Based on a series of recent papers, a powerful algorithm is reformulated for computing the maximal eigenpair of self-adjoint complex tridiagonal matrices. In parallel, the same problem in a particular case for computing the sub-maximal eigenpair is also introduced. The key ideas for each critical improvement are explained. To illustrate the present algorithm and compare it with the related algorithms, more than 10 examples are included.

We consider the polytope arising from a marked surface by flips of triangulations. D. D. Sleator, R. E. Tarjan, and W. P. Thurston [J. Amer. Math. Soc., 1988, 1(3): 647{681] studied the diameter of the associahedron, which is the polytope arising from a marked disc by flips of triangulations. They showed that every shortest path between two vertices in a face does not leave that face. We give a new method, which is different from the one used by V. Disarlo and H. Parlier [arXiv: 1411.4285] to establish the same non-leaving-face property for all unpunctured marked surfaces.

We establish the limiting weak type behaviors of Riesz transforms associated to the Bessel operators on ℝ^{+}; which are closely related to the best constants of the weak type (1; 1) estimates for such operators. Meanwhile, the corresponding results for Hardy-Littlewood maximal operator and fractional maximal operator in Bessel setting are also obtained.

Let $({{\displaystyle A}}_{i},|{{\displaystyle \varphi}}_{i,i+\mathrm{1}})$ be a generalized inductive system of a sequence (A_{i}) of unital separable C*-algebras, with $A={{\displaystyle \mathrm{lim}}}_{i\to \infty}({{\displaystyle A}}_{i},{{\displaystyle \varphi}}_{i,i+\mathrm{1}})$. Set ${{\displaystyle \varphi}}_{j,i}={{\displaystyle \varphi}}_{i-\mathrm{1},i}\mathrm{o}\cdots \mathrm{o}{{\displaystyle \varphi}}_{j+\mathrm{1},j+\mathrm{2}}\mathrm{o}{{\displaystyle \varphi}}_{j,j+\mathrm{1}}$ for all i>j: We prove that if ${{\displaystyle \varphi}}_{j,i}$ are order zero completely positive contractions for all j and i>j; and $L:=\mathrm{inf}\{\lambda |\lambda \in \sigma ({{\displaystyle \varphi}}_{j,i}({{\displaystyle \mathrm{1}}}_{{{\displaystyle A}}_{j}}))\}$ for all j and i>j}>0; where $\sigma ({{\displaystyle \varphi}}_{j,i}({{\displaystyle \mathrm{1}}}_{{{\displaystyle A}}_{j}}))$ is the spectrum of ${{\displaystyle \varphi}}_{j,i}({{\displaystyle \mathrm{1}}}_{{{\displaystyle A}}_{j}})$ ; then ${{\displaystyle \mathrm{lim}}}_{i\to \infty}(\mathrm{C}\mathrm{u}({{\displaystyle a}}_{i}),\mathrm{C}\mathrm{u}({{\displaystyle \varphi}}_{i,i+\mathrm{1}}))=\mathrm{C}\mathrm{u}(A)$ ; where Cu(A) is a stable version of the Cuntz semigroup of C*-algebra A: Let $({{\displaystyle A}}_{n},{{\displaystyle \varphi}}_{m,n})$ be a generalized inductive system of C*-algebras, with the ${{\displaystyle \varphi}}_{m,n}$ order zero completely positive contractions. We also prove that if the decomposition rank (nuclear dimension) of A_{n} is no more than some integer k for each n; then the decomposition rank (nuclear dimension) of A is also no more than k:

For a class of nonlinear elliptic boundary value problems including the von Kármán equations considered by D. M. Duc, N. L. Luc, L. Q. Nam, and T. T. Tuyen [Nonlinear Anal., 2003, 55: 951{968], we give a new proof of a corresponding theorem of three solutions via Morse theory instead of topological degree theory. Several bifurcation results for this class of boundary value problems are also obtained with Morse theory methods. In addition, for the von Kármán equations studied by A. Borisovich and J. Janczewska [Abstr. Appl. Anal., 2005, 8: 889{899], we prove a few of bifurcation results under Dirichlet boundary conditions based on the second named author's recent work about parameterized splitting theorems and bifurcations for potential operators.

Let be a function of homogeneous of degree zero and satisfy the cancellation condition on the unit sphere. Suppose that h is a radial function. Let ${{\displaystyle T}}_{h,\mathrm{\Omega},p}$ be the classical singular Radon transform, and let ${{\displaystyle T}}_{h,\mathrm{\Omega},{p}}^{\epsilon}$ be its truncated operator with rough kernels associated to polynomial mapping $p$ which is defined by ${{\displaystyle T}}_{h,\mathrm{\Omega},p}^{\epsilon}f(x)=\left|{{\displaystyle \int}}_{\left|y\right|\rangle \epsilon}f(x-p(y))h(\left|y\right|)\mathrm{\Omega}(y){\left|y\right|}^{-n}\mathrm{d}y\right|$. In this paper, we show that for any $\alpha \in (-\infty ,\infty )$ and $(p,q)$ satisfying certain index condition, the operator ${{\displaystyle T}}_{h,\mathrm{\Omega},p}^{\epsilon}$ enjoys the following convergence properties ${{\displaystyle \mathrm{lim}}}_{\epsilon \to \mathrm{0}}{\Vert {{\displaystyle T}}_{h,\mathrm{\Omega},p}^{\epsilon}f-{{\displaystyle T}}_{h,\mathrm{\Omega},p}f\Vert}_{{{\displaystyle \dot{F}}}_{\alpha}^{p,q}({\mathbb{R}}^{d})}=\mathrm{0}$ and ${{\displaystyle \mathrm{lim}}}_{\epsilon \to \mathrm{0}}{\Vert {{\displaystyle T}}_{h,\mathrm{\Omega},p}^{\epsilon}f-{{\displaystyle T}}_{h,\mathrm{\Omega},p}f\Vert}_{{{\displaystyle \dot{B}}}_{\alpha}^{p,q}({\mathbb{R}}^{d})}=\mathrm{0}$ provided that $\mathrm{\Omega}\in {L({\mathrm{log}}^{+}L)}^{\beta}({S}^{n-\mathrm{1}})$ for some $\beta \in (\mathrm{0},\mathrm{1}]$ or $\mathrm{\Omega}\in {H}^{\mathrm{1}}({S}^{n-\mathrm{1}})$, or $\mathrm{\Omega}\in ({{\displaystyle \cup}}_{\mathrm{1}\langle q\langle \infty}{{\displaystyle B}}_{q}^{(\mathrm{0},\mathrm{0})}({S}^{n-\mathrm{1}}))$.

The concept of a perfect coloring, introduced by P. Delsarte, generalizes the concept of completely regular code. We study the perfect 3-colorings (also known as the equitable partitions into three parts) on 6-regular graphs of order 9. A perfect n-colorings of a graph is a partition of its vertex set. It splits vertices into n parts ${{\displaystyle A}}_{\mathrm{1}},{{\displaystyle A}}_{\mathrm{2}},\mathrm{...},{{\displaystyle A}}_{n}$ such that for all $i,j\in \{\mathrm{1},\mathrm{2},\mathrm{...},n\}$, each vertex of A_{i} is adjacent to a_{ij} vertices of A_{j}. The matrix $A={{\displaystyle ({{\displaystyle a}}_{ij})}}_{n\times n}$ is called quotient matrix or parameter matrix. In this article, we start by giving an algorithm to find all different types of 6-regular graphs of order 9. Then, we classify all the realizable parameter matrices of perfect 3-colorings on 6-regular graphs of order 9.

Abundant exact interaction solutions, including lump-soliton, lumpkink, and lump-periodic solutions, are computed for the Hirota-Satsuma-Ito equation in (2+1)-dimensions, through conducting symbolic computations with Maple. The basic starting point is a Hirota bilinear form of the Hirota-Satsuma-Ito equation. A few three-dimensional plots and contour plots of three special presented solutions are made to shed light on the characteristic of interaction solutions.

We investigate a generalized (3+ 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its bilinear formalism and soliton solutions succinctly. Meanwhile, the first-order lump wave solution and second-order lump wave solution are well presented based on the corresponding two-soliton solution and four-soliton solution. Furthermore, two types of hybrid solutions are systematically established by using the long wave limit method. Finally, the graphical analyses of the obtained solutions are represented in order to better understand their dynamical behaviors.

We obtain the matrix-valued Schrödinger-type operators [H_{α,θ}] with Lipschitz potentials having no eigenvalues on the set {E: L(E)<δ_{C}_{,}_{d}(α,θ)}, where δ is an explicit function depending on the sampling function C(θ), dimension d, phase θ, and frequency α, and L(E) is the Lyapunov exponent.

We introduce a class of structured tensors, called generalized row strictly diagonally dominant tensors, and discuss some relationships between it and several classes of structured tensors, including nonnegative tensors, B-tensors, and strictly copositive tensors. In particular, we give estimations on upper and lower bounds of solutions to the tensor complementarity problem (TCP) when the involved tensor is a generalized row strictly diagonally dominant tensor with all positive diagonal entries. The main advantage of the results obtained in this paper is that both bounds we obtained depend only on the tensor and constant vector involved in the TCP; and hence, they are very easy to calculate.