We use the compactness theorem of continuous logic to give a new proof that $$L^r([0,1]; {\mathbb {R}})$$ isometrically embeds into $$L^p([0,1]; {\mathbb {R}})$$ whenever $$1 \le p \le r \le 2$$. We will also give a proof for the complex case. This will involve a new characterization of complex $$L^p$$ spaces based on Banach lattices.

Computability theory is used to evaluate the complexity of classifying various kinds of Lebesgue spaces and associated isometric isomorphism problems.

This uncommonly fine textbook of the modern theory of functions of a real variable is particularly well-suited for mathematically mature students in the fields of philosophy and foundations of mathematics, philosophy of physics, probability theory, and statistics. Those who wish to achieve first-hand acquaintance with the quantum theory will also need to have a grasp of the material presented in this book. The first chapter presents a capsule survey of topics in abstract set theory and algebra, including a discussion of (...) relations, functions, the axiom of choice, and the well-ordering theorem, and leading up to the construction of the fields of real and complex numbers. The second chapter is an introduction to topology and spaces of topological functions. The approach is set-theoretical. Chapter three deals with the theory of the Lebesgue integral and measure theory. This chapter is quite comprehensive, rigorous and embellished with many exercises for the student. Subsequent sections include the theories of function spaces, Banach spaces, abstract Hilbert spaces, and differentiation. This is clearly one of the more worthwhile texts in its field.—H. P. K. (shrink)

In the 1870s, Carl Neumann proposed the so-called method of the arithmetic mean for solving the Dirichlet problem on convex domains. Neumann’s approach was considered at the time to be a reliable existence proof, following Weierstrass’s criticism of the Dirichlet principle. However, in 1937 H. Lebesgue pointed out a serious gap in Neumann’s proof. Curiously, the erroneous argument once again involved confusion between the notions of infimum and minimum. The objective of this paper is to show that Lebesgue’s (...) sharp criticism of Neumann’s proof was only partially justified. (shrink)

We show that for an uncountable κ in a suitable Cohen real model for any family {A ν } ν<κ of sets of reals there is a σ-homomorphism h from the σ-algebra generated by Borel sets and the sets A ν into the algebra of Baire subsets of 2 κ modulo meager sets such that for all Borel B, B is meager iff h(B) = 0. The proof is uniform, works also for random reals and the Lebesgue measure, and (...) in this way generalizes previous results of Carlson and Solovay for the Lebesgue measure and of Kamburelis and Zakrzewski for the Baire property. (shrink)

In the Platonic work Alcibiades I, a divinely guided Socrates adopts the guise of a lover in order to divert Alcibiades from an unthinking political career. The contributors to this carefully focussed volume cover aspects of the background to the work; its arguments and the philosophical issues it raises; its relationship to other Platonic texts, and its subsequent history up to the time of the Neoplatonists. Despite its ancient prominence, the authorship of Alcibiades I is still unsettled; the essays and (...) two appendices, one historical and one stylometric, come together to suggest answers to this tantalising question. (shrink)

Geleitwort von Prof. P. Nicki -- Dr. Hermann Freund im Gespräch mit Prof. Dr. Dr. Hinderk M. Emrich -- Die Angst und der existentielle Sprung -- Die neue Zeit -- Der gedachte Gott -- Wie kommt das Ich zum Du? - Hysterie als Lebensform -- Was heisst, sich entscheiden? - Zur Theologie der Freiheit -- Re-formation der Reformation -- Das metaphysische und zugleich dämonische Wesen des Schlosses in Franz Kafkas Schloss-Roman -- Nachwort von Wolfgang Teichert: Epiphanie oder wo sprechende Philosophie (...) und Theologie aufhören müssen -- Literatur -- Verzeichnis der Mitwirkenden -- Abbildungsverzeichnis. (shrink)

Even well after his lifetime, Ibn Sina was renowned, not just in medicine or philosophy, but in other areas, especially in the Islamic world. In brief, he was an authority in the Islamic East, or an “auctoritas”. However, in the west, his work was massively influential in not only the medical education curricula, but also in the important, innovative doctrines in philosophy. The most fundamental sections of his major encyclopedia, al-Shifâ being translated into Latin as early as the 12th and (...) 13th centuries and spreading throughout universities dispersed this impact rapidly. Known as "the prince of physicians", Ibn Sina is the writer of the Canon of Medicine (al-Qānūn fi 'al-Tibb), which became a medical standard in the Christian west as well as the Islamic world. (shrink)

Manners and etiquettes of social life in the light of Islam.

This new edition emphasizes Emerson's philosophy and thoughts on such issues as freedom and fate; creativity and established culture; faith, experience, and evidence; the individual, God, and the world; unity and dualism; moral law, grace, and compensation; and wealth and success. Emerson's text has been fully annotated to explain difficult words and to clarify his references. The Introduction, Notes, Bibliography, Index, and Chronology of Emerson's life help the reader understand his distinctive outlook, his contributions to philosophy, and his place in (...) American culture and society. (shrink)

In assessing the value of any historical work it is necessary for the investigator to undertake the often frustrating and tedious task of Quellenforschung. In the case of the De Caesaribus the first substantial attempts began in Germany in 1873 and 1874 with the appearance of two important studies by A. Enmann and A. Cohn. Enmann sought to explain the mass of verbal similarities, numerous errors and shared idiosyncrasies to be found in Victor's De Caesaribus, Eutropius' Breviarium 7–10, and parts (...) of the Historia Augusta by postulating a common source which has subsequently come to be known as the Kaisergeschichte. He was so convinced by his findings that he stated , ‘Victor und Eutrop es beide als Haupt – wenn nicht als einzige Quelle benutzt haben’. Working quite independently Cohn's examination of the first eleven chapters led him to conclude that from the beginning of Augustus' sole reign to the death of Domitian Victor, Eutropius and the Epitomator followed a common source which he entitled Suetonius auctus, an account excerpted from Suetonius with some new material added. Enmann, however, had already described the Kaisergeschichte's early chapters in this way and the two were readily identified. One major problem remained. The traditional date of the completion of the HA was still believed to be 305/6; consequently if the K.G. had been used by its authors then the K.G. must have been written by that year. Nevertheless notable similarities between Victor and Eutropius continued apparently till 357 and Enmann therefore suggested a continuation of the K.G. until that date. (shrink)

After Mr. Stuart and Mr. Hardie I ought to be shy of speaking upon the Tragedies of Seneca. But Mr. Stuart and Mr. Hardie have stirred the dust that lay upon notes which I have had by me for some four years: and their papers encourage the hope that there is among English scholars some revival of interest in Seneca. I am afraid that I myself read Seneca for pleasure, with admiration for the justness of his moral sentiments and with (...) affection for his style. Yet I can hardly be blind to the opportunities which his text offers to those who practice the art of critical divination. (shrink)

A study of the place of human sexuality in religious systems indicates a possible universal stress on sexual attraction. This could be explained by using the theories of Richard Dawkins and other sociobiologists: the philandering male and the coy female express the best strategies for the survival of the “selfish gene.” Closer analysis of four religious systems throws doubt on these theories. In some systems the strategies are contradicted while in others there is stress on cooperative restraint rather than on (...) survival through selfish propagation. The principal objection to the sociobiological approach is its assumption of conflict between the sexes. (shrink)

Two minuscule codices carrying collections of grammatical and rhetorical treatises and extracts from such treatises, one written at Monte Cassino between A.D. 779 and 796 ), the other at Benevento towards the middle of the following century, contain among their uncial tituli the three words INCIPIT THVESTES VARII. There follows in both codices a twenty-four-word sentence stating the full name of Varius, the literary character of the Thyestes, an aesthetic judgement on the work, the date of a public performance in (...) a Roman theatre, and the price paid: Lucius Varius cognomento Rufus Thyesten tragoediam magna cura absolutam post Actiacam uictoriam Augusti ludis eius in scaena edidit pro qua fabula sestertium deciens accepit. (shrink)

The words 'true value' assume an existential relationship with the thing being measured. There is assumed to exist some aspect of the thing being measured that is independent of its relationship to the person who is interested in the result of the measurement. Yet measurement is a response to the need of an observer to know something about the real world. There is therefore an epistemological aspect to measurement. Some aspect of measurement has to do with the observer as a (...) knower of the result of the measurement. The paper considers electrical measurement. It proposes that these relationships correspond to two distinct kinds of measurement: one has the characteristic that there does exist such a thing as a true value, though it cannot be known: in the other kind, no such thing as true value even exists. The notion of true value is very useful, and for the kind of measurement where such a thing does not exist, the paper defines an alternative that has the advantage that it can actually be known under some circumstances in calibration. (shrink)

Die in diesem Buche entwickelte Staatslehre ist aus den Er gebnissen meiner Schrift: "die Lehre der Rechtssouveränität" (1906) hervorgegangen. Während jene Schrift insbesondere die Lehre von der Staatssouveränität kritisierte, bezweckt die jetzt vorliegende hauptsächlich, die positiven Grundlagen der ge genüberstehenden Lehre von der Rechtssouveränität auseinan derzusetzen und damit die moderne Staatsidee zum Ausdruck zu bringen. Auch diese Schrift wurde in der niederländischen Sprache, meiner Muttersprache, abgefasst, ist aber in eine den Auslän dern zugänglichere Sprache übersetzt worden, wofür ich auch diesmal (...) die deutsche Sprache wählte, weil die Ergebnisse spe ziell denjenigen der deutschen Staatswissenschaft gegenüber Stellung nehmen. Der Uebersetzung liegt das unter dem gleichen Titel in nie derländischer Sprache erschienene Werk, welches im Jahre 1915 veröffentlicht wurde, aber bereits vor dem Kriege druck in fertig war, zu grunde. Nachher veröffentlichte ich, gleichfalls niederländischer Sprache, eine Schrift: "Het Rechtsgezag" (1917), welche, anlässlich der hierzulande erschienenen Kriti ken, eine Verteidigung und Erläuterung der modernen Staats idee enthielt. Diese Schrift ist in der hier vorliegenden mitver arbeitet worden, demzufolge diese deutsche Ausgabe eine Zu sammenfassung der beiden obenerwähnten niederländischen Schriften bietet. H.K. LEIDEN, Mai 1919. INHALTSVERZEICHNIS. Seite EINLEITUNG. DIE MODERNE STAATS IDEE ERSTES KAPITEL STAATSAUTORITÄT UND RECHTSAUTORITÄT. 3 I. Der Gegensatz zwischen der alten und der neuen Staatsidee "... 3 11. Das Aufkommen der modernen Staatsidee. 6 111. Die Bedeutung der modernen Staatsidee 9 ZWEITES KAPITEL. (shrink)

We find a characterization of the covering number $cov({\mathbb R})$ , of the real line in terms of trees. We also show that the cofinality of $cov({\mathbb R})$ is greater than or equal to ${\mathfrak n}_\lambda$ for every $\lambda \in cov({\mathbb R}),$ where $\mathfrak n_\lambda \geq add({\mathcal L})$ ( $add( {\mathcal L})$ is the additivity number of the ideal of all Lebesgue measure zero sets) is the least cardinal number k for which the statement: $(\exists{\mathcal G}\in [^\omega \omega ]^{\leq (...) \lambda })(\forall{\mathcal F}\in [^\omega \omega ]^{\leq k})(\exists g\in{\mathcal G})(\exists h\in ^\omega \omega )(\forall f\in{\mathcal F})(\forall ^\infty n)(\exists u\in [g(n),g(n+1))(f(u)=h(u))$ fails. (shrink)

First Published in 1999. Routledge is an imprint of Taylor & Francis, an informa company.