Coefficients and values Additive structure Stone-Weierstrass Integer-valued polynomials on a subset Prime ideals Multiplicative properties Skolem properties Invertible ideals and the Picard group Integer-valued derivatives and finite differences Integer-valued rational functions Integer-valued polynomials in several indeterminates References List of symbols Index.

/pdf/integer-valued-polynomials-47pkncnklv.pdf

Integer-Valued Polynomials

We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our approach extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.

/pdf/polyhedral-divisors-and-algebraic-torus-actions-4gtcs2cynh.pdf

Polyhedral divisors and algebraic torus actions

1 About the course, and these notes.- Part I: Motivation and Examples.- 2 Making Some Things Precise.- 3 Free Groups.- 4 A Cook's Tour.- Part II: Basic Tools and Concepts.- 5 Ordered Sets, Induction, and the Axiom of Choice.- 6 Lattices, Closure Operators, and Galois Connections.- 7 Categories and Functors.- 8 Universal Constructions.- 9 Varieties of Algebras.- Part III: More on Adjunctions.- 10 Algebras, Coalgebras, and Adjunctions.- References.- List of Exercises.- Symbol Index.- Word and Phrase Index.

/pdf/an-invitation-to-general-algebra-and-universal-constructions-3851kak6h8.pdf

An Invitation to General Algebra and Universal Constructions

Free σ-products and noncommutatively slender groups

This is the first authored book to be dedicated to the new field of directed algebraic topology that arose in the 1990s, in homotopy theory and in the theory of concurrent processes. Its general aim can be stated as 'modelling non-reversible phenomena' and its domain should be distinguished from that of classical algebraic topology by the principle that directed spaces have privileged directions and directed paths therein need not be reversible. Its homotopical tools (corresponding in the classical case to ordinary homotopies, fundamental group and fundamental groupoid) should be similarly 'non-reversible': directed homotopies, fundamental monoid and fundamental category. Homotopy constructions occur here in a directed version, which gives rise to new 'shapes', like directed cones and directed spheres. Applications will deal with domains where privileged directions appear, including rewrite systems, traffic networks and biological systems. The most developed examples can be found in the area of concurrency.

/pdf/directed-algebraic-topology-models-of-non-reversible-worlds-3rgi7t9x6b.pdf

Directed algebraic topology : models of non-reversible worlds

Die Folgen ganzer Zahlen {an} (vom Typus ω) bilden auf Grund der Addition {an} + {bn} = {an + bn} eine abelsche Gruppe F. Diese Gruppe F und gewisse ihrer Untergruppen sollen im folgenden untersucht werden. Als erstes werden wir zeigen, dass jede abzahlbare Untergruppe von F eine freie abelsche Gruppe ist (Satz I; eine abelsche Gruppe heisst eine freie abelsche Gruppe, wenn sie eine solche Teilmenge B — Basis genannt — besitzt, dass jedes Gruppenelement J darstellbar ist in der Form \(\mathcal{J} = \sum\limits_{i = 1}^m {{n_i}{b_i}}\), ni ganz, bi ∈ B und wenn aus Σ nibi = 0, bi∈ B, bi ≠ b, fur i ≠ j folgt, dass ni = 0; da alle betrachteten Gruppen abelsch sind, werden wir statt «freie abelsche Gruppe» auch kurz «freie Gruppe» sagen.) Die Gruppe F selbst dagegen ist nicht frei. Wir werden namlich zeigen, dass F eine nicht freie Untergruppe der Machtigkeit ℵ1 besitzt (Satz II, der dasselbe auch fur gewisse Untergruppen von F aussagt); die F selbst betreffende Behauptung ist darin enthalten, da jede Untergruppe einer freien Gruppe frei ist (vgl. z. B. S. Lefschetz [1], S. 50).

Additive Gruppen von Folgen Ganzer Zahlen

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Additive Gruppen von Folgen Ganzer Zahlen