Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

Bounds relating the weakly connected domination number to the total domination number and the matching number

Let G=(V,E) be a connected graph. A dominating set S of G is a weakly connected dominating set of G if the subgraph (V,E∩(S×V)) of G with vertex set V that consists of all edges of G incident with at least one vertex of S is connected. The minimum cardinality of a weakly connected dominating set of G is the weakly connected domination number, denoted . A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. In this paper, we show that . Properties of connected graphs that achieve equality in these bounds are presented. We characterize bipartite graphs as well as the family of graphs of large girth that achieve equality in the lower bound, and we characterize the trees achieving equality in the upper bound. The number of edges in a maximum matching of G is called the matching number of G, denoted α^′(G). We also establish that , and show that for every tree T.     

Lower Bounds on the Size of Maximum Independent Sets and Matchings in Hypergraphs of Rank Three

In this paper, we study lower bounds on the size of a maximum independent set and a maximum matching in a hypergraph of rank three. The rank in a hypergraph is the size of a maximum edge in the hypergraph. A hypergraph is simple if no two edges contain exactly the same vertices. Let H be a hypergraph and let and be the size of a maximum independent set and a maximum matching, respectively, in H, where a set of vertices in H is independent (also called strongly independent in the literature) if no two vertices in the set belong to a common edge. Let H be a hypergraph of rank at most three and maximum degree at most three. We show that with equality if and only if H is the Fano plane. In fact, we show that if H is connected and different from the Fano plane, then and we characterize the hypergraphs achieving equality in this bound. Using this result, we show that that if H is a simple connected 3‐uniform hypergraph of order at least 8 and with maximum degree at most three, then and there is a connected 3‐uniform hypergraph H of order 19 achieving this lower bound. Finally, we show that if H is a connected hypergraph of rank at most three that is not a complete hypergraph on vertices, where denotes the maximum degree in H, then and this bound is asymptotically best possible. © 2012 Wiley Periodicals, Inc. J. Graph Theory     

Lorentz spaces with variable exponents

We introduce Lorentz spaces and with variable exponents. We prove several basic properties of these spaces including embeddings and the identity . We also show that these spaces arise through real interpolation between and . Furthermore, we answer in a negative way the question posed in 12 whether the Marcinkiewicz interpolation theorem holds in the frame of Lebesgue spaces with variable integrability.     

The early exercise region for Bermudan options on two underlyings

This paper investigates the early exercise region for Bermudan options on two underlying assets. We present a set of analytical validation results for the early exercise region which can be used as a means of validating pricing techniques. When all strike prices are identical we show the existence of an intersection point such that for any asset price pair below this point early exercise is always optimal. We develop an approximation to this point in the two asset put case. When the strike prices are not all equal, we show that three separate cases exist for the early exercise region. For a Bermudan put on two assets we present these cases and show that there exists a critical point in which the boundaries of the two asset early exercise region bifurcate. Comparisons are drawn between the Bermudan results presented and the corresponding American option results.     

Recent Developments in the Field of Inorganic Phosphors

Energy efficiency is in! New inorganic luminescent materials can help to increase energy efficiency when used in plasma display panels and white‐light‐emitting diodes (see color diagram; mixing the three emissions A–C produces any given point within the triangle). In mercury‐free fluorescent lamps these phosphors might contribute to environmental protection, and they provide better scintillation materials for medical diagnostics.

     

Tuning Human Serum Albumin (HSA) Hydrogels through Albumin Glycation

The changes of technological properties of albumin-based hydrogels induced by increasing degrees of post-translational modification of the protein are reported. Maillard-type modification of amino acids arginine and lysine of albumin is achieved through glyoxal as an α-dicarbonyl compound. The degrees of modification are fine-tuned using different molar ratios of glyoxal. Hydrogels are thermally induced by heating highly concentrated precursor solutions above the protein's denaturation temperature. While the post-translational modifications are determined and quantified with mass spectrometry, continuous-wave (CW) electron paramagnetic resonance (EPR) spectroscopy shed light on the protein fatty acid binding capacity and changes thereof in solution and in the gel state. The viscoelastic behavior is characterized as a measure of the physical strength of the hydrogels. On the nanoscopic level, the modified albumins in low concentration solution reveal lower binding capacities with increasing degrees of modification. On the contrary, in the gel state, the binding capacity remains constant at all degrees of modifications. This indicates that the loss of fatty acid binding capacity for individual albumin molecules is partially compensated by new binding sites in the gel state, potentially formed by modified amino acids. Such, albumin glycation offers a fine-tuning method of technological and nanoscopic properties of these gels.     

Programmable and Self-Healable Liquid Crystal Elastomer Actuators Based on Halogen Bonding

Shape-changing polymeric materials have gained significant attention in the field of bioinspired soft robotics. However, challenges remain in versatilizing the shape-morphing process to suit different tasks and environments, and in designing systems that combine reversible actuation and self-healing ability. Here, we report halogen-bonded liquid crystal elastomers (LCEs) that can be arbitrarily shape-programmed and that self-heal under mild thermal or photothermal stimulation. We incorporate halogen-bond-donating diiodotetrafluorobenzene molecules as dynamic supramolecular crosslinks into the LCEs and show that these relatively weak crosslinks are pertinent for their mechanical programming and self-healing. Utilizing the halogen-bonded LCEs, we demonstrate proof-of-concept soft robotic motions such as crawling and rolling with programmed velocities. Our results showcase halogen bonding as a promising, yet unexplored tool for the preparation of smart supramolecular constructs for the development of advanced soft actuators.     

Surface tension and fluid flow driven self-assembly of ordered ZnO nanorod films for high-performance field effect transistors

Colloidal nanorods of inorganic semiconductors are of interest for a range of optoelectronic devices. The ability to self-assemble these materials into ordered arrays by solution-processing techniques is crucial for achieving adequate device performance. Here we show that uniform ZnO nanorod films with defined nanorod alignment can be solution-deposited over large areas by controlling the surface energy of the nanorods through the choice of suitable ligands and by the fluid flow direction during growth. ZnO nanorods with long carbon chain ligands exhibit a smaller surface free energy than those with short carbon chain ligands resulting in better in-plane alignment and large domain sizes up to dozens of micrometers in spin-coated films. A model is presented to rationalize the observed self-assembly behavior. It is based on the formation of a lyotropic liquid crystalline phase on the surface of the liquid film which is facilitated by enhanced segregation of nanorods with low surface tension to the surface. Alignment of the nanorods is controlled by radial and vertical liquid flows in the drying solution. The ability to control the orientation of the nanorods and to achieve large domain size results in significant device performance improvement. Field-effect transistors with mobilities of up to 1.2-1.4 cm2/V.s are demonstrated in spin-coated, in-plane aligned ZnO nanorod films subject to postdeposition hydrothermal growth.     

Phane properties of [2.2]paracyclophane/dehydrobenzoannulene hybrids

A series of [2.2]paracyclophane/dehydrobenzo[14]annulene (PC/DBA) hybrids (hydrocarbons 5, 6, 9, 10 b, and 10 c), [2.2]paracyclophane/dehydro[14]annulene (PC/DA) hybrids (7 and 8) and suitable model systems (11, 12, and 33) has been synthesized. Comparison of the electronic absorption spectra in each series of compounds provides further insight into the global communication between the decks in the [2.2]paracyclophane unit.