Thetransmissioneigenvalueproblem,besidesitscriticalroleininversescattering problems, deserves special interest of its own due to the fact that the corresponding differen- tial operator is neither elliptic nor self-adjoint. In this paper, we provide a spectral analysis and propose a novel iterative algorithm for the computation of a few positive real eigen- values and the corresponding eigenfunctions of the transmission eigenvalue problem. Based on approximation using continuous finite elements, we first derive an associated symmetric quadratic eigenvalue problem (QEP) for the transmission eigenvalue problem to eliminate the nonphysical zero eigenvalues while preserve all nonzero ones. In addition, the derived QEP enables us to consider more refined discretization to overcome the limitation on the number of degrees of freedom. We then transform the QEP to a parameterized symmet- ric definite generalized eigenvalue problem (GEP) and develop a secant-type iteration for solving the resulting GEPs. Moreover, we carry out spectral analysis for various existence intervals of desired positive real eigenvalues, since a few lowest positive real transmission eigenvalues are of practical interest in the estimation and the reconstruction of the index of refraction. Numerical experiments show that the proposed method can find those desired smallest positive real transmission eigenvalues accurately, efficiently, and robustly.

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX − XD − AX + B = 0, with M ≡ [D,−C;−B,A] ∈ R(n1+n2)×(n1+n2) being a nonsingular M-matrix. In addition, A and D are sparselike, with the products A−1u, A−⊤u, D−1v, and D−⊤v computable in O(n) complexity (with n = max{n1, n2}), for some vectors u and v, and B, C are low ranked. The structure-preserving doubling algorithms (SDA) by Guo, Lin, and Xu [Numer. Math., 103 (2006), pp. 392–412] is adapted, with the appropriate applications of the Sherman–Morrison– Woodbury formula and the sparse-plus-low-rank representations of various iterates. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration and converges essentially quadratically. A detailed error analysis, on the effects of truncation of iterates with an explicit forward error bound for the approximate solution from the SDA, and some numerical results will be presented.

Starting from the roots of the equation 3^x + 4^x = 5^x discussed in the Tenth Grade, we utilized the method, the algebraic extension of the rational number eld, to produce the ways to judge whether the roots of the basic exponential equation with a form as a^x+b^x = c^x are rational or not. For equations with more than two terms on the left side, as is the equation a_1^x+a_2^x+
···

+a_n^x = d^x, the determination of whether the root was irrational was comparatively dicult.Therefore, we provided a prevalent method for the examination of the root of a three-term equation as well as a conclusion that if the equation doesn't have integer roots, the roots won't be a rational number with a denominator of two. Finally, based on the method, the algebraic extension of the rational number eld, we concluded that under special occasions, the root of the equation can't be some rational numbers with certain denominators.

Pell equation is an important research object in elementary number theory of indefinite equation. its form is simple, but it is rich in nature. Many number theory problems can be transformed into the problem of Pell equation’s solvability. However, the previous methods in determining the Pell equation’s solvability are sophisticated for calculation, which leads to the lack of efficiency. This paper gives new and more widely used methods to determine the solvability of Pell equation, including several necessary conditions, sufficient conditions and necessary and sufficient conditions.

I heard the concept of the ”equal‐sum point” in a math summer camps and found it very interesting. After searching on the Internet,I found that the ”equal‐sum point” was associated with Soddy point. Soddy point is a magic point in plane geometry, which was found by British physicist, chemist Frederick.Soddy. Some research about the Soddy point has been done in foreign countries, but the properties are not comprehensive. In China, Teacher HuasongHuang raised the concept of the “equal‐sum point” and the “equal‐difference point” and drew some properties, I found that these two points were very similar to the Soddy point, but they did not link the two points with Soddy point. In this article ,I will connect the Soddy point with the “equal‐sum point” and the “equal‐difference point” and study their properties more deeply and find some new discoveries.