Leaf‐Critical and Leaf‐Stable Graphs

The minimum leaf number ml(G) of a connected graph G is defined as the minimum number of leaves of the spanning trees of G if G is not hamiltonian and 1 if G is hamiltonian. We study nonhamiltonian graphs with the property for each or for each . These graphs will be called ‐leaf‐critical and l‐leaf‐stable, respectively. It is far from obvious whether such graphs exist; for example, the existence of 3‐leaf‐critical graphs (that turn out to be the so‐called hypotraceable graphs) was an open problem until 1975. We show that l‐leaf‐stable and l‐leaf‐critical graphs exist for every integer , moreover for n sufficiently large, planar l‐leaf‐stable and l‐leaf‐critical graphs exist on n vertices. We also characterize 2‐fragments of leaf‐critical graphs generalizing a lemma of Thomassen. As an application of some of the leaf‐critical graphs constructed, we settle an open problem of Gargano et al. concerning spanning spiders. We also explore connections with a family of graphs introduced by Grünbaum in correspondence with the problem of finding graphs without concurrent longest paths.     

Rough path properties for local time of symmetric α stable process

In this paper, we first prove that the local time associated with symmetric $\alpha$-stable processes is of bounded $p$-variation for any $p>\frac{2}{\alpha ?1}$ partly based on Barlow’s estimation of the modulus of the local time of such processes.  The fact that the local time is of bounded $p$-variation for any $p>\frac{2}{\alpha ?1}$ enables us to define the integral of the local time ${\int }_{?\infty }^{\infty }{?}_{?}^{\alpha ?1}f\left(x\right)d_x{L}_t^x$ as a Young integral for less smooth functions being of bounded $q$-variation with $1\le q<\frac{2}{3?\alpha }$. When $q\ge \frac{2}{3?\alpha }$, Young’s integration theory is no longer applicable. However, rough path theory is useful in this case. The main purpose of this paper is to establish a rough path theory for the integration with respect to the local times of symmetric $\alpha$-stable processes for $\frac{2}{3?\alpha }\le q<4$.     

On the Auslander–Reiten conjecture for Cohen–Macaulay rings and path algebras

In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and 𝒬 a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra R𝒬, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented.     

The Gauss-Newton Methods via Conjugate Gradient Path without Line Search Technique for Solving Nonlinear Systems

In this article, we propose the Gauss-Newton methods via conjugate gradient path for solving nonlinear systems. By constructing and solving a linearized model of the nonlinear systems, we obtain the iterative direction by employing the conjugate gradient path. In successive iterations, the approximate Jacobian of the nonlinear systems is updated by a Broyden formula to construct the conjugate path. The global convergence and local superlinear convergence rate of the proposed algorithms are established under some reasonable conditions. Finally, the numerical results are reported to show the effectiveness of the proposed algorithms.     

Koszulity of some path categories

We prove Koszulity of certain linear path categories obtained from connected graphs with some infinite directed walk. These categories can be viewed as locally quadratic dual to preprojective algebras.     

A Complete Solution to the Existence of ‐Cycle Frames of Type

The existence problem of a ‐cycle frame of type is now solved for any quadruple .     

Geometric analysis of anharmonic downward distortion following paths

A mathematical aspect of the anharmonic downward distortion following (ADDF) path is discussed. The ADDF method is utilized as an automated reaction path search method, which can explore transition state geometries on a potential energy surface from a potential minimum. We show that the maximum number of the ADD stationary paths intersecting the potential minimum is 2^{f + 1} ? 2 , where f denotes the degree of freedom of the system. We also show that the bifurcation of the ADD stationary path is essential to detect all the transition states connected from a given minimum. The ADDF computation is demonstrated for a H₂O molecule in which pitchfork bifurcation is observed.     

Construction of minimal mass blow-up solutions to rough nonlinear Schrödinger equations

We study the focusing mass-critical rough nonlinear Schrödinger equations, where the stochastic integration is taken in the sense of controlled rough path. In both dimensions one and two, the minimal mass blow-up solutions are constructed, which behave asymptotically like the pseudo-conformal blow-up solutions near the blow-up time. Furthermore, the global well-posedness is obtained if the mass of initial data is below that of the ground state. These results yield that the mass of ground state is exactly the threshold of global well-posedness and blow-up in the stochastic focusing mass-critical case. Similar results are also obtained for a class of nonlinear Schrödinger equations with lower order perturbations, particularly in the absence of the standard pseudo-conformal symmetry and the conservation law of energy.     

长光程共振式二氧化碳气体光声传感器研究

二氧化碳(CO₂)是植物光合作用的原材料,也是一种温室气体,其过量地排放会影响动植物的生态环境。在碳达峰、碳中和的背景下,研制高灵敏度的CO₂检测装置具有重要意义。为了监测大气环境中CO₂含量的变化,设计了一种长光程共振式CO₂气体光声传感器,并以此搭建了光声检测装置。以中心波长为2 004 nm的分布式反馈激光器(DFB)作为激发光源,激光射入由漫反射材料制成的球型吸收腔,在腔内多次反射以增加气体的吸收路径。吸收腔外部被两个高热传导率的铝制半球包裹,降低由池体吸收光能后产生的热噪声。吸收腔上耦合一根声学管,当其工作在一阶纵向共振模态时,光声信号被放大,在管子末端达到极大值。为了进一步增大光声信号,通过饱和加湿样品的方式来加快CO₂气体的弛豫速率,加湿后的样品产生的光声信号是干燥样品的2.1倍左右。使用一系列浓度的湿润CO₂样品标定光声检测装置,结果表明,光声信号与浓度之间呈现良好的线性关系。在此基础上,通过对标准气体的检测实验,验证了装置的准确性与稳定性...