The topological dynamics of semigroup actions
David
B.
Ellis;
Robert
Ellis;
Mahesh
Nerurkar
1279-1320
Abstract: In these notes we explore the fine structure of recurrence for semigroup actions, using the algebraic structure of compactifications of the acting semigroup.
On the tangent flow of a stochastic differential equation with fast drift
Richard
B.
Sowers
1321-1334
Abstract: We investigate the behavior of the tangent flow of a stochastic differential equation with a fast drift. The state space of the stochastic differential equation is the two-dimensional cylinder. The fast drift has closed orbits, and we assume that the orbit times vary nontrivially with the axial coordinate. Under a nondegeneracy assumption, we find the rate of growth of the tangent flow. The calculations involve a transformation introduced by Pinsky and Wihstutz.
Embedded minimal ends of finite type
Laurent
Hauswirth;
Joaquín
Pérez;
Pascal
Romon
1335-1370
Abstract: We prove that the end of a complete embedded minimal surface in $\mathbb{R} ^3$ with infinite total curvature and finite type has an explicit Weierstrass representation that only depends on a holomorphic function that vanishes at the puncture. Reciprocally, any choice of such an analytic function gives rise to a properly embedded minimal end $E$ provided that it solves the corresponding period problem. Furthermore, if the flux along the boundary vanishes, then the end is $C^0$-asymptotic to a Helicoid. We apply these results to proving that any complete embedded one-ended minimal surface of finite type and infinite total curvature is asymptotic to a Helicoid, and we characterize the Helicoid as the only simply connected complete embedded minimal surface of finite type in $\mathbb{R} ^3$.
Bloch constants in several variables
Huaihui
Chen;
P.
M.
Gauthier
1371-1386
Abstract: We give lower estimates for Bloch's constant for quasiregular holomorphic mappings. A holomorphic mapping of the unit ball $B^n$ into $\mathbf{C}^n$ is $K$-quasiregular if it maps infinitesimal spheres to infinitesimal ellipsoids whose major axes are less than or equal to $K$ times their minor axes. We show that if $f$ is a $K$-quasiregular holomorphic mapping with the normalization $1/12K^{1-1/n}.$ This result is best possible in terms of powers of $K.$ Also, we extend to several variables an analogous result of Landau for bounded holomorphic functions in the unit disk.
Effective estimates on the very ampleness of the canonical line bundle of locally Hermitian symmetric spaces
Sai-Kee
Yeung
1387-1401
Abstract: We study the problem about the very ampleness of the canonical line bundle of compact locally Hermitian symmetric manifolds of non-compact type. In particular, we show that any sufficiently large unramified covering of such manifolds has very ample canonical line bundle, and give estimates on the size of the covering manifold, which is itself a locally Hermitian symmetric manifold, in terms of geometric data such as injectivity radius or degree of coverings.
Representability is not decidable for finite relation algebras
Robin
Hirsch;
Ian
Hodkinson
1403-1425
Abstract: We prove that there is no algorithm that decides whether a finite relation algebra is representable. Representability of a finite relation algebra $\mathcal A$ is determined by playing a certain two player game $G({\mathcal A})$ over `atomic $\mathcal A$-networks'. It can be shown that the second player in this game has a winning strategy if and only if $\mathcal A$ is representable. Let $\tau$ be a finite set of square tiles, where each edge of each tile has a colour. Suppose $\tau$ includes a special tile whose four edges are all the same colour, a colour not used by any other tile. The tiling problem we use is this: is it the case that for each tile $T \in \tau$ there is a tiling of the plane ${\mathbb Z}\times {\mathbb Z}$ using only tiles from $\tau$ in which edge colours of adjacent tiles match and with $T$ placed at $(0,0)$? It is not hard to show that this problem is undecidable. From an instance of this tiling problem $\tau$, we construct a finite relation algebra $RA(\tau)$ and show that the second player has a winning strategy in $G(RA(\tau))$ if and only if $\tau$ is a yes-instance. This reduces the tiling problem to the representation problem and proves the latter's undecidability.
The information encoded in initial ideals
Gunnar
Fløystad;
Mark
L.
Green
1427-1453
Abstract: We consider homogeneous ideals $I$ and the initial ideal $\text{in}(I)$ for the revlex order. First we give a sequence of invariants computed from $I$ giving better and better ``approximations" to the initial ideal and ending in an equivalent description. Then we apply this to different settings in algebraic geometry to understand what information is encoded in the generic initial ideal of the ideal of a projective scheme. We also consider the higher initial ideals as defined in a paper by Fløystad. In particular, we show that giving the generic higher initial ideal of a space curve is equivalent to giving the generic initial ideal of a linked curve.
$K3$ surfaces of genus 8 and varieties of sums of powers of cubic fourfolds
Atanas
Iliev;
Kristian
Ranestad
1455-1468
Abstract: The main result of this paper is that the variety of presentations of a general cubic form $f$ in $6$ variables as a sum of $10$cubes is isomorphic to the Fano variety of lines of a cubic $4$-fold $F'$, in general different from $F=Z(f)$. A general $K3$ surface $S$ of genus $8$determines uniquely a pair of cubic $4$-folds: the apolar cubic $F(S)$ and the dual Pfaffian cubic $F'(S)$ (or for simplicity $F$ and $F'$). As Beauville and Donagi have shown, the Fano variety $\mathcal{F}_{F'}$ and $VSP(F,10)$. Furthermore, we show that $VSP(F,10)$ sets up a $(6,10)$ correspondence between $F'$ and
The automorphism groups of Kummer surfaces associated with the product of two elliptic curves
Jonghae
Keum;
Shigeyuki
Kondo
1469-1487
Abstract: We calculate the automorphism groups of several Kummer surfaces associated with the product of two elliptic curves. We give their generators explicitly.
A characteristic free approach to Brauer algebras
Steffen
König;
Changchang
Xi
1489-1505
Abstract: Brauer algebras arise in representation theory of orthogonal or symplectic groups. These algebras are shown to be iterated inflations of group algebras of symmetric groups. In particular, they are cellular (as had been shown before by Graham and Lehrer). This gives some information about block decomposition of Brauer algebras.
On the representation of unity by binary cubic forms
Michael
A.
Bennett
1507-1534
Abstract: If $F(x,y)$ is a binary cubic form with integer coefficients such that $F(x,1)$has at least two distinct complex roots, then the equation $F(x,y) = 1$possesses at most ten solutions in integers $x$ and $y$, nine if $F$ has a nontrivial automorphism group. If, further, $F(x,y)$ is reducible over $\mathbb{Z}[x,y]$, then this equation has at most $2$ solutions, unless $F(x,y)$ is equivalent under $GL_2(\mathbb{Z})$-action to either $x (x^2-xy-y^2)$ or $x (x^2-2y^2)$. The proofs of these results rely upon the method of Thue-Siegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations $F(x,y)=1$ for $F$ cubic and irreducible of positive discriminant $D_F \leq 10^6$. As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form $F(x,y)=m$ and to Mordell's equation $y^2=x^3+k$, where $m$ and $k$ are nonzero integers.
Shintani functions on $GL(2,{\mathbf{C}})$
Miki
Hirano
1535-1550
Abstract: In this paper, in analogy to the real case, we give a formulation of the Shintani functions on $GL(2,\mathbf{C})$, which have been studied by Murase and Sugano within the theory of automorphic $L$-functions. Also, we obtain the multiplicity one theorem for these functions and an explicit formula in a special case.
Anick's spaces and the double loops of odd primary Moore spaces
Stephen
D.
Theriault
1551-1566
Abstract: Several properties of Anick's spaces are established which give a retraction of Anick's $\Omega T_\infty$ off $\Omega^2P^{2np+1}(p^r)$ if $r\ge2$ and $p\ge5$. The proof is alternate to and more immediate than the two proofs of Neisendorfer's.
On Littlewood's boundedness problem for sublinear Duffing equations
Bin
Liu
1567-1585
Abstract: In this paper, we are concerned with the boundedness of all the solutions and the existence of quasi-periodic solutions for second order differential equations \begin{displaymath}x^{\prime\prime} + g(x) = e(t), \end{displaymath} where the 1-periodic function $e(t)$ is a smooth function and $g(x)$satisfies sublinearity: \begin{displaymath}{sign}(x)\cdot g(x)\to+\infty,\quad g(x)/x\to 0 \quad {as}\,\,\, \vert x\vert\to+\infty. \end{displaymath}
Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth
Avner
Friedman;
Fernando
Reitich
1587-1634
Abstract: In this paper we develop a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter $\varepsilon$. The technique is worked out primarily for a free boundary problem describing a model of a stationary tumor. We prove the existence of infinitely many branches of symmetry-breaking solutions which bifurcate from any given radially symmetric steady state; these asymmetric solutions are analytic jointly in the spatial variables and in $\varepsilon$.
On the lack of null-controllability of the heat equation on the half-line
Sorin
Micu;
Enrique
Zuazua
1635-1659
Abstract: We consider the linear heat equation on the half-line with a Dirichlet boundary control. We analyze the null-controllability problem. More precisely, we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the $L^2$ boundary control. We rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems. Next, the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials $\{e^{jt}\}_{j\geq1}$ in which the usual summability condition on the inverses of the eigenvalues does not hold. Roughly speaking, we prove that controllable data have Fourier coefficients that grow exponentially for large frequencies. This result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a Sobolev space of negative order may be driven to zero in an arbitrarily small time.
A similarity principle for complex vector fields and applications
S.
Berhanu;
J.
Hounie;
P.
Santiago
1661-1675
Abstract: This paper establishes a similarity principle for a class of non-elliptic, smooth complex vector fields in the plane. This principle is used to prove a uniqueness result for a nonlinear Cauchy problem.
Metrics and embeddings of generalizations of Thompson's group $F$
J.
Burillo;
S.
Cleary;
M.
I.
Stein
1677-1689
Abstract: The distance from the origin in the word metric for generalizations $F(p)$ of Thompson's group $F$ is quasi-isometric to the number of carets in the reduced rooted tree diagrams representing the elements of $F(p)$. This interpretation of the metric is used to prove that every $F(p)$ admits a quasi-isometric embedding into every $F(q)$, and also to study the behavior of the shift maps under these embeddings.
On composite formal power series
Jacques
Chaumat;
Anne-Marie
Chollet
1691-1703
Abstract: Let $F$ be a holomorphic map from ${\mathbb{C}}^{n}$ to ${\mathbb{C}}^{n}$ defined in a neighborhood of $0$ such that $F(0)=0$. If the Jacobian determinant of $F$ is not identically zero, P. M. Eakin et G. A. Harris proved the following result: any formal power series such that ${\mathcal{A}}\circ F$ is analytic is itself analytic. If the Jacobian determinant of $F$ is identically zero, they proved that the previous conclusion is no more true. The authors get similar results in the case of formal power series satifying growth conditions, of Gevrey type for instance. Moreover, the proofs here give, in the analytic case, a control of the radius of convergence of ${\mathcal{A}}$ by the radius of convergence of ${\mathcal{A}}\circ F$. RÉSUMÉ. Soit $F$ une application holomorphe de ${\mathbb{C}}^{n}$dans ${\mathbb{C}}^{n}$ définie dans un voisinage de $0$ et vérifiant $F(0)=0$. Si le jacobien de $F$ n'est pas identiquement nul au voisinage de $0$, P.M. Eakin et G.A. Harris ont établi le résultat suivant: toute série formelle ${\mathcal{A}}$ telle que ${\mathcal{A}}\circ F$ est analytique est elle-même analytique. Si le jacobien de $F$ est identiquement nul, ils montrent que la conclusion précédente est fausse. Les auteurs obtiennent des résultats analogues pour les séries formelles à croissance contrôlée, du type Gevrey par exemple. De plus, les preuves données ici permettent, dans le cas analytique, un contrôle du rayon de convergence de ${\mathcal{A}}$ par celui de ${\mathcal{A}}\circ F$.