In this survey article, we give an expository account of the recent developments on the Poncelet property for numerical ranges of the compressions of the shift $S(\phi)$. It can be considered as an updated and more advanced edition of the recent expository article published in the American Mathematical Monthly by the second author on this topic. The new information includes: (1) a simplified approach to the main results (generalizations of Poncelet, Brianchon--Ceva and Lucas--Siebeck theorems) in this area, (2) the recent discovery of Mirman refuting a previous conjecture on the coincidence of Poncelet curves and boundaries of the numerical ranges of finite-dimensional $S(\phi)$, and (3) some partial generalizations by the present authors of the above-mentioned results from the unitary-dilation context to the normal-dilation one and also from the finite-dimensional $S(\phi)$ to the infinite-dimensional.

In this paper, we consider the convergence rate of the deterministic Glimm scheme for general systems of hyperbolic conservation laws without assuming either genuine nonlinearity or linear degeneracy on the characteristic fields. It is shown that the convergence rate is $o(1)s^{\frac{1}{4}}|\ln s|$ compared to $o(1)s^{\frac{1}{2}}|\ln s|$ obtained in [3] for the case when the charateristic field is either genuinely nonlinear or linear degenerate. Here $s$ is the mesh size in the time direction.

In this paper, we estimate eigenvalues of the Laplace operator on surfaces of revolution. We first reduce our Laplace eigenvalue problems to the corresponding Sturm-Liouville eigenvalue problems. Two variational inequalities are then used to obtain lower-bound estimates for eigenvalues of the corresponding Sturm-Liouville problems. Based on the relationship between eigenvalues of the Laplace problems and the Sturm-Liouville problems, we obtain lower-bound estimates for eigenvalues of the mixed and Neumann problems of the Laplace operator (Theorem 1 and Theorem 2). Indeed, our estimate in the first case is optimal.

In this paper we study additive perturbations of a linear Volterra integral equation defined in a Banach space $X$ by means of $k$-regularized resolvent families. We give also a representation formula for the generator of such family, under certain conditions on the scalar kernel $k(t)$.

In this article, we prove an inequality between the ratio of the extended logarithmic means and the ratio of the exponential means. The proof is based on an inequality between logarithmic mean and one-parameter mean, which can be deduced from monotonicity of the extended mean values.

In this article, using Stirling’s formula, the series-expansion of digamma functions and other techniques, some inequalities and monotonicity concerning the ratio of gamma functions are obtained, several inequalities involving the geometric mean of natural numbers are deduced.

In [6] the authors proposed two open problems concerning the boundedness and the periodic nature of positive solutions of the nonlinear difference equation in the title. In this paper we prove a global covergence result and solve the open problems in the case $A \gt 1$.

In this paper, we propose the likelihood inference under the general response transformation model with heteroscedastic errors when the range of the response transformation is possibly different from the whole real line. Three commonly used families of response transformations are reviewed to illustrate the importance and applicability of the proposed model.

We show that the existence of a cubic polynomial systems having three real invariant straight lines forming a triangle with at least one limit cycle in the interior.

Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be locally convex spaces, $c(X)$ and $c(Y)$ the $X$-valued and $Y$-valued convergent sequence spaces, respectively, $A_{ij} \in L(X,Y)$ and $A=(A_{ij})$ an operator-valued infinite matrix. In this paper, we characterize the matrix $A=(A_{ij})$ which transforms $c(X)$ into $c(Y)$. As its applications, we introduce the chi function $\chi$ on locally convex spaces, and show that a conservative matrix is conull if and only if $\chi$ (A)=0$.

Let $D_n=\{(z_1,\dots, z_n)\in {\bf C}^n:\, |z_j|\lt 1,\,j=1,\dots,n\}$ be the unit polydisk in ${\bf C}^n$. The aim of this paper is to prove the boundedness of the generalized Ces\` aro operators ${\cal C}^{\vec\gamma}$ on $H^p(D_n)$ (Hardy) and ${\cal A}^{p,q}_{\vec \mu}(D_n)$ (the generalized Bergman) spaces, for $01$, $j=1,\dots,n$. Here $\vec \mu=(\mu_1,\dots,\mu_n)$ and each $\mu_j$ is a positive Borel measure on the interval $[0,1)$. Also we present a class of invariant spaces under the action of this operator.

Applying orthonormal wavelets, Meyer proved that all Calder\'on-\break Zygmund operators satisfying $T(1)=T^*(1)=0$ form an algebra. In this article the same result is proved on spaces of homogeneous type introduced by Coifman and Weiss [5]. Since there is no such an orthonormal wavelet on the general setting, we apply the discrete Calder\'on reproducing formula developed in [13] to approach.

Let $R$ be a prime ring with extended centroid $C$, $\rho$ a nonzero right ideal of $R$, $d$ a nonzero derivation of $R$, $f(X_1, \ldots, X_t)$ a multilinear polynomial over $C$, $a\in R$ and $n$ a fixed positive integer.

[(I)] If $ad(f(x_1, \ldots, x_t))^n=0$ ($d(f(x_1, \ldots, x_t))^na=0$) for all $x_1, \ldots, x_t$ $\in\rho$, then either $a\rho=0$ ($a=0$ resp.), $d(\rho)\rho=0$ or $\rho C = eRC$ for some idempotent $e$ in the socle of $RC$ such that $f(X_1, \ldots, X_t)$ is central-valued on $eRCe$.

[(II)] If $ad(f(x_1, \ldots, x_t))^n\in C$ ($d(f(x_1, \ldots, x_t))^na\in C$) for all $x_1, \ldots, x_t$ $\in\rho$ and $ad(f(y_1,\ldots,y_t))^n\ne 0$ $(d(f(y_1,\ldots,y_t))^na\ne 0)$ for some $y_1,\ldots,y_t\in\rho$, then either $f(\rho)\rho=0$ or $f(X_1, \ldots, X_t)$ is central-valued on $RC$ unless dim$_CRC=4$.

The wide diameter of a graph is a natural generalization of diameter in a graph when we take account of the connectivity of the graph. In this paper, we define the generalized wide diameter of a graph and show that every $k$-regular $k$-connected graph on $n$ vertices has generalized $k$-diameter at most $n/2$ and this upper bound cannot be improved when $n=4k-6+i(2k-4)$.