 Award ID(s):
 1700058
 NSFPAR ID:
 10060713
 Date Published:
 Journal Name:
 Advances in applied mathematics
 Volume:
 95
 ISSN:
 01968858
 Page Range / eLocation ID:
 95115
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta ' _{e_k} e_{n}$ , where $\Delta ' _{e_k}$ and $\Delta _{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $\operatorname {\mathrm {GL}}_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.more » « less

We conjecture a simple combinatorial formula for the Schur expansion of the Frobenius series of the Snmodules Rn,λ,s, which appear as the cohomology rings of the “∆Springer” varieties. These modules interpolate between the GarsiaProcesi modules Rµ (which are the type A Springer fiber cohomology rings) and the rings Rn,k defined by Haglund, Rhoades, and Shimozono in the context of the Delta Conjecture. Our formula directly generalizes the known cocharge formula for GarsiaProcesi modules and gives a new cocharge formula for the Delta Conjecture at t = 0, by introducing batterypowered tableaux that “store” extra charge in their battery. Our conjecture has been verified by computer for all n ≤ 10 and s ≤ ℓ(λ)+2, as well as for n ≤ 8 and s ≤ ℓ(λ)+7. We prove it holds for several infinite families of n,λ,s.more » « less

We prove that $\omega \Delta ^{\prime}_{e_{k}}e_{n}_{t=0}$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a HallLittlewood polynomial, and generalize this formula to the Frobenius series of all $\Delta $Springer modules. We use this to give an explicit Schur expansion in terms of the LascouxSchützenberger cocharge statistic on a new combinatorial object that we call a batterypowered tableau. Our proof is geometric, and shows that the $\Delta $Springer varieties of Levinson, Woo, and the second author are generalized Springer fibers coming from the partial resolutions of the nilpotent cone due to Borho and MacPherson. We also give alternative combinatorial proofs of our Schur expansion for several special cases, and give conjectural skewing formulas for the $t$ and $t^{2}$ coefficients of $\omega \Delta ^{\prime}_{e_{k}}e_{n}$.more » « less

In the context of metal particle catalysts, composition, shape, exposed facets, crystal structure, and atom distribution dictate activity. While techniques have been developed to control each of these parameters, there is no general method that allows one to optimize all parameters in the context of polyelemental systems. Herein, by combining a solidstate, Biinfluenced, highindex facet shape regulation strategy with thermal annealing, we achieve control over crystal structure and atom distribution on the exposed highindex facets, resulting in an unprecedentedly diverse library of chemically disordered and ordered multimetallic (Pt, Co, Ni, Cu, Fe, and Mn) tetrahexahedral (THH) nanoparticles. Density functional theory calculations show that surface Bi modification stabilizes the {210} highindex facets of the nanoparticles, regardless of their internal atomic ordering. Moreover, we find that the ordering transition temperatures for the nanoparticles are dependent on their composition, and, in the case of Pt_{3}Fe_{1}THH nanoparticles, increasing Ni substitution leads to an ordertodisorder transition at 900 °C. Finally, we have discovered that ordered intermetallic THH Pt_{1}Co_{1}nanocatalysts exhibit a catalytic performance superior to disordered THH Pt_{1}Co_{1}nanoparticles and commercial Pt/C catalysts toward methanol electrooxidation, highlighting the importance of crystal structure and atom distribution control on highindex facets in nanoscale catalysts.

null (Ed.)A quickest change detection problem is considered in a sensor network with observations whose statistical dependency structure across the sensors before and after the change is described by a decomposable graphical model (DGM). Distributed computation methods for this problem are proposed that are capable of producing the optimum centralized test statistic. The DGM leads to the proper way to collect nodes into local groups equivalent to cliques in the graph, such that a clique statistic which summarizes all the clique sensor data can be computed within each clique. The clique statistics are transmitted to a decision maker to produce the optimum centralized test statistic. In order to further improve communication efficiency, an ordered transmission approach is proposed where transmissions of the clique statistics to the fusion center are ordered and then adaptively halted when sufficient information is accumulated. This procedure is always guaranteed to provide the optimal change detection performance, despite not transmitting all the statistics from all the cliques. A lower bound on the average number of transmissions saved by ordered transmissions is provided and for the case where the change seldom occurs the lower bound approaches approximately half the number of cliques provided a well behaved distance measure between the distributions of the sensor observations before and after the change is sufficiently large. We also extend the approach to the case when the graph structure is different under each hypothesis. Numerical results show significant savings using the ordered transmission approach and validate the theoretical findings.more » « less