On context-free and Szilard languages

Szilard languages of context-free grammars are studied. Especially, classical pumping, generalized pumping, Sokolowski's criterion, and semilinearity are considered as possible distinguishing properties between context-free and Szilard languages.This work was supported by the Academy of Finland.     

On factorial states of operator algebras

Results for the factorial state space of a C¹-algebra A which are analogous to results of 11., 12., 572–612),Tomiyama and Takesaki (Tohôku Math. J. (2) 13 (1961), 498–523) for the pure state space. It is shown that A is prime if and only if the (type I) factorial states are dense in the state space. It follows that every factorial state is a w¹-limit of type I factorial states. The factorial state space of a von Neumann algebra is determined, and it is shown that if A is unital and acts non-degenerately on a Hilbert space then the factorial state space of the generated von Neumann algebra restricts precisely to the factorial state space of A. It is shown that the set of factorial states is w¹-compact if and only if A is unital, liminal and has Hausdorff primitive ideal space.     

On permutative grammars generating context-free languages

A grammar is said to be permutative if it has permutation productions of the formAB BA in addition to context-free productions. Szilard languages and label languages are studied as examples of languages generable by permutative grammars. Particularly, sufficient conditions for a permutative grammar to generate a context-free language are studied.This work was supported by the Academy of Finland.     

On dos languages and dos mappings

Roughly speaking,DOS systems formalize the notion of generatively deterministic context free grammars. We explore the containment relationships among the class of languages generated byDOS systems and other subclasses of the class of context free languages. Leaving the axiom of aDOS system unspecified yields aDOS scheme, which defines a mapping from words to languages over a given alphabet. We explore the algebraic properties ofDOS mappings and obtain an algebraic characterization of a fundamental subclass of theDOS mappings generated byDOS schemes which are propagating (non erasing) and have no cycles of derivability among letters of the alphabet. We apply this characterization to show that the mapping equivalence problem for propagatingDOS schemes is decidable.     

On possible growths of Toeplitz languages

We consider a new family of factorial languages whose subword complexity grows as Φ(n ^α ), where α is the only positive root of some transcendental equation. The asymptotic growth of the complexity function of these languages is studied by discrete and analytical methods, a corollary of the Wiener-Pitt theorem inclusive. The factorial languages considered are also languages of arithmetical factors of infinite words; so, we describe a new family of infinite words with an unusual growth of arithmetical complexity.     

On the irrationality of factorial series II

In this paper we give irrationality results for numbers of the form where the numbers an behave like a geometric progression for a while. The method is elementary, not using differentiation or integration. In particular, we derive elementary proofs of the irrationality of π and em for Gaussian integers m≠0.     

On the irrationality of factorial series III

In this paper we derive some irrationality and linear independence results for series of the form where is either a non-negative integer sequence with υ_n = o(log n/log log n) or a non-decreasing integer sequence with .     

On optimal two-level nonregular factorial split-plot designs

This article studies two-level nonregular factorial split-plot designs. The concepts of indicator function and aliasing are introduced to study such designs. The minimum G$G$-aberration criterion proposed by Deng and Tang (1999) [4] for two-level nonregular factorial designs is extended to the split-plot case. A method to construct the whole-plot and sub-plot parts is proposed for nonregular designs. Furthermore, the optimal split-plot schemes for 12$12$-, 16$16$-, 20$20$- and 24$24$-run two-level nonregular factorial designs are searched, and many such schemes are tabulated for practical use.     

On intermediate attractors

It is proved, for a four-moment model of a phonon gas system and the Dirac–Schwindler extension of the Maxwell system, that a Chapman correct restriction of a initial-boundary problem exists. The well-posedness condition is found in terms of algebraic relations for parameters of the problem and elements of the boundary matrix.     

On languages with non-homogeneous strings of quantifiers

It is shown that every linear string of quantifiers can be replaced by a well-ordered sequence of quantifiers. The results of this paper was part of the author’s Master Thesis, which was submitted to the Hebrew University in August, 1967. The work was done under the guidance of Professor H. Gaifman, whom I thank for his kind guidance, and his help to clarify the exposition in this article.     

On factorial double solids with simple double points

We study when double covers of P³ ramified along nodal surfaces are not Q-factorial. In particular, we describe all the Q-factorial double covers of P³ ramified along quartic surfaces with at most seven simple double points and sextic surfaces with at most 16 simple double points.