Let$X$⊂$V$be a closed embedding, with$V$∖$X$nonsingular. We define a constructible function$ψ$_{$X$,$V$}on$X$, agreeing with Verdier’s specialization of the constant function$1$_{$V$}when$X$is the zero-locus of a function on$V$. Our definition is given in terms of an embedded resolution of$X$; the independence of the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich–Karu–Matsuki–Włodarczyk. The main property of$ψ$_{$X$,$V$}is a compatibility with the specialization of the Chern class of the complement$V$∖$X$. With the definition adopted here, this is an easy consequence of standard intersection theory. It recovers Verdier’s result when$X$is the zero-locus of a function on$V$.

We reduced the unsolved problem of Evans triangles to the problem of nding rational points on a class of elliptic curves, the twisted Edwards curves En. We proved that the set of all Evans triangles with a given ratio is isomorphic to a quotient group of En(Q). After this, using Sage, we can solve the Evans problem for an amount of cases within the capacity of computer. Even without the help of a computer, we can still obtain a large class of new Evans triangles, and nd an eective way to test if the rank of an Edwards curve is positive or not.

We analyze spectral properties of the operator <i>H</i> = <i></i>²/<i>x</i>² <i></i>²/<i>y</i>² + <i></i>²<i>y</i>² <i>y</i>²<i>V</i>(<i>xy</i>) in <i>L</i>²(²), where <i></i> 0 and <i>V </i> 0 is a compactly supported and sufficiently regular potential. It is known that the spectrum of <i>H</i> depends on the one-dimensional Schrdinger operator <i>L</i> = <i>d</i>²/<i>dx</i>² + <i></i>² <i>V</i>(<i>x</i>) and it changes substantially as inf(<i>L</i>) switches sign. We prove that in the critical case, inf(<i>L</i>) = 0, the spectrum of <i>H</i> is purely essential and covers the interval [0, ). In the subcritical case, inf <i></i>(<i>L</i>) &gt; 0, the essential spectrum starts from <i></i> and there is a nonvoid discrete spectrum in the interval [0, <i></i>). We also derive a bound on the corresponding eigenvalue moments.

We propose a generalized multiscale finite element method (GMsFEM) based on clustering algorithm to study the elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage, we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In addition, we coarsen the corresponding random space through a clustering algorithm. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis. The new GMsFEM can be applied to multiscale SPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallest-scale of the solution. The new method offers considerable savings in solving multiscale SPDEs. Numerical results are

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