Subsmooth sets: Functional characterizations and related concepts
D.
Aussel;
A.
Daniilidis;
L.
Thibault
1275-1301
Abstract: Prox-regularity of a set (Poliquin-Rockafellar-Thibault, 2000), or its global version, proximal smoothness (Clarke-Stern-Wolenski, 1995) plays an important role in variational analysis, not only because it is associated with some fundamental properties as the local continuous differentiability of the function $\mbox{dist}\,(C;\cdot)$, or the local uniqueness of the projection mapping, but also because in the case where $C$is the epigraph of a locally Lipschitz function, it is equivalent to the weak convexity (lower-C$^{2}$ property) of the function. In this paper we provide an adapted geometrical concept, called subsmoothness, which permits an epigraphic characterization of the approximate convex functions (or lower-C$^{1}$ property). Subsmooth sets turn out to be naturally situated between the classes of prox-regular and of nearly radial sets. This latter class has been recently introduced by Lewis in 2002. We hereby relate it to the Mifflin semismooth functions.
An unusual self-adjoint linear partial differential operator
W.
N.
Everitt;
L.
Markus;
M.
Plum
1303-1324
Abstract: In an American Mathematical Society Memoir, published in 2003, the authors Everitt and Markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions, and the generation of self-adjoint differential operators in Sobolev Hilbert spaces. In the case when the differential expression has smooth coefficients on the closure of a bounded open region, in Euclidean space, and when the region has a smooth boundary, this theory leads to the construction of certain self-adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region. This present paper concerns the spectral properties of one of these unusual self-adjoint operators, sometimes called the ``Harmonic'' operator. The boundary value problems considered in the Memoir (see above) and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region; also the region is bounded and has a smooth boundary. Under these and some additional technical conditions it is shown in the Memoir, and emphasized in this present paper, that all the self-adjoint operators considered are explicitly determined on their domains by the partial differential expression; this property makes a remarkable comparison with the case of symmetric ordinary differential expressions. In the regular ordinary case the spectrum of all the self-adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation, and each eigenvalue is of finite multiplicity. Thus the essential spectrum of all these operators is empty. This spectral property extends to the present partial differential case for the classical Dirichlet and Neumann operators but not to the Harmonic operator. It is shown in this paper that the Harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane; thus the essential spectrum of this operator is not empty. Both the weak and strong formulations of the Harmonic boundary value problem are considered; these two formulations are shown to be equivalent. In the final section of the paper examples are considered which show that the Harmonic operator, defined by the methods of symplectic algebra, has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region.
Parameter-shifted shadowing property for geometric Lorenz attractors
Shin
Kiriki;
Teruhiko
Soma
1325-1339
Abstract: In this paper, we will show that any geometric Lorenz flow in a definite class satisfies the parameter-shifted shadowing property.
On adic genus and lambda-rings
Donald
Yau
1341-1348
Abstract: Sufficient conditions on a space are given which guarantee that the $K$-theory ring is an invariant of the adic genus. An immediate consequence of this result about adic genus is that for any positive integer $n$, the power series ring $\mathbf{Z} \lbrack \lbrack x_1, \ldots , x_n \rbrack \rbrack$ admits uncountably many pairwise non-isomorphic $\lambda$-ring structures.
Serre duality for non-commutative ${\mathbb{P}}^{1}$-bundles
Adam
Nyman
1349-1416
Abstract: Let $X$ be a smooth scheme of finite type over a field $K$, let $\mathcal{E}$ be a locally free $\mathcal{O}_{X}$-bimodule of rank $n$, and let $\mathcal{A}$ be the non-commutative symmetric algebra generated by $\mathcal{E}$. We construct an internal $\operatorname{Hom}$ functor, ${\underline{{\mathcal{H}}\textit{om}}_{\mathsf{Gr} \mathcal{A}}} (-,-)$, on the category of graded right $\mathcal{A}$-modules. When $\mathcal{E}$ has rank 2, we prove that $\mathcal{A}$ is Gorenstein by computing the right derived functors of ${\underline{{\mathcal{H}}\textit{om}}_{\mathsf{Gr} \mathcal{A}}} (\mathcal{O}_{X},-)$. When $X$ is a smooth projective variety, we prove a version of Serre Duality for ${\mathsf{Proj}} \mathcal{A}$ using the right derived functors of $\underset{n \to \infty}{\lim} \underline{\mathcal{H}\textit{om}}_{\mathsf{Gr} \mathcal{A}} (\mathcal{A}/\mathcal{A}_{\geq n}, -)$.
An iterative construction of Gorenstein ideals
C.
Bocci;
G.
Dalzotto;
R.
Notari;
M.
L.
Spreafico
1417-1444
Abstract: In this paper, we present a method to inductively construct Gorenstein ideals of any codimension $c.$ We start from a Gorenstein ideal $I$ of codimension $c$ contained in a complete intersection ideal $J$ of the same codimension, and we prove that under suitable hypotheses there exists a new Gorenstein ideal contained in the residual ideal $I : J.$ We compare some numerical data of the starting and the resulting Gorenstein ideals of the construction. We compare also the Buchsbaum-Eisenbud matrices of the two ideals, in the codimension three case. Furthermore, we show that this construction is independent from the other known geometrical constructions of Gorenstein ideals, providing examples.
Hyperpolygon spaces and their cores
Megumi
Harada;
Nicholas
Proudfoot
1445-1467
Abstract: Given an $n$-tuple of positive real numbers $(\alpha_1,\ldots,\alpha_n)$, Konno (2000) defines the hyperpolygon space $X(\alpha)$, a hyperkähler analogue of the Kähler variety $M(\alpha)$ parametrizing polygons in $\mathbb{R} ^3$with edge lengths $(\alpha_1,\ldots,\alpha_n)$. The polygon space $M(\alpha)$can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, $X(\alpha)$ is the hyperkähler quiver variety defined by Nakajima. A quiver variety admits a natural $\mathbb{C} ^*$-action, and the union of the precompact orbits is called the core. We study the components of the core of $X(\alpha)$, interpreting each one as a moduli space of pairs of polygons in $\mathbb{R} ^3$with certain properties. Konno gives a presentation of the cohomology ring of $X(\alpha)$; we extend this result by computing the $\mathbb{C} ^*$-equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.
Hardy space of exact forms on $\mathbb{R}^N$
Zengjian
Lou;
Alan
McIntosh
1469-1496
Abstract: We show that the Hardy space of divergence-free vector fields on $\mathbb{R}^{3}$ has a divergence-free atomic decomposition, and thus we characterize its dual as a variant of $BMO$. Using the duality result we prove a ``div-curl" type theorem: for $b$ in $L^{2}_{loc}(\mathbb{R}^{3}, \mathbb{R}^{3})$, $\sup \int b\cdot (\nabla u\times \nabla v) dx$ is equivalent to a $BMO$-type norm of $b$, where the supremum is taken over all $u, v\in W^{1,2}(\mathbb{R}^{3})$ with $\Vert\nabla u\Vert _{L^{2}}, \Vert\nabla v\Vert _{L^{2}}\le 1.$ This theorem is used to obtain some coercivity results for quadratic forms which arise in the linearization of polyconvex variational integrals studied in nonlinear elasticity. In addition, we introduce Hardy spaces of exact forms on $\mathbb{R}^N$, study their atomic decompositions and dual spaces, and establish ``div-curl" type theorems on $\mathbb{R}^N$.
Measurable Kac cohomology for bicrossed products
Saad
Baaj;
Georges
Skandalis;
Stefaan
Vaes
1497-1524
Abstract: We study the Kac cohomology for matched pairs of locally compact groups. This cohomology theory arises from the extension theory of locally compact quantum groups. We prove a measurable version of the Kac exact sequence and provide methods to compute the cohomology. We give explicit calculations in several examples using results of Moore and Wigner.
On the singular spectrum of Schrödinger operators with decaying potential
S.
Denisov;
S.
Kupin
1525-1544
Abstract: The relation between Hausdorff dimension of the singular spectrum of a Schrödinger operator and the decay of its potential has been extensively studied in many papers. In this work, we address similar questions from a different point of view. Our approach relies on the study of the so-called Krein systems. For Schrödinger operators, we show that some bounds on the singular spectrum, obtained recently by Remling and Christ-Kiselev, are optimal.
A sharp weak type $(p,p)$ inequality $(p>2)$ for martingale transforms and other subordinate martingales
Jiyeon
Suh
1545-1564
Abstract: If $(d_{n})_{n\geq 0}$ is a martingale difference sequence, $(\varepsilon_{n})_{n\geq 0}$ a sequence of numbers in $\{ 1,-1\}$, and $n$ a positive integer, then \begin{displaymath}P(\max _{0\leq m\leq n}\vert \sum_{k=0}^{m} \varepsilon_{k}d_... ...geq 1) \leq \alpha_{p}\Vert\sum_{k=0}^{n} d_{k}\Vert_{p}^{p}. \end{displaymath} Here $\alpha_{p}$ denotes the best constant. If $1\leq p\leq 2$, then $\alpha_{p}= 2/\Gamma(p+1)$ as was shown by Burkholder. We show here that $\alpha_p=p^{p-1}/2$ for the case $p > 2$, and that $p^{p-1}/2$ is also the best constant in the analogous inequality for two martingales $M$ and $N$ indexed by $[0,\infty)$, right continuous with limits from the left, adapted to the same filtration, and such that $[M,M]_t-[N,N]_t$ is nonnegative and nondecreasing in $t$. In Section 7, we prove a similar inequality for harmonic functions.
Persistence of lower dimensional tori of general types in Hamiltonian systems
Yong
Li;
Yingfei
Yi
1565-1600
Abstract: This work is a generalization to a result of J. You (1999). We study the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms. By introducing a modified linear KAM iterative scheme to deal with small divisors, we shall prove a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori.
Stable branching rules for classical symmetric pairs
Roger
Howe;
Eng-Chye
Tan;
Jeb
F.
Willenbring
1601-1626
Abstract: We approach the problem of obtaining branching rules from the point of view of dual reductive pairs. Specifically, we obtain a stable branching rule for each of $10$ classical families of symmetric pairs. In each case, the branching multiplicities are expressed in terms of Littlewood-Richardson coefficients. Some of the formulas are classical and include, for example, Littlewood's restriction rule as a special case.
The Tits boundary of a $\text{CAT}(0)$ 2-complex
Xiangdong
Xie
1627-1661
Abstract: We investigate the Tits boundary of $\text{CAT}(0)$ $2$-complexes that have only a finite number of isometry types of cells. In particular, we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the $2$-complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two $\text{CAT}(0)$ $2$-complexes are quasi-isometric, then the cores of their Tits boundaries are bi-Lipschitz.
On Bombieri's asymptotic sieve
Kevin
Ford
1663-1674
Abstract: If a sequence $(a_n)$ of non-negative real numbers has ``best possible'' distribution in arithmetic progressions, Bombieri showed that one can deduce an asymptotic formula for the sum $\sum_{n\le x} a_n \Lambda_k(n)$ for $k\ge 2$. By constructing appropriate sequences, we show that any weakening of the well-distribution property is not sufficient to deduce the same conclusion.
Powers in recurrence sequences: Pell equations
Michael
A.
Bennett
1675-1691
Abstract: In this paper, we present a new technique for determining all perfect powers in so-called Pell sequences. To be precise, given a positive nonsquare integer $D$, we show how to (practically) solve Diophantine equations of the form \begin{displaymath}x^2 - Dy^{2n} =1 \end{displaymath} in integers $x, y$ and $n \geq 2$. Our method relies upon Frey curves and corresponding Galois representations and eschews lower bounds for linear forms in logarithms. Along the way, we sharpen and generalize work of Cao, Af Ekenstam, Ljunggren and Tartakowsky on these and related questions.