Stability of a conditional Cauchy equation on a set of measure zero

In this paper, we prove the Hyers–Ulam stability theorem when \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy $$|f(x + y) - g(x) - h(y)| \leq \epsilon$$ in a set \({\Gamma \subset \mathbb{R}^{2}}\) of measure \({m(\Gamma) = 0}\) , which refines a previous result in Chung (Aequat Math 83:313–320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy the Pexider equation $$f(x + y) - g(x) - h(y) = 0$$ in \({\Gamma}\) , then the equation holds for all \({x, y \in \mathbb{R}}\) . Using our method of construction of the set, we can find a set \({\Gamma \subset \mathbb{R}^{2n}}\) of 2n-dimensional measure 0 and obtain the above result for the functions \({f, g, h : \mathbb{R}^{n} \to \mathbb{C}}\) .     

An example of nonuniqueness of the cauchy problem for the heat equation

We give an example of non trivial solution of the homogeneous Cauchy problem of the heat equation, which is, for each t, bounded in the space variables.     

A Pyrene-based Highly Selective Turn-on Fluorescent Sensor for Copper(II) Ion and its Application in Live Cell Imaging

A new pyrene derivative (chemosensor 1) containing a picolinohydrazide moiety exhibits high selectivity for Cu²⁺ ion detection in mixed aqueous media (CH₃OH:H₂O = 7:3). Significant fluorescence enhancement was observed with chemosensor 1 in the presence of Cu²⁺. However, the metal ions Ag⁺, Ca²⁺, Cd²⁺, Co²⁺, Cu²⁺, Fe²⁺, Fe³⁺, Hg²⁺, K⁺, Mg²⁺, Mn²⁺, Ni²⁺, Pb²⁺, and Zn²⁺ produced only minor changes in fluorescence for the system. The apparent association constant (K _a) of Cu²⁺ binding in chemosensor 1 was found to be 2.75*10³ M⁻¹. The maximum fluorescence enhancement caused by Cu²⁺ binding in chemosensor 1 was observed over the pH range 5–8. Moreover, by means of fluorescence microscopy experiments, it is demonstrated that 1 can be used as a fluorescent probe for detecting Cu²⁺ in living cells.     

High surface plasmon resonance sensitivity enabled by optical disks

We report a systematic, experimental, and theoretical investigation on the surface plasmon resonance (SPR) sensing using optical disks with different track pitches, including Blu-ray disk (BD), digital versatile disk (DVD), and compact disk (CD). Optical reflection measurements indicate that CD and DVD exhibit much higher SPR sensitivity than BD. Both experiments and finite-difference time-domain simulations reveal that the SPR sensitivity is significantly affected by the diffraction order of the SPR peaks and higher diffraction order results in lower sensitivity. Numerical simulations also show that very high sensitivity (～1600 nm per refractive index unit) is achievable by CDs.     

Flame speed and self-similar propagation of expanding turbulent premixed flames

In this Letter we present turbulent flame speeds and their scaling from experimental measurements on constant-pressure, unity Lewis number expanding turbulent flames, propagating in nearly homogeneous isotropic turbulence in a dual-chamber, fan-stirred vessel. It is found that the normalized turbulent flame speed as a function of the average radius scales as a turbulent Reynolds number to the one-half power, where the average radius is the length scale and the thermal diffusivity is the transport property, thus showing self-similar propagation. Utilizing this dependence it is found that the turbulent flame speeds from the present expanding flames and those from the Bunsen geometry in the literature can be unified by a turbulent Reynolds number based on flame length scales using recent theoretical results obtained by spectral closure of the transformed G equation.     

A novel synthesis of benzo[c]phenanthridine skeleton and biological evaluation of isoquinoline derivatives.

Benzo[c]phenanthridine skeleton was synthesized from easily available starting N-methyl-o-toluamide 2 and o-methylbenzonitrile 5 in 7 steps. Radical cyclization of styrene 10 afforded 6,11-dimethyl-6,11-dihydro-5H-indeno[1,2-c]isoquinolin-5-one 13. Most 3-arylisoquinolines have displayed strong activities against human tumor cell lines. Especially, indenoisoquinolinone 13 exhibited excellent cytotoxicity (IC50 = 0.002 microgram/ml; HCT 15).     

Characterizing the transmission properties of an ion funnel

The characteristic features of ion transmission properties in a radio-frequency (RF) electric-field driven ion funnel using the SIMION ion trajectory simulation package is presented. A user program applying the Douglas ion-neutral collisional drag coefficient model is incorporated to properly account for the ion focusing and transport effect of the background gas under the effect of the driving RF and a superimposed DC field. The simulated m/z transmission window compares favorably with the experimental results reported by Smith et al. RF amplitude and pressure dependence of experimentally observed m/z transmission windows are also examined, and an approximated effective potential model based on Gerlich's equation is proposed to interpret the low-m/z cutoff behavior. A modified ion funnel configuration is described.     

A DFT study on the mechanism of hydrosilylation of unsaturated compounds with neutral hydrido(hydrosilylene)tungsten complex

Recently, Tobita et al. reported stoichiometric hydrosilylation reactions of acetone and acetonitrile with neutral hydrido(hydrosilylene)tungsten complexes Cp'(CO)2(H)W=Si(H)[C(SiMe(3))(3)] (Cp' = Cp*, C(5)Me(4)Et). The mechanisms of the hydrosilylation reactions of unsaturated compounds (ketone and nitrile) with the tungsten complexes have been investigated with the B(3)LYP density functional theory method. Four possible reaction mechanisms were studied. The results of the calculations indicate that the hydrosilylation of acetone proceeds via a metal hydride migration mechanism proposed by Tobita et al., while the hydrosilylation of nitrile occurs through a silyl migration mechanism, analogous to the modified Chalk-Harrod mechanism. The [2(sigma)+2(pi)] additions of various CX (CX = C=O or CN) multiple bonds with the Si-H bonds in the neutral complexes have very high barriers although similar additions were found feasible in other related cationic complexes. All the hydrosilylation reactions studied here give stable tungsten-silylene or tungsten-silyl products, which are not easily converted into the starting hydrido(hydrosilylene)tungsten complexes when reacting with a hydrosilane substrate molecule. Therefore, we predict that hydrosilylation of acetonitrile and acetone catalyzed by these tungsten complexes is difficult to achieve.     

Prevention of UV radiation-induced premature skin aging in hairless mice by the novel compound Melanocin A

Repetitive exposure of the skin to UV radiation induces various harmful changes, such as thickening, wrinkle formation, inflammation and carcinogenesis. A variety of natural compounds and synthetic compounds have been studied to determine whether they can prevent UV-induced harmful effects. In this study, we investigated the effect of a novel compound, Melanocin A, which was isolated from Eupenicillium shearii F80695, on UV-induced premature skin aging. First, we studied the effect of Melanocin A on UV-induced matrix metalloproteinase (MMP)-9 expression in an immortalized human keratinocyte cell line, HaCaT, in vitro. Acute UV irradiation induced MMP-9 expression at both the mRNA and protein levels and Melanocin A suppressed this expression in a dose-dependent manner. We then investigated the effect of Melanocin A on UV-induced skin changes in hairless mice in vivo. Chronic exposure of hairless mouse dorsal skin to UV increased skin thickness and induced wrinkle formation and the gelatinase activities of MMP-2 and MMP-9. Moreover, Melanocin A significantly suppressed UV-induced morphologic skin changes and MMP-2 and MMP-9 expression. Taken together, these results show that Melanocin A can prevent the harmful effects of UV that lead to skin aging. Therefore, we suggest that Melanocin A should be viewed as a potential therapeutic agent for preventing and/or treating premature skin aging.     

NMR and FT-IR Investigation of Spinel LiMn2O4 Cathode Prepared by the Tartaric Acid Gel Process

The structure of the lithium manganese tartrate precursor and the synthesis mechanism of LiMn₂O₄ were investigated by FT-IR, NMR, TG/DSC, and XRD in this study. The results of FT-IR and ⁷Li and ¹³C NMR measurements revealed that lithium ions bond with carboxylic acid ligands and the O–H stretching modes of tartaric acid. Manganese ion bonds only with carboxylic acid. Lithium and manganese ions were trapped homogeneously on an atomic scale throughout the precursor. Such a structure eliminates the need for long-range diffusion during the formation of lithium manganese oxides. Therefore, spinel LiMn₂O₄ was synthesized at temperatures as low as 300°C. In this work, the electrochemical properties of Li/Li_xMn₂O₄ were studied. It is clear that the discharge curves exhibit two pseudo plateaus as the LiMn₂O₄ is fired to higher temperatures. The discharge capacity of LiMn₂O₄ increases from 84 to 117 mAh/g as the calcination temperature increases from 300 to 500°C. The LiMn₂O₄ powders calcined at low temperatures with a high specific surface area and an average valence of manganese exhibit a better cycle life.