Proofs and algebra in al-Fārisī's commentary

The Persian mathematician al-Fārisī (late thirteenth century) wrote a commentary on a practical arithmetic book as a means of giving techniques associated with mental reckoning a foundation in proofs modeled on those in Euclid's number theory books. One problem with this intercultural project is the incompatibility of Euclidean and Arabic numbers, while another is the occasional inadequacy of Euclid's mode of representing numbers via lines labeled with letters. Like others, al-Fārisī found a partial solution to the former by identifying fractions with ratios of integers, and for the latter he turned to the algebra of polynomials to work through one proof. To properly interpret this proof, Arabic algebra is situated in its contemporary mathematical context.     

A quantitative version of Kirk's fixed point theorem for asymptotic contractions

In [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645-650], W.A. Kirk introduced the notion of asymptotic contractions and proved a fixed point theorem for such mappings. Using techniques from proof mining, we develop a variant of the notion of asymptotic contractions and prove a quantitative version of the corresponding fixed point theorem.     

Authority and whole-class proving in high school geometry: The case of Ms. Finley

Students’ experiences with proving in schools often lead them to see proof as a static product rather than a negotiated process that can help students justify and make sense of mathematical ideas. We investigated how authority manifested in whole-class proving episodes within Ms. Finley’s high school geometry classroom. We designed a coding scheme that helped us identify the proving actions and interactions that occurred during whole-class proving and how Ms. Finley and her students contributed to those processes. By considering the authority over proof initiation, proof construction, and proof validation, the episodes illustrate how whole-class proving interactions might relate to students’ potential development (or maintenance) of authoritative proof schemes. In particular, the authority of the teacher and textbook limited students’ opportunities to engage collectively in proving and sometimes allowed invalid arguments to be accepted in the public discourse. We offer suggestions for research and practice with respect to authority and proof instruction.     

Rate of Metastability for Bruck'S Iteration of Pseudocontractive Mappings in Hilbert Space

We give a quantitative version of the strong convergence of Bruck's iteration schema for Lipschitzian pseudocontractions in Hilbert space.     

A rate of convergence for asymptotic contractions

In [P. Gerhardy, A quantitative version of Kirk's fixed point theorem for asymptotic contractions, J. Math. Anal. Appl. 316 (2006) 339-345], P. Gerhardy gives a quantitative version of Kirk's fixed point theorem for asymptotic contractions. This involves modifying the definition of an asymptotic contraction, subsuming the old definition under the new one, and giving a bound, expressed in the relevant moduli and a bound on the Picard iteration sequence, on how far one must go in the iteration sequence to at least once get close to the fixed point. However, since the convergence to the fixed point needs not be monotone, this theorem does not provide a full rate of convergence. We here give an explicit rate of convergence for the iteration sequence, expressed in the relevant moduli and a bound on the sequence. We furthermore give a characterization of asymptotic contractions on bounded, complete metric spaces, showing that they are exactly the mappings for which every Picard iteration sequence converges to the same point with a rate of convergence which is uniform in the starting point.     

A Cut‐free Proof System for Bounded Metric Temporal Logic Over a Dense Time Domain

We present a complete and cut‐free proof‐system for a fragment of MTL, where modal operators are only labelled by bounded intervals with rational endpoints.     

Duplication of directed graphs and exponential blow up of proofs

We develop a combinatorial model to study the evolution of graphs underlying proofs during the process of cut elimination. Proofs are two-dimensional objects and differences in the behavior of their cut elimination can often be accounted for by differences in their two-dimensional structure. Our purpose is to determine geometrical conditions on the graphs of proofs to explain the expansion of the size of proofs after cut elimination. We will be concerned with exponential expansion and we give upper and lower bounds which depend on the geometry of the graphs. The lower bound is computed passing through the notion of universal covering for directed graphs.

In this paper we present ground material for the study of cut elimination and structure of proofs in purely combinatorial terms. We develop a theory of duplication for directed graphs and derive results on graphs of proofs as corollaries.     

The proof theoretic strength of the Steinitz exchange theorem

We show that the logical theory QLA proves the Cayley-Hamilton theorem from the Steinitz exchange theorem together with a strengthening of the linear independence principle. Since QLA is a fairly weak theory (in the sense that its quantifier-free fragment, LA, translates into tautologies with TC⁰-Frege proofs—when restricted to the field Q of the rationals), it follows that the proof complexity of matrix algebra can be distilled to the Steinitz exchange theorem.     

Some logical metatheorems with applications in functional analysis

In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. Here `uniform' means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, CAT(0)-spaces, normed linear spaces, uniformly convex spaces, as well as inner product spaces.

     

EIN NEUER BEWEIS UND EINE VERSCHÄRFUNG FÜR DEN REDUKTIONSTYP ∀∃∀∞(0, 1) MIT EINER ANWENDUNG AUF DIE SPEKTRALE DARSTELLUNG VON PRÄDIKATEN

A simultaneous semantical and syntactical reduction is given for the satisfiability respectively finite satisfiability of first order formulas. We choose ???∞(0, 1) as conservative reduction class and allow only formulas out of ???∞(0, 1) having a simple set theoretical model if they are satisfiable at all. With the same method we get a spectral representation of any ?-ary enumerable respectively coenumerable predicate by a formula out of ???∞(?, 1).