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In order to improve the security and increase the discharge burnup of the reactor, nuclear materials UN and U3Si2 have been considered for their higher densities and breeding ratios compared with uranium dioxide. A specific modeling of Tristructural isotropic particles with UN and U3Si2 as nuclear fuels was developed, and its failure probability was evaluated under irradiation conditions typical of a Molten Salt Reactor. The dimensional changes of the fuel particle kernel and layers were given. The stresses in the pyrolytic carbon and silicon carbide layers caused by the kernel gas pressure under normal operating conditions are calculated alone with the failure probability of the coated particle. The finding of this study will facilitate the development of new coated particulate fuels as well as support fuel parameters for design and construction of nuclear reactors. [ABSTRACT FROM AUTHOR]

For each positive integer n, let Qn denote the Boolean lattice of dimension n. For posets P , P ′ , define the poset Ramsey number R (P , P ′) to be the least N such that for any red/blue coloring of the elements of QN, there exists either a subposet isomorphic to P with all elements red, or a subposet isomorphic to P ′ with all elements blue. Axenovich and Walzer introduced this concept in Order (2017), where they proved R(Q2,Qn) ≤ 2n + 2 and R(Qn,Qm) ≤ mn + n + m. They later proved 2n ≤ R(Qn,Qn) ≤ n2 + 2n. Walzer later proved R(Qn,Qn) ≤ n2 + 1. We provide some improved bounds for R(Qn,Qm) for various n , m ∈ ℕ . In particular, we prove that R(Qn,Qn) ≤ n2 − n + 2, R (Q 2 , Q n) ≤ 5 3 n + 2 , and R (Q 3 , Q n) ≤ ⌈ 37 16 n + 55 16 ⌉ . We also prove that R(Q2,Q3) = 5, and R (Q m , Q n) ≤ m − 1 + 2 m + 1 n + 1 3 m + 2 for all n > m ≥ 4. [ABSTRACT FROM AUTHOR]

We propose a directed acyclic hypergraph framework for a probabilistic graphical model that we call Bayesian hypergraphs. The space of directed acyclic hypergraphs is much larger than the space of chain graphs. Hence Bayesian hypergraphs can model much finer factorizations than Bayesian networks or LWF chain graphs and provide simpler and more computationally efficient procedures for factorizations and interventions. Bayesian hypergraphs also allow a modeler to represent causal patterns of interaction such as Noisy-OR graphically (without additional annotations). We introduce global, local and pairwise Markov properties of Bayesian hypergraphs and prove under which conditions they are equivalent. We also extend the causal interpretation of LWF chain graphs to Bayesian hypergraphs and provide corresponding formulas and a graphical criterion for intervention. [ABSTRACT FROM AUTHOR]

Many graphs arising in various information networks exhibit the "power law" behavior –the number of vertices of degree k is proportional to k[sup -β] for some positive β.We show that if β > 2.5, the largest eigenvalue of a random power law graph is almost surely$ (1+ o(1))\sqrt{m} $$*FORMULAINSPRINGER$ where m is the maximum degree. Moreover, the klargest eigenvalues of a random power law graph with exponent β have power law distribution with exponent2β if the maximum degree is sufficiently large, where k is a function depending on β, mand d, the average degree. When 2 < β < 2.5, the largest eigenvalue is heavily concentrated at cm[sup 3-β] for some constant c depending on β and the average degree. This resultfollows from a more general theorem which shows that the largest eigenvalue of a random graph with a given expected degree sequence is determined by m, the maximum degree, and$ \tilde{d} $$*FORMULAINSPRINGER$,the weighted average of the squares of the expected degrees. We show that the k-th largest eigenvalue is almost surely [ABSTRACT FROM AUTHOR]

We consider a family of random graphs with a given expected degree sequence. Each edge is chosen independently with probability proportional to the product of the expected degrees of its endpoints. We examine the distribution of the sizes/volumes of the connected components which turns out depending primarily on the average degree d and the second-order average degree d[sup ~] . Here d[sup ~] denotes the weighted average of squares of the expected degrees. For example, we prove that the giant component exists if the expected average degree d is at least 1, and there is no giant component if the expected second-order average degree d[sup ~] is at most 1. Examples are given to illustrate that both bounds are best possible. [ABSTRACT FROM AUTHOR]

David Vogan gave programmatic conjectures about the Dixmier’s map and he made two conjectures that induction may be independent of the choice of parabolic group used and the sheets of orbit data are conjugated or disjointed[1]. In our previous paper, we gave a geometric version of the parabolic induction of the geometric orbit datum (i.e. orbit covers), and proved Vogan’s first conjecture for geometric orbit datum: the parabolic induction of the geometric orbit datum is independent of the choice of parabolic group. In this paper, we will prove the other Vogan’s conjecture, that is, the sheets are conjugated or disjointed for classical semisimple complex groups. [ABSTRACT FROM AUTHOR]

Discusses the sheets and rigid orbit covers of exceptional Lie groups. Induced orbit covers and sheets; Levi subgroups of exceptional groups; Proving the conjecture that the sheets of an exceptional Lie group are disjoint on the basis of the calculating result.