this paper we develop the basic infinitesimal deformation theory of abelian categories. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian categories. We show that various basic properties are preserved under flat deformations, and we construct several equivalences between deformation problems.
On a class of special linear systems of $\mathbb{P}^3$
Antonio
Laface;
Luca
Ugaglia
5485-5500
Abstract: In this paper we deal with linear systems of $\mathbb{P}^3$ through fat points. We consider the behavior of these systems under a cubic Cremona transformation that allows us to produce a class of special systems which we conjecture to be the only ones.
Fourier expansions of functions with bounded variation of several variables
Leonardo
Colzani
5501-5521
Abstract: In the first part of the paper we establish the pointwise convergence as $t\rightarrow +\infty$ for convolution operators $\int_{\mathbb{R}^{d}}t^{d}K\left( ty\right) \varphi (x-y)dy$ under the assumptions that $\varphi (y)$ has integrable derivatives up to an order $\alpha$ and that $\left\vert K(y)\right\vert \leq c\left( 1+\left\vert y\right\vert \right) ^{-\beta }$ with $\alpha +\beta >d$. We also estimate the Hausdorff dimension of the set where divergence may occur. In particular, when the kernel is the Fourier transform of a bounded set in the plane, we recover a two-dimensional analog of the Dirichlet theorem on the convergence of Fourier series of functions with bounded variation. In the second part of the paper we prove an equiconvergence result between Fourier integrals on euclidean spaces and expansions in eigenfunctions of elliptic operators on manifolds, which allows us to transfer some of the results proved for Fourier integrals to eigenfunction expansions. Finally, we present some examples of different behaviors between Fourier integrals, Fourier series and spherical harmonic expansions.
Singularities of linear systems and the Waring problem
Massimiliano
Mella
5523-5538
Abstract: The Waring problem for homogeneous forms asks for additive decomposition of a form $f$ into powers of linear forms. A classical problem is to determine when such a decomposition is unique. In this paper we answer this question when the degree of $f$ is greater than the number of variables. To do this we translate the algebraic statement into a geometric one concerning the singularities of linear systems of $\mathbb{P}^n$ with assigned singularities.
Blaschke- and Minkowski-endomorphisms of convex bodies
Markus
Kiderlen
5539-5564
Abstract: We consider maps of the family of convex bodies in Euclidean $d$-dimensional space into itself that are compatible with certain structures on this family: A Minkowski-endomorphism is a continuous, Minkowski-additive map that commutes with rotations. For $d\ge 3$, a representation theorem for such maps is given, showing that they are mixtures of certain prototypes. These prototypes are obtained by applying the generalized spherical Radon transform to support functions. We give a complete characterization of weakly monotonic Minkowski-endomorphisms. A corresponding theory is developed for Blaschke-endomorphisms, where additivity is now understood with respect to Blaschke-addition. Using a special mixed volume, an adjoining operator can be introduced. This operator allows one to identify the class of Blaschke-endomorphisms with the class of weakly monotonic, non-degenerate and translation-covariant Minkowski-endomorphisms. The following application is also shown: If a (weakly monotonic and) non-trivial endomorphism maps a convex body to a homothet of itself, then this body must be a ball.
Varieties with small discriminant variety
Antonio
Lanteri;
Roberto
Muñoz
5565-5585
Abstract: Let $X$ be a smooth complex projective variety, let $L$ be an ample and spanned line bundle on $X$, $V\subseteq H^{0}(X,L)$ defining a morphism $\phi _{V}:X \to \mathbb{P}^{N}$ and let $\mathcal{D}(X,V)$ be its discriminant locus, the variety parameterizing the singular elements of $\vert V\vert$. We present two bounds on the dimension of $\mathcal{D}(X,V)$ and its main component relying on the geometry of $\phi _{V}(X) \subset \mathbb{P}^{N}$. Classification results for triplets $(X,L,V)$ reaching the bounds as well as significant examples are provided.
Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains
Zdzislaw
Brzezniak;
Yuhong
Li
5587-5629
Abstract: We introduce a notion of an asymptotically compact (AC) random dynamical system (RDS). We prove that for an AC RDS the $\Omega$-limit set $\Omega_B(\omega)$ of any bounded set $B$ is nonempty, compact, strictly invariant and attracts the set $B$. We establish that the $2$D Navier Stokes Equations (NSEs) in a domain satisfying the Poincaré inequality perturbed by an additive irregular noise generate an AC RDS in the energy space $\mathrm{H}$. As a consequence we deduce existence of an invariant measure for such NSEs. Our study generalizes on the one hand the earlier results by Flandoli-Crauel (1994) and Schmalfuss (1992) obtained in the case of bounded domains and regular noise, and on the other hand the results by Rosa (1998) for the deterministic NSEs.
Corrigendum to ``West's problem on equivariant hyperspaces and Banach-Mazur compacta''
Sergey
Antonyan
5631-5633
Corrections to ``Involutions fixing $\mathbb{RP}^{\text{odd}}\sqcup P(h,i)$, II''
Bo
Chen;
Zhi
Lü
5635-5638