Symmetry of properly embedded special Weingarten surfaces in $\mathbf{H}^3$
Ricardo
Sa Earp;
Eric
Toubiana
4693-4711
Abstract: In this paper we prove some existence and uniqueness results about special Weingarten surfaces in hyperbolic space.
Homogeneous spaces with invariant projectively flat affine connections
Hirohiko
Shima
4713-4726
Abstract: We characterize invariant projectively flat affine connections in terms of affine representations of Lie algebras, and show that a homogeneous space admits an invariant projectively flat affine connection if and only if it has an equivariant centro-affine immersion. We give a correspondence between semi-simple symmetric spaces with invariant projectively flat affine connections and central-simple Jordan algebras.
Warped products of metric spaces of curvature bounded from above
Chien-Hsiung
Chen
4727-4740
Abstract: In this work we extend the idea of warped products, which was previously defined on smooth Riemannian manifolds, to geodesic metric spaces and prove the analogue of the theorems on spaces with curvature bounded from above.
A Schwarz lemma for multivalued functions and distortion theorems for Bloch functions with branch points
Ian
Graham;
David
Minda
4741-4752
Abstract: We give a version of the Schwarz lemma for multivalued mappings between hyperbolic plane regions. As in the original work of Nehari on this subject, the derivative must remain bounded near the branch points. Our version of the distance-decreasing principle represents a considerable strengthening of previous results. We apply it to the study of Bloch functions with branch points of specified order. We obtain upper and lower estimates for $|f'|$, an upper estimate for $|f|$, and a lower estimate for the radius of the largest schlicht disk in the image of $f$ centered at $f(0)$. We also obtain some results requiring estimates of second order derivatives of $f$.
Capacity convergence results and applications to a Berstein-Markov inequality
T.
Bloom;
N.
Levenberg
4753-4767
Abstract: Given a sequence $\{E_{j}\}$ of Borel subsets of a given non-pluripolar Borel set $E$ in the unit ball $B$ in $\mathbf{C}^{N}$ with $E \subset \subset B$, we show that the relative capacities $C(E_{j})$ converge to $C(E)$ if and only if the relative (global) extremal functions $u_{E_{j}}^{*}$ ($V_{E_{j}}^{*}$) converge pointwise to $u_{E}^{*}$ ($V_{E}^{*}$). This is used to prove a sufficient mass-density condition on a finite positive Borel measure with compact support $K$ in $\mathbf{C}^{N}$ guaranteeing that the pair $(K,\mu )$ satisfy a Bernstein-Markov inequality. This implies that the $L^{2}-$orthonormal polynomials associated to $\mu$ may be used to recover the global extremal function $V_{K}^{*}$.
On the coefficients of Jacobi sums in prime cyclotomic fields
F.
Thaine
4769-4790
Abstract: Let $p\geq 5$ and $q=pf+1$ be prime numbers, and let $s$ be a primitive root mod $q$. For $1\leq n\leq p-2$, denote by $J_{n}$ the Jacobi sum $-\sum _{k=2}^{q-1}\zeta _p ^{\, \text{ind}_{s}(k)+n\, \text{ind}_{s}(1-k)}$. We study the integers $d_{n,k}$ such that $J_{n}=\sum _{k=0}^{p-1}d_{n,k}\zeta _p ^{k}$ and $\sum _{k=0}^{p-1}d_{n,k}=1$. We give a list of properties that characterize these coefficients. Then we show some of their applications to the study of the arithmetic of $\mathbb {Z} [\zeta _p +\zeta _p ^{-1}]$, in particular to the study of Vandiver's conjecture. For $m\in \mathbb {Z}-q\mathbb {Z}$, let $\rho _{n}(m)$ be the number of distinct roots of $X^{n+1}-X^{n}+m$ in $\mathbb {Z}/q\mathbb {Z}$. We show that $d_{n,k}=f-\sum _{a=0}^{f-1}\rho _{n}(s^{k+pa})$. We give closed formulas for the numbers $d_{1,k}$ and $d_{2,k}$ in terms of quadratic and cubic power residue symbols mod $q$.
Möbius-like groups of homeomorphisms of the circle
Natasa
Kovacevic
4791-4822
Abstract: An orientation preserving homeomorphism of $S^1$ is Möbius-like if it is conjugate in $Homeo(S^1{1})$ to a Möbius transformation. Our main result is: given a (noncyclic) group $G\hookrightarrow Homeo_{+}(S^{1})$ whose every element is Möbius-like, if $G$ has at least one global fixed point, then the whole group $G$ is conjugate in $Homeo(S^1{1})$ to a Möbius group if and only if the limit set of $G$ is all of $S^1$. Moreover, we prove that if the limit set of $G$ is not all of $S^1$, then after identifying some closed subintervals of $S^1$ to points, the induced action of $G$ is conjugate to an action of a Möbius group. Said differently, $G$ is obtained from a group which is conjugate to a Möbius group, by a sort of generalized Denjoy's insertion of intervals. In this case $G$ is isomorphic, as a group, to a Möbius group. This result has another interpretation. Namely, we prove that a group $G$ of orientation preserving homeomorphisms of $\boldsymbol{R}$ whose every element can be conjugated to an affine map (i.e., a map of the form $x \mapsto ax + b$) is just the conjugate of a group of affine maps, up to a certain insertion of intervals. In any case, the group structure of $G$ is the one of an affine group.
Examples of Möbius-like groups which are not Möbius groups
Natasa
Kovacevic
4823-4835
Abstract: In this paper we give two basic constructions of groups with the following properties: (a) $G \hookrightarrow {Homeo_{+}(S^{1})}$, i.e., the group $G$ is acting by orientation preserving homeomorphisms on ${S^{1}}$; (b) every element of $G$ is Möbius-like; (c) ${L(G)}= {S^{1}}$, where ${L(G)}$ denotes the limit set of $G$; (d) $G$ is discrete; (e) $G$ is not a conjugate of a Möbius group. Both constructions have the same basic idea (inspired by Denjoy): we start with a Möbius group $H$ (of a certain type) and then we change the underlying circle upon which $H$ acts by inserting some closed intervals and then extending the group action over the new circle. We denote this new action by $\overline{H}$. Now we form a new group $G$ which is generated by all of $\overline{H}$ and an additional element $g$ whose existence is enabled by the inserted intervals. This group $G$ has all the properties (a) through (e).
Homology of the universal covering of a co-H-space
Norio
Iwase;
Shiroshi
Saito;
Toshio
Sumi
4837-4846
Abstract: The problem 10 posed by Tudor Ganea is known as the Ganea conjecture on a co-H-space, a space with co-H-structure. Many efforts are devoted to show the Ganea conjecture under additional assumptions on the given co-H-structure. In this paper, we show a homological property of co-H-spaces in a slightly general situation. As a corollary, we get the Ganea conjecture for spaces up to dimension 3.
Steepest descent evolution equations: asymptotic behavior of solutions and rate of convergence
R.
Cominetti;
O.
Alemany
4847-4860
Abstract: We study the asymptotic behavior of the solutions of evolution equations of the form $\dot u(t)\in -\partial f(u(t),r(t))$, where $f(\cdot,r)$ is a one-parameter family of approximations of a convex function $f(\cdot)$ we wish to minimize. We investigate sufficient conditions on the parametrization $r(t)$ ensuring that the integral curves $u(t)$ converge when $t\rightarrow\infty$ towards a particular minimizer $u_\infty$ of $f$. The speed of convergence is also investigated, and a result concerning the continuity of the limit point $u_\infty$ with respect to the parametrization $r(\cdot)$ is established. The results are illustrated on different approximation methods. In particular, we present a detailed application to the logarithmic barrier in linear programming.
Boundary value problems on infinite intervals
Jan
Andres;
Grzegorz
Gabor;
Lech
Górniewicz
4861-4903
Abstract: We present two methods, both based on topological ideas, to the solvability of boundary value problems for differential equations and inclusions on infinite intervals. In the first one, related to the rich family of asymptotic problems, we generalize and extend some statements due to the Florence group of mathematicians Anichini, Cecchi, Conti, Furi, Marini, Pera, and Zecca. Thus, their conclusions for differential systems are as well true for inclusions; all under weaker assumptions (for example, the convexity restrictions in the Schauder linearization device can be avoided). In the second, dealing with the existence of bounded solutions on the positive ray, we follow and develop the ideas of Andres, Górniewicz, and Lewicka, who considered periodic problems. A special case of these results was previously announced by Andres. Besides that, the structure of solution sets is investigated. The case of l.s.c. right hand sides of differential inclusions and the implicit differential equations are also considered. The large list of references also includes some where different techniques (like the Conley index approach) have been applied for the same goal, allowing us to envision the full range of recent attacks on the problem stated in the title.
Complex symplectic geometry with applications to ordinary differential operators
W.
N.
Everitt;
L.
Markus
4905-4945
Abstract: Complex symplectic spaces, and their Lagrangian subspaces, are defined in accord with motivations from Lagrangian classical dynamics and from linear ordinary differential operators; and then their basic algebraic properties are established. After these purely algebraic developments, an Appendix presents a related new result on the theory of self-adjoint operators in Hilbert spaces, and this provides an important application of the principal theorems.
Behavior near the boundary of positive solutions of second order parabolic equations. II
E.
B.
Fabes;
M.
V.
Safonov;
Yu
Yuan
4947-4961
Abstract: A boundary backward Harnack inequality is proved for positive solutions of second order parabolic equations in non-divergence form in a bounded cylinder $Q=\Omega \times \left(0,T\right)$ which vanish on $\partial _xQ=\partial\Omega\times \left(0,T\right)$, where $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^n$. This inequality is applied to the proof of the Hölder continuity of the quotient of two positive solutions vanishing on a portion of $\partial _xQ.$
Lower bounds for the absolute value of random polynomials on a neighborhood of the unit circle
S.
V.
Konyagin;
W.
Schlag
4963-4980
Abstract: Let $T(x)=\sum _{j=0}^{n-1}\pm e^{ijx}$ where $\pm$ stands for a random choice of sign with equal probability. The first author recently showed that for any $\epsilon>0$ and most choices of sign, $\min _{x\in[0,2\pi)}|T(x)|<n^{-1/2+\epsilon}$, provided $n$ is large. In this paper we show that the power $n^{-1/2}$ is optimal. More precisely, for sufficiently small $\epsilon>0$ and large $n$ most choices of sign satisfy $\min _{x\in[0,2\pi)}|T(x)|> \epsilon n^{-1/2}$. Furthermore, we study the case of more general random coefficients and applications of our methods to complex zeros of random polynomials.
Optimal individual stability estimates for $C_0$-semigroups in Banach spaces
Volker
Wrobel
4981-4994
Abstract: In a previous paper we proved that the asymptotic behavior of a $C_0$-semigroup is completely determined by growth properties of the resolvent of its generator and geometric properties of the underlying Banach space as described by its Fourier type. The given estimates turned out to be optimal. The method of proof uses complex interpolation theory and reflects the full semigroup structure. In the present paper we show that these uniform estimates have to be replaced by weaker ones, if individual initial value problems and local resolvents are considered because the full semigroup structure is lacking. In a different approach this problem has also been studied by Huang and van Neerven, and a part of our straightforward estimates can be inferred from their results. We mainly stress upon the surprising fact that these estimates turn out to be optimal. Therefore it is not possible to obtain the optimal uniform estimates mentioned above from individual ones. Concerning Hardy-abscissas, individual orbits and their local resolvents behave as badly as general vector valued functions and their Laplace-transforms. This is in strict contrast to the uniform situation of a $C_0$-semigroup itself and the resolvent of its generator where a simple dichotomy holds true.
Conformal iterated function systems with applications to the geometry of continued fractions
R.
Daniel
Mauldin;
Mariusz
Urbanski
4995-5025
Abstract: In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior. We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries.
When almost multiplicative morphisms are close to homomorphisms
Huaxin
Lin
5027-5049
Abstract: It is shown that approximately multiplicative contractive positive morphisms from $C(X)$ (with dim $X\le 2$) into a simple $C^*$-algebra $A$ of real rank zero and of stable rank one are close to homomorphisms, provided that certain $K$-theoretical obstacles vanish. As a corollary we show that a homomorphism $h: C(X)\to A$ is approximated by homomorphisms with finite dimensional range, if $h$ gives no $K$-theoretical obstacle.
On the Normal Subgroups of $G_2(A)$
Douglas
L.
Costa;
Gordon
E.
Keller
5051-5088
Abstract: We give a characterization theorem for the $E(A)$-normalized subgroups of $G_2(A)$, where $A$ is any commutative ring. This is the last of the simple Chevalley-Demazure group-schemes for which such a theorem is lacking.
*-polynomial identities of matrices with the transpose involution: The low degrees
Alain
D'Amour;
Michel
Racine
5089-5106
Abstract: In this paper, we investigate $*$-polynomial identities of minimal degree for the algebra of $n\times n$ matrices over a field, where $n<5$ and $*$ is the transpose involution. We first present some basic generators, and then proceed to show that all other minimal degree identities can be derived from those.