On the minimal free resolution of $n+1$ general forms
J.
Migliore;
R.
M.
Miró-Roig
1-36
Abstract: Let $R = k[x_1,\dots,x_n]$ and let $I$ be the ideal of $n+1$generically chosen forms of degrees $d_1 \leq \dots \leq d_{n+1}$. We give the precise graded Betti numbers of $R/I$ in the following cases: $n=3$; $n=4$ and $\sum_{i=1}^5 d_i$ is even; $n=4$, $\sum_{i=1}^{5} d_i$ is odd and $d_2 + d_3 + d_4 < d_1 + d_5 + 4$; $n$ is even and all generators have the same degree, $a$, which is even; $(\sum_{i=1}^{n+1} d_i) -n$ is even and $d_2 + \dots + d_n < d_1 + d_{n+1} + n$; $(\sum_{i=1}^{n+1} d_i) - n$ is odd, $n \geq 6$ is even, $d_2 + \dots+d_n < d_1 + d_{n+1} + n$ and $d_1 + \dots + d_n - d_{n+1} - n \gg 0$. We give very good bounds on the graded Betti numbers in many other cases. We also extend a result of M. Boij by giving the graded Betti numbers for a generic compressed Gorenstein algebra (i.e., one for which the Hilbert function is maximal, given $n$ and the socle degree) when $n$ is even and the socle degree is large. A recurring theme is to examine when and why the minimal free resolution may be forced to have redundant summands. We conjecture that if the forms all have the same degree, then there are no redundant summands, and we present some evidence for this conjecture.
Iitaka's fibrations via multiplier ideals
Shigeharu
Takayama
37-47
Abstract: We give a new characterization of Iitaka's fibration of algebraic varieties associated to line bundles. Introducing an ``intersection number'' of line bundles and curves by using the notion of multiplier ideal sheaves, Iitaka's fibration can be regarded as a ``numerically trivial fibration'' in terms of this intersection theory.
Nondegenerate multidimensional matrices and instanton bundles
Laura
Costa;
Giorgio
Ottaviani
49-55
Abstract: In this paper we prove that the moduli space of rank $2n$ symplectic instanton bundles on ${\mathbb{P} ^{2n+1}}$, defined from the well-known monad condition, is affine. This result was not known even in the case $n=1$, where by Atiyah, Drinfeld, Hitchin, and Manin in 1978 the real instanton bundles correspond to self-dual Yang Mills $Sp(1)$-connections over the $4$-dimensional sphere. The result is proved as a consequence of the existence of an invariant of the multidimensional matrices representing the instanton bundles.
Inverse problem for upper asymptotic density
Renling
Jin
57-78
Abstract: For a set $A$ of natural numbers, the structural properties are described when the upper asymptotic density of $2A+\{0,1\}$achieves the infimum of the upper asymptotic densities of all sets of the form $2B+\{0,1\}$, where the upper asymptotic density of $B$ is greater than or equal to the upper asymptotic density of $A$. As a corollary, we prove that if the upper asymptotic density of $A$ is less than $1$and the upper asymptotic density of $2A+\{0,1\}$ achieves the infimum, then the lower asymptotic density of $A$ must be $0$.
Abelian groups with layered tiles and the sumset phenomenon
Renling
Jin;
H.
Jerome
Keisler
79-97
Abstract: We prove a generalization of the main theorem in Jin, The sumset phenomenon, about the sumset phenomenon in the setting of an abelian group with layered tiles of cell measures. Then we give some applications of the theorem for multi-dimensional cases of the sumset phenomenon. Several examples are given in order to show that the applications obtained are not vacuous and cannot be improved in various directions. We also give a new proof of Shnirel'man's theorem to illustrate a different approach (which uses the sumset phenomenon) to some combinatorial problems.
On the Jacobi group and the mapping class group of $S^3\times S^3$
Nikolai
A.
Krylov
99-117
Abstract: The paper contains a proof that the mapping class group of the manifold $S^3\times S^3$ is isomorphic to a central extension of the (full) Jacobi group $\Gamma^J$by the group of 7-dimensional homotopy spheres. Using a presentation of the group $\Gamma^J$ and the $\mu$-invariant of the homotopy spheres, we give a presentation of this mapping class group with generators and defining relations. We also compute the cohomology of the group $\Gamma^J$ and determine 2-cocycles that correspond to the mapping class group of $S^3\times S^3$.
A higher Lefschetz formula for flat bundles
Moulay-Tahar
Benameur
119-142
Abstract: In this paper, we prove a fixed point formula for flat bundles. To this end, we use cyclic cocycles which are constructed out of closed invariant currents. We show that such cyclic cocycles are equivariant with respect to isometric longitudinal actions of compact Lie groups. This enables us to prove fixed point formulae in the cyclic homology of the smooth convolution algebra of the foliation.
Levi foliations in pseudoconvex boundaries and vector fields that commute approximately with $\bar\partial$
Emil
J.
Straube;
Marcel
K.
Sucheston
143-154
Abstract: Boas and Straube proved a general sufficient condition for global regularity of the $\bar\partial$-Neumann problem in terms of families of vector fields that commute approximately with $\bar\partial$. In this paper, we study the existence of these vector fields on a compact subset of the boundary whose interior is foliated by complex manifolds. This question turns out to be closely related to properties of interest from the point of view of foliation theory.
On hypersphericity of manifolds with finite asymptotic dimension
A.
N.
Dranishnikov
155-167
Abstract: We prove the following embedding theorems in the coarse geometry: \begin{theorem1}Every metric space$X$\space with bounded geometry whose asympt... ... embedding into the product of$n+1$\space locally finite trees. \end{theorem1} \begin{theorem2}Every metric space$X$\space with bounded geometry whose asympt... ...ding into a non-positively curved manifold of dimension$2n+2$. \end{theorem2} The Corollary is used in the proof of the following. \begin{theorem3}For every uniformly contractible manifold$X$whose asymptotic d... ...\mathbf{R}^{n}$\space is integrally hyperspherical for some$n$. \end{theorem3} Theorem B together with a theorem of Gromov-Lawson implies the result, previously proven by G. Yu (1998), which states that an aspherical manifold whose fundamental group has a finite asymptotic dimension cannot carry a metric of positive scalar curvature. We also prove that if a uniformly contractible manifold $X$ of bounded geometry is large scale uniformly embeddable into a Hilbert space, then $X$ is stably integrally hyperspherical.
Biharmonic lifts by means of pseudo-Riemannian submersions in dimension three
Miguel
A.
Javaloyes Victoria;
Miguel
A.
Meroño Bayo
169-176
Abstract: We study the total lifts of curves by means of a submersion $\pi:M_s^3\rightarrow B_r^2$ that satisfy the condition $\Delta H=\lambda H$analyzing, in particular, the cases in which the submersion has totally geodesic fibres or integrable horizontal distribution. We also consider in detail the case $\lambda=0$ (biharmonic lifts). Moreover, we obtain a biharmonic lift in ${\mathbb R}^3$ by means of a Riemannian submersion that has non-constant mean curvature, getting so a counterexample to the Chen conjecture for ${\mathbb R}^3$ with a non-flat Riemannian metric.
Integration of multivalued operators and cyclic submonotonicity
Aris
Daniilidis;
Pando
Georgiev;
Jean-Paul
Penot
177-195
Abstract: We introduce a notion of cyclic submonotonicity for multivalued operators from a Banach space $X$ to its dual. We show that if the Clarke subdifferential of a locally Lipschitz function is strictly submonotone on an open subset $U$ of $X$, then it is also maximal cyclically submonotone on $U$, and, conversely, that every maximal cyclically submonotone operator on $U$ is the Clarke subdifferential of a locally Lipschitz function, which is unique up to a constant if $U$ is connected. In finite dimensions these functions are exactly the lower C$^{1}$ functions considered by Spingarn and Rockafellar.
Linear parabolic equations with strong singular potentials
Jerome
A.
Goldstein;
Qi
S.
Zhang
197-211
Abstract: Using an extension of a recent method of Cabré and Martel (1999), we extend the blow-up and existence result in the paper of Baras and Goldstein (1984) to parabolic equations with variable leading coefficients under almost optimal conditions on the singular potentials. This problem has been left open in Baras and Goldstein. These potentials lie at a borderline case where standard theories such as the strong maximum principle and boundedness of weak solutions fail. Even in the special case when the leading operator is the Laplacian, we extend a recent result in Cabré and Martel from bounded smooth domains to unbounded nonsmooth domains.
Uniqueness for the determination of sound-soft defects in an inhomogeneous planar medium by acoustic boundary measurements
Luca
Rondi
213-239
Abstract: We consider the inverse problem of determining shape and location of sound-soft defects inside a known planar inhomogeneous and anisotropic medium through acoustic imaging at low frequency. In order to determine the defects, we perform acoustic boundary measurements, with prescribed boundary conditions of different types. We prove that at most two, suitably chosen, measurements allow us to uniquely determine multiple defects under minimal regularity assumptions on the defects and the medium containing them. Finally, we treat applications of these results to the case of inverse scattering.
Another way to say harmonic
Michael
G.
Crandall;
Jianying
Zhang
241-263
Abstract: It is known that solutions of $-\Delta_\infty u=-\sum_{i,j=1}^nu_{x_i} u_{x_j}u_{x_ix_j}=0$, that is, the $\infty$-harmonic functions, are exactly those functions having a comparison property with respect to the family of translates of the radial solutions $G(x)=a\vert x\vert$. We establish a more difficult linear result: a function in ${\mathbb R^n}$ is harmonic if it has the comparison property with respect to sums of $n$ translates of the radial harmonic functions $G(x)=a\vert x\vert^{2-n}$ for $n\not=2$ and $G(x)=b\ln(\vert x\vert)$ for $n=2$. An attempt to generalize these results for $-\Delta_\infty u=0$ ($p=\infty$) and $-\Delta u=0$ ($p=2$) to the general $p$-Laplacian leads to the fascinating discovery that certain sums of translates of radial $p$-superharmonic functions are again $p$-superharmonic. Mystery remains: the class of $p$-superharmonic functions so constructed for $p\not\in\{2,\infty\}$ does not suffice to characterize $p$-subharmonic functions.
On graphic Bernstein type results in higher codimension
Mu-Tao
Wang
265-271
Abstract: Let $\Sigma$ be a minimal submanifold of $\mathbb{R} ^{n+m}$ that can be represented as the graph of a smooth map $f:\mathbb{R} ^n\mapsto\mathbb{R} ^m$. We apply a formula that we derived in the study of mean curvature flow to obtain conditions under which $\Sigma$ must be an affine subspace. Our result covers all known ones in the general case. The conditions are stated in terms of the singular values of $df$.
Matrix-weighted Besov spaces
Svetlana
Roudenko
273-314
Abstract: Nazarov, Treil and Volberg defined matrix $A_p$ weights and extended the theory of weighted norm inequalities on $L^p$ to the case of vector-valued functions. We develop some aspects of Littlewood-Paley function space theory in the matrix weight setting. In particular, we introduce matrix- weighted homogeneous Besov spaces $\dot{B}^{\alpha q}_p(W)$and matrix-weighted sequence Besov spaces $\dot{b}^{\alpha q}_p(W)$, as well as $\dot{b}^{\alpha q}_p(\{A_Q\})$, where the $A_Q$ are reducing operators for $W$. Under any of three different conditions on the weight $W$, we prove the norm equivalences $\Vert \vec{f} \,\Vert_{\dot{B}^{\alpha q}_p(W)} \approx \Vert \{ \vec{s}_Q \}... ... q}_p(W)} \approx \Vert \{ \vec{s}_Q \}_Q \Vert_{\dot{b}^{\alpha q}_p(\{A_Q\})}$, where $\{ \vec{s}_Q \}_Q$ is the vector-valued sequence of $\varphi$-transform coefficients of $\vec{f}$. In the process, we note and use an alternate, more explicit characterization of the matrix $A_p$ class. Furthermore, we introduce a weighted version of almost diagonality and prove that an almost diagonal matrix is bounded on $\dot{b}^{\alpha q}_p(W)$ if $W$ is doubling. We also obtain the boundedness of almost diagonal operators on $\dot{B}^{\alpha q}_p(W)$ under any of the three conditions on $W$. This leads to the boundedness of convolution and non-convolution type Calderón-Zygmund operators (CZOs) on $\dot{B}^{\alpha q}_p(W)$, in particular, the Hilbert transform. We apply these results to wavelets to show that the above norm equivalence holds if the $\varphi$-transform coefficients are replaced by the wavelet coefficients. Finally, we construct inhomogeneous matrix-weighted Besov spaces ${B}^{\alpha q}_p(W)$ and show that results corresponding to those above are true also for the inhomogeneous case.
The space $H^1$ for nondoubling measures in terms of a grand maximal operator
Xavier
Tolsa
315-348
Abstract: Let $\mu$ be a Radon measure on $\mathbb{R} ^d$, which may be nondoubling. The only condition that $\mu$ must satisfy is the size condition $\mu(B(x,r))\leq C\,r^n$, for some fixed $0<n\leq d$. Recently, some spaces of type $BMO(\mu)$ and $H^1(\mu)$ were introduced by the author. These new spaces have properties similar to those of the classical spaces ${BMO}$ and $H^1$defined for doubling measures, and they have proved to be useful for studying the $L^p(\mu)$ boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space $H^1(\mu)$ in terms of a maximal operator $M_\Phi$ is given. It is shown that $f$ belongs to $H^1(\mu)$ if and only if $f\in L^1(\mu)$, $\int f\, d\mu=0$ and $M_\Phi f \in L^1(\mu)$, as in the usual doubling situation.
Fullness, Connes' $\chi $-groups, and ultra-products of amalgamated free products over Cartan subalgebras
Yoshimichi
Ueda
349-371
Abstract: Ultra-product algebras associated with amalgamated free products over Cartan subalgebras are investigated. As applications, their Connes' $\chi$-groups are computed in terms of ergodic theory, and also we clarify what condition makes them full factors (i.e., their inner automorphism groups become closed).
Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains
Richard
F.
Bass;
Edwin
A.
Perkins
373-405
Abstract: We consider the operator \begin{displaymath}\sum_{i,j=1}^d \sqrt{x_ix_j}\gamma_{ij}(x) \frac{\partial^2}... ...\partial x_j}+\sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i}\end{displaymath} acting on functions in $C_b^2(\mathbb{R}^d_+)$. We prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on $\gamma_{ij}$ and $b_i$. In contrast to previous work, the $b_i$ need only be nonnegative on the boundary rather than strictly positive, at the expense of the $\gamma_{ij}$and $b_i$ being Hölder continuous. Applications to super-Markov chains are given. The proof follows Stroock and Varadhan's perturbation argument, but the underlying function space is now a weighted Hölder space and each component of the constant coefficient process being perturbed is the square of a Bessel process.
Proper actions on cohomology manifolds
Harald
Biller
407-432
Abstract: Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most $2$, then the group is effectively a Lie group.