The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics

Prudent walks are special self-avoiding walks that never take a step towards an already occupied site, and \emph{$k$-sided prudent walks} (with $k=1,2,3,4$) are, in essence, only allowed to grow along $k$ directions. Prudent polygons are prudent walks that return to a point adjacent to their starting point. Prudent walks and polygons have been previously enumerated by length and perimeter (Bousquet-M\'elou, Schwerdtfeger; 2010). We consider the enumeration of \emph{prudent polygons} by \emph{area}. For the 3-sided variety, we find that the generating function is expressed in terms of a $q$-hypergeometric function, with an accumulation of poles towards the dominant singularity. This expression reveals an unusual asymptotic structure of the number of polygons of area $n$, where the critical exponent is the transcendental number $\log_23$ and and the amplitude involves tiny oscillations. Based on numerical data, we also expect similar phenomena to occur for 4-sided polygons. The asymptotic methodology involves an original combination of Mellin transform techniques and singularity analysis, which is of potential interest in a number of other asymptotic enumeration problems.Comment: 27 pages, 6 figure

On consecutive pattern-avoiding permutations of length 4, 5 and beyond

We review and extend what is known about the generating functions for consecutive pattern-avoiding permutations of length 4, 5 and beyond, and their asymptotic behaviour. There are respectively, seven length-4 and twenty-five length-5 consecutive-Wilf classes. D-finite differential equations are known for the reciprocal of the exponential generating functions for four of the length-4 and eight of the length-5 classes. We give the solutions of some of these ODEs. An unsolved functional equation is known for one more class of length-4, length-5 and beyond. We give the solution of this functional equation, and use it to show that the solution is not D-finite. For three further length-5 c-Wilf classes we give recurrences for two and a differential-functional equation for a third. For a fourth class we find a new algebraic solution. We give a polynomial-time algorithm to generate the coefficients of the generating functions which is faster than existing algorithms, and use this to (a) calculate the asymptotics for all classes of length 4 and length 5 to significantly greater precision than previously, and (b) use these extended series to search, unsuccessfully, for D-finite solutions for the unsolved classes, leading us to conjecture that the solutions are not D-finite. We have also searched, unsuccessfully, for differentially algebraic solutions.Comment: 23 pages, 2 figures (update of references, plus web link to enumeration data). Minor update. Typos corrected. One additional referenc

Marine Geodesy in the Department of Defense (USA)

Marine geodesy during and since World War II has occupied a significant place in the military whenever precise positioning or distance measurements were required. Applications of geodetic principles in coastal and deep ocean marine environments have been of perhaps greater importance to the Naval Oceanographic Office (NAVOCEANO) and the Defense Mapping Agency (DMA) in DoD. The Army Corps o f Engineers depends also on accurate geodetic control in the conduct of hydrographic survey and dredging operations in United States harbors and inland waterways. Accurate absolute positions are critical for Air Force global operations using electronic navigation systems referenced to the World Geodetic System (WGS). Even the Department of State depends on DMA and the National Ocean Service (NOS) for Doppler-derived island positions and computed geodesics to determine Exclusive Economic Zone (EEZ) 200-mile limits from continental baselines and median lines between U.S. and foreign territories. Other navigational problems demanding precise geodetic control are associated with radioactive waste disposal and missile-firing evaluations. The aim of this paper, therefore, is to review major marine geodetic requirements, relevant applications, and major developing systems, some o f which are being developed for DMA through the Naval Ocean Research and Development Activity (NORDA)

Compressed self-avoiding walks, bridges and polygons

We study various self-avoiding walks (SAWs) which are constrained to lie in the upper half-plane and are subjected to a compressive force. This force is applied to the vertex or vertices of the walk located at the maximum distance above the boundary of the half-space. In the case of bridges, this is the unique end-point. In the case of SAWs or self-avoiding polygons, this corresponds to all vertices of maximal height. We first use the conjectured relation with the Schramm-Loewner evolution to predict the form of the partition function including the values of the exponents, and then we use series analysis to test these predictions.Comment: 29 pages, 6 figure

PER UNIT COSTS TO OWN AND OPERATE FARM MACHINERY

Entropy and jackknife estimation procedures were used to find that custom rates are 20.3% lower than the true cost to own and operate machinery for an average size Kansas farm. A method was then developed to estimate a farms total machinery costs with which to benchmark machinery costs.Farm Management,

Forecasting Sensorimotor Adaptability from Baseline Inter-Trial Correlations

One of the greatest challenges for sensorimotor adaptation to the spaceflight environment is the large variability in symptoms, and corresponding functional impairments, from one crewmember to the next. This renders preflight training and countermeasure development difficult, as a "one-size-fits-all" approach is inappropriate. Therefore, it would be highly advantageous to know ahead of time which crewmembers might have more difficulty adjusting to the novel g-levels inherent to spaceflight. This information could guide individually customized countermeasures, which would enable more efficient use of crew time and provide better outcomes. The principal aim of this work is to look for baseline performance metrics that relate to locomotor adaptability. We propose a novel hypothesis that considers baseline inter-trial correlations, the trial-to-trial fluctuations ("noise") in motor performance, as a predictor of individual adaptive capabilities

How might stress contribute to increased risk for schizophrenia in children with chromosome 22q11.2 deletion syndrome?

The most common human microdeletion occurs at chromosome 22q11.2. The associated syndrome (22q11.2DS) has a complex and variable phenotype with a high risk of schizophrenia. While the role of stress in the etiopathology of schizophrenia has been under investigation for over 30 years (Walker et al. 2008), the stress–diathesis model has yet to be investigated in children with 22q11.2DS. Children with 22q11.2DS face serious medical, behavioral, and socioemotional challenges from infancy into adulthood. Chronic stress elevates glucocorticoids, decreases immunocompetence, negatively impacts brain development and function, and is associated with psychiatric illness in adulthood. Drawing knowledge from the extant and well-developed anxiety and stress literature will provide invaluable insight into the complex etiopathology of schizophrenia in people with 22q11.2DS while suggesting possible early interventions. Childhood anxiety is treatable and stress coping skills can be developed thereby improving quality of life in the short-term and potentially mitigating the risk of developing psychosis

Analysis of existing mathematics textbooks for use in secondary schools.

Thesis (Ed.M.)--Boston University Thesis (M.A.)--Boston Universit

A numerical adaptation of SAW identities from the honeycomb to other 2D lattices

Recently, Duminil-Copin and Smirnov proved a long-standing conjecture by Nienhuis that the connective constant of self-avoiding walks on the honeycomb lattice is $\sqrt{2+\sqrt{2}}.$ A key identity used in that proof depends on the existence of a parafermionic observable for self-avoiding walks on the honeycomb lattice. Despite the absence of a corresponding observable for SAW on the square and triangular lattices, we show that in the limit of large lattices, some of the consequences observed on the honeycomb lattice persist on other lattices. This permits the accurate estimation, though not an exact evaluation, of certain critical amplitudes, as well as critical points, for these lattices. For the honeycomb lattice an exact amplitude for loops is proved.Comment: 21 pages, 7 figures. Changes in v2: Improved numerical analysis, giving greater precision. Explanation of why we observe what we do. Extra reference