Minimal kernels of Dirac operators along maps

Let M be a closed spin manifold and let N be a closed manifold. For maps and Riemannian metrics g on M and h on N, we consider the Dirac operator of the twisted Dirac bundle . To this Dirac operator one can associate an index in . If M is 2‐dimensional, one gets a lower bound for the dimension of the kernel of out of this index. We investigate the question whether this lower bound is obtained for generic tupels .     

Continuum versus discrete networks,graph Laplacians,and reproducing kernel Hilbert spaces

Motivated by applications to machine learning, we construct a reversible and irreducible Markov chain whose state space is a certain collection of measurable sets of a chosen l.c.h. space $\mathfrak{X}$. We study the resulting network (connected undirected graph), including transience, Royden and Riesz decompositions, and kernel factorization. We describe a construction for Hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of $L^2$ functions by functions of finite energy. The latter has applications to machine learning (for Markov random fields, for example).     

The Bergman kernel for the Vekua equation

The paper extends some well-known results for analytic functions onto solutions of the Vekua equation $\partial \overline{{}_z}W=aW+b\overline{W}$ regarding the existence and construction of the Bergman kernel and of the corresponding Bergman projection operator.     

The Szegö kernel on a class of non-compact CR manifolds of high codimension

We generalize Nagel’s formula for the Szegö kernel and use it to compute the Szegö kernel on a class of non-compact CR manifolds whose tangent space decomposes into one complex direction and several totally real directions. We also discuss the control metric on these manifolds and relate it to the size of the Szegö kernel.     

高精度半导体激光打标机F-θ镜头设计

为保证半导体激光打标机F-θ镜头的扫描质量，实现系统像高与扫描角的线性变化，需对F-θ镜头给予一定的畸变量，并使其满足等晕条件。分析F-θ镜头工作原理及像差要求，根据1 064 nm半导体激光打标机的光源成像要求选择合适的玻璃材料，合理分配每片透镜的光焦度，以保证等晕成像；根据F-θ镜头线性成像要求，计算系统总畸变量为1.6%，系统总畸变量为系统的实际桶形畸变与相对畸变量之和；在光学系统优化设计时，引入这两项优化参数，优化过程中观察系统成像变化情况。设计结果表明:系统MTF曲线接近衍射极限，F-θ镜头相对畸变小于0.36%，各视场均方根半径均小于艾里斑直径，并且整个系统70%的能量集中在直径为16 μm的圆内，系统总畸变量为1.58%，满足设计指标要求。     

Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.     

Scalable Total Syntheses and Structure–Activity Relationships of Haouamines A,B, and Their Derivatives as Stable Formate Salts

Haouamines A, B, and their derivatives were synthesized via Suzuki–Miyaura coupling and three key cyclization reactions as follows: the newly developed palladium(0)-catalyzed arylative cyclization of phenylalanine-derived alkyne–aldehydes with 2-bromoarylboronic acid (an “anti-Wacker”-type cyclization); BF₃ ⋅ OEt₂-promoted Friedel–Crafts-type cyclization of symmetrical electron-rich aromatic rings adjacent to a tertiary allylic alcohol leading to the indeno-tetrahydropyridine skeleton; and (cyanomethyl)trimethylphosphonium iodide-mediated macrocyclization of amino alcohols to afford aza-paracyclophane precursors. The palladium-catalyzed reduction of mono- and di-triflate intermediates in the later stages enabled the alteration of both the position and number of hydroxyl groups on the C-ring. The instability of haouamine B was dramatically improved by salt formation with formic acid. An unambiguous evaluation of the cytotoxicity of the prepared haouamine derivative formates with and without hydroxyl groups at different positions on the C-ring indicated that the catechol structure in haouamine B produced weak cytotoxicity.     

LITTLEWOOD-PALEY THEOREM FOR SCHR(O)DINGER OPERATORS

Let H be a Schr(o)dinger operator on Rn. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.     

Application of holomorphic invariants in reproducing kernel

We use holomorphic invariants to calculate the Bergman kernel for generalized quasi-homogeneous Reinhardt-Hartogs domains. In addition, we present a complete orthonormal basis for the Bergman space on bounded Reinhardt-Hartogs domains.