Homomesies on permutations -- an analysis of maps and statistics in the FindStat database

In this paper, we perform a systematic study of permutation statistics and bijective maps on permutations in which we identify and prove 122 instances of the homomesy phenomenon. Homomesy occurs when the average value of a statistic is the same on each orbit of a given map. The maps we investigate include the Lehmer code rotation, the reverse, the complement, the Foata bijection, and the Kreweras complement. The statistics studied relate to familiar notions such as inversions, descents, and permutation patterns, and also more obscure constructs. Beside the many new homomesy results, we discuss our research method, in which we used SageMath to search the FindStat combinatorial statistics database to identify potential homomesies

Fuzzy Family Ties: Familial Similarity Between Melodic Contours of Different Cardinalities

All melodies have shape: a pattern of ascents, descents, and plateaus that occur as music moves through time. This shape—or contour—is one of a melody’s defining characteristics. Music theorists such as Michael Friedmann (1985), Robert Morris (1987), Elizabeth Marvin (1987), and Ian Quinn (1997) have developed models for analyzing contour, but only a few compare contours with different numbers of notes (cardinalities), and fewer still compare entire families of contours. Since these models do not account for familial relations between different-sized contours, they apply only to a limited musical repertoire, and therefore it seems unlikely that they reflect how listeners perceive melodic shape. This dissertation introduces a new method for evaluating familial similarities between related contours, even if the contours have different cardinalities. My Familial Contour Membership model extends theories of contour transformation by using fuzzy set theory and probability. I measure a contour’s degree of familial membership by examining the contour’s transformational pathway and calculating the probability that each move in the pathway is shared by other family members. Through the potential of differing alignments along these pathways, I allow for the possibility that pathways may be omitted or inserted within a contour that exhibits familial resemblance, despite its different cardinality. Integrating variable cardinality into contour similarity relations more adequately accounts for familial relationships between contours, opening up new possibilities for analytical application to a wide variety of repertoires. I examine familial relationships between variants of medieval plainchant, and demonstrate how the sensitivity to familial variation illuminated by fuzzy theoretical models can contribute to our understanding of musical ontology. I explain how melodic shape contributes to motivic development and narrative creation in Brahms’s “Regenlied” Op. 59, No. 3, and the related Violin Sonata No. 1, Op. 78. Finally, I explore how melodic shape is perceived within the repetitive context of melodic phasing in Steve Reich’s The Desert Music. Throughout each study, I show that a more flexible attitude toward cardinality can open contour theory to more nuanced judgments of similarity and familial membership, and can provide new and valuable insights into one of music’s most fundamental elements

Parity Theorems for Combinatorial Statistics

A q-generalization Gn(q) of a combinatorial sequence Gn which reduces to that sequence when q = 1 is obtained by q-counting a statistic defined on a sequence of finite discrete structures enumerated by Gn. In what follows, we evaluate Gn(−1) for statistics on several classes of discrete structures, giving both algebraic and combinatorial proofs. For the latter, we define appropriate sign-reversing involutions on the associated structures. We shall call the actual algebraic result of such an evaluation at q = −1 a parity theorem (for the statistic on the associated class of discrete structures). Among the structures we study are permutations, binary sequences, Laguerre configurations, derangements, Catalan words, and finite set partitions. As a consequence of our results, we obtain bijective proofs of congruences involving Stirling, Catalan, and Bell numbers. In addition, we modify the ideas used to construct the aforementioned sign-reversing involutions to furnish bijective proofs of combinatorial identities involving sums with alternating signs

The role of aggregates in the thermal stability of Mg-PSZ refractories for vacuum induction melting

Mg-PSZ refractories used as vacuum induction melting crucibles are particle-reinforced composites with aggregate and matrix phases comprising fused zirconia. Three commercial varieties were cycled eight times to service temperatures and their microstructural and thermomechanical evolution investigated, with focus placed on the aggregate populations. Two refractories, with large aggregates of similar size, were found to retain stiffness after cycling but in the refractory containing aggregates with high stabiliser levels, reaction between the stabiliser and Al and Si impurities produced secondary phases. Volume changes accompanying formation of these phases, and subsequent thermal expansion mismatches, led to aggregate break-up with consequent reductions in refractory toughness and strength. Secondary phases developed only rarely in the aggregates (with lower levels of stabiliser) of the second refractory. These aggregates remained intact and the refractory retained its toughness and strength. A third refractory contained small, unstabilised aggregates in a stabilised matrix and the strain mismatches that ensued during polymorphic transformation damaged microstructural interfaces. Refractory stiffness halved within eight cycles and toughness and strength were lost. All three refractories displayed R-curve behaviour and quasi-stable fracture curves were observed during bend tests. The study shows that when using fused zirconia aggregates to design refractories, engineers need to i) limit stabiliser concentrations - a difference of just ±1 wt% Mg (in the presence of impurity elements) may determine whether secondary phase formation occurs and ii) eliminate alumina and silica impurities when possible through substitution of zircon sand with baddeleyite as the source for fused zirconia.Open Acces

Modelling and Structural Studies of a Gelling Polysaccharide: Agarose.

This thesis details work carried out over a period of three years on the two gelling carbohydrates agarose and carrageenan. The major part of the work deals with agarose. Two approaches have been used which yield information from different angles; these are the experimental (laboratory) and the simulation (computational) approaches. There is a large field of interest in gelling carbohydrates from the point of view of the food industry. Their extraordinary ability to form stable gels and emulsions incorporating other food ingredients makes them important in many deserts and dairy products. In the present work, models for agarose and carrageenan carbohydrates were developed using structural x-ray data and related carbohydrate literature. The models were treated with two different solvent simulation methods. It was found that the inclusion of individual solvent molecules (the closest approximation to a real solution) was extremely uneconomical when the demands on computing time were taken into account, and in fact the long term outcome of the simulation was the same for both methods. Inclusion of solvent simply reduces diffusion rates and the time constant for chain flexing. Gel permeation chromatography and differential scanning calorimetry were used to prepare samples of agarose molecules of known size, and to probe temperature dependent phase transitions. This work was done at the UNILEVER laboratories at Colworth House, Sharnbrook in Bedfordshire. It was found that only molecules longer than fifteen residues displayed the molecular ordering transition typical of agarose polymer, and a value for the enthalpy of the transition of -1.5kcal per mole of residues was measured. It was predicted that in agarose itself, helical regions of a size of approximately 40 residues should exist. Simulations were then done on several agarose molecules of different sizes in order to parallel the experimental work. The differences in energy between molecules in various conformations were compared. These results were also related to helix-coil transition theory. The modelling predicts an enthalpy per mole of residues for the agarose coil to helix transition of approximately -2kcal, and indicates that single agarose coils may be of some importance in agarose gel structure. The work illustrates the difficulty in modelling such complex systems, and in fact it remains impossible to observe agarose molecules undergoing the transition between a random coil and a helical conformation

Gaussian Process applied to modeling the dynamics of a deformable material

En aquesta tesi, establirem la base teòrica d'alguns algoritmes de reducció de la dimensionalitat com el GPLVM i la seva aplicació a la reproducció d'una sèrie temporal d'observables amb el GPDM i la seva generalització amb control CGPDM. Finalment, anem a introduir un nou model computacionalment més eficient, el MoCGPDM aplicant una barreja d'experts. L'última secció consistirà en afinar el model i comparar-lo amb el model previ.En esta tesis, estableceremos la base teórica de algunos algoritmos de reducción de la dimensionalidad como el GPLVM y su aplicación a la reproducción de una serie temporal de observables con el GPDM y su generalización con control CGPDM. Finalmente, vamos a introducir un nuevo modelo computacionálmente más eficiente, el MoCGPDM aplicando una mezcla de expertos. La última sección consistirá en afinar el modelo y compararlo con el modelo previo.In this thesis, we establish the theoretical basis of some dimensional reduction algorithms like the GPLVM and their application to the reproduction of a time series of observable data with the GPDM and its generalization with control CGPDM. Finally, we are going to introduce a new more time efficient model MoCGPDM applying a mixture of experts. And the final section will consist in fine-tuning the model and compare it to the previous model