Generalized conjugate function theorems for solutions of first-order elliptic systems on the plane
Chung Ling
Yu
1-35
Abstract: Our essential aim is to generalize Privoloff's theorem, Schwarz reflection principle, Kolmogorov's theorem and the theorem of M. Riesz for conjugate functions to the solutions of differential equations in the $z = x + iy$ plane of the following elliptic type: $\displaystyle \frac{{\partial u}} {{\partial x}} - \frac{{\partial v}} {{\parti... ...\partial u}} {{\partial y}} + \frac{{\partial v}} {{\partial x}} = cu + dv + g.$ ( (M)) Theorem 1. Let the coefficients of (M) be Hölder continuous on $ \left\vert z \right\vert \leqslant 1$. Let $(u, v)$ be a solution of (M) in $\left\vert z \right\vert < 1$. If u is continuous on $\left\vert z \right\vert \leqslant 1$ and Hölder continuous with index $\alpha$ on $\left\vert z \right\vert = 1$, then $ (u, v)$ is Höolder continuous with index $\alpha$ on $\left\vert z \right\vert \leqslant 1$. Theorem 2. Let the coefficients of (M) be continuous on $ \left\vert z \right\vert \leqslant 1$ and satisfy the condition $\displaystyle \int_0^y {b(x,t)dt + \int_0^x {d(t,y)dt = \int_0^y {b(0,t)dt + \int_0^x {d(t,0)dt}}}}$ ( (N)) for $\left\vert z \right\vert \leqslant 1$. And let ${\left\Vert f \right\Vert _p} = {\sup _{0 \leqslant r < 1}}\{ (1/2\pi )\int_{ ... ...i }^\pi {{{\left\vert {f(r{e^{i\theta }})} \right\vert}^p}d\theta {\} ^{1/p}}}$. Then to each p, $0 < p < \infty$, there correspond two constants ${A_p}$ and ${B_p}$ such that \begin{displaymath}\begin{array}{*{20}{c}} {{{\left\Vert v \right\Vert}_p} \leqs... ...Vert u \right\Vert}_1} + {B_p},} & {0 < p < 1,} \end{array} \end{displaymath} hold for every solution $ (u, v)$ of (M) in $ \left\vert z \right\vert < 1$ with $v (0) = 0$. If $ f \equiv g \equiv 0$, the theorem holds for ${B_p} = 0$. Furthermore, if b and d do not satisfy the condition (N) in $ \left\vert z \right\vert \leqslant 1$, then we can relax the condition $ v (0) = 0$, and still have the above inequalities. Theorem 3. Let the coefficients of (M) be analytic for x, y in $ \left\vert z \right\vert < 1$. Let $(u, v)$ be a solution of (M) in $ \{ \left\vert z \right\vert < 1\} \cap \{ y > 0\} $. If u is continuous in $\{ \left\vert z \right\vert < 1\} \cap \{ y \geqslant 0\}$ and analytic on $\{ - 1 < x < 1\} $, then $ (u,v)$ can be continued analytically across the boundary $\{ - 1 < x < 1\} $. Furthermore, if the coefficients and u satisfy some further boundary conditions, then $(u, v)$ can be continued analytically into the whole of $\{ \left\vert z \right\vert < 1\}$.
Existence theorems for Pareto optimization; multivalued and Banach space valued functionals
L.
Cesari;
M. B.
Suryanarayana
37-65
Abstract: Existence theorems are obtained for optimization problems where the cost functional takes values in an ordered Banach space. The order is defined in terms of a closed convex cone in the Banach space; and in this connection, several relevant properties of cones are studied and they are shown to coincide in the finite dimensional case. The notion of a weak (Pareto) extremum of a subset of an ordered Banach space is then introduced. Existence theorems are proved for extrema for Mayer type as well as Lagrange type problems-in a manner analogous to and including those with scalar valued cost. The side conditions are in the form of general operator equations on a class of measurable functions defined on a finite measure space. Needed closure and lower closure theorems are proved. Also, several analytic criteria for lower closure are provided. Before the appendix, several illustrative examples are given. In the appendix, a criterion (different from the one used in main text) is given and proved, for the Pareto optimality of an element.
On the boundary behaviour of functions orthogonal to invariant subspaces
Kar Koi
Leung
67-84
Abstract: In this paper we study the relationship between the boundary behaviour of functions in the orthogonal subspace of the invariant subspace $\varphi {H^2}$ and the boundary behaviour of the inner function $\varphi$ itself.
Two-dimensional $\varepsilon $-isometries
D. G.
Bourgin
85-102
Abstract: An affirmative answer to the antipodal $ \varepsilon$-isometry conjecture is established for 2-dimensional Banach spaces.
Application of the sector condition to the classification of sub-Markovian semigroups
Martin L.
Silverstein
103-146
Abstract: Let ${p_{t}}$, $t > 0$, be a strongly continuous submarkovian semigroup on a real Hilbert space ${L^2}(X, m)$. The measure m is assumed to be excessive and the ${L^2}$ generator A is assumed to satisfy an estimate (the sector condition) which permits the application of Dirichlet spaces (not necessarily symmetric). Other submarkovian semigroups $P_t^ \sim$ with the same local generator and cogenerator and relative to which m is again excessive are classified in terms of generators for processes which live on a suitable boundary.
The Hanf number of the first order theory of Banach spaces
Saharon
Shelah;
Jacques
Stern
147-171
Abstract: In this paper, we discuss the possibility of developing a nice i.e. first order theory for Banach spaces: the restrictions on the set of sentences for recent compactness arguments applied to Banach spaces as well as for other model-theoretic results are both natural and necessary; without them we essentially get a second order logic with quantification over countable sets. Especially, the Hanf number for sets of sentences of the first order theory of Banach spaces is exactly the Hanf number for the second order logic of binary relations (with the second order quantifiers ranging over countable sets).
On the exceptional central simple non-Lie Mal'cev algebras
Renate
Carlsson
173-184
Abstract: Malcev algebras belong to the class of binary Lie algebras. Any Lie algebra is a Malcev algebra. In this paper we show that for each seven-dimensional central simple non-Lie Malcev algebra any finite dimensional Malcev module is completely reducible also for positive characteristics. This contrasts with each modular semisimple Lie algebra. As a consequence we get that the classical structure theory for characteristic zero is valid also in the modular case if semisimplicity is replaced by $ {G_1}$-separability. The Wedderburn principal theorem is proved for Malcev algebras.
The isotopy problem for Jordan matrix algebras
Holger P.
Petersson
185-197
Abstract: Conditions are given which are necessary and sufficient for two members of a certain class of Jordan matrix algebras to be isotopic. The main ingredient of these conditions is McCrimmon's notion of isotopy for alternative algebras.
Rational fibrations, minimal models, and fibrings of homogeneous spaces
Stephen
Halperin
199-224
Abstract: Sullivan's theory of minimal models is used to study a class of maps called rational fibrations, which contains most Serre fibrations. It is shown that if the total space has finite rank and the fibre has finite dimensional cohomology, then both fibre and base have finite rank. This is applied to prove that certain homogeneous spaces cannot be the total space of locally trivial bundles. In addition two main theorems are proved which exhibit a close relation between the connecting homomorphism of the long exact homotopy sequence, and certain properties of the cohomology of fibre and base.
Analytically decomposable operators
Ridgley
Lange
225-240
Abstract: The author introduces the notion of an analytically decomposable operator which generalizes the decomposable operator due to C. Foias in that the spectral decompositions of the underlying Banach space (1) admit a wider class of invariant subspaces called ``analytically invariant'' and (2) span the space only densely. It is shown that analytic decomposability is stable under the functional calculus, direct sums and restrictions to certain kinds of invariant subspaces, as well as perturbation by commuting scalar operators. It is fundamental for many of these results that every analytically decomposable operator has the single-valued extension property. An extensive investigation of analytically invariant subspaces is given. The author shows by example that this class is distinct from those of spectral maximal and hyperinvariant subspaces, but he further shows that analytically invariant subspaces have many useful spectral properties. Some applications of the general theory are made. For example, it is shown that under certain restrictions an analytically decomposable operator is decomposable.
$R$-separation of variables for the four-dimensional flat space Laplace and Hamilton-Jacobi equations
E. G.
Kalnins;
Willard
Miller
241-261
Abstract: All A-separable orthogonal coordinate systems for the complex equations $\Sigma_{i = 1}^4 {{\partial _{ii}}\Psi = 0}$ and $\Sigma_{i = 1}^4 {{{({\partial _i}W)}^2} = 0}$ are classified and it is shown that these equations separate in exactly the same systems.
Algebraic structures for $\bigoplus \sum \sb{n\geq 1}L\sp{2}(Z/n)$ compatible with the finite Fourier transform
L.
Auslander;
R.
Tolimieri
263-272
Abstract: Let ${Z / n}$ denote the integers $ \bmod\,n$ and let ${\mathcal{F}_n}$ denote the finite Fourier transform on ${L^2}({Z / n})$. We let $ \oplus \Sigma {{\mathcal{F}_n}} = F$ operate on $\oplus \Sigma {L^2}({Z / n})$ and show that $\oplus \Sigma {L^2}({Z / n})$ can be given a graded algebra structure (with no zero divisors) such that $\mathcal{F}(fg) = \mathcal{F}(f)\mathcal{F}(g)$. We do this by establishing a natural isomorphism with the algebra of theta functions with period i. In addition, we find all algebra structures on $ \oplus \Sigma {L^2}({Z / n})$ satisfying the above condition.
An asymptotic double commutant theorem for $C\sp{\ast} $-algebras
Donald W.
Hadwin
273-297
Abstract: An asymptotic version of von Neumann's double commutant theorem is proved in which $ {C^{\ast}}$-algebras play the role of von Neumann algebras. This theorem is used to investigate asymptotic versions of similarity, reflexivity, and reductivity. It is shown that every nonseparable, norm closed, commutative, strongly reductive algebra is selfadjoint. Applications are made to the study of operators that are similar to normal (subnormal) operators. In particular, if T is similar to a normal (subnormal) operator and $\pi$ is a representation of the ${C^{\ast}}$-algebra generated by t, then $ \pi (T)$ is similar to a normal (subnormal) operator.
Some $3$-manifolds which admit Klein bottles
Paik Kee
Kim
299-312
Abstract: Consider a closed, orientable, irreducible 3-manifold M with $ \left\vert {{\pi _1}(M)} \right\vert < \infty$, in which a Klein bottle can be embedded. We present a classification of the spaces M and show that, if $ {\pi _1}(M)$ is cyclic, then M is homeomorphic to a lens space. Note that all surfaces of even genus can be embedded in each space M. We also classify all free involutions on lens spaces whose orbit spaces contain Klein bottles.
The eigenvalue spectrum as moduli for flat tori
Scott
Wolpert
313-321
Abstract: A flat torus T carries a natural Laplace Beltrami operator. It is a conjecture that the spectrum of the Laplace Beltrami operator determines T modulo isometries. We prove that, with the exception of a subvariety in the moduli space of flat tori, this conjecture is true. A description of the subvariety is given.