A note on the Hamilton–Waterloo problem with C8-factors and Cm-factors

In this paper, we almost completely solve the Hamilton–Waterloo problem with $C_8$-factors and $C_m$-factors where the number of vertices is a multiple of $8m$.     

Dyck paths with a first return decomposition constrained by height

We study the enumeration of Dyck paths having a first return decomposition with special properties based on a height constraint. We exhibit new restricted sets of Dyck paths counted by the Motzkin numbers, and we give a constructive bijection between these objects and Motzkin paths. As a byproduct, we provide a generating function for the number of Motzkin paths of height $k$ with a flat (resp. with no flats) at the maximal height.     

Depth and Stanley depth of the edge ideals of square paths and square cycles

In this paper, we compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square path on n vertices. We also compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square cycle on n vertices, when n≡0,3,4( mod 5), and give tight bounds when n≡1,2( mod 5). We also prove a conjecture of Herzog presented in [5 Herzog, J. (2013). A survey on Stanley depth. In: Bigatti, M. A., Gimenez, P., Sáenz-de-Cabezón, E., eds. Monomial Ideals, Computations and Applications. Lecture Notes in Mathematics, Vol. 2083. Heidelberg: Springer, pp. 3–45. https://arxiv.org/pdf/1702.00781.pdf.[Crossref] , [Google Scholar]], for the edge ideals of square paths and square cycles.     

Extremal graphs for the k‐flower

The Turán number of a graph H, , is the maximum number of edges in any graph of order n that does not contain an H as a subgraph. A graph on vertices consisting of k triangles that intersect in exactly one common vertex is called a k‐fan, and a graph consisting of k cycles that intersect in exactly one common vertex is called a k‐flower. In this article, we determine the Turán number of any k‐flower containing at least one odd cycle and characterize all extremal graphs provided n is sufficiently large. Erdős, Füredi, Gould, and Gunderson determined the Turán number for the k‐fan. Our result is a generalization of their result. The addition aim of this article is to draw attention to a powerful tool, the so‐called progressive induction lemma of Simonovits.     

Decomposing planar cubic graphs

The 3‐Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2‐regular subgraph and a matching. We show that this conjecture holds for the class of connected plane cubic graphs.     

On globally solving the extended trust-region subproblems

With the help of the newly developed technique—second order cone (SOC) constraints to strengthen the SDP relaxation of the extended trust-region subproblem (eTRS), we modify two recent SDP relaxation based branch and bound algorithms for solving eTRS. Numerical experiments on some types of problems show that the new algorithms run faster for finding the global optimal solutions than the SDP relaxation based algorithms.     

梁方程时间依赖全局吸引子的存在性

研究了梁方程时间依赖吸引子的存在性,在非线性项f满足临界增长条件时,基于时间依赖全局吸引子的存在性定理,应用先验估计和算子分解方法验证了系数参数与时间t有关时,梁方程对应的过程族{U(t,τ)}的渐近紧性,从而得到梁方程时间依赖全局吸引子的存在性及正则性.     

Numerical Investigation of Sound Generation due to Laminar Flow Past Elliptic Cylinders

Numerical investigation of sound generation due to unsteady laminar flow past elliptic cylinders has been carried out using direct numerical simulation $(DNS)$ approach at a free-stream Mach number of $0.2$. Effects of aspect ratio $(0.6\le AR\le 1.0)$ and Reynolds number $(100\le Re \le 160)$ on the characteristics of radiated sound fields are analyzed. Two-dimensional compressible fluid flow equations are solved on a refined grid using high resolution dispersion relation preserving $(DRP)$ schemes. Using present $DNS$ data, equivalent noise sources as given by various acoustic analogies are evaluated. Amplitudes and frequencies associated with these noise sources are further related to characteristics of disturbance pressure fields. Disturbance pressure fields are intensified with increase in Reynolds number and aspect ratio. Thus, radiated sound power increases with increase in Reynolds number and aspect ratio. Among various cases studied here, minimum and maximum values of radiated sound power are found at $Re=120$ & $AR=0.6$ and $Re=160$ & $AR=1.0$, respectively. Directivity patterns show that the generated sound fields are dominated by the lift dipole for all cases. Next, proper orthogonal decomposition $(POD)$ technique has been implemented for decomposing disturbance pressure fields. The $POD$ modes associated with the lift and the drag dipoles have been identified. $POD$ analyses also clearly display that the radiated sound fields are dominated by the lift dipole only. Further, acoustic and hydrodynamic modes obtained using Doak's decomposition method have confirmed the patterns of radiated sound field intensities.     

Parallel Smoothed Aggregation Multilevel Schwarz Preconditioned Newton-Krylov Algorithms for Poisson-Boltzmann Problems

We study a multilevel Schwarz preconditioned Newton-Krylov algorithm to solve the Poisson-Boltzmann equation with applications in multi-particle colloidal simulation. The smoothed aggregation-type coarse mesh space is introduced in collaboration with the one-level Schwarz method as a composite preconditioner for accelerating the convergence of a Krylov subspace method for solving the Jacobian system at each Newton step. The important feature of the proposed solution algorithm is that the geometric mesh information needed for constructing the multilevel preconditioner is the same as the one-level Schwarz method on the fine mesh. Other components, such as the definition of the coarse mesh, all the mesh transfer operators, and the coarse mesh problem, are taken care of by the Trillinos/ML packages of the Sandia National Laboratories in the United States. After algorithmic parameter tuning, we show that the proposed smoothed aggregation multilevel Newton-Krylov-Schwarz (NKS) algorithm numerically outperforms than smoothed aggregation multigrid method and one-level version of the NKS algorithm with satisfactory parallel performances up to a few thousand cores. Besides, we investigate how the electrostatic forces between particles for the separation distance depend on the radius of spherical colloidal particles and valence ratios of cation and anion in a cubic system.     

极分解定理的注记

设H和K是Hilbert空间.首先,对给定的算子T∈B（H,K）,刻画了集合U_T={U∈B（H,K）：U是部分等距算子并且T=U（T^*T）^1/2}. 其次,对部分等距算子U∈B（H,K）,还给出集合T_U={T∈B（H,K）：N（T）=N（U）,R（T）=R（U）,T=U（T^*T）^1/2}的刻画. 最后,作为主要结果的应用得到了相关结论.