The first terms in the expansion of the Bergman kernel in higher degrees

We establish the cancellation of the first $2j$ terms in the diagonal asymptotic expansion of the restriction to the $(0,2j)$-forms of the Bergman kernel associated to the spin${}^c$ Dirac operator on high tensor powers of a positive line bundle twisted by a (non necessarily holomorphic) complex vector bundle, over a compact K\"{a}hler manifold. Moreover, we give a local formula for the first and the second (non-zero) leading coefficients

Atiyah-Patodi-Singer index and domain-wall eta invariants

In this paper we establish a formula, expressing the generalized Atiyah-Patodi-Singer index in terms of eta invariants of domain-wall massive Dirac operators, without assuming that the Dirac operator on the boundary is invertible. Compared with the original Atiyah-Patodi-Singer index theorem, this formula has the advantage that no global spectral projection boundary conditions appear. Our main tool is an asymptotic gluing formula for eta invariants proved by using a splitting principle developed by Douglas and Wojciechowski in adiabatic limit. The eta invariant splits into a contribution from the interior, one from the boundary, and an error term vanishing in the adiabatic limit process.Comment: 38 pages, 10 figures; minor revision, typos fixe

A Galerkin boundary node method and its convergence analysis

AbstractThe boundary node method (BNM) exploits the dimensionality of the boundary integral equation (BIE) and the meshless attribute of the moving least-square (MLS) approximations. However, since MLS shape functions lack the property of a delta function, it is difficult to exactly satisfy boundary conditions in BNM. Besides, the system matrices of BNM are non-symmetric.A Galerkin boundary node method (GBNM) is proposed in this paper for solving boundary value problems. In this approach, an equivalent variational form of a BIE is used for representing the governing equation, and the trial and test functions of the variational formulation are generated by the MLS approximation. As a result, boundary conditions can be implemented accurately and the system matrices are symmetric. Total details of numerical implementation and error analysis are given for a general BIE. Taking the Dirichlet problem of Laplace equation as an example, we set up a framework for error estimates of GBNM. Some numerical examples are also given to demonstrate the efficacity of the method

A projection iterative algorithm boundary element method for the Signorini problem

International audienceWe propose a projection iterative algorithm based on a fixed point equation for solving a certain class of Signorini problem. The satisfaction of the Signorini boundary conditions is verified in a projection iterative manner, and at each iterative step, an elliptic mixed boundary value problem is solved by a boundary element method which is suitable for any domain. We prove the convergence of the algorithm by the property of projection. The advantage of this algorithm is that it is easy to be implemented and converge quickly. Some numerical results show the accuracy and effectiveness of the algorithm

Adiabatic limit, Witten deformation and analytic torsion forms

We consider a smooth fibration equipped with a flat complex vector bundle and a hypersurface cutting the fibration into two pieces. Our main result is a gluing formula relating the Bismut-Lott analytic torsion form of the whole fibration to that of each piece. This result confirms a conjecture proposed in a conference in Goettingen in 2003. Our approach combines an adiabatic limit along the normal direction of the hypersurface and a Witten type deformation on the flat vector bundle.Comment: 76 page

Analysis and Countermeasures of College Students’ Cheating in Examinations: Taking X as an Example

The phenomenon of College Students’ cheating in examinations is still serious. This paper takes X University as an example to study the phenomenon of cheating in colleges. Based on the statistical analysis of 396 students punished for cheating in exams from 2011 to 2017, it is found that there is a concentrated outbreak of cheating in exams, and that sophomores and juniors are more likely to cheat in exams with paper tapes. It is suggested that we should strengthen honesty and credit education, formulate and revise relevant supporting documents of examination management to strengthen the sense of responsibility of the academy, strengthen the education and guidance of freshmen, reform the way of curriculum assessment as well as build up a qualified invigilator team to prevent cheating in examinations