This means that for all n ≥ 2, there exists no first-order sentence Fn in the language Ln = (+,−,×,=, 0, 1, In) satisfying the following property: for any real algebraic variety V of R, if we interpret the n-ary predicate In by membership to V then R |= Fn if and only if V is irreducible. Our proof will imply that irreducibility remains undefinable even if we add the exponential function to Ln....

Let us recall that Raphael Robinson’s Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms [13, 18]. We will generalize the semantictableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π1 sentence, valid in the standard model of the Natural Numbers a...

In this paper we present the fundamental approaches of financial economics to valuation. Three methods are demonstrated by which financial economists account for risk. We illustrate how these methods relate to one another and how they can be applied in the valuation of risky corporate bonds, guaranteed investment contracts (GICs) with and without interest rate contingencies, and whole life insu...

Bjorn Poonen discussed the “recognition problem” for finitely generated rings (and fields). That is, given two finitely generated commutative rings A and B, presented in terms of generators and relations, is there a decision procedure to determine whether or not these rings are isomorphic (this being, one would think, a basic issue for algebraic geometry!). Of course, if one drops the requireme...

Taking five Anglo-Saxon countries that have relatively similar backgrounds and tax systems – Australia, Canada, New Zealand, the UK and the USA – we see that the shares of the very richest exhibit a strikingly similar pattern, falling in the three decades after World War II, before rising sharply from the mid-1970s onwards. The share of the top 1 per cent is highly correlated across AngloSaxon ...

O F ALL the materials associated with dental practice and technic, those containing gypsum products are probably used most extensively. In keeping with this fact, a great deal of research has been directed toward problems involving the physical behavior of these materials. It is felt, however, that knowledge is lacking on some of the more fundamental behavior characteristics. The manifestation ...

We study initial cuts of models of weak two-sorted Bounded Arithmetics with respect to the strength of their theories and show that these theories are stronger than the original one. More explicitly we will see that polylogarithmic cuts of models of V are models of VNC by formalizing a proof of Nepomnjascij’s Theorem in such cuts. This is a strengthening of a result by Paris and Wilkie. We can ...

Let R be a real closed field. An integer part I for R is a discretely ordered subring such that for every r ∈ R, there exists an i ∈ I so that i ≤ r < i+ 1. Mourgues and Ressayre [11] showed that every real closed field has an integer part. In [6], it is shown that for a countable real closed field R, the integer part obtained by the procedure of Mourgues and Ressayre is ∆ωω (R). We would like ...

Some new finite lattices (for example, A/4 , M7, and the hexagon lattice) are shown to be isomorphic to the lattice of elementary substructures of a model of Peano Arithmetic. The set of elementary substructures of a model JV of Peano Arithmetic forms a lattice Lt(yf), the substructure lattice of jV. It is unknown whether there are finite lattices that are not isomorphic to any substructure lat...

The Employee Benefit Research Institute (EBRI) and the Investment Company Institute (ICI) have been collaborating for the past two years to collect data on participants in 401(k) plans. This effort, known as the EBRI/ICI Participant-Directed Retirement Plan Data Collection Project, has obtained data for 401(k) plan participants from certain of EBRI and ICI sponsors and members serving as plan r...