In this article, we establish Minkowski, Brunn-Minkowski and Aleksandrov-Fenchel type inequalities for differences of quermass- and dual quermassintegrals of mixed Blaschke-Minkowski and mixed radial Blaschke-Minkowski homomorphisms.

In this manuscript, we obtain some common fixed point results of two pairs having the common limit range property in the setting of integral type contraction in the framework of symmetric (semi-metric) spaces. Moreover, we extend our results from two pairs of self-mappings to four finite families of self mappings to get common fixed points. Our results improve and extend a host of previously known results. Further, we establish some illustrative examples to show the validity of the main results.

In this paper, we study the existence of nontrivial periodic solutions for the second order Hamiltonian systems $ \ddot u(t)+\nabla F(t,u(t))=0$, where $F(t,x)$ is either nonquadratic or superquadratic as $|u|\mathbb{R}ightarrow \infty$. Furthermore, if $F(t,x)$ is even in $x$, we prove the existence of infinitely many periodic solutions for the general Hamiltonian systems $ \ddot u(t)+A(t)u(t)+\nabla F(t,u(t))=0$, where $A(\cdot)$ is a continuous $T$-periodic symmetric matrix. Our theorems mainly improve the recent result of Tang and Jiang [X.H. Tang, J. Jiang, Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems, Comput. Math. Appl. 59 (2010) 3646-3655].

We prove a spinorial characterization of surfaces isometrically immersed into the $4$-dimensional product spaces $\mathbb{M}^3(c)\times{\mathbb R}$ and $\mathbb{M}^2(c)\times{\mathbb R}^2$, where $\mathbb{M}^n(c)$ is the $n$-dimensional real space form of curvature $c$.

In this paper we study piecewise linear (PWL) vector fields $$ F(x,y)=\left\{\begin{array}{l} F^+(x,y)\quad $if$ \quad x\geq0, \\ F^-(x,y)\quad $if$ \quad x\leq0,\end{array}\right. $$ where $\mathrm{x}=(x,y)\in \mathbb{R}^2$, $F^{+}(\mathrm{x})=A^{+}\mathrm{x}+b^{+}$ and $F^{-}(\mathrm{x})=A^{-}\mathrm{x}+b^{-}$, $A^{+}=(a_{ij}^{+})$ and $A^{-}=(a_{ij}^{-})$ are $(2\times2)$ constant matrices, $b^{+}=(b_{1}^{+},b_{2}^{+})\in\mathbb{R}^2$ and $b^{-}=(b_{1}^{-},b_{2}^{-})\in\mathbb{R}^2$ are constant vectors in $\mathbb{R}^2$. We suppose that the equilibrium points are saddle or focus in each half-plane. We establish a correspondence between the PWL vector fields and vectors formed by some of the following parameters: sets on $\Sigma$ (crossing, sliding or escaping), kind of equilibrium (real or virtual), intersection of manifolds with $\Sigma$, stability and orientation of the focus. Such vectors are called configurations. We reduce the number of configurations by an equivalent relation. Besides, we analyze for which configurations the corresponding PWL vector fields can have or not closed sliding poly-trajectories.

For any finite group $G$, we define the notion of a \emph{Bredon homotopy action} of $G$, modelled on the diagram of fixed point sets \ensuremath{(\mathbf{X}\sp{H})\sb{H\leq G}} for a $G$-space $\mathbf{X}$, together with a pointed homotopy action of the group \ensuremath{N\sb{G}H/H} on \ensuremath{\mathbf{X}\sp{H}/(\bigcup\sb{H<K} \mathbf{X}\sp{K}).} We then describe a procedure for constructing a suitable diagram \ensuremath{\underline{\mbox{X}}:{\EuScript O}_{G}\op\to{\EuScript Top}} from this data, by solving a sequence of elementary lifting problems. If successful, we obtain a $G$-space \ensuremath{\mathbf{X}'}\ realizing the given homotopy information, determined up to Bredon $G$-homotopy type. Such lifting methods may also be used to understand other homotopy questions about group actions, such as transferring a $G$-action along a map \ensuremath{f:\mathbf{X}\to \mathbf{Y}. }

The objective of this work is to give some relationship between the Favard spaces and the $p$-admissibility (resp. $(p,q)$-admissibility) of unbounded control operators for linear (resp; bilinear) systems in Banach spaces. For linear case, this enables to give a simple identification of the space of the $1-$admissible control operators in Banach space and it enables us to extend the result of Weiss [29] (for $p=1$) on reflexive Banach spaces to a general situation. This result is applied to boundary control systems. The results obtained for bilinear systems generalize those given in Idrissi [16] and Berrahmoune [2] and are applied to diffusion equations of fractional order time distributed order.

We apply Hermitean circulant $(2 \times 2)$ matrix function theoretic results for direct account of the jump and Dirichlet type boundary value problems for solutions of perturbed Hermitean matrix Dirac equations in the case where the boundary of the considered domains is fractal.

A Hamiltonian cycle system of the complete graph minus a $1$--factor $K_{2v}-I$ (briefly, an HCS$(2v)$) is {\it $2$-pyramidal} if it admits an automorphism group of order $2v-2$ fixing two vertices. In spite of the fact that the very first example of an HCS$(2v)$ is very old and 2-pyramidal, a thorough investigation of this class of HCSs is lacking. We give first evidence that there is a strong relationship between 2-pyramidal HCS$(2v)$ and {\it$1$-rotational} Hamiltonian cycle systems of the complete graph $K_{2v-1}$. Then, as main result, we determine the full automorphism group of every 2-pyramidal HCS$(2v)$. This allows us to obtain an exponential lower bound on the number of non-isomorphic $2$-pyramidal HCS$(2v)$.

The Hilbert-Schmidtness of composition operators acting between the classical Hilbert Hardy space and the Dirichlet space is known. We here consider boundedness and compactness of composition operators acting between their spaces.