We consider linear elliptic equations $-\Delta u+q\left(x\right)u=\lambda u+f$ in bounded Lipschitz domains $D\subset {ℝ}^N$ with mixed boundary conditions $\partial u/\partial n=\sigma \left(x\right)\lambda u+g$ on $\partial D$. The main feature of this boundary value problem is the appearance of $\lambda$ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient $\sigma \left(x\right)$. We study positivity principles and anti-maximum principles. One of our main results states that if $\sigma$ is somewhere negative, $q\ge 0$ and ${\int }_{D}q\left(x\right)dx>0$ then there exist two eigenvalues ${\lambda }_{-1}$, ${\lambda }_1$ such the positivity...

Let $a=(a_1,a_2,...,a_n)$ be a nonincreasing sequence of positive real numbers. Denote by $S=\{1,2,...,n\}$ the index set and by $J_k=\{I=\{r_1,r_2,...,r_k\}$, $1\le r_1<r_2<\cdots <r_k\le n\}$ the set of all subsets of $S$ of cardinality $k$, $1\le k\le n-1$. In addition, denote by $a_I=a_{r_1}+a_{r_2}+\cdots +a_{r_k}$, $1\le k\le n-1$, $1\le r_1<r_2<\cdots <r_k\le n$, the sum of $k$ arbitrary elements of sequence $a$, where $a_{I_1}=a_1+a_2+\cdots +a_k$ and $a_{I_n}=a_{n-k+1}+a_{n-k+2}+\cdots +a_n$. We consider bounds of the quantities $R{S}_k\left(a\right)=a_{I_1}/a_{I_n}$, $L{S}_k\left(a\right)=a_{I_1}-a_{I_n}$ and $S_{k,\alpha }\left(a\right)={\sum }_{I\in J_k}{a}_I^{\alpha }$ in terms of $A={\sum }_{i=1}^n{a}_i$ and $B={\sum }_{i=1}^n{a}_i^2$. Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.

Let $A$ be a bounded linear operator in a complex separable Hilbert space $\mathscr{H}$, and $S$ be a selfadjoint operator in $\mathscr{H}$. Assuming that $A-S$ belongs to the Schatten-von Neumann ideal ${𝒮}_p$ $(p>1),$ we derive a bound for ${\sum }_k{|\mathrm{R}{\lambda }_k\left(A\right)-{\lambda }_k\left(S\right)|}^p$, where ${\lambda }_k\left(A\right)$ $(k=1,2,\cdots )$ are the eigenvalues of $A$. Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. In addition, in the case $p=2$ we refine the Weyl inequality between the real parts of the eigenvalues of $A$ and the eigenvalues...

Let $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

A vector $x$ is said to be an eigenvector of a square max-min matrix $A$ if $A\otimes x=x$. An eigenvector $x$ of $A$ is called the greatest $𝐗$-eigenvector of $A$ if $x\in 𝐗=\{x;\underline{x}\le x\le \overline{x}\}$ and $y\le x$ for each eigenvector $y\in 𝐗$. A max-min matrix $A$ is called strongly $𝐗$-robust if the orbit $x,A\otimes x,A^2\otimes x,\cdots$ reaches the greatest $𝐗$-eigenvector with any starting vector of $𝐗$. We suggest an $O\left(n^3\right)$ algorithm for computing the greatest $𝐗$-eigenvector of $A$ and study the strong $𝐗$-robustness. The necessary and sufficient conditions for strong $𝐗$-robustness are introduced...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...