Completely bounded norms of right module maps

It is well-known that if T is a $D_m-D_n$ bimodule map on the m × n complex matrices, then T is a Schur multiplier and $6T{6}_{\mathit{cb}}=6T6$. If n = 2 and T is merely assumed to be a right D₂-module map, then we show that $6T{6}_{\mathit{cb}}=6T6$. However, this property fails if m ? 2 and n ? 3. For m ? 2 and n = 3, 4 or n ? m² we give examples of maps T attaining the supremum$C(m\text{,}n)=\sup T{6}_{\mathit{cb}}:T\text{a right}{D}_n\text{-module map on}{M}_{m\text{,}n}\text{with}6T6\le 1\}\text{,}$we show that $C(m\text{,}{m}^2)=\sqrt{m}$ and succeed in finding sharp results for C(m, n) in certain other cases. As a consequence, if H is an infinite-dimensional Hilbert space and D is a masa in B(H), then there is a bounded right D-module map on K(H) which is not completely bounded.     

The inverse of nonsymmetric two-level Toeplitz operator matrices

The Gohberg–Semencul formula allows one to express the entries of the inverse of a Toeplitz matrix using only a few entries (the first row and the first column) of the inverse matrix, under some nonsingularity condition. In this paper we will provide a two variable generalization of the Gohberg–Semencul formula in the case of a nonsymmetric two-level Toeplitz matrix with a symbol of the form $f(z_1\text{,}{z}_2)=\frac{1}{\overline{P(z_1\text{,}{z}_2)}Q(z_1\text{,}{z}_2)}$ where $P(z_1\text{,}{z}_2)$ and $Q(z_1\text{,}{z}_2)$ are stable polynomials of two variables. We also consider the case of operator valued two-level Toeplitz matrices. In addition, we propose an equation solver involving two-level Toeplitz matrices. Numerical results are included.     

Thirteen limit cycles for a class of Hamiltonian systems under seven-order perturbed terms

In this paper we study the existence, number and distribution of limit cycles of the perturbed Hamiltonian system:$\begin{array}{cc}\hfill & x^{\prime }=4y({\mathit{abx}}^2-{\mathit{by}}^2+1)+\epsilon x\left({\mathit{ux}}^n+{\mathit{vy}}^n-b\frac{\beta +1}{\mu +1}{x}^{\mu }{y}^{\beta }-{\mathit{ux}}^2-\lambda \right)\hfill \\ \hfill & y^{\prime }=4x({\mathit{ax}}^2-{\mathit{aby}}^2-1)+\epsilon y({\mathit{ux}}^n+{\mathit{vy}}^n+{\mathit{bx}}^{\mu }{y}^{\beta }-{\mathit{vy}}^2-\lambda )\hfill \end{array}$where μ + β = n, 0 < a < b < 1, 0 < ε ≪ 1, u, v, λ are the real parameters and n = 2k, k an integer positive.Applying the Abelian integral method [Blows TR, Perko LM. Bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems. SIAM Rev 1994;36:341–76] in the case n = 6 we find that the system can have at least 13 limit cycles.Numerical explorations allow us to draw the distribution of limit cycles.     

On topological and geometric (194) configurations

An $\left(n_k\right)$ configuration is a set of  $n$ points and  $n$ lines such that each point lies on  $k$ lines while each line contains  $k$ points. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. The existence and enumeration of $\left(n_k\right)$ configurations for a given  $k$ has been subject to active research. A current front of research concerns geometric $\left(n_4\right)$ configurations: it is now known that geometric $\left(n_4\right)$ configurations exist for all  $n\ge 18$, apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of $\left(19_4\right)$ configurations: we obtain all topological $\left(19_4\right)$ configurations among which none are geometrically realizable.     

Dimension group of a non-negative integer irreducible matrix

In this paper we are concerned with a non-negative integer and irreducible matrix $A:{\mathbb{Z}}^d\to {\mathbb{Z}}^d$. The main contribution is to prove that if the matrix satisfies certain spectral and algebraic constraints, the cone:$C=\{v\in {\mathbb{Z}}^d/\exists n⩾0\mathrm{and}{A}^{n}v⩾0\}\subset {\mathbb{Z}}^d$is defined by linear maps ${\varphi }_0\text{,}\dots \text{,}{\varphi }_{k-1}:{\mathbb{Z}}^d\to \mathbb{R}$, in the sense that v ∈ C is equivalent to, ϕ_l(v) ⩾ 0 for all l = 0, … , k − 1 (where k is the index of cyclicity of the irreducible matrix). This result allows us to characterize the dimension group generated by the matrix, it is a subgroup of ${\mathbb{R}}^k$ endowed with an order induced by the positive cone of ${\mathbb{R}}^k$.     

ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR GENERALIZED BURGERS EQUATION

This paper is concerned with the stability of the rarefaction wave for the Burgers equationwhere 0 ≤ a < 1/4p (q is determined by (2.2)). Roughly speaking, under the assumption that u_ < u , the authors prove the existence of the global smooth solution to the Cauchy problem (I), also find the solution u(x, t) to the Cauchy problem (I) satisfying sup |u(x, t) -uR(x/t)| → 0 as t → ∞, where uR(x/t) is the rarefaction wave of the non-viscous Burgersequation ut f(u)x = 0 with Riemann initial data u(x, 0) =     

A satellite of the grand Furuta inequality and its application

The grand Furuta inequality has the following satellite (SGF;$t\in [0\text{,}1]$), given as a mean theoretic expression:$A?B>0\text{,}t\in [0\text{,}1]?A^{-r+t}{\#}_{\frac{1-t+r}{(p-t)s+r}}(A^t{?}_s{B}^p)?B\text{for}r?t\text{;}p\text{,}s?1\text{,}$where ${\#}_{\alpha }$ is the $\alpha$-geometric mean and ${?}_s$ ($s?[0\text{,}1]$) is a formal extension of ${\#}_{\alpha }$. It is shown that (SGF; $t\in [0\text{,}1]$) has the Löwner–Heinz property, i.e. (SGF; $t=1$) implies (SGF;t) for every $t\in [0\text{,}1]$. Furthermore, we show that a recent further extension of (GFI) by Furuta himself has also the Löwner–Heinz property.     

Positive solution of fourth order ordinary differential equation with four-point boundary conditions

In this work, the authors consider the fourth order nonlinear ordinary differential equation$u^{\left(4\right)}\left(t\right)=f(t,u\left(t\right))\text{,}0<t<1\text{,}$ with the four-point boundary conditions $u\left(0\right)=u\left(1\right)=0\text{,}$$a{u}^{″}\left({\xi }_1\right)-b{u}^{‴}\left({\xi }_1\right)=0\text{,}c{u}^{″}\left({\xi }_2\right)+d{u}^{‴}\left({\xi }_2\right)=0\text{,}$ where $0\le {\xi }_1<{\xi }_2\le 1$. By means of the upper and lower solution method and fixed point theorems, some results on the existence of positive solutions to the above four-point boundary value problem are obtained.