Extension of Bernstein's theorem for polynomial maps of complex Banach spaces

Bernstein's inequality (1) will be extended to polynomial maps of arbitrary complex Banach spaces; cf. (3). We also give conditions for equality in (3). The extension of Bernstein's inequality to the case of polynomial maps of ℂ into ℂ was given byTung [4] and [5].     

A topological proof of the bott periodicity theorems

Summary A proof is given of theBott periodicity theorems using only well known techniques of algebrdbc topology. To Enrico Bompiani on his scientific Jubilee. This research was done in part while the first author held a Sloan Foundation Fellowship and the second author held a Senior National Science Foundation Fellowhip. It was also supported in part by the National Science Foundation under contract number G-10369.     

On thermodynamics of directed continuous media

Zusammenfassung Die Thermodynamik der richtungsorientierten Medien wird mit Hilfe der Clausius-Duhem-Ungleichheit und des Prinzips der materiellen Objektivität untersucht. Ein besonderer Fall der Materialsymmetrie wird diskutiert.

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Acknowledgment. This work was sponsored by the National Science Foundation. The author thanksS. J. Allen andC. N. DeSilva for many helpful discussions and is especially grateful to Prof.DeSilva for guidance.     

Which of the Cancellative Semigroups are Groups?

f : S ⇒ S such that the range of the function s maps to (s, ^ f(s)) is finite.     

A stability property of the densest circle packing

A new stability property of the densest circle packing in the plane is proved. This property is related to a conjecture ofL. Fejes Tóth.Research of this author was partially supported by Hungarian National Foundation for Scientific Research grant no. 1812.     

On Cartan's theoremsA andB in several complex variables

Summary New proofs ofCartan's fundamental theorems are given here, based on the notion of a syzygetic sheaf of modules over a sheaf of rings; the latter is a sheaf admitting a finite resolution by free sheaves. To enrico bompiani on his scientific jubilee The author holds anAlfred P. Sloan Fellowship.     

Fourier multipliers on compact groups

In a recent paperIan Inglis gives some sufficient conditions for a function on a totally disconnected compact Abelian group to be anL ^p Fourier multiplier. His proof depends on an interpolation theorem ofE. M. Stein. In this note we prove a generalization ofInglis' theorem. Our result is deduced from a factorization theorem, the proof of which is elementary, and a standard multiplier theorem.This work was done while the second-named author held a visiting appointment at the University of Washington. He wishes to thank ProfessorsE. Hewitt andR. R. Phelps for making the visit possible.     

On groups of hyperbolic length

We investigate the recently introduced notion of rotation numbers for periodic orbits of interval maps. We identify twist orbits, that is those orbits that are the simplest ones with given rotation number. We estimate from below the topological entropy of a map having an orbit with given rotation number. Our estimates are sharp: there are unimodal maps where the equality holds. We also discuss what happens for maps with larger modality. In the Appendix we present a new approach to the problem of monotonicity of entropy in one-parameter families of unimodal maps. This work was partially done during the first author’s visit to IUPUI (funded by a Faculty Research Grant from UAB Graduate School) and his visit to MSRI (the research at MSRI funded in part by NSF grant DMS-9022140) whose support the first author acknowledges with gratitude. The second author was partially supported by NSF grant DMS-9305899, and his gratitude is as great as that of the first author.     

Two-letter group codes that preserve aperiodicity of inverse finite automata

We construct group codes over two letters (i.e., bases of subgroups of a two-generated free group) with special properties. Such group codes can be used for reducing algorithmic problems over large alphabets to algorithmic problems over a two-letter alphabet. Our group codes preserve aperiodicity of inverse finite automata. As an application we show that the following problems are PSpace-complete for two-letter alphabets (this was previously known for large enough finite alphabets): The intersection-emptiness problem for inverse finite automata, the aperiodicity problem for inverse finite automata, and the closure-under-radical problem for finitely generated subgroups of a free group. The membership problem for 3-generated inverse monoids is PSpace-complete. Both authors were supported in part by NSF grant DMS-9970471. The first author was also supported in part by NSF grant CCR-0310793. The second author acknowledges the support of the Excellency Center, “Group Theoretic Methods for the Study of Algebraic Varieties” of the Israeli Science Foundation.     

Thomsen surfaces in affine 4-space

Surfaces which are both affine and Euclidean minimal are called Thomsen surfaces. In ³, these surfaces have been completely classified byBarthel, Volkmer andHaubitz. A similar problem, in the Lorentzian 3-space was solved byMagid. In the present paper, we study Thomsen surfaces in ⁴ and show that these surfaces are affine equivalent to the complex paraboloid.The author is a Senior Research Assistant of the National Fund, for Scientific Research (Belgium). This work was done while the author visited Brown University (Providence, USA) in April 93. He would like to thank Professors T. Cecil, M. Magid, and K. Nomizu for their hospitality.     

Über die verallgemeinerte Form der Lagrangeschen Gleichungen,welche auch die Behandlung von nicht-holonomen mechanischen Systemen gestattet

Summary A generalized form of the equations of motion of a rheonomic-holonomic mechanical system is proved. As special cases the equations ofNielsen andTzenoff are obtained. By a method of the last author the form ofAppell's equations is derived, in which the kinetic energy, but not the energy of acceleration, appears. The generalized equations can be extended to the case of non-holonomical systems.     

Einige Eigenschaften monotoner Matrizen

In [5]Ky Fan derived several theorems on real square matrices from well-known covering theorems on simplexes given byE. Sperner [13] andB. Knaster-C. Kuratowski-S. Mazurkiewicz [9]. In this note some theorems on monotone matrices and nonsingularM-matrices are presented which extend some ofFan's results. The proofs will be based on certain topological theorems on simplexes which were recently established by the author in [12] and which include the mentioned covering theorems.     

Zur Kennzeichnung der Euklidischen Norm

Ohne Zusammenfassung Leon Lichtenstein in memoriamThe author is indebted to the Office of Naval Research for a research grant at Indiana University.     

Bemerkung zu einem Hauptachsenproblem in der Plastizitätstheorie

Summary In an ideally plastic material any principal axis of the stress tensor is also a principal axis of the strain rate tensor. It is shown here that in plastic flow also the reverse is true. Besides, the validity of either statement is discussed for materials obeyingPrager's hardening rule or its modification proposed by the author.     

On the harmonic summability of double Fourier series

Summary The author studies a problem relating Harmonic Summability of doubleFourier serie analogous to theGergen's criteria for convergence.     

On the regularity of solutions of hyperbolic nonlinear equations

Summary Some regularity results for the solutions of certain hyperbolic nonlinear equations are given. This paper was written while the author was visiting Department of Mathematics, Sir GeorgeWilliams University, Montreal (Canada). Entrata in Redazione il 22 dicembre 1971.     

A theorem on the absolute Cesàro summability of factored fourier series

Summary In this paper, the author proves a theorem on the absoluteCesàro summability of factoredFourier series. His theorem extends a theorem ofMatsumoto and generalizes a theorem ofPrasad andBhatt.     

Valiron summability of the derived fourier series and its coniugate series

Summary Valiron [12] obtained very extensive generalisations ofBorel’s method. In this paper the author has consideredValiron summability. The results proved are equivalent toBorel’s for k=1.     

On the determinant of an asymmetric hyperbolic region

Summary A main problem in the geometry of numbers is the evaluation of the so-called determinant of various regions. The author derives a new estimate for the determinant of a certain two-dimensional region bounded by two hyperbolas and applies his result to a problem in the theory of automorphic star bodies. dedicated to Prof.B. Segre.     

Counting non-isomorphic three-connected planar maps

A counting formula for the number of non-isomorphic planar maps with m edges was obtained by V. A. Liskovets, and non-isomorphic 2-connected planar maps were counted by Liskovets and the author. R. W. Robinson's generalization of Polya's counting theory can be applied to these formulae to count, in polynomial time, non-isomorphic planar maps satisfying various sets of restrictions. Here two sets of planar maps are counted: maps with given 2-connected components, and 3-connected maps.