Union Closed Set Conjecture and Maximum Dicut in Connected Digraph

In this dissertation, we study the following two topics, i.e., the union closed set conjecture and the maximum edges cut in connected digraphs. The union-closed-set-conjecture-topic goes as follows. A finite family of finite sets is {\it union closed} if it contains the union of any two sets in it. Let $X_{\mathcal{F}}=\cup_{F\in\mathcal{F}}F$. A union closed family of sets is {\it separating} if for any two distinct elements in $\mathcal{F}$, there is a set in $\mathcal{F}$ containing one of them, but not the other and there does not exist an element which is contained in every set of it. Note that any union closed family $\mathcal{F}$ is a poset with set inclusion as the partial order relation. A separating union closed family $\mathcal{F}$ is {\it irreducible} ({\it normalized}) if $|X_{\mathcal{F}}|$ is the minimum (maximum, resp.) with respect to the poset structure of $\mathcal{F}$. In the part of dissertation related to this topic, we develop algorithms to transfer any given separating union closed family to a/an normalized/irreducible family without changing its poset structure. We also study properties of these two extremal union closed families in connection with the {\it Union Closed Sets Conjecture} of Frankl. Our result may lead to potential full proof of the union closed set conjecture and several other conjectures. The part of the dissertation related to the maximum edge cuts in connected digraphs goes as follows. In a given digraph $D$, a set $F$ of edges is defined to be a {\it directed cut} if there is a nontrivial partition $(X,Y)$ of $V(D)$ such that $F$ consists of all the directed edges from $X$ to $Y$. The maximum size of a directed cut in a given digraph $D$ is denoted by $\Lambda (D)$, and we let $\mathcal{D}(1,1)$ be the set of all digraphs $D$ such that $d^{+}(v)=1$ or $d^{-}(v)=1$ for every vertex $v$ in $D$. In this part of dissertation, we prove that $\Lambda (D) \geq \frac{3}{8}(|E(D)|-1)$ for any connected digraph $D\in\mathcal{D}(1,1)$, which provides a positive answer to a problem of Lehel, Maffray, and Preissmann. Additionally, we consider triangle-free digraphs in $\mathcal{D}(1,1)$ and answer their another question

Transformation & Countermeasures on Business Operation Models in Uncertain Environment

In an uncertain environment, esp. facing a market transformed from sellers’ market to buyers’ market, each enterprise focus not just on competitors as traditionally, but on external system sources such as customer and government and inner system sources such as employee and material etc. As the globalization-economy, the market emphases and the influence of a government to an enterprise changed, and the importance of knowledge management in an enterprise’s inner system emerged. Under this situation, the author thinks that we should not view an enterprise as a simple machine or just an open system any more, while a complex and conformity concept should be applied to research its operation model. The traditional business operation model (BOM) is a closed produce - sell model. It focuses planning and decision, which means there should be a plan first then do it. In fact, the traditional BOM is a kind of initiative-passive model, the corresponding main management theory is management process and its function. While the modern BOM is an open perceive - response model. It emphases knowledge and intelligence, which means perceived something first and then feedback. Essentially, it is a passive-initiative model and the main relevant management theory is competition intelligence and knowledge management. Based on these analysis, the author think that relative to traditional businesses try to keep stabilization during a proceeding phase, i.e. quantitative change, the modern businesses should transform the perceived changes to profitable responses, i.e. qualitative change. We can use a loop course of induction - explanation - decision - running - again induction to explain it. In addition, facing the complexity, the management should pay more attention to situation and harmony , but not just to order and control

Resonant optical trapping of Janus nanoparticles in plasmonic nanoaperture

Controlled trapping of light absorbing nanoparticles with low-power optical tweezers is crucial for remote manipulation of small objects. This study takes advantage of the synergetic effects of tightly confined local fields of plasmonic nanoaperture, self-induced back-action of nanoparticles, and resonant optical trapping method to demonstrate enhanced manipulation of Janus nanoparticles in metallic nanohole aperture. We theoretically demonstrate that displacement of Au-coated Janus nanoparticles toward plasmonic nanoaperture and proper orientation of the metal coating give rise to enhanced near-field intensity and pronounced optical force. We also explore the effect of resonant optical trapping by employing dual laser system, where an on-resonant green laser excites the metal-coated nanoparticle whereas an off-resonant near-infrared laser plays trapping role. It is found that, at optimum nanoparticle configuration, the resonant optical trapping technique can result in three-fold enhancement of optical force, which is attributed to excitation of surface plasmon resonance in Janus nanoparticles. The findings of this study might pave the way for low power optical manipulation of light-absorbing nanoparticles with possible applications in nanorobotics and drug delivery