What If Quantum Theory Violates All Mathematics?

It is shown by using a rather elementary argument in Mathematical Logic that if indeed, quantum theory does violate the famous Bell Inequalities, then quantum theory must inevitably also violate all valid mathematical statements, and in particular, such basic algebraic relations like 0 = 0, 1 = 1, 2 = 2, 3 = 3, … and so on … An interest in that result is due to the following three alternatives which it imposes upon both Physics and Mathematics: Quantum Theory is inconsistent. Quantum Theory together with Mathematics are inconsistent. Mathematics is inconsistent. In this regard one should recall that, up until now, it is not known whether Mathematics is indeed consistent.

On Sumsets of Multisets in $\mathbb{Z}_p^m$

For a sequence $A$ of given length $n$ contained in $\mathbb{Z}_p^2$ we study how many distinct subsums $A$ must have when $A$ is not "wasteful" by containing too many elements in same subgroup. Martin, Peilloux and Wong have made a conjecture for a sharp lower bound and established it when $n$ is not too large whereas Peng has previously established the conjecture for large $n$. In this note we build on these earlier works and add an elementary argument leading to the conjecture for every $n$.Martin, Peilloux and Wong also made a more general conjecture for sequences in $\mathbb{Z}_p^m$. Here we show that the special case $n = mp-1$ of this conjecture implies the whole conjecture and that the conjecture is equivalent to a strong version of the additive basis conjecture of Jaeger, Linial, Payan and Tarsi.

Constructional ‘scene encoding’ and acquisition: Mothers’ use of argument structure constructions in English child-directed speech

Construction-based language models assume that grammar is meaningful and learnable from experience. Focusing on five of the most elementary argument structure constructions of English, a large-scale corpus study of child-directed speech (CDS) investigates exactly which meanings/functions are associated with these patterns in CDS, and whether they are indeed specially indicated to children by their caretakers (as suggested by previous research, cf. Goldberg, Casenhiser and Sethuraman 2004). Collostructional analysis (Stefanowitsch and Gries 2003) is employed to uncover significantly attracted verb-construction combinations, and attracted pairs are classified semantically in order to systematise the attested usage patterns of the target constructions. The results indicate that the structure of the input may aid learners in making the right generalisations about constructional usage patterns, but such scaffolding is not strictly necessary for construction learning: not all argument structure constructions are coherently semanticised to the same extent (in the sense that they designate a single schematic event type of the kind envisioned in Goldberg’s [1995] ‘scene encoding hypothesis’), and they also differ in the extent to which individual semantic subtypes predominate in learners’ input

ELLIPTIC GENERA OF COMPLETE INTERSECTIONS

We propose a new definition of the elliptic genera for complete intersections, not necessarily nonsingular, in projective spaces. We also prove they coincide with the expressions obtained from Landau–Ginzburg model by an elementary argument.

Traces for Star Products on the Dual of a Lie Algebra

In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the ad-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.

The noncommutativity of random and generic extensions

Throughout, M will denote a transitive model of ZFC. Using the terms “random” and “generic” in the sense of [1], one may ask whether there can exist real numbers x and y such that x is generic over M[y] and y is random over M[x]. We shall see below by an elementary argument that this is not possible, and so, in a crude sense at least, random and generic extensions do not commute. This does not however rule out the possibility of a weaker commutativity. Let B be the complete Boolean algebra (in M) for adjoining a random real followed by a generic real and C be the complete Boolean algebra for adjoining a generic real followed by a random real. Then it still might be the case that B and C are isomorphic. This also fails, though, and we shall prove this by establishing the following combinatorial properties of MB and MC:butIn addition this will show that C cannot be embedded as a complete subalgebra of B.The property satisfied by B is reminiscent of calibre ℵ1 [2]. B would have calibre if we could replace “infinite” by “uncountable”, and this occurs if Martin's Axiom holds in M. To obtain the nonisomorphism of B and C in general necessitated looking at the weaker property.