We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V |= “ZFC + Ω is the least inaccessible limit of measurable limits of supercompact cardinals + f : Ω → 2 is a function”, then there is a partial ordering P ∈ V so that for...

Suppose is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal function L (x) = P k2Z c k exp(?(x ? k) 2), x 2 R, satisfying the interpolatory conditions L (j) = 0j , j 2 Z. The paper considers the Gaussian cardinal interpolation operator as a linear mapping from`p (Z) into L p (R), 1 p < 1, and in particular, its behaviour as ! 0 +. It is sh...

This paper deals with the question whether the assumption that for every inaccessible cardinal κ there is a well-order of H(κ+) definable over the structure 〈H(κ+),∈〉 by a formula without parameters is consistent with the existence of (large) large cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the i-function and construct a c...

A mutual coordination of size between developing arteries and veins is essential for establishing proper connections between these vessels and, ultimately, a functional vasculature; however, the cellular and molecular regulation of this parity is not understood. Here, we demonstrate that the size of the developing dorsal aorta and cardinal vein is reciprocally balanced. Mouse embryos carrying g...

It is shown that if every real has a sharp and every subset of ω1 is constructible from a real, then there is an inner model with a Woodin cardinal. The following theorem was produced at the AIM meeting “Descriptive Inner Model Theory,” June 02–06, 2014, in a working group whose participants were the authors listed above. The authors would like to thank AIM for their generous hospitality. Theor...

Stimuli from one family of complex motions are defined by their spiral pitch, where cardinal axes represent signed expansion and rotation. Intermediate spirals are represented by intermediate pitches. It is well established that vision contains mechanisms that sum over space and direction to detect these stimuli (Morrone et al., Nature 376 (1995) 507) and one possibility is that four cardinal m...