一般拓扑空间上的KKM定理,匹配定理,重合点定理

引进没有任何凸结构的拓扑空间上的广义R-KKM映射的定义,并利用古典的KKM原理得到一般拓扑空间上的KKM型定理,然后利用该结果得到Lassonde型匹配定理和Klee型相交定理,最后作为应用给出非紧的拓扑空间上不动点定理和重合点存在定理.推广和改进了文献中的相应结果.     

T(2,2,n)∪(∪i∈A C_i)的匹配等价图类

利用图的匹配多项式及其最大实数根的性质完整刻画了T(2,2,n)∪(∪i∈A Ci)(n≥3,A是大于等于3的整数组成的可重集)的匹配等价图类.     

Measurement of nonlinear coefficient of a highly nonlinear fiber based on peak four-wave mixing signal generation conditions

We introduce a new measurement method of the nonlinear coefficient (NC) of a highly nonlinear fiber based on a four-wave mixing (FWM) technique. The NC along with the fiber's zero-dispersion wavelength (ZDW) was determined from the precise measurement of the pump power dependent optimum pump frequency for peak FWM signal generation conditions. The measurement errors of the NC and ZDW values determined with this method were less than 4.6% and 0.051 nm, respectively.     

Reflective vertical cavity semiconductor saturable absorber for functional operations with thermal limitations and saturable index change

A numerical simulation on absorptive bi-stability and resonance wavelength shift is carried out in a vertical-cavity semiconductor saturable absorber consisting of a Fabry-Perot cavity embedded with quantum-wells (QWs) material with due consideration of its nonlinear index change and thermal effects. Necessary equations are derived and discussions are made for optimization of cavity parameters required for all-optical logic operations. For InGaAs/InP quantum wells, the resonant wavelength is blue-shifted by 4 nm and red-shifted by 16 nm for input intensities of 0.15I_S and 2.2I_S, respectively where I_S represents the saturation intensity. Faithful NAND operation with extinction over 85% is possible for a signal intensity change (ΔI_in) of 0.5I_S in a sample with saturated nonlinear index (n_2S) of 0.0012, unit small-signal absorption (α₀d) and non-saturable absorption (α_nsd) of 0.025. Whereas, the NOR operation is obtained with extinction over 80% for ΔI_in = 1.0I_S in a sample with n_2S = 0.0012, α₀d = 1.0 and α_nsd = 0.0375, respectively.     

On the complexity of the dominating induced matching problem in hereditary classes of graphs

The dominating induced matching problem, also known as efficient edge domination, is the problem of determining whether a graph has an induced matching that dominates every edge of the graph. This problem is known to be NP-complete. We study the computational complexity of the problem in special graph classes. In the present paper, we identify a critical class for this problem (i.e., a class lying on a “boundary” separating difficult instances of the problem from polynomially solvable ones) and derive a number of polynomial-time results. In particular, we develop polynomial-time algorithms to solve the problem for claw-free graphs and convex graphs.     

1-factor covers of regular graphs

We consider minimal 1-factor covers of regular multigraphs, focusing on those that are 1-factorizations. In particular, we classify cubic graphs such that every minimal 1-factor cover is also a 1-factorization, and also classify simple regular bipartite graphs with this property. For r>3, we show that there are finitely many simple r-regular graphs such that every minimal 1-factor cover is also a 1-factorization.     

On distance-3 matchings and induced matchings

For a finite undirected graph G=(V,E) and positive integer k≥1, an edge set M⊆E is a distance-k matching if the pairwise distance of edges in M is at least k in G. For k=1, this gives the usual notion of matching in graphs, and for general k≥1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k=2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G) by Cameron and strong matching by Golumbic and Laskar in various papers.Finding a maximum induced matching is NP-complete even on very restricted bipartite graphs and on claw-free graphs but it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L(G)² of the line graph L(G) of G which, by a result of Cameron, is chordal for any chordal graph G.We show that, unlike for k=2, for a chordal graph G, L(G)³ is not necessarily chordal, and finding a maximum distance-3 matching, and more generally, finding a maximum distance-(2k+1) matching for k≥1, remains NP-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-k matching problem can be solved in polynomial time for every k≥1. Moreover, we obtain various new results for maximum induced matchings on subclasses of claw-free graphs.     

On forcing matching number of boron-nitrogen fullerene graphs

Let G be a graph that admits a perfect matching M. A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G. The smallest cardinality of forcing sets of M is called the forcing number of M. Computing the minimum forcing number of perfect matchings of a graph is an NP-complete problem. In this paper, we consider boron-nitrogen (BN) fullerene graphs, cubic 3-connected plane bipartite graphs with exactly six square faces and other hexagonal faces. We obtain the forcing spectrum of tubular BN-fullerene graphs with cyclic edge-connectivity 3. Then we show that all perfect matchings of any BN-fullerene graphs have the forcing number at least two. Furthermore, we mainly construct all seven BN-fullerene graphs with the minimum forcing number two.     

All-optical wavelength tuning in Ti:PPLN Šolc filter via the photorefractive effect

We demonstrate fine-wavelength tuning of a periodically poled Ti:LiNbO₃ (Ti:PPLN) Šolc filter by use of the photorefractive (PR) effect. The center wavelength of a Ti:PPLN Šolc filter is controlled by the PR effect, which is induced by second-harmonic generation (SHG). The refractive index change caused by the PR effect is calculated from the measured shift in center wavelength from the Ti:PPLN Šolc filter. The experimental results show that all-optical fine-wavelength tuning in a Ti:PPLN Šolc filter is possible by the use of a second-order nonlinear effect and that a Ti:PPLN waveguide has much higher resistance to PR damage than a Ti:LiNbO₃ waveguide. The measured wavelength tuning rate as a function of the pump beam power is about − 0.033 nm/mW.     

Colored conical emission in BBO crystal induced by intense femtosecond pulses

We present the experimental and theoretical study of colored conical emission (CCE) during intense femtosecond pulses propagating in a β-barium borate (BBO) crystal. CCE revealed special nonlinear dynamics of the strong coupling between the fundamental and second harmonic pulses due to the spatiotemporal modulation instability (MI). The underlying physics of CCE involve three three-photon processes, according to which the beam angular spectra was calculated on the phase-matching conditions and accords with the experimental observation.