Functions of real variables. Functions of a complex variable.

Mode of access: Internet.

The theory of functions of a complex variable. [Lectures] New York University, Institute for Mathematics and Mechanics.

"First edition 1948. Second printing 1952."

Elements of the theory of functions of a complex variable with especial reference to the methods of Riemann,

Translation of Elemente der Theorie der Funktionen einer complexen veränderlichen Grösse.

Theory of functions of a complex variable,

Available on demand as hard copy or computer file from Cornell University Library.

Theory of functions of a complex variable /

Mode of access: Internet.

The Bonenblust-Hille inequality for homogeneous polynomials is hypercontractive

The Bohnenblust-Hille inequality says that the $\ell^{\frac{2m}{m+1}}$ -norm of the coefficients of an $m$-homogeneous polynomial $P$ on $\Bbb{C}^n$ is bounded by $\| P \|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$ denotes the supremum norm on the polydisc $\mathbb{D}^n$. The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be $C^m$ for some $C>1$. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc $\mathbb{D}^n$ behaves asymptotically as $\sqrt{(\log n)/n}$ modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies $\bigl\{ \log n: n \text{a positive integer} \le N\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$.

A lower bound in Nehari's theorem on the polydisc

By theorems of Ferguson and Lacey ($d=2$) and Lacey and Terwilleger ($d>2$), Nehari's theorem is known to hold on the polydisc $\D^d$ for $d>1$, i.e., if $H_\psi$ is a bounded Hankel form on $H^2(\D^d)$ with analytic symbol $\psi$, then there is a function $\varphi$ in $L^\infty(\T^d)$ such that $\psi$ is the Riesz projection of $\varphi$. A method proposed in Helson's last paper is used to show that the constant $C_d$ in the estimate $\|\varphi\|_\infty\le C_d \|H_\psi\|$ grows at least exponentially with $d$; it follows that there is no analogue of Nehari's theorem on the infinite-dimensional polydisc.

Special functions of Weyl groups and their continuous and discrete orthogonality

Cette thèse s'intéresse à l'étude des propriétés et applications de quatre familles des fonctions spéciales associées aux groupes de Weyl et dénotées $C$, $S$, $S^s$ et $S^l$. Ces fonctions peuvent être vues comme des généralisations des polynômes de Tchebyshev. Elles sont en lien avec des polynômes orthogonaux à plusieurs variables associés aux algèbres de Lie simples, par exemple les polynômes de Jacobi et de Macdonald. Elles ont plusieurs propriétés remarquables, dont l'orthogonalité continue et discrète. En particulier, il est prouvé dans la présente thèse que les fonctions $S^s$ et $S^l$ caractérisées par certains paramètres sont mutuellement orthogonales par rapport à une mesure discrète. Leur orthogonalité discrète permet de déduire deux types de transformées discrètes analogues aux transformées de Fourier pour chaque algèbre de Lie simple avec racines des longueurs différentes. Comme les polynômes de Tchebyshev, ces quatre familles des fonctions ont des applications en analyse numérique. On obtient dans cette thèse quelques formules de <>, pour des fonctions de plusieurs variables, en liaison avec les fonctions $C$, $S^s$ et $S^l$. On fournit également une description complète des transformées en cosinus discrètes de types V--VIII à $n$ dimensions en employant les fonctions spéciales associées aux algèbres de Lie simples $B_n$ et $C_n$, appelées cosinus antisymétriques et symétriques. Enfin, on étudie quatre familles de polynômes orthogonaux à plusieurs variables, analogues aux polynômes de Tchebyshev, introduits en utilisant les cosinus (anti)symétriques.

Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka–Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930s, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview chapter we present three classes of properties: (1) density property, (2) flexibility, and (3) Oka–Forstnerič. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.

Major and minor threshold functions of six variables /

Issued also as thesis (M.S.) University of Illinois.

Role of peripheral afference during acquisition of a complex coordination task

It has long been supposed that the interference observed in certain patterns of coordination is mediated, at least in part, by peripheral afference from the moving limbs. We manipulated the level of afferent input, arising from movement of the opposite limb, during the acquisition of a complex coordination task. Participants learned to generate flexion and extension movements of the right wrist, of 75degrees amplitude, that were a quarter cycle out of phase with a 1-Hz sinusoidal visual reference signal. On separate trials, the left wrist either was at rest, or was moved passively by a torque motor through 50degrees, 75degrees or 100degrees, in synchrony with the reference signal. Five acquisition sessions were conducted on successive days. A retention session was conducted I week later. Performance was initially superior when the opposite limb was moved passively than when it was static. The amplitude and frequency of active movement were lower in the static condition than in the driven conditions and the variation in the relative phase relation across trials was greater than in the driven conditions. In addition, the variability of amplitude, frequency and the relative phase relation during each trial was greater when the opposite limb was static than when driven. Similar effects were expressed in electromyograms. The most marked and consistent differences in the accuracy and consistency of performance (defined in terms of relative phase) were between the static condition and the condition in which the left wrist was moved through 50degrees. These outcomes were exhibited most prominently during initial exposure to the task. Increases in task performance during the acquisition period, as assessed by a number of kinematic variables, were generally well described by power functions. In addition, the recruitment of extensor carpi radialis (ECR), and the degree of co-contraction of flexor carpi radialis and ECR, decreased during acquisition. Our results indicate that, in an appropriate task context, afferent feedback from the opposite limb, even when out of phase with the focal movement, may have a positive influence upon the stability of coordination.

Desmin: molecular interactions and putative functions of the muscle intermediate filament protein

Desmin is the intermediate filament (IF) protein occurring exclusively in muscle and endothelial cells. There are other IF proteins in muscle such as nestin, peripherin, and vimentin, besides the ubiquitous lamins, but they are not unique to muscle. Desmin was purified in 1977, the desmin gene was characterized in 1989, and knock-out animals were generated in 1996. Several isoforms have been described. Desmin IFs are present throughout smooth, cardiac and skeletal muscle cells, but can be more concentrated in some particular structures, such as dense bodies, around the nuclei, around the Z-line or in costameres. Desmin is up-regulated in muscle-derived cellular adaptations, including conductive fibers in the heart, electric organs, some myopathies, and experimental treatments with drugs that induce muscle degeneration, like phorbol esters. Many molecules have been reported to associate with desmin, such as other IF proteins (including members of the membrane dystroglycan complex), nebulin, the actin and tubulin binding protein plectin, the molecular motor dynein, the gene regulatory protein MyoD, DNA, the chaperone alphaB-crystallin, and proteases such as calpain and caspase. Desmin has an important medical role, since it is used as a marker of tumors' origin. More recently, several myopathies have been described, with accumulation of desmin deposits. Yet, after almost 30 years since its identification, the function of desmin is still unclear. Suggested functions include myofibrillogenesis, mechanical support for the muscle, mitochondrial localization, gene expression regulation, and intracellular signaling. This review focuses on the biochemical interactions of desmin, with a discussion of its putative functions.

Exploring Rac GTPase regulation : the molecular mechanisms governing the DOCK180 and ELMO interaction and the role of this complex in Rac-mediated cell migration

Les protéines DOCK180 et ELMO coopèrent ensemble biochimiquement et génétiquement afin d’activer la GTPase Rac1 lors de plusieurs évènements biologiques. Toutefois, le rôle que jouent ces protéines dans la signalisation par Rac est encore mal compris. Nous émettons l’hypothèse que Dock180 agit comme activateur de Rac, alors que ELMO est requis pour l’intégration de la signalisation de Rac plutôt que son activation per se. Nous postulons que ELMO agit comme signal de localisation intracellulaire afin de restreindre de façon spatio-temporelle la signalisation de Rac en aval de Dock180, et/ou que ELMO agit comme protéine d’échafaudage entre Rac et ses effecteurs pour amplifier la migration cellulaire. Dans l’objectif nº 1, nous démontrons que le domaine PH atypique de ELMO1 est le site d’interaction principal entre cette protéine et DOCK180. De plus, nous démontrons que la liaison entre ELMO et DOCK180 n’est pas nécessaire pour l’activation de Rac, mais est plutôt essentielle pour faciliter la réorganisation du cytosquelette induite par l’activation de Rac en aval de Dock180. Ces résultats impliquent que ELMO pourrait jouer des rôles additionnels dans la signalisation par Rac. Dans l’objectif nº 2, nous avons découvert l’existence d’une homologie structurelle entre ELMO et un module d’autorégulation de la formine Dia1, et avons identifié trois nouveaux domaines dans la protéine ELMO : les domaines RBD, EID et EAD. De façon analogue à Dia1, nous avons découvert que ELMO à l’état basal est autoinhibé grâce à des intéractions intramoléculaires. Nous proposons que l’état d’activation des protéines ELMO est régulé de façon similaire aux formines de la famille Dia, c’est-à-dire grâce à des interactions avec d’autres protéines. Dans l’objectif nº 3, nous identifions un domaine RBD polyvalent chez ELMO. Ce domaine possède une double spécificité pour les GTPases de la famille Rho et Arf. Nous avons découvert que Arl4A agit comme signal de recrutement membranaire pour le module ELMO/DOCK180/Rac. Nos résultats nous permettent de supposer que d’autres GTPases pourraient être impliquées dans l’activation et la localisation de cette voie de signalisation. Nous concluons qu’à l’état basal, ELMO et DOCK180 forment un complexe dans lequel ELMO est dans sa conformation autoinhibée. Bien que le mécanisme d’activation de ELMO ne soit pas encore bien compris, nous avons découvert que, lorsqu’il y a stimulation cellulaire, certaines GTPases liées au GTP peuvent intéragir avec le domaine RBD de ELMO pour relâcher les contacts intramoléculaires et/ou localiser le complexe à la membrane. Ainsi, les GTPases peuvent servir d’ancrage au complexe ELMO/DOCK180 pour assurer une regulation spatiotemporelle adequate de l’activation et de la signalisation de Rac.

C(k)-solvability near the characteristic set for a class of planar complex vector fields of infinite type

This paper deals with semi-global C(k)-solvability of complex vector fields of the form L = partial derivative/partial derivative t + x(r) (a(x) + ib(x))partial derivative/partial derivative x, r >= 1, defined on Omega(epsilon) = (-epsilon, epsilon) x S(1), epsilon > 0, where a and b are C(infinity) real-valued functions in (-epsilon, epsilon). It is shown that the interplay between the order of vanishing of the functions a and b at x = 0 influences the C(k)-solvability at Sigma = {0} x S(1). When r = 1, it is permitted that the functions a and b of L depend on the x and t variables, that is, L = partial derivative/partial derivative t + x(a(x, t) + ib(x, t))partial derivative/partial derivative x, where (x, t) is an element of Omega(epsilon).