A q-weighted version of the Robinson-Schensted algorithm

We introduce a q-weighted version of the Robinson-Schensted (column insertion) algorithm which is closely connected to q-Whittaker functions (or Macdonald polynomials with t=0) and reduces to the usual Robinson-Schensted algorithm when q=0. The q-insertion algorithm is `randomised', or `quantum', in the sense that when inserting a positive integer into a tableau, the output is a distribution of weights on a particular set of tableaux which includes the output which would have been obtained via the usual column insertion algorithm. There is also a notion of recording tableau in this setting. We show that the distribution of weights of the pair of tableaux obtained when one applies the q-insertion algorithm to a random word or permutation takes a particularly simple form and is closely related to q-Whittaker functions. In the case $0\le q<1$, the q-insertion algorithm applied to a random word also provides a new framework for solving the q-TASEP interacting particle system introduced (in the language of q-bosons) by Sasamoto and Wadati (1998) and yields formulas which are equivalent to some of those recently obtained by Borodin and Corwin (2011) via a stochastic evolution on discrete Gelfand-Tsetlin patterns (or semistandard tableaux) which is coupled to the q-TASEP process. We show that the sequence of P-tableaux obtained when one applies the q-insertion algorithm to a random word defines another, quite different, evolution on semistandard tableaux which is also coupled to the q-TASEP process

Mesoporous silica encapsulated metal nanoparticles in catalysis

Nanosized particles can demonstrate dramatic performance in comparison to bulk materials in heterogeneous catalysis, due to their high density of under coordinate sites associated with altered electronic properties. The structural and compositional design of bimetallic nanoparticles can further afford the precise control of activity and selectivity via the geometric and electronic effects of secondary metals. Intermetallic compounds are one of the special alloys with defined stoichiometry and ordered crystal structure, which exemplifies them as ideal model catalysts for structure-property studies in catalysis. Direct colloidal synthesis of intermetallic nanoparticles requires the presence of organic capping agents, which limits the thermal stability of nanoparticles and complicates their surface structures. Mesoporous silica shells can be used to encapsulate monometallic and bimetallic nanoparticles with high sinter-resistance for high-temperature treatment, and enables their applications for harsh reaction conditions and fundamental mechanism studies. Several examples of silica-encapsulated nanoparticles have been demonstrated in this thesis to study their catalytic properties

Robinson-Schensted algorithms and quantum stochastic double product integrals

This thesis is divided into two parts. In the first part (Chapters 1, 2, 3) various Robinson-Schensted (RS) algorithms are discussed. An introduction to the classical RS algorithm is presented, including the symmetry property, and the result of the algorithm Doob h-transforming the kernel from the Pieri rule of Schur functions h when taking a random word [O'C03a]. This is followed by the extension to a q-weighted version that has a branching structure, which can be alternatively viewed as a randomisation of the classical algorithm. The q-weighted RS algorithm is related to the q-Whittaker functions in the same way as the classical algorithm is to the Schur functions. That is, when taking a random word, the algorithm Doob h-transforms the Hamiltonian of the quantum Toda lattice where h are the q-Whittaker functions. Moreover, it can also be applied to model the q-totally asymmetric simple exclusion process introduced in [SW98]. Furthermore, the q-RS algorithm also enjoys a symmetry property analogous to that of the symmetry property of the classical algorithm. This is proved by extending Fomin's growth diagram technique [Fom79, Fom88, Fom94, Fom95], which covers a family of the so-called branching insertion algorithms, including the row insertion proposed in [BP13]. In the second part (Chapters 4, 5) we work with quantum stochastic analysis. First we introduce the basic elements in quantum stochastic analysis, including the quantum probability space, the momentum and position Brownian motions [CH77], and the relation between rotations and angular momenta via the second quantisation, which is generalised to a family of rotation-like operators [HP15a]. Then we discuss a family of unitary quantum causal stochastic double product integrals E, which are expected to be the second quantisation of the continuous limit W of a discrete double product of aforementioned rotation-like operators. In one special case, the operator E is related to the quantum Levy stochastic area, while in another case it is related to the quantum 2-d Bessel process. The explicit formula for the kernel of W is obtained by enumerating linear extensions of partial orderings related to a path model, and the combinatorial aspect is closely related to generalisations of the Catalan numbers and the Dyck paths. Furthermore W is shown to be unitary using integrals of the Bessel functions

A New Dynamic Path Planning Approach for Unmanned Aerial Vehicles

Dynamic path planning is one of the key procedures for unmanned aerial vehicles (UAV) to successfully fulfill the diversified missions. In this paper, we propose a new algorithm for path planning based on ant colony optimization (ACO) and artificial potential field. In the proposed algorithm, both dynamic threats and static obstacles are taken into account to generate an artificial field representing the environment for collision free path planning. To enhance the path searching efficiency, a coordinate transformation is applied to move the origin of the map to the starting point of the path and in line with the source-destination direction. Cost functions are established to represent the dynamically changing threats, and the cost value is considered as a scalar value of mobile threats which are vectors actually. In the process of searching for an optimal moving direction for UAV, the cost values of path, mobile threats, and total cost are optimized using ant optimization algorithm. The experimental results demonstrated the performance of the new proposed algorithm, which showed that a smoother planning path with the lowest cost for UAVs can be obtained through our algorithm. (PDF) A New Dynamic Path Planning Approach for Unmanned Aerial Vehicles. Available from: https://www.researchgate.net/publication/328765418_A_New_Dynamic_Path_Planning_Approach_for_Unmanned_Aerial_Vehicles [accessed Nov 20 2018]