Web matrices : structural properties and generating combinatorial identities

In this paper we present new results for the combinatorics of web diagrams and web worlds. These are discrete objects that arise in the physics of calculating scattering amplitudes in non-abelian gauge theories. Web-colouring and web-mixing matrices (collectively known as web matrices) are indexed by ordered pairs of web-diagrams and contain information relating the number of colourings of the first web diagram that will produce the second diagram. We introduce the black diamond product on power series and show how it determines the web-colouring matrix of disjoint web worlds. Furthermore, we show that combining known physical results with the black diamond product gives a new technique for generating combinatorial identities. Due to the complicated action of the product on power series, the resulting identities appear highly non-trivial. We present two results to explain repeated entries that appear in the web matrices. The first of these shows how diagonal web matrix entries will be the same if the comparability graphs of their associated decomposition posets are the same. The second result concerns general repeated entries in conjunction with a flipping operation on web diagrams. We present a combinatorial proof of idempotency of the web-mixing matrices, previously established using physical arguments only. We also show how the entries of the square of the web-colouring matrix can be achieved by a linear transformation that maps the standard basis for formal power series in one variable to a sequence of polynomials. We look at one parameterized web world that is related to indecomposable permutations and show how determining the web-colouring matrix entries in this case is equivalent to a combinatorics on words problem

On the inverse image of pattern classes under bubble sort

Let B be the operation of re-ordering a sequence by one pass of bubble sort. We completely answer the question of when the inverse image of a principal pattern class under B is a pattern class.Comment: 11 page

Improved model of the triple system V746 Cas that has a bipolar magnetic field associated with the tertiary

V746 Cas is known to be a triple system composed of a close binary with an alternatively reported period of either 25.4d or 27.8d and a third component in a 62000d orbit. The object was also reported to exhibit multiperiodic light variations with periods from 0.83d to 2.50d, on the basis of which it was classified as a slowly pulsating B star. Interest in further investigation of this system was raised by the detection of a variable magnetic field. Analysing spectra from four instruments, earlier published radial velocities, and several sets of photometric observations, we arrived at the following conclusions: (1) The optical spectrum is dominated by the lines of the B-type primary (Teff1~16500(100) K), contributing 70% of the light in the optical region, and a slightly cooler B tertiary (Teff3~13620(150) K). The lines of the low-mass secondary are below our detection threshold; we estimate that it could be a normal A or F star. (2) We resolved the ambiguity in the value of the inner binary period and arrived at a linear ephemeris of T_super.conj.=HJD 2443838.78(81)+25.41569(42)xE. (3) The intensity of the magnetic field undergoes a~sinusoidal variation in phase with one of the known photometric periods, namely 2.503867(19)d, which we identify with the rotational period of the tertiary. (4) The second photometric 1.0649524(40)d period is identified with the rotational period of the B-type primary, but this interpretation is much less certain and needs further verification. (5) If our interpretation of photometric periods is confirmed, the classification of the object as a slowly pulsating B star should be revised. (6) Applying an N-body model to different types of available observational data, we constrain the orbital inclination of the inner orbit to ~60 deg to 85 deg even in the absence of eclipses, and estimate the probable properties of the triple system and its components.Comment: Accepted for publication in Astronomy and Astrophysic

Random tree growth by vertex splitting

We study a model of growing planar tree graphs where in each time step we separate the tree into two components by splitting a vertex and then connect the two pieces by inserting a new link between the daughter vertices. This model generalises the preferential attachment model and Ford's $\alpha$-model for phylogenetic trees. We develop a mean field theory for the vertex degree distribution, prove that the mean field theory is exact in some special cases and check that it agrees with numerical simulations in general. We calculate various correlation functions and show that the intrinsic Hausdorff dimension can vary from one to infinity, depending on the parameters of the model.Comment: 47 page

Diagonally Neighbour Transitive Codes and Frequency Permutation Arrays

Constant composition codes have been proposed as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication. In particular, a certain class of constant composition codes called frequency permutation arrays have been suggested as ideal, in some sense, for these purposes. In this paper we characterise a family of neighbour transitive codes in Hamming graphs in which frequency permutation arrays play a central rode. We also classify all the permutation codes generated by groups in this family

Emerging technologies for sustainable irrigation: Selected papers from the 2015 ASABE and IA irrigation symposium

Citation: Lamm, F. R., Stone, K. C., Dukes, M. D., Howell, T. A., Sr., Robbins, J. W. D., Jr., & Mecham, B. Q. (2016). Emerging technologies for sustainable irrigation: Selected papers from the 2015 ASABE and IA irrigation symposium. Transactions of the Asabe, 59(1), 155-161. doi:10.13031/trans.59.11706This article is an introduction to the "Emerging Technologies in Sustainable Irrigation: A Tribute to the Career of Terry Howell, Sr." Special Collection in this issue of Transactions of the ASABE and the next issue of Applied Engineering in Agriculture, consisting of 16 articles selected from 62 papers and presentations at the joint irrigation symposium of ASABE and the Irrigation Association (IA), which was held in November 2015 in Long Beach, California. The joint cooperation on irrigation symposia between ASABE and IA can be traced back to 1970, and this time period roughly coincides with the career of Dr. Howell. The cooperative symposia have offered an important venue for discussion of emerging technologies that can lead to sustainable irrigation. This most recent symposium is another point on the continuum. The articles in this Special Collection address three major topic areas: evapotranspiration measurement and determination, irrigation systems and their associated technologies, and irrigation scheduling and water management. While these 16 articles are not inclusive of all the important advances in irrigation since 1970, they illustrate that continued progress occurs by combining a recognition of the current status with the postulation of new ideas to advance our understanding of irrigation engineering and science. The global food and water challenges will require continued progress from our portion of the scientific community. This article serves to introduce and provide a brief summary of the Special Collection. © 2016 American Society of Agricultural and Biological Engineers