Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers

This is the first of three papers that develop structures which are counted by a "parabolic" generalization of Catalan numbers. Fix a subset R of {1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes are determined by R. These are the "inverses" of (parabolic) multipermutations whose multiplicities are determined by R. The standard forms of the ordered partitions are refered to as "R-permutations". The notion of 312-avoidance is extended from permutations to R-permutations. Let lambda be a partition of N such that the set of column lengths in its shape is R or R union {n}. Fix an R-permutation pi. The type A Demazure character (key polynomial) in x_1, .., x_n that is indexed by lambda and pi can be described as the sum of the weight monomials for some of the semistandard Young tableau of shape lambda that are used to describe the Schur function indexed by lambda. Descriptions of these "Demazure" tableaux developed by the authors in earlier papers are used to prove that the set of these tableaux is convex in Z^N if and only if pi is R-312-avoiding if and only if the tableau set is the entire principal ideal generated by the key of pi. These papers were inspired by results of Reiner and Shimozono and by Postnikov and Stanley concerning coincidences between Demazure characters and flagged Schur functions. This convexity result is used in the next paper to deepen those results from the level of polynomials to the level of tableau sets. The R-parabolic Catalan number is defined to be the number of R-312-avoiding permutations. These special R-permutations are reformulated as "R-rightmost clump deleting" chains of subsets of {1,..,n} and as "gapless R-tuples"; the latter n-tuples arise in multiple contexts in these papers.Comment: 20 pp with 2 figs. Identical to v.3, except for the insertion of the publication data for the DMTCS journal (dates and volume/issue/number). This is one third of our "Parabolic Catalan numbers ..", arXiv:1612.06323v

Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials)

Fix an integer partition lambda that has no more than n parts. Let beta be a weakly increasing n-tuple with entries from {1,..,n}. The flagged Schur function indexed by lambda and beta is a polynomial generating function in x_1, .., x_n for certain semistandard tableaux of shape lambda. Let pi be an n-permutation. The type A Demazure character (key polynomial, Demazure polynomial) indexed by lambda and pi is another such polynomial generating function. Reiner and Shimozono and then Postnikov and Stanley studied coincidences between these two families of polynomials. Here their results are sharpened by the specification of unique representatives for the equivalence classes of indexes for both families of polynomials, extended by the consideration of more general beta, and deepened by proving that the polynomial coincidences also hold at the level of the underlying tableau sets. Let R be the set of lengths of columns in the shape of lambda that are less than n. Ordered set partitions of {1,..,n} with block sizes determined by R, called R-permutations, are used to describe the minimal length representatives for the parabolic quotient of the nth symmetric group specified by the set {1,..,n-1}\R. The notion of 312-avoidance is generalized from n-permutations to these set partitions. The R-parabolic Catalan number is defined to be the number of these. Every flagged Schur function arises as a Demazure polynomial. Those Demazure polynomials are precisely indexed by the R-312-avoiding R-permutations. Hence the number of flagged Schur functions that are distinct as polynomials is shown to be the R-parabolic Catalan number. The projecting and lifting processes that relate the notions of 312-avoidance and of R-312-avoidance are described with maps developed for other purposes.Comment: 27 pages, 2 figures. Identical to v.2, except for the insertion of the publication data for the DMTCS journal (dates and volume/issue/number). This is two-thirds of our preprint "Parabolic Catalan numbers count flagged Schur functions; Convexity of tableau sets for Demazure characters", arXiv:1612.06323v

Dynamic Aspects of Earnings Mobility

This paper proposes an econometric methodology to deal with life cycle earnings and mobility among discrete earnings classes. First, we use panel data on male log earnings to estimate an earnings function with permanent and serially correlated transitory components due to both measured and unmeasured variables. Assuming that the error components are normally distributed, we develop statements for the probability that an individual's earnings will fall into a particular but arbitrary time sequence of poverty states. Using these statements, we illustrate the implications of our earnings model for poverty dynamics and compare our approach to Markov chain models of income mobility.

Occupational Learning, Financial Knowledge, and the Accumulation of Retirement Wealth

This study explores the relationship between general human capital investment, financial knowledge, occupational spillovers, and the accumulation of wealth in a primarily descriptive manner. Drawing upon human capital theory and following previous related work by Delavande, Rohwedder and Willis (2008), we hypothesized that individuals with daily exposure to financial knowledge through their occupation would benefit by having greater financial knowledge that would translate into greater wealth accumulation than individuals who do not enjoy such spillovers from their occupation. Using data from the Cognitive Economics Study and the Health and Retirement Study, we find strong evidence that individuals in financial occupations tend to have greater financial knowledge and moderate evidence that they also have greater wealth accumulation.

The Economic Consequences of Widowhood

We analyzed the economic consequences of a husband’s death using events that occurred between the first two waves of the HRS and AHEAD studies. We compared poverty transitions against published results from Social Security’s Retirement History Survey of the 1970s. Widowhood remains an important risk factor for transition into poverty, although somewhat less so than twenty years ago. Women over age 65 (AHEAD) are less likely to experience severe economic changes than women under age 61 (HRS). Several factors account for the age differences: the declining importance of husband’s earnings with age, the rising importance of Social Security benefits, and the occasionally large out-of-pocket medical expenses associated with husband’s death before Medicare eligibility. The greater economic impact of widowhood at younger ages is consistent with our cross-section evidence that poverty rates rise with duration of widowhood but are only weakly associated with age.

A Quarter Century, Not a Raised Voice

This dedication was in recognition of Wilbert D. Ham’s retirement from the University of Kentucky College of Law faculty. Wilber D. Ham served on the faculty from 1949 until 1986

Enhancing IS Education with Flexible Teaching and Learning

Flexible teaching and learning is not a new concept, but it is one that we, as educators, do not focus on enough. Designing and delivering innovative, exciting and relevant learning experiences is needed if we are to make our classes good learning experiences. Information systems (IS) educators deal with technology every day, yet we are sometimes the first ones to forget how to use it in the classroom. Educators must recognize the importance of increasing student control over and active participation in their own learning. This Special Issue of the Journal of Information Systems Education looks at flexible teaching and learning in the IS classroom. We present eight papers on flexible teaching and learning, dealing with both the face-to-face and online classroom environments. We hope that the ideas presented in these papers will foster your thinking in using flexible teaching techniques. In the end, flexible teaching and learning focuses on improving student learning, a goal that we all aspire to in our classrooms

Shocked Molecular Hydrogen in the 3C 326 Radio Galaxy System

The Spitzer spectrum of the giant FR II radio galaxy 3C 326 is dominated by very strong molecular hydrogen emission lines on a faint IR continuum. The H2 emission originates in the northern component of a double-galaxy system associated with 3C 326. The integrated luminosity in H2 pure-rotational lines is 8.0E41 erg/s, which corresponds to 17% of the 8-70 micron luminosity of the galaxy. A wide range of temperatures (125-1000 K) is measured from the H2 0-0 S(0)-S(7) transitions, leading to a warm H2 mass of 1.1E9 Msun. Low-excitation ionic forbidden emission lines are consistent with an optical LINER classification for the active nucleus, which is not luminous enough to power the observed H2 emission. The H2 could be shock-heated by the radio jets, but there is no direct indication of this. More likely, the H2 is shock-heated in a tidal accretion flow induced by interaction with the southern companion galaxy. The latter scenario is supported by an irregular morphology, tidal bridge, and possible tidal tail imaged with IRAC at 3-9 micron. Unlike ULIRGs, which in some cases exhibit H2 line luminosities of comparable strength, 3C 326 shows little star-formation activity (~0.1 Msun/yr). This may represent an important stage in galaxy evolution. Starburst activity and efficient accretion onto the central supermassive black hole may be delayed until the shock-heated H2 can kinematically settle and coolComment: 27 pages, 7 figures, accepted for publication in the Astrophysical Journa

Going with the Flow : Tunable Flow-Induced Polymer Mechanochemistry

Mechanical forces can drive chemical transformations in polymers, directing reactions along otherwise inaccessible pathways, providing exciting possibilities for developing smart, responsive materials. The state-of-the-art test for solution-based polymer mechanochemistry development is ultrasonication. However, this does not accurately model the forces that will be applied during device fabrication using processes such as 3D printing or spray coating. Here, a step is taken toward predictably translating mechanochemistry from molecular design to manufacturing by demonstrating a highly controlled nozzle flow setup in which the shear forces being delivered are precisely tuned. The results show that solvent viscosity, fluid strain rate, and the nature of the breaking bond can be individually studied. Importantly, it is shown that the influence of each is different to that suggested by ultrasonication (altered quantity of chain breakage and critical polymer chain length). Significant development is presented in the understanding of polymer bond breakage during manufacturing flows to help guide design of active components that trigger on demand. Using an anthracene-based mechanophore, the triggering of a fluorescence turn-on is demonstrated through careful selection of the flow parameters. This work opens the avenue for programmed chemical transformations during inline manufacturing processes leading to tunable, heterogeneous final products from a single source material. © 2020 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei