Finding any given 2‐factor in sparse pseudorandom graphs efficiently

Given an $n$‐vertex pseudorandom graph $G$ and an $n$‐vertex graph $H$ with maximum degree at most two, we wish to find a copy of $H$ in $G$, that is, an embedding $\phi :V\left(H\right)\to V\left(G\right)$ so that $\phi \left(u\right)\phi \left(v\right)\in E\left(G\right)$ for all $uv\in E\left(H\right)$. Particular instances of this problem include finding a triangle‐factor and finding a Hamilton cycle in $G$. Here, we provide a deterministic polynomial time algorithm that finds a given $H$ in any suitably pseudorandom graph $G$. The pseudorandom graphs we consider are $\left(p,\lambda \right)$‐bijumbled graphs of minimum degree which is a constant proportion of the average degree, that is, $\mathrm{\Omega }\left(pn\right)$. A $\left(p,\lambda \right)$‐bijumbled graph is characterised through the discrepancy property: $|e\left(A,B\right)?p|A||B||<\lambda \sqrt{|A||B|}$ for any two sets of vertices $A$ and $B$. Our condition $\lambda =O(p^{2}n/\log n)$ on bijumbledness is within a log factor from being tight and provides a positive answer to a recent question of Nenadov. We combine novel variants of the absorption‐reservoir method, a powerful tool from extremal graph theory and random graphs. Our approach builds on our previous work, incorporating the work of Nenadov, together with additional ideas and simplifications.     

Counting restricted orientations of random graphs

We count orientations of avoiding certain classes of oriented graphs. In particular, we study , the number of orientations of the binomial random graph in which every copy of is transitive, and , the number of orientations of containing no strongly connected copy of . We give the correct order of growth of and up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly improves a result of Allen, Kohayakawa, Mota, and Parente. We also discuss the problem for a single forbidden oriented graph, and state a number of open problems and conjectures.     

Synthesis and mechanochemical properties of biobased ABCBA-type pentablock copolymers comprising poly-d-lactide (A), poly-l-lactide (B) and poly(1,2-propylene succinate) (C)

The ABCBA pentablock copolymers (p-d -l -PPS) comprising poly(d -lactide) (PDLA: A), poly(l -lactide) (PLLA: B) and poly(propylene succinate) (PPS: C) were successfully synthesized by two-step ring-opening polymerization (ROP) of d - and l -lactide using a dihydroxy-terminated PPS as a macro-initiator. The pentablock copolymers revealed the high stereocomplex (sc) crystallinity, thermal stability and elastomeric property in their solution-cast films. It was found that the T_g was found to be proportional to the PPS content, whereas the T_m was proportional to their average block length. The thermal resistivity of the copolymer films was found to be as high as 202°C owing to their sc formation. The copolymers also showed improved stereocomplexibility compared to the enantiomeric mixtures of triblock copolymers (PLLA-PPS-PLLA and PDLA-PPS-PDLA) having similar PLLA and PDLA chain lengths. These pentablock copolymers can afford thermoplastic elastomers or flexible plastic materials having a 100% bio-based content, showing high heat-resistive property.     

Supratransmission in discrete one-dimensional lattices with the cubic–quintic nonlinearity

Nonlinear Dynamics - We numerically analyzed the supratransmission phenomenon in the discrete nonlinear Schrödinger equation with the cubic–quintic nonlinearity. It has been reported...     

The size Ramsey number of short subdivisions of bounded degree graphs

For graphs G and F, write if any coloring of the edges of G with colors yields a monochromatic copy of the graph F. Suppose is obtained from a graph S with s vertices and maximum degree d by subdividing its edges h times (that is, by replacing the edges of S by paths of length h + 1). We prove that there exists a graph G with no more than edges for which holds, provided that . We also extend this result to the case in which Q is a graph with maximum degree d on q vertices with the property that every pair of vertices of degree greater than 2 are distance at least h + 1 apart. This complements work of Pak regarding the size Ramsey number of “long subdivisions” of bounded degree graphs.     

Integrating over quiver variety and BPS/CFT correspondence

Letters in Mathematical Physics - We show the vertex operator formalism for the quiver gauge theory partition function and the qq-character of the highest weight module on quiver, both associated...     

Effect of the Ethanol/BTC Ratio on the Methane Uptake of Mechanochemically Synthesized MOF-199

We report on a detailed textural analysis of mechanochemically synthesized MOF-199 including N₂ adsorption-desorption and CO₂ adsorption isotherms data at 77 K and 273 K (up to atmospheric pressure), respectively, and CH₄ adsorption data at 298 K (up to 35 bar). We used the isotherm adsorption data to determine the micropore volume of the MOF-199 structures, to establish their methane uptake capacity and to understand how these properties depended on the Ethanol/BTC ratio used during the synthesis. The maximum methane uptake capacity for our specimens was recorded at 130 v/v at 35 bars. These results open an avenue for a better understanding of alternative manufacturing processes of MOF structures for gas storage applications.     

Enzymatic Hydrolysis-Responsive Supramolecular Hydrogels Composed of Maltose-Coupled Amphiphilic Ureas

Maltose is a ubiquitous disaccharide produced by the hydrolysis of starch. Amphiphilic ureas bearing hydrophilic maltose moiety were synthesized via the following three steps: I) construction of urea derivatives by the condensation of 4-nitrophenyl isocyanate and alkylamines, II) reduction of the nitro group by hydrogenation, and III) an aminoglycosylation reaction of the amino group and the unprotected maltose. These amphiphilic ureas functioned as low molecular weight hydrogelators, and the mixtures of the amphipathic ureas and water formed supramolecular hydrogels. The gelation ability largely depended on the chain length of the alkyl group of the amphiphilic urea; amphipathic urea having a decyl group had the highest gelation ability (minimum gelation concentration=0.4 mM). The physical properties of the supramolecular hydrogels were evaluated by measuring their thermal stability and dynamic viscoelasticity. These supramolecular hydrogels underwent gel-to-sol phase transition upon the addition of α-glucosidase as a result of the α-glucosidase-catalyzed hydrolysis of the maltose moiety of the amphipathic urea.     

Formulation of Aluminum Chloride Phthalocyanine in Pluronic™ P‐123 and F‐127 Block Copolymer Micelles: Photophysical properties and Photodynamic Inactivation of Microorganisms

Aluminum Chloride Phthalocyanine (AlPcCl) can be used as a photosensitizer (PS) for Photodynamic Inactivation of Microorganisms (PDI). The AlPcCl showed favorable characteristics for PDI due to high quantum yield of singlet oxygen (Φ_Δ) and photostability. Physicochemical properties and photodynamic inactivation of AlPcCl incorporated in polymeric micelles of tri‐block copolymer (P‐123 and F‐127) against microorganisms Staphylococcus aureus, Escherichia coli and Candida albicans were investigated in this work. Previously, it was observed that the AlPcCl undergoes self‐aggregation in F‐127, while in P‐123 the PS is in a monomeric form suitable for PDI. Due to the self‐aggregation of AlPcCl in F‐127, this formulation did not show any effect on these microorganisms. On the other hand, AlPcCl formulated in P‐123 was effective against S. aureus and C. albicans and the death of microorganisms was dependent on the PS concentration and illumination time. Additionally, it was found that the values of PS concentration and illumination time to eradicate 90% of the initial population of microorganisms (IC₉₀ and D₉₀, respectively) were small for the AlPcCl in P‐123, showing the effectiveness of this formulation for PDI.     

Phase structure change and ER effect in liquid crystalline polymer/dimethylsiloxane blends

The mechanism of the electrorheological (ER) effect in two types of liquid crystalline polymer (LCP)/dimethylsiloxane (DMS) blends was investigated by rheological measurements and by structure observation under electric field and shear flow. The results show that the phase structures of these immiscible blends can be categorized into slipping (low viscosity) and non-slipping (high viscosity) states. In the non-slipping state, higher viscosity LCP domains connect the electrodes. In the slipping state, on the other hand, LCP domains do not connect the electrodes and the shear is mainly confined in the lower viscosity DMS domains. The ER effect (electrically induced viscosity increase) originates from the electrically induced slipping to non-slipping transition. In one of the blends, the ER effect occurs only at high shear rate, since this blend is in non-slipping state even under no field if the shear rate is low. Received: 29 April 1997 Accepted: 3 November 1997