Infima of d.r.e. degrees

Lachlan observed that the infimum of two r.e. degrees considered in the r.e. degrees coincides with the one considered in the D₂⁰{\Delta_2^0} degrees. It is not true anymore for the d.r.e. degrees. Kaddah proved in (Ann Pure Appl Log 62(3):207–263, 1993) that there are d.r.e. degrees a, b, c and a 3-r.e. degree x such that a is the infimum of b, c in the d.r.e. degrees, but not in the 3-r.e. degrees, as a < x < b, c. In this paper, we extend Kaddah’s result by showing that such a structural difference occurs densely in the r.e. degrees. Our result immediately implies that the isolated 3-r.e. degrees are dense in the r.e. degrees, which was first proved by LaForte.     

On the r.e. predecessors of d.r.e. degrees

Let d be a Turing degree containing differences of recursively enumerable sets (d.r.e.sets) and R[d] be the class of less than d r.e. degrees in whichd is relatively enumerable (r.e.). A.H.Lachlan proved that for any non-recursive d.r.e. d R[d] is not empty. We show that the r.e. degree defined by Lachlan for a d.r.e.set d is just the minimum degree in which D is r.e. Then we study for a given d.r.e. degree d class R[d] and show that there exists a d.r.e.d such that R d] has a minimum element 0. The most striking result of the paper is the existence of d.r.e. degrees for which R[d] consists of one element. Finally we prove that for some d.r.e. d R[d] can be the interval [a,b] for some r.e. degrees a,b, a b d. Received: 17 January 1996     

On a conjecture of W.E. Clark and M.E.H. Ismail

W.E. Clark and M.E.H. Ismail conjectured the inequality

Monotone factors of I.I.D. processes

LetB(p) andB(q) be Bernoulli shifts on {0, 1,...,d - 1}^ℤ. Ifh(p)>h(q), it is a classical theorem of Sinai that there is a factor map takingB(p) toB(q). If, in addition,p stochastically dominatesq, we can ask whether there is such a factor map ϕ which is monotone: ϕ(x) _i≤x_i for each coordinatei of almost every pointx. Here we show that there is a monotone finitary code fromB(p) toB(q) in the case whereB(q) is a shift on two symbols.     

The Influence of P. A. Florensky on N. N. Luzin

The study of the early history of the Moscow school of the theory of functions has been greatly enhanced by the discovery of correspondence between one of its founders, N. N. Luzin, and the priest, theologian, philosopher, and scientist P. A. Florensky. The correspondence reveals that Luzin experienced a profound spiritual crisis in 1905 when his materialist worldview collapsed. This crisis continued for three years and was finally resolved when Luzin had a decisive encounter with the religious philosophy of Florensky. After this, Luzin's interest in mathematics gradually revived until he was able, by 1909, to commit himself to a career in mathematics.Copyright 1998 Academic Press.Les études concernant l'histoire des débuts de l'école moscovite de la théorie des functions ont été grandement éclairées par la découverte d'une correspondance entre l'un de ses fondateurs, N. N. Luzin, et P. A. Florenski, un prêtre, théologien, philosophe, et savant. Cette correspondance révèle que Luzin a traversé une crise spirituelle profonde à partir de 1905, lorsque sa perspective matérialiste du monde s'est effondrée. Cette crise a duré trois ans, et a trouvé sa résolution dans la rencontre décisive de Luzin avec la philosophie religieuse de Florenski. Par la suite, l'intérêt de Luzin pour les mathématiques s'est progressivement ravivé, jusqu'à ce qu'il soit en mesure de décider, en 1909, de consacrer sa vie aux mathématiques.Copyright 1998 Academic Press.[formula]Copyright 1998 Academic Press.MSC 1991 subject classification: 01A30, 01A60, 01A80, 03-03     

Tightness of products of i.i.d. random matrices

Summary In this paper we present a necessary and sufficient condition for tightness of products of i.i.d. finite dimensional random nonnegative matrices. We give an example illustrating the use of our theorem and treat completely the case of 2×2 matrices. We also describe stationary solutions of the linear equationy _n=X_ny_n–1, n>0, in (R ^d )⁺, whereX ₁,X ₂,... are i.i.d.d×d nonnegative matrices.     

S.A.G.B.I. bases for rings of formal modular seminvariants

We use the theory of S.A.G.B.I. bases to construct a generating set for the ring of invariants for the four and five dimensional indecomposable modular representations of a cyclic group of prime order. We observe that for the four dimensional representation the ring of invariants is generated in degrees less than or equal to 2p–3, and for the five dimensional representation the ring of invariants is generated in degrees less than or equal to 2p–2. Received: January 22, 1997     

On a problem of C. C. Chen and D. E. Daykin

Let α(k, p, h) be the maximum number of vertices a complete edge-colored graph may have with no color appearing more than k times at any vertex and not containing a complete subgraph on p vertices with no color appearing more than h times at any vertex. We prove that $\text{α(k, p, h) ≤ h + 1 + (k ? 1){(p ? h ? 1) × (h}{}^{\mathrm{p}}\text{+ 1)}}{}^{\mathrm{<mtext>1</mtext><mtext>h</mtext>}}$ and obtain a stronger upper bound for α(k, 3, 1). Further, we prove that a complete edge-colored graph with n vertices contains a complete subgraph on p vertices in which no two edges have the same color if

$\text{(}\text{n}\text{3}\text{)>(}\text{p}\text{3}\text{)}\text{Σ}\text{i=1}\text{t}\text{(}\text{e}{}_{\mathrm{i}}\text{2}\text{)}$

where e_i is the number of edges of color i, 1 ≤ i ≤ t.     

The mathematics of F. J. Almgren, Jr.

Frederick Justin Almgren, Jr, one of the world’s leading geometric analysts and a pioneer in the geometric calculus of variations, died on February 5, 1997 at the age of 63 as a result of myelodysplasia. Throughout his career, Almgren brought great geometric insight, technical power, and relentless determination to bear on a series of the most important and difficult problems in his field. He solved many of them and, in the process, discovered ideas which turned out to be useful for many other problems. This article is a more-or-less chronological survey of Almgren’s mathematical research. (Excerpts from this article appeared in the December 1997 issue of theNotices of the American Mathematical Society.) Almgren was also an outstanding educator, and he supervised the thesis work of nineteen PhD students; the 1997 volume 6 issue of the journalExperimental Mathematics is dedicated to Almgren and contains reminiscences by two of his PhD students and by various colleagues. A general article about Almgren’s life appeared in the October 1997Notices of the American Mathematical Society [MD]. See [T3]for a brief biography.     

On a Method of J. H. C. Whitehead

In this article which is partly a continuation of Goldstein (1999 Goldstein , R. Z. ( 1999 ). The length and thickness of words in a free group . Proc. Amer. Math. Soc. 127 ( 10 ): 2857 – 2863 . [CSA] [CROSSREF]  [Google Scholar]), we give some necessary and sufficient conditions for an element of a free group to be primitive. For certain words x, we also give a lower bound on the length of words obtained from x by an automorphism. In the last section we give another proof of the Primitivity Theorem recently proven by Lee, using ideas from the first part of the article.     

Temperature dependence of S.C.L.C. in anthracene crystals

Zusammenfassung Aus der Temperaturabhängigkeit des Löcherstroms in Anthrazen-Einkristallen wird die Aktivierungsenergie bei verschiedenen Dunkelstromwerten gemessen. Die Aktivierungsenergien liegen zwischen 0,65 eV und 0,95 eV für Ströme von 10^–10 bzw. 10^–12 Ampère bei Zimmertemperatur.Ferner wird für mehrere Temperaturwerte die Aktivierung der in Haftstellen sitzenden Ladungsträger mit Hilfe von Licht verschiedener Intensität untersucht.     

Construction of the functional model of B. S.-Nagy-C. Foias

Methodological paper: quick and simple construction of the Nagy—Foias functional model, using only elementary information about Hilbert spaces and operators on them. A formula is derived for a characteristic function. The account is based on an idea of B. S. Pavlov (Izv. Akad. Nauk SSSR, Ser. Mat.,39, No. 1, 123–148 (1975)) on symmetrization of the notation for a model space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 16–23, 1977.     

Numerical treatment of O.D.Es.: The theory ofA-methods

Summary Almost all commonly used methods for O.D.Es. and their most miscellaneous compositions areA-methods, i.e. they can be reduced toz _o=;z _j =Az _j–1 +h(x _j–1 ,z _j–1 ,z _j ;h),z _j ^s ,A(s,s),j=1(1)m. This paper presents a general theory forA-methods and discusses its practical consequences. An analysis of local discretization error (l.d.e.) accumulation results in a general order criterium and reveals which part of the l.d.e. effectively influences the global error. This facilitates the comparison of methods and generalizes considerably the concept of error constants. It is shown, as a consequence, that the global error cannot be safely controlled by the size of the l.d.e. and that the conventional error control may fail in important cases. Furthermore, Butcher's effective order methods, the concept of Nordsieck forms, and Gear's interpretation of lineark-step schemes as relaxation methods are generalized. The stability of step changing is shortly discussed.     

On the stability of couette-benard M.H.D. flows of binary mixtures

Summary We study a class of Couette-Benard M.H.D. flows of a binary mixture in a plane slant layer. By using the theory of semigroups of linear bounded transformations, we prove that the initial-value problem forfinite perturbations has one and only one solution belonging to a suitable Banach space. Finally, we show the inconditionated asymptotical stability in norm of such a solution.

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Sunto Si studia una classe di moti magnetoidrodinamici alla Couette-Benard di una miscela fluida binaria in uno strato piano obliquo e, con le tecniche della teoria dei semigruppi di trasformazioni lineari limitate, si dimostra, in un opportuno spazio di Banach, l’esistenza e l’unicità della soluzione del problema di evoluzione per perturbazionifinite e l’incondizionata stabilità asintotica in norma di tali moti.

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Entrata in Redazione il 19 Ottobre 1974.

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Work performed under the auspices of the National Research Council (C.N.R., Gruppo Nazionale per la Fisica Matematica).     

Proof of two conjectures of H. Brezis and L.A. Peletier

In this paper, we consider the problem: –u=N(N–2)u ^p– , u>0 on ; u=0 on , where is a smooth and bounded domain inR ^N, N3, p= , and >0. We prove a conjecture of H. Brezis and L.A. Peletier about the asymptotic behaviour of solutions of this problem which are minimizing for the Sobolev inequality as goes to zero. We give similar results concerning the related problem: –u=N(N–2)u^p+u, u>0 on ; u=0 on , for N is larger than 4.     

A class of three-replicate three-associate p.b.i.b. designs

Summary Partially balanced incomplete block (p.b.i.b.) designs of two and more associate classes have been constructed by Saha and Das [9] through the use of confounded designs for 2 ⁿ factorals. By dualising one series of two-associated p.b.i.b. designs given by them, a class of three-replicate three-associate p.b.i.b. designs have been obtained. This can be taken to be a new class of designs in the sense that they are not included by Nair [7] in his table of three-replicate three-associate p.b.i.b. designs.     

N.M.R. and I.R. studies of 1, 1, 1 trifluoroacetone in proton donor solvents

Interactions of 1, 1, 1 trifluoroacetone with proton donor solvents of the type RH [where R=HO, CH₃S, CH₃CH₂CH₂CH₂S, CH₃O, (CH₃)₂N, (C₂H₅)₂N] have been investigated using NMR and infra-red techniques. Evidence of the formation of addition products of the type CF₃.C(OH)R.CH₃ where OH group is hydrogen-bonded with the fluorine of the CF₃ group has been obtained.