Commutation methods applied to the mKdV-equation
F.
Gesztesy;
W.
Schweiger;
B.
Simon
465-525
Abstract: An explicit construction of solutions of the modified Korteweg-de Vries equation given a solution of the (ordinary) Korteweg-de Vries equation is provided. Our theory is based on commutation methods (i.e., $N = 1$ supersymmetry) underlying Miura's transformation that links solutions of the two evolution equations.
Ramsey theorems for knots, links and spatial graphs
Seiya
Negami
527-541
Abstract: An embedding $ f:G \to {{\mathbf{R}}^3}$ of a graph $G$ into $ {{\mathbf{R}}^3}$ is said to be linear if each edge $f(e)\quad (e \in E(G))$ is a straight line segment. It will be shown that for any knot or link type $ k$, there is a finite number $R(k)$ such that every linear embedding of the complete graph ${K_n}$ with at least $R(k)$ vertices $ (n \geqslant R(k))$ in $ {{\mathbf{R}}^3}$ contains a knot or link equivalent to $k$.
Ultra-irreducibility of induced representations of semidirect products
Henrik
Stetkær
543-554
Abstract: Let the Lie group $ G$ be a semidirect product, $G = SK$, of a connected, closed, normal subgroup $ S$ and a closed subgroup $ K$. Let $\Lambda$ be a nonunitary character of $ S$, and let ${K_\Lambda }$ be its stability subgroup in $ K$. Let ${I^{\Lambda \mu }}$, for any irreducible representation $\mu$ of $ {K_\Lambda }$, denote the representation $ {I^{\Lambda \mu }}$ of $ G$ induced by the representation $\Lambda \mu$ of $S{K_\Lambda }$. The representation spaces are subspaces of the distributions. We show that ${I^{\Lambda \mu }}$ is ultra-irreducible when the corresponding Poisson transform is injective, and find a sufficient condition for this injectivity.
Local singularities such that all deformations are tangentially flat
Bernd
Herzog
555-601
Abstract: We give a criterion for a local ring $({B_0},{\mathfrak{n}_0})$ containing a field to have only tangentially flat deformations. Various examples of such local rings are constructed.
Construction of units in integral group rings of finite nilpotent groups
Jürgen
Ritter;
Sudarshan K.
Sehgal
603-621
Abstract: Let $U$ be the group of units of the integral group ring of a finite group $G$. We give a set of generators of a subgroup $B$ of $U$. This subgroup is of finite index in $ U$ if $G$ is an odd nilpotent group. We also give an example of a $2$-group such that $B$ is of infinite index in $U$.
The Selberg trace formula. VIII. Contribution from the continuous spectrum
M. Scott
Osborne;
Garth
Warner
623-653
Abstract: The purpose of this paper is to isolate the contribution from the continuous spectrum to the Selberg trace formula.
A generalized Berele-Schensted algorithm and conjectured Young tableaux for intermediate symplectic groups
Robert A.
Proctor
655-692
Abstract: The Schensted and Berele algorithms combinatorially mimic the decompositions of ${ \otimes ^k}V$ with respect to $ {\operatorname{GL} _N}$ and ${\operatorname{Sp} _{2n}}$. Here we present an algorithm which is a common generalization of these two algorithms. "Intermediate symplectic groups" ${\operatorname{Sp} _{2n,m}}$ are defined. These groups interpolate between ${\operatorname{GL} _N}$ and ${\operatorname{Sp} _N}$. We conjecture that there is a decomposition of $ { \otimes ^k}V$ with respect to ${\operatorname{Sp} _{2n,m}}$ which is described by the output of the new algorithm.
On the growth of solutions of $f''+gf'+hf=0$
Simon
Hellerstein;
Joseph
Miles;
John
Rossi
693-706
Abstract: Suppose $ g$ and $h$ are entire functions with the order of $ h$ less than the order of $ g$. If the order of $ g$ does not exceed $\tfrac{1} {2}$, it is shown that every (necessarily entire) nonconstant solution $f$ of the differential equation
Hankel operators on the Bergman space of bounded symmetric domains
Ke He
Zhu
707-730
Abstract: Let $\Omega$ be a bounded symmetric domain in $ {\mathbb{C}^n}$ with normalized volume measure $dV$. Let $P$ be the orthogonal projection from ${L^2}(\Omega ,dV)$ onto the Bergman space $L_a^2(\Omega )$ of holomorphic functions in ${L^2}(\Omega ,dV)$. Let $\overline P$ be the orthogonal projection from ${L^2}(\Omega ,dV)$ onto the closed subspace of antiholomorphic functions in ${L^2}(\Omega ,dV)$. The "little" Hankel operator $ {h_f}$ with symbol $ f$ is the operator from $L_a^2(\Omega )$ into ${L^2}(\Omega ,dV)$ defined by ${h_f}g = \overline P (fg)$. We characterize the boundedness, compactness, and membership in the Schatten classes of the Hankel operators ${h_f}$ in terms of a certain integral transform of the symbol $f$. These characterizations are further studied in the special cases of the open unit ball and the poly-disc in $ {\mathbb{C}^n}$.
The Frobenius-Perron operator on spaces of curves
P.
Góra;
A.
Boyarsky
731-746
Abstract: Let $\tau :{R^2} \to {R^2}$ be a diffeomorphism which leaves a compact set $A$ invariant. Let $B \subset A$ be such that $ \tau$ can map out of $ B$. Assume that $ \tau$ has a hyperbolic fixed point $p$ in $B$. Let $ \mathcal{C}$ be a space of smooth curves in $B$. We define a normalized Frobenius-Perron operator on the vector bundle of Lipschitz continuous functions labelled by the curves in $\mathcal{C}$, and use it to prove the existence of a unique, smooth conditionally invariant measure $ \mu$ on a segment $ {V^u}$ of the unstable manifold of $p$. A formula for the computation of ${f^{\ast}}$, the density of $\mu$, is derived, and $\mu ({\tau ^{ - 1}}{V^u})$ is shown to be equal to the reciprocal of the maximal modulus eigenvalue of the Jacobian of $\tau$ at $p$.
A cohomological approach to the Brauer-Long group and the groups of Galois extensions and strongly graded rings
S.
Caenepeel;
M.
Beattie
747-775
Abstract: Let $G$ be a finite abelian group, and $ R$ a commutative ring. The Brauer-Long group $\operatorname{BD} (R,G)$ is described by an exact sequence $\displaystyle 1 \to {\operatorname{BD} ^s}(R,G) \to \operatorname{BD} (R,G)\xrightarrow{\beta }\operatorname{Aut} (G \times {G^{\ast}})(R)$ where $ {\operatorname{BD} ^s}(R,G)$ is a product of étale cohomology groups, and Im $\beta$ is a kind of orthogonal subgroup of $ \operatorname{Aut} (G \times {G^{\ast}})(R)$. This sequence generalizes some other well-known exact sequences, and restricts to two split exact sequences describing Galois extensions and strongly graded rings.
Relative Frobenius of plane singularities
D.
Daigle
777-791
Abstract: In view of the well-known conjecture concerning the classification of lines in the affine plane in characteristic $p > 0$, it is desirable to understand how the characteristic pairs of an irreducible algebroid plane curve are affected by the relative Frobenius. This paper determines the relation between the characteristic sequences $[x,y]$ and $[x,{y^p}]$, where $x$ and $y$ are formal power series in one variable with coefficients in a field of characteristic $p > 0$.
Prescribing curvature on compact surfaces with conical singularities
Marc
Troyanov
793-821
Abstract: We study the Berger-Nirenberg problem on surfaces with conical singularities, i.e. we discuss conditions under which a function on a Riemann surface is the Gaussian curvature of some conformal metric with a prescribed set of singularities of conical types.
Univalence criteria and quasiconformal extensions
J. M.
Anderson;
A.
Hinkkanen
823-842
Abstract: Let $f$ be a locally univalent meromorphic function in the unit disk $\Delta$. Recently, Epstein obtained a differential geometric proof for the fact that if $f$ satisfies an inequality involving a suitable real-valued function $\sigma$, then $f$ is univalent in $\Delta$ and has a quasiconformal extension to the sphere. We give a more classical proof for this result by means of an explicit quasiconformal extension, and obtain generalizations of the result under suitable conditions even if $\sigma$ is allowed to be complex-valued and $ \Delta$ is replaced by a quasidisk.
On pseudo-differentiability
Roberto
Cominetti
843-865
Abstract: We present some new relations between the pseudo-derivatives and parabolic epiderivatives recently introduced by Rockafellar, and also its infinite dimensional counterparts. Significant extensions of the most important known results are proven, which further clarify the range of applicability of this new theory.
Linearization of bounded holomorphic mappings on Banach spaces
Jorge
Mujica
867-887
Abstract: The main result in this paper is the following linearization theorem. For each open set $U$ in a complex Banach space $E$, there is a complex Banach space ${G^\infty }(U)$ and a bounded holomorphic mapping $ {g_U}:U \to {G^\infty }(U)$ with the following universal property: For each complex Banach space $F$ and each bounded holomorphic mapping $ f:U \to F$, there is a unique continuous linear operator ${T_f}:{G^\infty }(U) \to F$ such that ${T_f} \circ {g_U} = f$. The correspondence $f \to {T_f}$ is an isometric isomorphism between the space $ {H^\infty }(U;F)$ of all bounded holomorphic mappings from $U$ into $F$, and the space $L({G^\infty }(U);F)$ of all continuous linear operators from $ {G^\infty }(U)$ into $ F$. These properties characterize ${G^\infty }(U)$ uniquely up to an isometric isomorphism. The rest of the paper is devoted to the study of some aspects of the interplay between the spaces $ {H^\infty }(U;F)$ and $L({G^\infty }(U);F)$.
On certain partial differential operators of finite odd type
A. Alexandrou
Himonas
889-900
Abstract: Let $P$ be a linear partial differential operator of order $m \geqslant 1$ with real-analytic coefficients defined in $\Omega$, an open set of ${\mathbb{R}^n}$, and let $\gamma$ be in the cotangent space of $ \Omega$ minus the zero section. If $P$ is of odd finite type $k$ and if the Hörmander numbers are $ 1 = {k_1} < {k_2},{k_2}$ odd, then $P$ is analytic hypoelliptic at $ \gamma$. These operators are not semirigid.
The existence of generalized isothermal coordinates for higher-dimensional Riemannian manifolds
Jian Guo
Cao
901-920
Abstract: We shall show that, for any given point $p$ on a Riemannian manifold $(M,{g^0})$, there is a pointwise conformal metric $g = \Phi {g^0}$ in which the $ g$-geodesic sphere centered at $p$ with radius $r$ has constant mean curvature $1/r$ for all sufficiently small $ r$. Furthermore, the exponential map of $g$ at $p$ is a measure preserving map in a small ball around $p$.