A moment approach to analyze zeros of triangular polynomial sets
Jean
B.
Lasserre
1403-1420
Abstract: Let $I=\langle g_1,\ldots, g_n\rangle$ be a zero-dimensional ideal of $\mathbb{R}[x_1,\ldots ,x_n]$ such that its associated set $\mathbb{G}$ of polynomial equations $g_i(x)=0$ for all $i=1,\ldots ,n$ is in triangular form. By introducing multivariate Newton sums we provide a numerical characterization of polynomials in $\sqrt{I}$. We also provide a necessary and sufficient (numerical) condition for all the zeros of $\mathbb{G}$ to be in a given set $\mathbb{K}\subset \mathbb{C}^n$, without explicitly computing the zeros. In addition, we also provide a necessary and sufficient condition on the coefficients of the $g_i$'s for $\mathbb{G}$ to have (a) only real zeros, (b) to have only real zeros, all contained in a given semi-algebraic set $\mathbb{K}\subset\mathbb{R}^n$. In the proof technique, we use a deep result of Curto and Fialkow (2000) on the $\mathbb{K}$-moment problem, and the conditions we provide are given in terms of positive definiteness of some related moment and localizing matrices depending on the $g_i$'s via the Newton sums of $\mathbb{G}$. In addition, the number of distinct real zeros is shown to be the maximal rank of a related moment matrix.
Covering a compact set in a Banach space by an operator range of a Banach space with basis
V.
P.
Fonf;
W.
B.
Johnson;
A.
M.
Plichko;
V.
V.
Shevchyk
1421-1434
Abstract: A Banach space $X$ has the approximation property if and only if every compact set in $X$ is in the range of a one-to-one bounded linear operator from a space that has a Schauder basis. Characterizations are given for $\mathcal{L}_p$spaces and quotients of $\mathcal{L}_p$ spaces in terms of covering compact sets in $X$ by operator ranges from $\mathcal{L}_p$ spaces. A Banach space $X$is a $\mathcal{L}_1$ space if and only if every compact set in $X$ is contained in the closed convex symmetric hull of a basic sequence which converges to zero.
Sharp dimension estimates of holomorphic functions and rigidity
Bing-Long
Chen;
Xiao-Yong
Fu;
Le
Yin;
Xi-Ping
Zhu
1435-1454
Abstract: Let $M^n$ be a complete noncompact Kähler manifold of complex dimension $n$ with nonnegative holomorphic bisectional curvature. Denote by $\mathcal{O}_d(M^n)$ the space of holomorphic functions of polynomial growth of degree at most $d$ on $M^n$. In this paper we prove that $\displaystyle dim_{\mathbb{C}}{\mathcal{O}}_d(M^n)\leq dim_{\mathbb{C}}{\mathcal{O}}_{[d]}(\mathbb{C}^n),$ for all $d>0$, with equality for some positive integer $d$ if and only if $M^n$ is holomorphically isometric to $\mathbb{C}^n$. We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.
Low-dimensional homogeneous Einstein manifolds
Christoph
Böhm;
Megan
M.
Kerr
1455-1468
Abstract: We show that compact, simply connected homogeneous spaces up to dimension $11$ admit homogeneous Einstein metrics.
Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces
Marcin
Bownik;
Kwok-Pun
Ho
1469-1510
Abstract: Weighted anisotropic Triebel-Lizorkin spaces are introduced and studied with the use of discrete wavelet transforms. This study extends the isotropic methods of dyadic $\varphi$-transforms of Frazier and Jawerth (1985, 1989) to non-isotropic settings associated with general expansive matrix dilations and $A_\infty$ weights. In close analogy with the isotropic theory, we show that weighted anisotropic Triebel-Lizorkin spaces are characterized by the magnitude of the $\varphi$-transforms in appropriate sequence spaces. We also introduce non-isotropic analogues of the class of almost diagonal operators and we obtain atomic and molecular decompositions of these spaces, thus extending isotropic results of Frazier and Jawerth.
Local theta correspondence for small unitary groups
Shu-Yen
Pan
1511-1535
Abstract: In this paper we give an explicit parameterization of the local theta correspondence of supercuspidal representations for the reductive dual pairs $({\rm U}_1(F),{\rm U}_1(F))$, $({\rm U}_1(F),{\rm U}_{1,1}(F))$, $({\rm U}_1(F),{\rm U}_{2}(F))$, and $({\rm U}_1(F),{\rm U}_{1,2}(F))$ of unitary groups over a nonarchimedean local field $F$ of odd residue characteristic.
Algebra of dimension theory
Jerzy
Dydak
1537-1561
Abstract: The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory, this algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes $\dim(A)$ of graded groups $A$. There are two geometric interpretations of these equivalence classes: 1) For pointed CW complexes $K$ and $L$, $\dim(H_\ast(K))=\dim(H_\ast(L))$ if and only if the infinite symmetric products $SP(K)$ and $SP(L)$ are of the same extension type (i.e., $SP(K)\in AE(X)$ iff $SP(L)\in AE(X)$ for all compact $X$). 2) For pointed compact spaces $X$ and $Y$, $\dim(\mathcal{H}^{-\ast}(X))=\dim(\mathcal{H}^{-\ast}(Y))$if and only if $X$ and $Y$ are of the same dimension type (i.e., $\dim_G(X)=\dim_G(Y)$ for all Abelian groups $G$). Dranishnikov's version of the Hurewicz Theorem in extension theory becomes $\dim(\pi_\ast(K))=\dim(H_\ast(K))$ for all simply connected $K$. The concept of cohomological dimension $\dim_A(X)$of a pointed compact space $X$ with respect to a graded group $A$ is introduced. It turns out $\dim_A(X) \leq 0$ iff $\dim_{A(n)}(X)\leq n$for all $n\in\mathbf{Z}$. If $A$ and $B$ are two positive graded groups, then $\dim(A)=\dim(B)$ if and only if $\dim_A(X)=\dim_B(X)$for all compact $X$.
Average size of $2$-Selmer groups of elliptic curves, I
Gang
Yu
1563-1584
Abstract: In this paper, we study a class of elliptic curves over $\mathbb{Q}$ with $\mathbb{Q}$-torsion group ${\mathbb{Z}}_{2}\times\mathbb{Z}_{2}$, and prove that the average order of the $2$-Selmer groups is bounded.
Stable geometric dimension of vector bundles over even-dimensional real projective spaces
Martin
Bendersky;
Donald
M.
Davis;
Mark
Mahowald
1585-1603
Abstract: In 1981, Davis, Gitler, and Mahowald determined the geometric dimension of stable vector bundles of order $2^e$ over $RP^{n}$ if $n$ is even and sufficiently large and $e\ge75$. In this paper, we use the Bendersky-Davis computation of $v_1^{-1}\pi _*(SO(m))$ to show that the 1981 result extends to all $e\ge5$ (still provided that $n$ is sufficiently large). If $e\le4$, the result is often different due to anomalies in the formula for $v_1^{-1}\pi_*(SO(m))$ when $m\le8$, but we also determine the stable geometric dimension in these cases.
On the Cohen-Macaulay property of multiplicative invariants
Martin
Lorenz
1605-1617
Abstract: We investigate the Cohen-Macaulay property for rings of invariants under multiplicative actions of a finite group $\mathcal{G}$. By definition, these are $\mathcal{G}$-actions on Laurent polynomial algebras $\Bbbk[x_1^{\pm 1},\dots,x_n^{\pm 1}]$that stabilize the multiplicative group consisting of all monomials in the variables $x_i$. For the most part, we concentrate on the case where the base ring $\Bbbk$ is $\mathbb{Z}$. Our main result states that if $\mathcal{G}$ acts non-trivially and the invariant ring $\mathbb{Z} [x_1^{\pm 1},\dots,x_n^{\pm 1}]^\mathcal{G}$ is Cohen-Macaulay, then the abelianized isotropy groups ${\mathcal{G}}_m^{{ab}}$ of all monomials $m$ are generated by the bireflections in $\mathcal{G}_m$ and at least one ${\mathcal{G}}_m^{{ab}}$ is non-trivial. As an application, we prove the multiplicative version of Kemper's $3$-copies conjecture.
Innately transitive subgroups of wreath products in product action
Robert
W.
Baddeley;
Cheryl
E.
Praeger;
Csaba
Schneider
1619-1641
Abstract: A permutation group is innately transitive if it has a transitive minimal normal subgroup, which is referred to as a plinth. We study the class of finite, innately transitive permutation groups that can be embedded into wreath products in product action. This investigation is carried out by observing that such a wreath product preserves a natural Cartesian decomposition of the underlying set. Previously we classified the possible embeddings in the case where the plinth is simple. Here we extend that classification and identify several different types of Cartesian decompositions that can be preserved by an innately transitive group with a non-abelian plinth. These different types of decompositions lead to different types of embeddings of the acting group into wreath products in product action. We also obtain a full characterisation of embeddings of innately transitive groups with diagonal type into such wreath products.
The geometry of symplectic pairs
G.
Bande;
D.
Kotschick
1643-1655
Abstract: We study the geometry of manifolds carrying symplectic pairs consisting of two closed $2$-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.
Height uniformity for integral points on elliptic curves
Su-ion
Ih
1657-1675
Abstract: We recall the result of D. Abramovich and its generalization by P. Pacelli on the uniformity for stably integral points on elliptic curves. It says that the Lang-Vojta conjecture on the distribution of integral points on a variety of logarithmic general type implies the uniformity for the numbers of stably integral points on elliptic curves. In this paper we will investigate its analogue for their heights under the assumption of the Vojta conjecture. Basically, we will show that the Vojta conjecture gives a naturally expected simple uniformity for their heights.
Parallel focal structure and singular Riemannian foliations
Dirk
Töben
1677-1704
Abstract: We give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds. We show that this is a singular Riemannian foliation with complete orthogonal transversals. For this object we construct an action on the transversals that generalizes the Weyl group action for polar actions.
Poisson structures on complex flag manifolds associated with real forms
Philip
Foth;
Jiang-Hua
Lu
1705-1714
Abstract: For a complex semisimple Lie group $G$ and a real form $G_0$ we define a Poisson structure on the variety of Borel subgroups of $G$ with the property that all $G_0$-orbits in $X$ as well as all Bruhat cells (for a suitable choice of a Borel subgroup of $G$) are Poisson submanifolds. In particular, we show that every non-empty intersection of a $G_0$-orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.
Multiple homoclinic orbits in conservative and reversible systems
Ale
Jan
Homburg;
Jürgen
Knobloch
1715-1740
Abstract: We study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows. We consider the existence of two homoclinic orbits in the bellows configuration, where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times. In conservative systems one finds one parameter families of suspended horseshoes, parameterized by the level of the first integral. A somewhat similar picture occurs in reversible systems, with two homoclinic orbits that are both symmetric. The lack of a first integral implies that complete horseshoes do not exist. We provide a description of orbits that necessarily do exist. A second possible configuration in reversible systems occurs if a non-symmetric homoclinic orbit exists and forms a bellows together with its symmetric image. We describe the nonwandering set in an unfolding. The nonwandering set is shown to simultaneously contain one-parameter families of periodic orbits, hyperbolic periodic orbits of different index, and heteroclinic cycles between these periodic orbits.
On polynomial-factorial diophantine equations
Daniel
Berend;
Jørgen
E.
Harmse
1741-1779
Abstract: We study equations of the form $P(x)=n!$ and show that for some classes of polynomials $P$ the equation has only finitely many solutions. This is the case, say, if $P$ is irreducible (of degree greater than 1) or has an irreducible factor of ``relatively large" degree. This is also the case if the factorization of $P$ contains some ``large" power(s) of irreducible(s). For example, we can show that the equation $x^{r}(x+1)=n!$ has only finitely many solutions for $r\ge 4$, but not that this is the case for $1\le r\le 3$ (although it undoubtedly should be). We also study the equation $P(x)=H_{n}$, where $(H_{n})$ is one of several other ``highly divisible" sequences, proving again that for various classes of polynomials these equations have only finitely many solutions.
Automorphisms of Coxeter groups
Patrick
Bahls
1781-1796
Abstract: We compute ${\rm Aut}(W)$ for any even Coxeter group whose Coxeter diagram is connected, contains no edges labeled 2, and cannot be separated into more than 2 connected components by removing a single vertex. The description is given explicitly in terms of the given presentation for the Coxeter group and admits an easy characterization of those groups $W$ for which ${\rm Out}(W)$ is finite.
On the correlations of directions in the Euclidean plane
Florin
P.
Boca;
Alexandru
Zaharescu
1797-1825
Abstract: Let ${\mathcal{R}}^{(\nu )}_{(x,y),Q}$ denote the repartition of the $\nu$-level correlation measure of the finite set of directions $P_{(x,y)}P$, where $P_{(x,y)}$ is the fixed point $(x,y)\in [0,1)^{2}$ and $P$ is an integer lattice point in the square $[-Q,Q]^{2}$. We show that the average of the pair correlation repartition ${\mathcal{R}}^{(2)}_{(x,y),Q}$ over $(x,y)$ in a fixed disc ${\mathbb{D}}_{0}$ converges as $Q\rightarrow \infty$. More precisely we prove, for every $\lambda \in {\mathbb{R}}_{+}$ and $0<\delta <\frac{1}{10}$, the estimate \begin{displaymath}\frac{1}{\operatorname{Area} ({\mathbb{D}}_{0})} \iint \limi... ...1}{10}+\delta }) \qquad \text{\rm as$Q\rightarrow \infty$.} \end{displaymath} We also prove that for each individual point $(x,y)\in [0,1)^{2}$, the $6$-level correlation ${\mathcal{R}}^{(6)}_{(x,y),Q}(\lambda )$diverges at any point $\lambda \in {\mathbb{R}}^{5}_{+}$ as $Q\rightarrow \infty$, and we give an explicit lower bound for the rate of divergence.
On the hyperbolicity of the period-doubling fixed point
Daniel
Smania
1827-1846
Abstract: We give a new proof of the hyperbolicity of the fixed point for the period-doubling renormalization operator using the local dynamics near a semi-attractive fixed point (in a Banach space) and the theory of holomorphic motions. We also give a new proof of the exponential contraction of the Feigenbaum renormalization operator in the hybrid class of the period-doubling fixed point: our proof uses the non-existence of invariant line fields in the period-doubling tower (C. McMullen), the topological convergence (D. Sullivan), and a new infinitesimal argument.
Crystals of type $D_n^{(1)}$ and Young walls
Hyeonmi
Lee
1847-1867
Abstract: We give a new realization of arbitrary level perfect crystals and arbitrary level irreducible highest weight crystals of type $D_n^{(1)}$, in the language of Young walls. We refine the notions of splitting of blocks and slices that have appeared in our previous works, and these play crucial roles in the construction of crystals. The perfect crystals are realized as the set of equivalence classes of slices, and the irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls which, in turn, are concatenations of slices.