Symplectic subspaces of symplectic Grassmannians

Let $V$ be a non-degenerate symplectic space of dimension $2n$ over the field $\mathbb{F}$ and for a natural number $l<n$ denote by $C_l\left(V\right)$ the incidence geometry whose points are the totally isotropic $l$-dimensional subspaces of $V$. Two points $U,W$ of $C_l\left(V\right)$ will be collinear when $W\subset U^{\perp }$ and $dim(U\cap W)=l-1$ and then the line on $U$ and $W$ will consist of all the $l$-dimensional subspaces of $U+W$ which contain $U\cap W$. The isomorphism type of this geometry is denoted by $C_{n,l}\left(\mathbb{F}\right)$. When $\mathrm{char}\left(\mathbb{F}\right)\ne 2$ we classify subspaces $S$ of $C_l\left(\mathbb{F}\right)$ where $S\cong C_{m,k}\left(\mathbb{F}\right)$.     

On exact controllability and complete stabilizability for linear systems

This work is concerned with the relations between exact controllability and complete stabilizability for linear systems in Hilbert spaces. We give an affirmative answer to the open problem posed by Rabah and Karrakchou [R. Rabah, J. Karrakchou, Exact controllability and complete stabilizability for linear systems in Hilbert spaces, Appl. Math. Lett. 10 (1997) 35–40]. More precisely, if the $C_0$-semigroup $S\left(t\right)$ generated by $A$ is surjective and the pair $(A,B)$ with a bounded operator $B$ is completely stabilizable, then $(A,B)$ is exactly controllable without any additional condition.     

Interpolating sequences in mean

This paper deals with interpolating sequences ${\left(z_n\right)}_n$ for two spaces of holomorphic functions $f$ in the unit disk $\mathbb{D}$ in $\mathbb{C}$: those that are bounded and those that satisfy a Lipschitz condition $|f\left(z\right)?f\left(w\right)|\le c|z?w{|}^{\alpha }$, $0<\alpha \le 1$. Given a sequence of values ${\left(w_n\right)}_n$ in a certain target space, we look for a function $f$ interpolating ‘in mean”, that is, with $(f\left(z_1\right)+?+f\left(z_n\right))∕n=w_n$, $n\ge 1$. We obtain target spaces when we prescribe that the corresponding interpolating sequences be the uniformly separated ones or the union of two uniformly separated ones.     

On pα-dual spaces of generalized difference sequence spaces

In this paper, we compute the $p\alpha$-, $p\beta$- and $p\gamma$-duals of the sequence spaces ${?}_{\infty }\left(k^m\right),c\left(k^m\right),c_0\left(k^m\right)$, the $pN$-dual of the sequence spaces ${\Delta }_{v,r}^m\left({?}_{\infty }\right),{\Delta }_{v,r}^m\left(c\right)$ and the $p\alpha$-dual of the sequence spaces ${\Delta }_{v,r}^m\left({?}_{\infty }\right),{\Delta }_{v,r}^m\left(c\right)$ and ${\Delta }_{v,r}^m\left(c_0\right)$.     

Triangular C1-bialgebra defined as the direct sum of matrix algebras

Let $M_{*}(\mathbf{C})$ denote the ${\mathrm{C}}^1$-algebra defined as the direct sum of all matrix algebras $\{M_n(\mathbf{C}):n?1\}$. It is known that $M_{*}(\mathbf{C})$ has a non-cocommutative comultiplication ${\mathrm{\Delta }}_{\phi }$. From a certain set of transformations of integers, we construct a universal R-matrix R of the ${\mathrm{C}}^1$-bialgebra $(M_{*}(\mathbf{C})\text{,}{\mathrm{\Delta }}_{\phi })$ such that the quasi-cocommutative ${\mathrm{C}}^1$-bialgebra $(M_{*}(\mathbf{C})\text{,}{\mathrm{\Delta }}_{\phi }\text{,}R)$ is triangular. Furthermore, it is shown that certain linear Diophantine equations are corresponded to the Yang–Baxter equations of R.     

Approximation numbers for composition operators on spaces of entire functions

We study the degree of compactness of composition operators $C_{\phi }$ acting on weighted Hilbert spaces of entire functions, which include (i) the space of entire Dirichlet series, (ii) the space of entire power series, and (iii) the Fock space (we must have $\phi \left(z\right)=az+b$, and it is known that $C_{\phi }$ is compact if and only if $\left|a\right|<1$). More precisely, the sequence $\left(a_n\right)$ of approximation numbers of $C_{\phi }$ is investigated: for (i), we give the exact formula for $\left(a_n\right)$, while for (ii) and (iii) we give upper and lower estimates for $a_n$, showing that $a_n$ behaves like ${\left|a\right|}^n$ up to a subexponential factor. In particular, $C_{\phi }$ belongs to all Schatten classes $S_p,p>0$ as soon as it is compact.     

Degree Ramsey numbers for even cycles

Let $H\stackrel{s}{?}G$ denote that any $s$-coloring of $E\left(H\right)$ contains a monochromatic $G$. The degree Ramsey number of a graph $G$, denoted by $R_{\Delta }(G,s)$, is $min\{\Delta \left(H\right):H\stackrel{s}{?}G\}$. We consider degree Ramsey numbers where $G$ is a fixed even cycle. Kinnersley, Milans, and West showed that $R_{\Delta }(C_{2k},s)\ge 2s$, and Kang and Perarnau showed that $R_{\Delta }(C_4,s)=\Theta \left(s^2\right)$. Our main result is that $R_{\Delta }(C_6,s)=\Theta \left(s^{3∕2}\right)$ and $R_{\Delta }(C_{10},s)=\Theta \left(s^{5∕4}\right)$. Additionally, we substantially improve the lower bound for $R_{\Delta }(C_{2k},s)$ for general $k$.     

Y spaces and global smooth solution of fractional Navier–Stokes equations with initial value in the critical oscillation spaces

For fractional Navier–Stokes equations and critical initial spaces X, one used to establish the well-posedness in the solution space which is contained in $C({\mathbb{R}}_{+},X)$. In this paper, for heat flow, we apply parameter Meyer wavelets to introduce Y spaces $Y^{m,\beta }$ where $Y^{m,\beta }$ is not contained in $C({\mathbb{R}}_{+},{\dot{B}}_{\infty }^{1?2\beta ,\infty })$. Consequently, for $\frac{1}{2}<\beta <1$, we establish the global well-posedness of fractional Navier–Stokes equations with small initial data in all the critical oscillation spaces. The critical oscillation spaces may be any Besov–Morrey spaces ${({\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\mathbb{R}}^n))}^n$ or any Triebel–Lizorkin–Morrey spaces ${({\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\mathbb{R}}^n))}^n$ where $1\le p,q\le \infty ,0\le {\gamma }_2\le \frac{n}{p},{\gamma }_1?{\gamma }_2=1?2\beta$. These critical spaces include many known spaces. For example, Besov spaces, Sobolev spaces, Bloch spaces, Q-spaces, Morrey spaces and Triebel–Lizorkin spaces etc.     

Limit properties of monotone matrix functions

The basic objects in this paper are monotonically nondecreasing $n\times n$ matrix functions $D(·)$ defined on some open interval $?=(a\text{,}b)$ of $\mathbb{R}$ and their limit values $D(a)$ and $D(b)$ at the endpoints a and b which are, in general, selfadjoint relations in ${\mathbb{C}}^n$. Certain space decompositions induced by the matrix function $D(·)$ are made explicit by means of the limit values $D(a)$ and $D(b)$. They are a consequence of operator inequalities involving these limit values and the notion of strictness (or definiteness) of monotonically nondecreasing matrix functions. This treatment provides a geometric approach to the square-integrability of solutions of definite canonical systems of differential equations.