‎It is well known that different kinds of expressions of modulus of‎ ‎convexity are essentially based on two geometrical propositions‎. ‎For‎ ‎one of the propositions we first present a new proof by the‎ ‎Hahn-Banach theorem and intermediate value theorem‎, ‎then give some‎ ‎corollaries to it which lead to some new expressions of modulus of‎ ‎convexity‎.

‎A singular value inequality for sums and products of Hilbert space operators‎ ‎is given‎. ‎This inequality generalizes several recent singular value‎ ‎inequalities‎, ‎and includes that if $A$‎, ‎$B$‎, ‎and $X$ are positive operators‎ ‎on a complex Hilbert space $H$‎, ‎then ‎\begin{equation*}‎ ‎s_{j}\left( A^{^{1/2}}XB^{^{1/2}}\right) \leq \frac{1}{2}\left\Vert‎ ‎X\right\Vert \text{ }s_{j}\left( A+B\right) \text{, ‎\‎ ‎}j=1,2,\cdots\text{,}‎ ‎\end{equation*} ‎which is equivalent to‎ ‎ \begin{equation*}‎ ‎s_{j}\left( A^{^{1/2}}XA^{^{1/2}}-B^{^{1/2}}XB^{^{1/2}}\right) \leq‎ ‎\left\Vert X\right\Vert s_{j}\left( A\oplus B\right) \text{, ‎\ }j=1,2,\cdots ‎\text{.}‎ ‎\end{equation*}‎ ‎ Other singular value inequalities for sums and products of operators are‎ ‎presented‎. ‎Related arithmetic-geometric mean inequalities are also‎ ‎discussed‎.

‎We define the $p$-power quasiconcave function and show‎ ‎relationships between $p$-power quasiconcave fundamental function and‎ ‎r.i‎. ‎spaces like Lorentz space and Marcinkiewicz space‎.

‎The sequence spaces $\ell_\infty(p)$‎, ‎$c(p)$ and $c_0(p)$ were‎ ‎introduced and studied by Maddox [Proc‎. ‎Cambridge Philos‎. ‎Soc‎. ‎64‎ ‎(1968)‎, ‎335-340]‎. ‎In the present paper‎, ‎we introduce the sequence‎ ‎spaces $\ell_\infty(B,p)$‎, ‎$c(B,p)$ and $c_0(B,p)$ of non-absolute‎ ‎type which are derived by the triple band matrix $B(r,s,t)$ and is‎ ‎proved that the spaces $\ell_\infty(B,p)$‎, ‎$c(B,p)$ and $c_0(B,p)$‎ ‎are paranorm isomorphic to the spaces $\ell_\infty(p)$‎, ‎$c(p)$ and‎ ‎$c_0(p)$; respectively‎. ‎Besides this‎, ‎the $\alpha$-‎, ‎$\beta$‎- ‎and‎ ‎$\gamma$-duals of the spaces $\ell_\infty(B,p)$‎, ‎$c(B,p)$ and‎ ‎$c_0(B,p)$ are computed and the bases of the spaces $c(B,p)$ and‎ ‎$c_0(B,p)$ are constructed‎. ‎Finally‎, ‎the matrix mappings from the‎ ‎sequence spaces $\lambda(B,p)$ to a given sequence space $\mu$ and‎ ‎from the sequence space $\mu$ to the sequence space $\lambda(B,p)$‎ ‎are characterized‎, ‎where $\lambda\in\{\ell_\infty,c,c_0\}$‎.

‎In this paper‎, ‎we first give the concept of properly distributed‎ ‎sequence‎, ‎and prove that it is almost convergent with F-limit‎ ‎expressed as a formal integral‎. ‎Basing on these‎, ‎we review the work‎ ‎of Feng and Li‎, ‎which is shown to be a special case of our‎ ‎generalized theory‎. ‎Then we generalize Banach limit to Banach limit‎ ‎functional‎, ‎which is the minimum requirement to characterize strong‎ ‎almost convergence for bounded sequences in normed vector space‎. ‎With this machinery‎, ‎we show that Hajdukovi\'{c}'s almost‎ ‎convergence and quasi-almost convergence are both equivalent to our‎ ‎strong almost convergence‎.

‎We give an extension of the refined Jensen's operator inequality for‎ ‎$n-$tuples of self-adjoint operators‎, ‎unital $n-$tuples of positive‎ ‎linear mappings and real valued continuous convex functions with‎ ‎conditions on the spectra of the operators‎. ‎We also study the order‎ ‎among quasi-arithmetic means under similar conditions‎.

‎An operator $T\in B(H)$ is called quasi $*$-paranormal if‎ ‎$||T^{*}Tx||^{2}\leq||T^{3}x|||Tx||$ for all $x\in H$‎. ‎If $\mu$ is‎ ‎an isolated point of the spectrum of $T$‎, ‎then the Riesz idempotent‎ ‎$E$ of $T$ with respect to $\mu$ is defined by‎ ‎$$E:= {1\over {2\pi i}}\int_{\partial D}(\mu I‎ - ‎T)^{-1}d\mu,$$‎ ‎where $D$ is a closed disk centered at $\mu$ which contains no other‎ ‎points of the spectrum of $T$‎. ‎Stampfli [Trans‎. ‎Amer‎. ‎Math‎. ‎Soc.‎, ‎117 (1965)‎, ‎469-476]‎, ‎showed that if $T$ satisfies the growth‎ ‎condition $G_{1}$‎, ‎then $E$ is self-adjoint and $E(H)=N(T-\mu)$‎. ‎Recently‎, ‎Uchiyama and Tanahashi [Integral Equations and Operator‎ ‎Theory‎, ‎55 (2006)‎, ‎145-151] obtained Stampfli's result for‎ ‎paranormal operators‎. ‎In general even though $T$ is a paranormal‎ ‎operator‎, ‎the Riesz idempotent $E$ of $T$ with respect to $\mu \in ‎{\rm iso\,}\sigma(T)$ is not necessary self-adjoint‎. ‎In this paper‎ ‎$2\times 2$ matrix representation of a quasi $*$-paranormal operator‎ ‎is given‎. ‎Using this representation we show that if $E$ is the Riesz‎ ‎idempotent for a nonzero isolated point $\lambda_{0}$ of the‎ ‎spectrum of a quasi $*$-paranormal operator $T$‎, ‎then $E$ is‎ ‎self-adjoint if and only if the null space of $T-\lambda_{0}$‎ ‎satisfies $N(T-\lambda_{0})\subseteq‎ ‎N(T^{*}-\overline{\lambda_{0}})$‎. ‎Other related results are also‎ ‎given‎.

‎We survey the recent results concerning the applications of power‎ ‎series method to the study of Hyers-Ulam stability of differential‎ ‎equations‎.

‎The $k$-rank numerical range $\Lambda_{k}(A)$ is expressed via an intersection of any countable family of numerical ranges $\{F(M^{*}_{\nu}AM_{\nu})\}_{\nu\in\mathbb{N}}$ with respect to $n\times (n-k+1)$ isometries $M_{\nu}$‎. ‎This implication for $\Lambda_{k}(A)$ provides further elaboration of the $k$-rank numerical radii of $A$‎.

‎The aim of this paper is to show that any continuous‎ ‎$*$-homomorphism of $L^1(\C^n)$(with twisted convolution as‎ ‎multiplication) into $\CB(L^2(\Rn))$ is essentially a Weyl‎ ‎transform‎. ‎From this we deduce a similar characterisation for the‎ ‎group Fourier transform on the Heisenberg group‎, ‎in terms of‎ ‎convolution‎.

‎We show that‎, ‎in complex interpolation‎, ‎an operator function that is compact‎ ‎on one side of the interpolation scale will be compact for all proper‎ ‎interpolating‎ ‎spaces if the right hand side $(Y^0,Y^1)$ is reduced to a single space‎. ‎A corresponding result‎, ‎in restricted generality‎, ‎is shown if the left hand side‎ ‎$(X^0,X^1)$ is reduced to a single space‎. ‎These results are derived from the fact that a holomorphic operator valued‎ ‎function on an open subset of $\mathbb{C}$ which‎ ‎takes values in the compact operators on part of the boundary is in fact compact‎ ‎operator valued‎.

‎In this paper‎, ‎using the atomic theory of the Herz-type Hardy spaces‎ ‎with variable exponent‎, ‎we give their wavelet characterization by‎ ‎means of some discrete tent spaces with variable exponent at the‎ ‎origin‎.

‎By means of weight functions and Hadamard's inequality‎, ‎a new half-discrete‎ ‎Mulholland-type inequality with a best constant factor is given‎. ‎A best‎ ‎extension with parameters‎, ‎some equivalent forms‎, ‎the operator expressions‎ ‎as well as some particular cases are also considered‎.

‎The fixed point method‎, ‎which is the second most popular technique‎ ‎of proving the Hyers-Ulam stability of functional equations‎, ‎was‎ ‎used for the first time in 1991 by J.A‎. ‎Baker who applied a variant‎ ‎of Banach's fixed point theorem to obtain the stability of a‎ ‎functional equation in a single variable‎. ‎However‎, ‎most authors‎ ‎follow Radu's approach and make use of a theorem of Diaz and‎ ‎Margolis‎. ‎The main aim of this survey is to present applications of‎ ‎different fixed point theorems to the theory of the Hyers-Ulam‎ ‎stability of functional equations‎.