On the M\"obius Function and Topology of General Pattern Posets

We introduce a formal definition of a pattern poset which encompasses several previously studied posets in the literature. Using this definition we present some general results on the M\"obius function and topology of such pattern posets. We prove our results using a poset fibration based on the embeddings of the poset, where embeddings are representations of occurrences. We show that the M\"obius function of these posets is intrinsically linked to the number of embeddings, and in particular to so called normal embeddings. We present results on when topological properties such as Cohen-Macaulayness and shellability are preserved by this fibration. Furthermore, we apply these results to some pattern posets and derive alternative proofs of existing results, such as Bj\"orner's results on subword order.Comment: 28 Page

The economics of defence in France and the UK

France and the UK face similar geostrategic circumstances: both were once Great Powers and still retain their positions among the five permanent members of the UN Security Council. During the Cold War both were dwarfed by the super-powers and were thus extremely sensitive about their status: what the French called their grandeur and the British called their seat at the top table. Despite their strategic similarities, they have differed in many of their defence policy choices and in particular how they balanced their strategic aspirations with their limited financial resources. Thus a comparison of British and French defence policies provides a revealing case study of military choices

A Hubble Space Telescope Survey of X-ray Luminous Galaxy Clusters: Gravitationally Lensed Arcs and EROs

We are conducting a systematic lensing survey of X-ray luminous galaxy clusters at z~0.2 using the Hubble Space Telescope and large ground-based telescopes. We summarize initial results from our survey, including a measurement of the inner slope of the mass profile of A383, and a search for gravitationally lensed Extremely Red Objects.Comment: 4 pages, 2 figures. To appear in "A New Era in Cosmology" (ASP Conference Proceedings), eds. T. Shanks and N. Metcalf

Monitor assures availability and quality of communication channels

System monitors a communication channel for proper circuit parameters and energizes an alarm if these parameters do not fall within allowable limits. It comprises a monitor-signal transmitter at the transmitting end of the channel and a monitor-signal receiver at the receiving end

On the M\"obius Function of Permutations With One Descent

The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the M\"obius function of intervals $[1,\pi]$ in this poset, for any permutation $\pi$ with at most one descent. We compute the M\"obius function as a function of the number and positions of pairs of consecutive letters in $\pi$ that are consecutive in value. As a result of this we show that the M\"obius function is unbounded on the poset of all permutations. We show that the M\"obius function is zero on any interval $[1,\pi]$ where $\pi$ has a triple of consecutive letters whose values are consecutive and monotone. We also conjecture values of the M\"obius function on some other intervals of permutations with at most one descent

Intervals of Permutations with a Fixed Number of Descents are Shellable

The set of all permutations, ordered by pattern containment, is a poset. We present an order isomorphism from the poset of permutations with a fixed number of descents to a certain poset of words with subword order. We use this bijection to show that intervals of permutations with a fixed number of descents are shellable, and we present a formula for the M\"obius function of these intervals. We present an alternative proof for a result on the M\"obius function of intervals $[1,\pi]$ such that $\pi$ has exactly one descent. We prove that if $\pi$ has exactly one descent and avoids 456123 and 356124, then the intervals $[1,\pi]$ have no nontrivial disconnected subintervals; we conjecture that these intervals are shellable

Computer program for fitting low-order polynomial splines by method of least squares

FITLOS is computer program which implements new curve fitting technique. Main program reads input data, calls appropriate subroutines for curve fitting, calculates statistical analysis, and writes output data. Method was devised as result of need to suppress noise in calibration of multiplier phototube capacitors