arXiv subjects

Zongsheng Gao

Publications and source records attributed to Zongsheng Gao.

6 recordsLinked to original sources

arXiv2013

A novel weighting scheme for random $k$-SAT

Consider a random $k$-CNF formula $F_{k}(n, rn)$ with $n$ variables and $rn$ clauses. For every truth assignment $σ\in \{0, 1\}^{n}$ and every clause $c=\ell_{1}\vee\cdots\vee\ell_{k}$, let $d=d(σ, c)$ be the number of satisfied literal occurrences in $c$ under $σ$. For fixed $β>-1$ and $λ>0$, we take $ω(σ, c)=0$, if $d=0$; $ω(σ, c)=λ(1+β)$, if $d=1$ and $ω(σ, c)=λ^{d}$, if $d>1$. Applying the above weighting scheme, we get that if $F_{k}(n, rn)$ is unsatisfiable with probability tending to one as $n\rightarrow\infty$, then $r\geq2.83, 8.09, 18.91, 40.81, 84.87$ for $k=3, 4, 5, 6$ and $7,$ respectively.

cs.DM↗

arXiv2013

A sufficient condition on operator order for strictly positive operators

Let $A_{1}$, $A_{2}$, $...$, $A_{k}$ be strictly positive operators on a Hilbert space. This note is to show a sufficient condition of $A_{k}\geq A_{k-1}\geq\geq A_{3}\geq A_{2}\geq A_{1}$, which extends the related result before.

math.FA↗

arXiv2011

Characterizations of operator order for k strictly positive operators

Let $A_{i}\ (i=1, 2, ..., k)$ be bounded linear operators on a Hilbert space. This paper aims to show characterizations of operator order $A_{k}\geq A_{k-1}\geq...\geq A_{2}\geq A_{1}>0$ in terms of operator inequalities. Afterwards, an application of the characterizations is given to operator equalities due to Douglas's majorization and factorization theorem.

math.FA↗

arXiv2011

Further development of positive semidefinite solutions of the operator equation $\sum_{j=1}^{n}A^{n-j}XA^{j-1}=B$

In \cite{Positive semidefinite solutions}, T. Furuta discusses the existence of positive semidefinite solutions of the operator equation $\sum_{j=1}^{n}A^{n-j}XA^{j-1}=B$. In this paper, we shall apply Grand Furuta inequality to study the operator equation. A generalized special type of $B$ is obtained due to \cite{Positive semidefinite solutions}.

math.FA↗

arXiv2011

Operator inequalities dealing with operator equations

In this paper, we study the existence of solutions of some kinds of operator equations via operator inequalities. First, we investigate characterizations of operator order $A\geqslant B >0$ and chaotic operator order log $A \geqslant$ log $B$ for positive definite operators $A$, $B$ in terms of operator equations, and generalize the results in \cite{CSLin}. Then, we introduce applications of complete form of Furuta inequality in operator equations. Some kinds of operator equations are researched and related characterizations of solutions are proved.

math.FA↗

arXiv2009

The Second Main Theorem Concerning Small Algebroid Functions

In this paper, we firstly give the definition of meromorphic function element and algebroid mapping. We also construct the algebroid function family in which the arithmetic, differential operations is closed. On basis of these works, we firstly proved the Second Main Theorem concerning small algebroid functions for v-valued algebroid functions.

math.CV↗