Univalent harmonic functions
W.
Hengartner;
G.
Schober
1-31
Abstract: Several families of complex-valued, univalent, harmonic functions are studied from the point of view of geometric function theory. One class consists of mappings of a simply-connected domain onto an infinite horizontal strip with a normalization at the origin. Extreme points and support points are determined, as well as sharp estimates for Fourier coefficients and distortion theorems. Next, mappings in $ \left\vert z \right\vert > 1$ are considered that leave infinity fixed. Some coefficient estimates, distortion theorems, and covering properties are obtained. For such mappings with real boundary values, many extremal problems are solved explicitly.
Completely reducible operators that commute with compact operators
Shlomo
Rosenoer
33-40
Abstract: It is shown that if $ T$ is a completely reducible operator on a Banach space and $TK = KT$, where $K$ is an injective compact operator with a dense range, then $T$ is a scalar type spectral operator. Other related results are also obtained.
Applications of the covering lemma for sequences of measures
W.
Mitchell
41-58
Abstract: We present several applications of the covering lemma for the core model for sequences of measures, including characterizations of the large cardinal strength necessary to make the filter of closed, unbounded subsets of ${\omega _1}$ an ultrafilter or to change the cofinality of a regular cardinal, and a characterization of the minimal inner model containing an arbitrary elementary embedding.
On the values at integers of the Dedekind zeta function of a real quadratic field
David
Kramer
59-79
Abstract: In 1976 Shintani gave a decomposition of the Dedekind zeta function, $\zeta \kappa (s)$, of a totally real number field into a finite sum of functions, each given by a Dirichlet series whose meromorphic continuation assumes rational values at negative integers. He obtained a formula for these values, thereby giving an expression for $ \zeta \kappa ( - n),$, $n = 0,\,1,\,2, \ldots$. Earlier, Zagier had studied the special case of $\zeta (A,\,s)$, the narrow ideal class zeta function for a real quadratic field. He decomposes $\zeta (A,\,s)$ into ${\Sigma _A}{Z_Q}(s)$, where ${Z_Q}(s)$ is given as a Dirichlet series associated to a binary quadratic form $Q(x,\,y) = a{x^2} + bxy + c{y^2}$, and the summation is over a canonically given finite cycle of ``reduced'' quadratic forms associated to a narrow ideal class $A$. He then obtains a formula for ${Z_Q}( - n)$ as a rational function in the coefficients of the form $Q$. Since the denominator of $\zeta (A,\, - n)$ is known not to depend on the class $A$, whereas the coefficients of reduced forms attain arbitrarily large values, it is natural to ask whether the rational function in Zagier's formula might be replaced by a polynomial. In this paper such a result is obtained. For example, Zagier gives $\displaystyle 15120\zeta (A, - 2) = \sum\limits_A {\frac{{{b^5} - 10a{b^3}c + 3... ...}} + \frac{{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{c^3}}} - 21b(a + c){\text{ }}$ while our result is $\displaystyle 15120\zeta (A,\, - 2) = \frac{1} {2}\left( {\mathop \sum \limits_... ...mits_{\theta A} } \right)\,(60{a^2} - 117ab + 76ac + 38{b^2} - 117bc + 60{c^2})$ , where $ \theta$ is the narrow ideal class consisting of principal ideals generated by elements of negative norm. Starting with a representation of ${Z_Q}(1 + n)$ due to Shanks and Zagier for $n = 1,\,2,\,3, \ldots$ as a certain transcendental function of the coefficients of $ Q$, we also obtain the result that $ \zeta (A,\,1 + n)$ is given as the same sum of reduced quadratic forms as in the formula for $ \zeta (A,\, - n)$, times the appropriate ``gamma factor.'' This gives a new proof of the functional equation of $\zeta (A,\,s)$ at integer values of $ s$, and suggests the possibility that one might be able to prove the functional equation for all $s$ by finding some relation between $ {Z_Q}(s)$ and ${Z_Q}(1 - s)$. So far we have not found such a relation.
Concavity of solutions of the porous medium equation
Philippe
Bénilan;
Juan Luis
Vázquez
81-93
Abstract: We consider the problem $\displaystyle \left( {\text{P}} \right)\quad \quad \left\{ {\begin{array}{*{20}... ...\in {\mathbf{R}},\,t > 0} {{\text{for}}\,x \in {\mathbf{R}}} \end{array}$ where $m > 1$ and ${u_0}$ is a continuous, nonnegative function that vanishes outside an interval $(a,\,b)$ and such that $(u_0^{m - 1})'' \leq - C \leq 0$ in $ (a,\,b)$. Using a Trotter-Kato formula we show that the solution conserves the concavity in time: for every $t > 0,\,u(x,t)$ vanishes outside an interval $\Omega (t) = ({}_{\zeta 1}(t),\,{}_{\zeta 2}(t))$ and $\displaystyle {({u^{m - 1}})_{xx}} \leq - \frac{C} {{1 + C(m(m + 1)/(m - 1))t}}$ in $\Omega (t)$. Consequently the interfaces $x{ = _{\zeta i}}(t)$, $i = 1,\,2$, are concave curves. These results also give precise information about the large time behavior of solutions and interfaces.
Dimension de Hausdorff des ensembles de z\'eros et d'interpolation pour $A\sp \infty(D)$
Jacques
Chaumat;
Anne-Marie
Chollet
95-114
Abstract: Soit $D$ un domaine borné strictement pseudoconvexe dans $ {{\mathbf{C}}^n}$ à frontière régulière $\partial D$ et soit $ {A^\infty }(D)$ la classe des fonctions holomorphes dans $D$, indéfiniment dérivables dans $ \overline D$. Un sous-ensemble compact $E$ de $ \partial D$ est un ensemble de zéros pour $ {A^\infty }(D)$ s'il existe une fonction de $ {A^\infty }(D)$ s'annulant seulement sur $E$. C'est un ensemble d'interpolation d'ordre infini pour ${A^\infty }(D)$ si, pour toute fonction $f$ de classe $ {C^\infty }$ dans ${{\mathbf{C}}^n}$ telle que $\overline \partial f$ soit plate sur $ E$, il existe une fonction $ F$ de ${A^\infty }(D)$ telle que $F - f$ soit plate sur $E$. On construit ici des ensembles de dimension de Hausdorff $n$. Ce résultat est le meilleur possible dans le cas d'ensembles totalement réels. Le point de vue utilisé pour montrer qu'un sous-ensemble $ E$ de $\partial D$ est d'interpolation d'ordre infini pour ${A^\infty }(D)$ est de vérifier qu'il a la propriété de division par ${A^\infty }(D)$, c'est-á-dire, que, pour toute famille de fonctions ${({f_i})_{i \in {\text{N}}}}$ de ${C^\infty }(\overline D )$, plates sur $E$, il existe une fonction $F$ de $ {A^\infty }(D)$, plate sur $ E$ et nulle seulement sur $ E$ et une famille de fonctions $ {({k_i})_{i \in {\text{N}}}}$ de $ {C^\infty }(\overline D )$, plates sur $E$, telles que l'on ait, pour tout $i$ dans $ {\mathbf{N}}$, ${f_i} = F{k_i}$.
The structure of the critical set in the mountain pass theorem
Patrizia
Pucci;
James
Serrin
115-132
Abstract: We show that the critical set generated by the Mountain Pass Theorem of Ambrosetti and Rabinowitz must have a well-defined structure. In particular, if the underlying Banach space is infinite dimensional then either the critical set contains a saddle point of mountain-pass type, or the set of local minima intersects at least two components of the set of saddle points. Related conclusions are also established for the finite dimensional case, and when other special conditions are assumed. Throughout the paper, no hypotheses of nondegeneracy are required on the critical set.
Persistence of form and the value group of reducible cubics
P. D. T. A.
Elliott
133-143
Abstract: It is proved that the values of $ x({x^2} + c)$, $c \ne 0$, at positive integers, multiplicatively generate the positive rationals. Analogs in rational function fields are obtained.
Regressive partition relations for infinite cardinals
András
Hajnal;
Akihiro
Kanamori;
Saharon
Shelah
145-154
Abstract: The regressive partition relation, which turns out to be important in incompleteness phenomena, is completely characterized in the transfinite case. This work is related to Schmerl $ \left[ {\mathbf{S}} \right]$, whose characterizations we complete.
Construction of group actions on four-manifolds
Allan L.
Edmonds
155-170
Abstract: It is shown that any cyclic group of odd prime order acts on any closed, simply connected topological $4$-manifold, inducing the identity on integral homology. The action is locally linear except perhaps at one isolated fixed point. In the case of primes greater than three a more careful argument is used to show that the action can be constructed to be locally linear.
Countably generated Douglas algebras
Keiji
Izuchi
171-192
Abstract: Under a certain assumption of $f$ and $g$ in $ {L^\infty }$ which is considered by Sarason, a strong separation theorem is proved. This is available to study a Douglas algebra $[{H^\infty },\,f]$ generated by ${H^\infty }$ and $f$. It is proved that (1) ball $(B/{H^\infty } + C)$ does not have exposed points for every Douglas algebra $B$, (2) Sarason's three functions problem is solved affirmatively, (3) some characterization of $ f$ for which $[{H^\infty },\,f]$ is singly generated, and (4) the $M$-ideal conjecture for Douglas algebras is not true.
A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schr\"odinger equations
O.
Kavian
193-203
Abstract: We consider solutions to $i{u_t} = \Delta u + {\left\vert u \right\vert^{p - 1}}u$, $u(0) = {u_0}$, where $x$ belongs to a smooth domain $ \Omega \subset {{\mathbf{R}}^N}$, and we prove that under suitable conditions on $ p$, $N$ and ${u_0} \in {H^2}(\Omega ) \cap H_0^1(\Omega )$, $ {\left\Vert {\nabla u(t)} \right\Vert _{{L^2}}}$ blows up in finite time. The range of $p$'s for which blowing-up occurs depends on whether $ \Omega$ is starshaped or not. Examples of blowing-up under Neuman or periodic boundary conditions are given. On considère des solutions de $i{u_t} = \Delta u + {\left\vert u \right\vert^{p - 1}}u$, $u(0) = {u_0}$, où la variable d'espace $x$ appartient à un domaine régulier $ \Omega \subset {{\mathbf{R}}^N}$, et on prouve que sous des conditions adéquates sur $p$, $N$ et ${u_0} \in {H^2}(\Omega ) \cap H_0^1(\Omega )$, $ {\left\Vert {\nabla u(t)} \right\Vert _{{L^2}}}$ explose en temps fini. Les valeurs de $p$ pour lesquelles l'explosion a lieu dépend de la forme de l'ouvert $\Omega$ (en fait $\Omega$ étoilé ou non). On donne également des exemples d'explosion sous des conditions de Neuman ou périodiques au bord.
Category and group rings in homotopy theory
William J.
Ralph
205-223
Abstract: It frequently arises in algebraic topology that a function $\beta :G \to H$, between two groups, is not a homomorphism. We show that in many standard situations $ \beta$ induces a group homomorphism $\overline \beta :{\mathbf{Z}}(G)/{\mathcal{A}^d} \to H$, where $ {\mathcal{A}^d}$ is a power of the augumentation ideal in the group ring ${\mathbf{Z}}(G)$. A typical example is $ \beta :[X,\,Y] \to [{S^2}X,\,{S^2}Y]$ where $Y$ is some $H$-group, in which case $d$ can be taken to be $1 + {\text{cat}}\,X$.
Geometric theory of extremals in optimal control problems. I. The fold and Maxwell case
I.
Kupka
225-243
Abstract: The behavior of the extremal curves in optimal control theory is much more complex than that of their namesakes in the classical calculus of variations. Here we analyze the simplest instances of singular behavior of these extremals. Among others, in sharp contrast to the classical case, a $ {C^0}$-limit of extremals may not be an extremal. In the simplest cases (elliptic fold and Maxwell points) of this occurrence, the limits are trajectories of a new vector field. A special case of this field showed up in some work of V. I. Arnold. Results related to ours have been obtained in low dimension by I. Ekeland.
Nonharmonic Fourier series and spectral theory
Harold E.
Benzinger
245-259
Abstract: We consider the problem of using functions ${g_n}(x): = exp(i{\lambda _n}x)$ to form biorthogonal expansions in the spaces ${L^p}( - \pi ,\,\pi )$, for various values of $ p$. The work of Paley and Wiener and of Levinson considered conditions of the form $\left\vert {{\lambda _n} - n} \right\vert \leq \Delta (p)$ which insure that $ \{ {g_n}\}$ is part of a biorthogonal system and the resulting biorthogonal expansions are pointwise equiconvergent with ordinary Fourier series. Norm convergence is obtained for $p = 2$. In this paper, rather than imposing an explicit growth condition, we assume that $ \{ {\lambda _n} - n\}$ is a multiplier sequence on ${L^p}( - \pi ,\,\pi )$. Conditions are given insuring that $\{ {g_n}\}$ inherits both norm and pointwise convergence properties of ordinary Fourier series. Further, ${\lambda _n}$ and ${g_n}$ are shown to be the eigenvalues and eigenfunctions of an unbounded operator $ \Lambda$ which is closely related to a differential operator, $i\Lambda$ generates a strongly continuous group and $ - {\Lambda ^2}$ generates a strongly continuous semigroup. Half-range expansions, involving ${\text{cos}}{\lambda _n}x$ or ${\text{sin}}{\lambda _n}x$ on $(0,\,\pi )$ are also shown to arise from linear operators which generate semigroups. Many of these results are obtained using the functional calculus for well-bounded operators.
Growth properties of functions in Hardy fields
Maxwell
Rosenlicht
261-272
Abstract: This paper continues the author's earlier work on the notion of rank in a Hardy field. Further results are given on functions in Hardy fields of finite rank, including extensions of Hardy's results on the rates of growth of his logarithmico-exponential functions.
On the behavior near the crest of waves of extreme form
C. J.
Amick;
L. E.
Fraenkel
273-298
Abstract: The angle $ \phi$ which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation that does not fit (as far as we know) into the theory of compact operators on Banach spaces. It has been proved only recently that solutions exist and that (as Stokes suggested in 1880) these solutions represent waves with sharp crests of included angle $2\pi /3$. In this paper we use the integral equation, known properties of solutions and the technique of the Mellin transform to obtain the asymptotic expansion $\displaystyle ( * )\qquad \phi (s) = \frac{\pi } {6} + \sum\limits_{n = 1}^k {{a_n}{s^{{\mu _n}}} + o({s^{{\mu _k}}})} \quad {\text{as}}\,s \downarrow 0$ , to arbitrary order; the coordinate $s$ is related to distance from the crest as measured by the velocity potential rather than by length. The first few (and probably all) of the exponents $ {\mu _n}$ are transcendental numbers. We are unable to evaluate the coefficients $ {a_n}$ explicitly, but define some in terms of global properties of $\phi$, and the others in terms of earlier coefficients. It is proved in [8] that $ {a_1} < 0$, and follows here that ${a_2} > 0$. The derivation of (*) includes an assumption about a question in number theory; if that assumption should be false, logarithmic terms would enter the series at very large values of $ n$.
The asymptotic behavior near the crest of waves of extreme form
J. B.
McLeod
299-302
Abstract: The angle which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation. It has been proved only recently that solutions exist and that (as Stokes suggested in 1880) these solutions represent waves with sharp crests of included angle $ \frac{2} {3}\pi$. Amick and Fraenkel have investigated the asymptotic behavior of the free surface near the crest and obtained an asymptotic expansion for this behavior, but are unable to say whether the leading term in this expansion has a nonzero coefficient (and so whether it is in fact the leading term or not). The present paper shows that the coefficient is nonzero and determines its sign.
On inductive limits of matrix algebras of holomorphic functions
Justin
Peters
303-318
Abstract: Let $\mathfrak{A}$ be a UHF algebra and $ \mathcal{A}({\mathbf{D}})$ the disk algebra. If $\mathfrak{A} = {\left[ {{ \cup _{n \geq 1}}{\mathfrak{A}_n}} \right]^ - }$ and $\alpha$ is a product-type automorphism of $\mathfrak{A}$ which leaves each ${\mathfrak{A}_n}$ invariant, then $ \alpha$ defines an embedding $\displaystyle \mathfrak{A}_n \otimes \mathcal{A}({\mathbf{D}}) \stackrel{\imath_n}{\hookrightarrow} {\mathfrak{A}_{n + 1}} \otimes \mathcal{A}({\mathbf{D}})$ . The inductive limit of this system is a Banach algebra whose maximal ideal space is closely related to that of the disk algebra if the Connes spectrum $ \Gamma (\alpha )$ is finite.
First layer formulas for characters of ${\rm SL}(n,{\bf C})$
John R.
Stembridge
319-350
Abstract: Some problems concerning the decomposition of certain characters of $ SL(n,\,{\mathbf{C}})$ are studied from a combinatorial point of view. The specific characters considered include those of the exterior and symmetric algebras of the adjoint representation and the Euler characteristic of Hanlon's so-called ``Macdonald complex.'' A general recursion is given for computing the irreducible decomposition of these characters. The recursion is explicitly solved for the first layer representations, which are the irreducible representations corresponding to partitions of $ n$. In the case of the exterior algebra, this settles a conjecture of Gupta and Hanlon. A further application of the recursion is used to give a family of formal Laurent series identities that generalize the (equal parameter) $q$-Dyson Theorem.
Best constants in norm inequalities for the difference operator
Hans G.
Kaper;
Beth E.
Spellman
351-372
Abstract: Let $ \xi = {({\xi _m})_{m \in {\mathbf{Z}}}}$ be an arbitrary element of the sequence space ${l^\infty }({\mathbf{Z}})$, and let $ \Delta$ be the difference operator on ${l^\infty }({\mathbf{Z}}):\Delta \xi = {({\xi _{m + 1}} - {\xi _m})_{m \in {\mathbf{Z}}}}$. The object of this investigation is the best possible value $\displaystyle C(n,\,k) = {\operatorname{sup}}\{ {Q_{n,k}}(\xi ):\xi \in {l^\infty }({\mathbf{Z}}),\,{\Delta ^n}\xi \ne 0\}$ of the quotient $\displaystyle {Q_{n,k}}(\xi ) = \frac{{\left\Vert {{\Delta ^k}\xi } \right\Vert... ...i \right\Vert}^{(n - k)/n}}{{\left\Vert {{\Delta ^n}\xi } \right\Vert}^{k/n}}}}$ , where $n = 2,\:3, \ldots$; $k = 1, \ldots ,\,n - 1$. It is shown that $ C(n,\,k)$ is at least equal to the corresponding constant $K(n,\,k)$, determined by Kolmogorov [Moscov. Gos. Univ. Uchen. Zap. Mat. 30 (1939), 3-13; Amer. Math. Soc. Transl. (1) 2 (1962), 233-243] for the differential operator $D$ on ${L^\infty }({\mathbf{R}})$, and exactly equal to $K(n,\,k)$ if $k = n - 1$. Lower bounds for $C(n,\,k)$ are derived that show that $ C(n,\,k)$ is generally greater than $K(n,\,k)$. The values of $C(n,\,k)$, $ k = 1, \ldots ,\,n - 1$, are computed for $ n = 2, \ldots ,5$.
On embedding of group rings of residually torsion free nilpotent groups into skew fields
A.
Eizenbud;
A. I.
Lichtman
373-386
Abstract: It is proven that the group ring of an amalgamated free product of residually torsion free nilpotent groups is a domain and can be embedded in a skew field. This is a generalization of J. Lewin's theorem, proven for the case of free groups. Our proof is based on the study of the Malcev-Neumann power series ring $K\left\langle G \right\rangle$ of a residually torsion free nilpotent group $G$. It is shown that its subfield $ D$, generated by the group ring $KG$, does not depend on the order of $G$ for many kinds of orders and the study of $D$ can be reduced in some sense to the case when $ G$ is nilpotent.
Integration on noncompact supermanifolds
Mitchell J.
Rothstein
387-396
Abstract: We note that the Berezin integral, which is ill-defined for noncompact supermanifolds, is a distribution with support on the underlying manifold. This leads to the discovery of correction terms in the Berezinian transformation law and thereby eliminates the boundary ambiguities.
$K$-theory and multipliers of stable $C\sp \ast$-algebras
J. A.
Mingo
397-411
Abstract: The main theorem is that if $A$ is a $C^{\ast}$-algebra with a countable approximate identity consisting of projections, then the unitary group of $ M(A \otimes K)$ is contractible. This gives a realization, via the index map, of ${K_0}(A)$ as components in the set of Fredholm operators on ${H_A}$.
The dimension of closed sets in the Stone-\v Cech compactification
James
Keesling
413-428
Abstract: In this paper properties of compacta $K$ in $ \beta X\backslash X$ are studied for Lindelöf spaces $X$. If ${\operatorname{dim}}\,K = \infty$, then there is a mapping $f:K \to {T^c}$ such that $f$ is onto and every mapping homotopic to $ f$ is onto. This implies that there is an essential family for $K$ consisting of $c$ disjoint pairs of closed sets. It also implies that if $K = \cup \left\{ {{K_\alpha }\vert\alpha < c} \right\}$ with each $ {K_\alpha }$ closed, then there is a $\beta$ such that $ {\operatorname{dim}}\,{K_\beta } = \infty$. Assume $K$ is a compactum in $\beta X\backslash X$ as above. Then if $ {\operatorname{dim}}\,K = n$, there is a closed set $K'$ in $K$ such that ${G_\delta }$-set in $K'$ contains an $n$-dimensional compactum. This holds for $ n$ finite or infinite. If ${\operatorname{dim}}\,K = n$ and $K = \cup \left\{ {{K_\alpha }\vert\alpha < {\omega _1}} \right\}$ with each ${K_\alpha }$ closed, then there must be a $ \beta$ such that $ {\operatorname{dim}}\,{K_\beta } = n$.
Erratum to: ``Bounds on the dimension of variations of Hodge structure''
James A.
Carlson
429