Liouville type theorems for fourth order elliptic equations in a half plane
Avner
Friedman;
Juan
J. L.
Velázquez
2537-2603
Abstract: Consider an elliptic equation $\omega \Delta\varphi -\Delta ^2\varphi =0$ in the half plane $\{(x,\,y),\,-\infty <x<\infty ,\,y>0\}$ with boundary conditions $\varphi =\varphi _y=0$ if $y=0,\,x>0$ and $B_j\varphi =0$ if $y=0,\,x<0$ where $B_j$ $(j=2,3)$ are second and third order differential operators. It is proved that if $Re\,\omega \geq0,\,\omega \neq0$ and, for some $\varepsilon >0$, $|\varphi |\leq Cr^\alpha$ if $r=\sqrt {x^2+y^2}\to \infty ,\quad |\varphi |\leq Cr^\beta$ if $r\to 0$ where $\alpha =n+\frac {1}{2}-\varepsilon \,,\quad \beta=n+\frac {1}{2}+\varepsilon$ for some nonnegative integer $n$, then $\varphi \equiv0$. Results of this type are also established in case $\omega =0$ under different conditions on $\alpha$ and $\beta$; furthermore, in one case $B_3\varphi$ has a lower order term which depends nonlocally on $\varphi$. Such Liouville type theorems arise in the study of coating flow; in fact, they play a crucial role in the analysis of the linearized version of this problem. The methods developed in this paper are entirely different for the two cases (i) $Re\,\omega \geq0,\,\omega \neq0$ and (ii) $\omega =0$; both methods can be extended to other linear elliptic boundary value problems in a half plane.
Analytic subgroups of $t$-modules
Robert
Tubbs
2605-2617
Abstract: In this paper we study the structure of analytic subgroups and of $t$-submodules of $t$-modules.
The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems
Di
Dong;
Yiming
Long
2619-2661
Abstract: In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.
Verma type modules of level zero for affine Lie algebras
Viatcheslav
Futorny
2663-2685
Abstract: We study the structure of Verma type modules of level zero induced from non-standard Borel subalgebras of an affine Kac-Moody algebra. For such modules in ``general position'' we describe the unique irreducible quotients, construct a BGG type resolution and prove the BGG duality in certain categories. All results are extended to generalized Verma type modules of zero level.
Generalized Weil's reciprocity law and multiplicativity theorems
András
Némethi
2687-2697
Abstract: Fix a one-dimensional group variety $G$ with Euler-characteristic $\chi (G)=0$, and a quasi-projective variety $Y$, both defined over $\bold {C}$. For any $f\in Hom(Y,G)$ and constructible sheaf ${\cal F}$ on $Y$, we construct an invariant $c_{{\cal F}}(f)\in G$, which provides substantial information about the topology of the fiber-structure of $f$ and the structure of ${\cal F}$ along the fibers of $f$. Moreover, $c_{{\cal F}}:Hom(Y,G)\to G$ is a group homomorphism.
Isomorphism of lattices of recursively enumerable sets
Todd
Hammond
2699-2719
Abstract: Let $\omega = \{\,0,1,2,\ldots \,\}$, and for $A\subseteq \omega$, let $\mathcal E^A$ be the lattice of subsets of $\omega$ which are recursively enumerable relative to the ``oracle'' $A$. Let $(\mathcal E^A)^*$ be $\mathcal E^A/\mathcal I$, where $\mathcal I$ is the ideal of finite subsets of $\omega$. It is established that for any $A,B\subseteq \omega$, $(\mathcal E^A)^*$ is effectively isomorphic to $(\mathcal E^B)^*$ if and only if $A'\equiv _T B'$, where $A'$ is the Turing jump of $A$. A consequence is that if $A'\equiv _T B'$, then $\mathcal E^A\cong \mathcal E^B$. A second consequence is that $(\mathcal E^A)^*$ can be effectively embedded into $(\mathcal E^B)^*$ preserving least and greatest elements if and only if $A'\leq _T B'$.
A combinatorial correspondence related to Göllnitz' (big) partition theorem and applications
Krishnaswami
Alladi
2721-2735
Abstract: In recent work, Alladi, Andrews and Gordon discovered a key identity which captures several fundamental theorems in partition theory. In this paper we construct a combinatorial bijection which explains this key identity. This immediately leads to a better understanding of a deep theorem of Göllnitz, as well as Jacobi's triple product identity and Schur's partition theorem.
Specification on the interval
Jérôme
Buzzi
2737-2754
Abstract: We study the consequences of discontinuities on the specification property for interval maps. After giving a necessary and sufficient condition for a piecewise monotonic, piecewise continuous map to have this property, we show that for a large and natural class of families of such maps (including the $\beta$-transformations), the set of parameters for which the specification property holds, though dense, has zero Lebesgue measure. Thus, regarding the specification property, the general case is at the opposite of the continuous case solved by A.M. Blokh (Russian Math. Surveys 38 (1983), 133-134) (for which we give a proof).
Tarski's finite basis problem via $\mathbf A(\mathcal T)$
Ross
Willard
2755-2774
Abstract: R. McKenzie has recently associated to each Turing machine ${\mathcal T}$ a finite algebra $\mathbf {A} ({\mathcal T})$ having some remarkable properties. We add to the list of properties, by proving that the equational theory of $\mathbf {A}({\mathcal T})$ is finitely axiomatizable if ${\mathcal T}$ halts on the empty input. This completes an alternate (and simpler) proof of McKenzie's negative answer to A. Tarski's finite basis problem. It also removes the possibility, raised by McKenzie, of using $\mathbf {A}({\mathcal T})$ to answer an old question of B. Jónsson.
Symmetric Gibbs measures
Karl
Petersen;
Klaus
Schmidt
2775-2811
Abstract: We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures-a version of de Finetti's theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, `canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.
Invariant cocycles, random tilings and the super-$K$ and strong Markov properties
Klaus
Schmidt
2813-2825
Abstract: We consider $1$-cocycles with values in locally compact, second countable abelian groups on discrete, nonsingular, ergodic equivalence relations. If such a cocycle is invariant under certain automorphisms of these relations, we show that the skew product extension defined by the cocycle is ergodic. As an application we obtain an extension of many recent results of the author and K. Petersen to higher-dimensional shifts of finite type, and prove a transitivity result concerning rearrangements of certain random tilings.
A homotopy classification of certain 7-manifolds
Bernd
Kruggel
2827-2843
Abstract: This paper gives a homotopy classification of Wallach spaces and a partial homotopy classification of closely related spaces obtained by free $S^1$-actions on $SU(3)$ and on $S^3\times S^5$.
On the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank
Haseo
Ki
2845-2870
Abstract: We show that the Denjoy rank and the Zalcwasser rank are incomparable. We construct for any countable ordinal $\alpha$ differentiable functions $f$ and $g$ such that the Zalcwasser rank and the Kechris-Woodin rank of $f$ are $\alpha +1$ but the Denjoy rank of $f$ is 2 and the Denjoy rank and the Kechris-Woodin rank of $g$ are $\alpha +1$ but the Zalcwasser rank of $g$ is 1. We then derive a theorem that shows the surprising behavior of the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank.
Euler-Lagrange and Hamiltonian formalisms in dynamic optimization
Alexander
Ioffe
2871-2900
Abstract: We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of Euler-Lagrange and Hamiltonian formalisms. The principal new results are: an extension of the recently discovered form of the Euler-Weierstrass condition to nonconvex valued differential inclusions, and a new Hamiltonian condition for convex valued inclusions. In both cases additional attention was given to weakening Lipschitz type requirements on the set-valued mapping. The central role of the Euler type condition is emphasized by showing that both the new Hamiltonian condition and the most general form of the Pontriagin maximum principle for equality constrained control systems are consequences of the Euler-Weierstrass condition. An example is given demonstrating that the new Hamiltonian condition is strictly stronger than the previously known one.
On the Faber coefficients of functions univalent in an ellipse
E.
Haliloglu
2901-2916
Abstract: Let $E$ be the elliptical domain \begin{displaymath}E=\{x+iy: \frac {x^{2}}{(5/4)^{2}}+ \frac {y^{2}}{(3/4)^{2}}<1 \}.\end{displaymath} Let $S(E)$ denote the class of functions $F(z)$ analytic and univalent in $E$ and satisfying the conditions $F(0)=0$ and $F'(0)=1$. In this paper, we obtain global sharp bounds for the Faber coefficients of the functions $F(z)$ in certain related classes and subclasses of $S(E).$
The local dimensions of the Bernoulli convolution associated with the golden number
Tian-You
Hu
2917-2940
Abstract: Let $X_1,X_2,\dotsc$ be a sequence of i.i.d. random variables each taking values of 1 and $-1$ with equal probability. For $1/2<\rho <1$ satisfying the equation $1-\rho -\dotsb -\rho ^s=0$, let $\mu$ be the probability measure induced by $S=\sum _{i=1}^\infty \rho ^iX_i$. For any $x$ in the range of $S$, let \begin{displaymath}d(\mu ,x)=\lim _{r\to 0^+}\log \mu([x-r,x+r])/\log r\end{displaymath} be the local dimension of $\mu$ at $x$ whenever the limit exists. We prove that \begin{displaymath}\alpha ^*=-\frac {\log 2}{\log \rho}\quad \text{and}\quad \alpha _*=-\frac {\log \delta }{s\log \rho}-\frac {\log 2}{\log \rho},\end{displaymath} where $\delta =(\sqrt {5}-1)/2$, are respectively the maximum and minimum values of the local dimensions. If $s=2$, then $\rho$ is the golden number, and the approximate numerical values are $\alpha ^*\approx 1.4404$ and $\alpha _*\approx 0.9404$.
Bloch constants of bounded symmetric domains
Genkai
Zhang
2941-2949
Abstract: Let $D_{1}$ and $D_{2}$ be two irreducible bounded symmetric domains in the complex spaces $V_{1}$ and $V_{2}$ respectively. Let $E$ be the Euclidean metric on $V_{2}$ and $h$ the Bergman metric on $V_{1}$. The Bloch constant $b(D_{1}, D_{2})$ is defined to be the supremum of $E(f^{\prime }(z)x, f^{\prime }(z)x)^{\frac {1}{2}}/h_{z}(x, x)^{1/2}$, taken over all the holomorphic functions $f: D_{1}\to D_{2}$ and $z\in D_{1}$, and nonzero vectors $x\in V_{1}$. We find the constants for all the irreducible bounded symmetric domains $D_{1}$ and $D_{2}$. As a special case we answer an open question of Cohen and Colonna.
An isometry theorem for quadratic differentials on Riemann surfaces of finite genus
Nikola
Lakic
2951-2967
Abstract: Assume both $X$ and $Y$ are Riemann surfaces which are subsets of compact Riemann surfaces $X_1$ and $Y_1,$ respectively, and that the set $ X_1 - X$ has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials on $X$ and $Y$ are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows that every biholomorphic map from the Teichmüller space of $X$ onto the Teichmüller space of $Y$ is induced by some quasiconformal map of $X$ onto $Y$. Consequently we can find an uncountable set of Riemann surfaces whose Teichmüller spaces are not biholomorphically equivalent.
Subgroups of finite soluble groups inducing the same permutation character
Norberto
Gavioli
2969-2980
Abstract: In this paper there are found necessary and sufficient conditions that a pair of solvable finite groups, say $G$ and $K$, must satisfy for the existence of a solvable finite group $L$ containing two isomorphic copies of $G$ and $H$ inducing the same permutation character. Also a construction of $L$ is given as an iterated wreath product, with respect to their actions on their natural modules, of finite one-dimensional affine groups.