Aspects of singular cofinality

We study properties of closure operators of singular cofinality, and introduce several ZFC sufficient and equivalent conditions for the existence of antichain sequences in posets of singular cofinality. We also notice that the Proper Forcing Axiom implies the Milner-Sauer conjecture

Partially Ordered Sets and Their Invariants

We investigate how much information cardinal invariants can give on the structure of the ordered set on which they are de�ned. We consider the basic de�nitions of an ordered set and see how they are related to one another. We generalize some results on cardinal invariants for ordered sets and state some useful characterizations. We investigate how cardinal invariants can in uence the existence of some special suborderings. We generalize some results on the Dilworth and Sierpinski theorems and explore the conjecture of Miller and Sauer. We address some open problems on dominating numbers. We investigate Model Games to �nd some characterizations on the cardinality of a set.

Temperature and spatial connectivity drive patterns in freshwater macroinvertebrate diversity across the Arctic

Warming in the Arctic is predicted to change freshwater biodiversity through loss of unique taxa and northward range expansion of lower latitude taxa. Detecting such changes requires establishing circumpolar baselines for diversity, and understanding the primary drivers of diversity. We examined benthic macroinvertebrate diversity using a circumpolar dataset of &gt;1,500 Arctic lake and river sites. Rarefied α diversity within catchments was assessed along latitude and temperature gradients. Community composition was assessed through region-scale analysis of β diversity and its components (nestedness and turnover), and analysis of biotic–abiotic relationships. Rarefied α diversity of lakes and rivers declined with increasing latitude, although more strongly across mainland regions than islands. Diversity was strongly related to air temperature, with the lowest diversity in the coldest catchments. Regional dissimilarity was highest when mainland regions were compared with islands, suggesting that connectivity limitations led to the strongest dissimilarity. High contributions of nestedness indicated that island regions contained a subset of the taxa found in mainland regions. High Arctic rivers and lakes were predominately occupied by Chironomidae and Oligochaeta, whereas Ephemeroptera, Plecoptera, and Trichoptera taxa were more abundant at lower latitudes. Community composition was strongly associated with temperature, although geology and precipitation were also important correlates. The strong association with temperature supports the prediction that warming will increase Arctic macroinvertebrate diversity, although low diversity on islands suggests that this increase will be limited by biogeographical constraints. Long-term harmonised monitoring across the circumpolar region is necessary to detect such changes to diversity and inform science-based management.</p

A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees

Building on early work by Stevo Todorcevic, we describe a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem: Let $\kappa$ be any infinite regular cardinal, let $\xi$ be any ordinal such that $2^{\left|\xi\right|} < \kappa$, and let $k$ be any natural number. Then $$\text{non-$\left(2^{<\kappa}\right)$-special tree } \to \left(\kappa + \xi \right)^2_k.$$ This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal $(2^{<\kappa})^+$, the simplest example of a non-$(2^{<\kappa})$-special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem: Let $\kappa$ be any infinite regular cardinal, let $\xi$ be any ordinal such that $2^{\left|\xi\right|} < \kappa$, and let $k$ be any natural number. Let $P$ be a partially ordered set such that $P \to (2^{<\kappa})^1_{2^{<\kappa}}$. Then $$P \to \left(\kappa + \xi \right)^2_k.$$Comment: Submitted to Acta Mathematica Hungaric

The character of topological groups, via Pontryagin-van Kampen duality

The Birkhoff-Kakutani Theorem asserts that a topological group is metrizable if and only if it has countable character. We develop and apply tools for the estimation of the character for a wide class of nonmetrizable topological groups. We consider abelian groups whose topology is determined by a countable cofinal family of compact sets. These are precisely the closed subgroups of Pontryagin-van Kampen duals of metrizable abelian groups, or equivalently, complete abelian groups whose dual is metrizable. By investigating these connections, we show that also in these cases, the character can be estimated, and that it is determined by the weights of the compact subsets of the group, or of quotients of the group by compact subgroups. It follows, for example, that the density and the local density of an abelian metrizable group determine the character of its dual group. Our main result applies to the more general case of closed subgroups of Pontryagin-van Kampen duals of abelian ˇCech-complete groups. Even in the special case of free abelian topological groups, our results extend a number of results of Nickolas and Tkachenko, which were proved using laborious elementary methods. In order to obtain concrete estimations, we establish a natural bridge between the studied concepts and pcf theory, which allows the direct application of several major results from that theory. We include an introduction to these results, their use, and their limitations

A Galvin-Hajnal theorem for generalized cardinal characteristics

We prove that a variety of generalized cardinal characteristics, including meeting numbers, the reaping number, and the dominating number, satisfy an analogue of the Galvin-Hajnal theorem, and hence also of Silver's theorem, at singular cardinals of uncountable cofinality.Comment: 18 page

The character of topological groups, via bounded systems, Pontryagin--van Kampen duality and pcf theory

The Birkhoff--Kakutani Theorem asserts that a topological group is metrizable if and only if it has countable character. We develop and apply tools for the estimation of the character for a wide class of nonmetrizable topological groups. We consider abelian groups whose topology is determined by a countable cofinal family of compact sets. These are the closed subgroups of Pontryagin--van Kampen duals of \emph{metrizable} abelian groups, or equivalently, complete abelian groups whose dual is metrizable. By investigating these connections, we show that also in these cases, the character can be estimated, and that it is determined by the weights of the \emph{compact} subsets of the group, or of quotients of the group by compact subgroups. It follows, for example, that the density and the local density of an abelian metrizable group determine the character of its dual group. Our main result applies to the more general case of closed subgroups of Pontryagin--van Kampen duals of abelian \v{C}ech-complete groups. In the special case of free abelian topological groups, our results extend a number of results of Nickolas and Tkachenko, which were proved using combinatorial methods. In order to obtain concrete estimations, we establish a natural bridge between the studied concepts and pcf theory, that allows the direct application of several major results from that theory. We include an introduction to these results and their use.Comment: Minor corrections. Final version before journal editin

Drones and Support for the Use of Force

Drones and Support for the Use of Force utilizes experimental research to analyze the effects of combat drones on Americans’ support for the use of force. The authors develop expectations drawn from social science theory and then assess these conjectures using a series of survey experiments. Their findings—that drones have had important but nuanced effects on support for the use of force—have implications for democratic control of military action and civil-military relations, and provide insight into how the development and proliferation of current and future military technologies influence the domestic politics of foreign policy