Normal subgroups in the group of column-finite infinite matrices

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{GL}(V) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

Global Solution of Semi-infinite Programs with Existence Constraints

We consider what we term existence-constrained semi-infinite programs. They contain a finite number of (upper-level) variables, a regular objective, and semi-infinite existence constraints. These constraints assert that for all (medial-level) variable values from a set of infinite cardinality, there must exist (lower-level) variable values from a second set that satisfy an inequality. Existence-constrained semi-infinite programs are a generalization of regular semi-infinite programs, possess three rather than two levels, and are found in a number of applications. Building on our previous work on the global solution of semi-infinite programs (Djelassi and Mitsos in J Glob Optim 68(2):227–253, 2017), we propose (for the first time) an algorithm for the global solution of existence-constrained semi-infinite programs absent any convexity or concavity assumptions. The algorithm is guaranteed to terminate with a globally optimal solution with guaranteed feasibility under assumptions that are similar to the ones made in the regular semi-infinite case. In particular, it is assumed that host sets are compact, defining functions are continuous, an appropriate global nonlinear programming subsolver is used, and that there exists a Slater point with respect to the semi-infinite existence constraints. A proof of finite termination is provided. Numerical results are provided for the solution of an adjustable robust design problem from the chemical engineering literature.

MADNESS IN VECTOR SPACES

We consider maximal almost disjoint families of block subspaces of countable vector spaces, focusing on questions of their size and definability. We prove that the minimum infinite cardinality of such a family cannot be decided in ZFC and that the “spectrum” of cardinalities of mad families of subspaces can be made arbitrarily large, in analogy to results for mad families on ω. We apply the author’s local Ramsey theory for vector spaces [32] to give partial results concerning their definability.

Cantor’s theorem

Cantor’s theorem states that the cardinal number (‘size’) of the set of subsets of any set is greater than the cardinal number of the set itself. So once the existence of one infinite set has been proved, sets of ever increasing infinite cardinality can be generated. The philosophical interest of this result lies (1) in the foundational role it played in Cantor’s work, prior to the axiomatization of set theory, (2) in the similarity between its proof and arguments which lead to the set-theoretic paradoxes, and (3) in controversy between intuitionist and classical mathematicians concerning what exactly its proof proves.

Infinitely productive language can arise from chance under communicative pressure

Human communication is unparalleled in the animal kingdom. The key distinctive feature of our language is productivity: we are able to express an infinite number of ideas using a limited set of words. Traditionally, it has been argued or assumed that productivity emerged as a consequence of very specific, innate grammatical systems. Here we formally develop an alternative hypothesis: productivity may have rather solely arisen as a consequence of increasing the number of signals (e.g. sentences) in a communication system, under the additional assumption that the processing mechanisms are algorithmically unconstrained. Using tools from algorithmic information theory, we examine the consequences of two intuitive constraints on the probability that a language will be infinitely productive. We prove that under maximum entropy assumptions, increasing the complexity of a language will not strongly pressure it to be finite or infinite. In contrast, increasing the number of signals in a language increases the probability of languages that have—in fact—infinite cardinality. Thus, across evolutionary time, the productivity of human language could have arisen solely from algorithmic randomness combined with a communicative pressure for a large number of signals.

CATEGORICITY OF MODULAR AND SHIMURA CURVES

We describe a model-theoretic setting for the study of Shimura varieties, and study the interaction between model theory and arithmetic geometry in this setting. In particular, we show that the model-theoretic statement of a certain${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$-sentence having a unique model of cardinality$\aleph _{1}$is equivalent to a condition regarding certain Galois representations associated with Hodge-generic points. We then show that for modular and Shimura curves this${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$-sentence has a unique model in every infinite cardinality. In the process, we prove a new characterisation of the special points on any Shimura variety.

First-order and counting theories of ω-automatic structures

The logic extends first-order logic by a generalized form of counting quantifiers (“the number of elements satisfying … belongs to the set C”). This logic is investigated for structures with an injectively ω-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable [6]. It is shown that, as in the case of automatic structures [21], also modulo-counting quantifiers as well as infinite cardinality quantifiers (“there are many elements satisfying …”) lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of that contains only effective quantifiers is shown to be decidable and an elementary algorithm for this decision is presented. Both assumptions (ω-automaticity and bounded degree) are necessary for this result to hold.

Relative purity over noetherian rings

In this note we are going to show that if M is a left module over a left noetherian ring R of the infinite cardinality λ ≥ |R|, then its injective hull E(M) is of the same size. Further, if M is an injective module with |M| ≥ (2λ)+ and K ≤ M is its submodule such that |M/K| ≤ λ, then K contains an injective submodule L with |M/L| ≤ 2λ. These results are applied to modules which are torsionfree with respect to a given hereditary torsion theory and generalize the results obtained by different methods in author’s previous papers: [A note on pure subgroups, Contributions to General Algebra 12. Proceedings of the Vienna Conference, June 3–6, 1999, Verlag Johannes Heyn, Klagenfurt, 2000, pp. 105–107], [Pure subgroups, Math. Bohem. 126 (2001), 649–652].

How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson's arithmetic q

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 semantic-tableaux 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 and denoted as V. such that if α is any finite consistent extension of Q + V then α will be unable to prove its Semantic Tableaux consistency. The same result will also apply to axiom systems α with infinite cardinality when these infinite-sized axiom systems satisfy a minor additional constraint, called the Conventional Encoding Property.Our formalism will also imply that the semantic-tableaux version of the Second Incompleteness Theorem generalizes for the axiom system IΣ0, as well as for all its natural extensions. (This answers an open question raised twenty years ago by Paris and Wilkie [15].)