On Fraïssé’s conjecture for linear orders of finite Hausdorff rank

We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is φ₂(0), the first fixed point of the ε-function. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in  +“φ₂(0) is well-ordered” and, over , implies  +“φ₂(0) is well-ordered”.     

A LOCAL PROPERTY OF DERIVATIONS ON JORDAN–BANACH ALGEBRAS

Let D be a (possibly unbounded) derivation on a complex Jordan–Banach algebra J. We show that if a ∈ J and D ² a = 0, then Da is quasinilpotent.     

The Merrifield–Simmons Conjecture Holds for Bipartite Graphs

Let be a graph and the number of independent (vertex) sets of G. Then the Merrifield–Simmons conjecture states that the sign of the term only depends on the parity of the distance of the vertices in G. We prove that the conjecture holds for bipartite graphs by considering a generalization of the term, where vertex subsets instead of vertices are deleted.     

Probabilistic properties of number fields

In general a bound on number theoretic invariants under the Generalized Riemann Hypothesis (GRH) for the Dedekind zeta function of a number field K   is much stronger than an unconditional one. In this article, we consider three invariants; the residue of _ζK(s)${\zeta }_K(s)$ at s=1$s=1$, the logarithmic derivative of Artin L-function attached to K   at s=1$s=1$, and the smallest prime which does not split completely in K. We obtain bounds on them just as good as the bounds under GRH except for a density zero set of number fields.     

Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions

Suppose is a sequence of polynomials orthogonal with respect to the moment functional τ=σ+ν, where σ is a classical moment functional (Jacobi, Laguerre, Hermite) and ν is a point mass distribution with finite support. In this paper, we develop a new method for constructing a differential equation having as eigenfunctions.     

关于分组化验模型的一个猜想

运用概率和微分学基本理论推导出分组化验法的适用范围,给出最佳分组的存在性证明,并提出最佳分组人数的一个猜想.     

Thomassen's conjecture implies polynomiality of 1‐Hamilton‐connectedness in line graphs

A graph G is 1‐Hamilton‐connected if G?x is Hamilton‐connected for every x∈V(G), and G is 2‐edge‐Hamilton‐connected if the graph G+ X has a hamiltonian cycle containing all edges of X for any X?E⁺(G) = {xy| x, y∈V(G)} with 1≤|X|≤2. We prove that Thomassen's conjecture (every 4‐connected line graph is hamiltonian, or, equivalently, every snark has a dominating cycle) is equivalent to the statements that every 4‐connected line graph is 1‐Hamilton‐connected and/or 2‐edge‐Hamilton‐connected. As a corollary, we obtain that Thomassen's conjecture implies polynomiality of both 1‐Hamilton‐connectedness and 2‐edge‐Hamilton‐connectedness in line graphs. Consequently, proving that 1‐Hamilton‐connectedness is NP‐complete in line graphs would disprove Thomassen's conjecture, unless P = NP. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 241–250, 2012     

Combinatorial expressions for F-polynomials in classical types

We give combinatorial formulas for F-polynomials in cluster algebras of classical types in terms of the weighted paths in certain directed graphs. As a consequence we prove the positivity of F-polynomials in cluster algebras of classical types.