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41.
Let x=(x',x')with x'∈Rk and x'∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x'-symmetric function u∈C0^∞(Ω)with∫Ω|■u(x)|^N-a|x'|^-adx≤1,the following uniform inequality holds1/∫Ω|x|^-adx∫Ωe^βa|u|N-a/N-a-1|x'|^-adx≤C,whereβa=(N-a)(2πN/2Γ(k-a/2)Γ(k/2)/Γ(k/2)r(N-a/2))1/N-a-1.Furthermore,βa can not be replaced by any greater number.As the application,we obtain some weighted Trudinger–Moser inequalities for x-symmetric function on Grushin space. 相似文献
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We consider the random‐cluster model (RCM) on with parameters p∈(0,1) and q ≥ 1. This is a generalization of the standard bond percolation (with edges open independently with probability p) which is biased by a factor q raised to the number of connected components. We study the well‐known Fortuin‐Kasteleyn (FK)‐dynamics on this model where the update at an edge depends on the global geometry of the system unlike the Glauber heat‐bath dynamics for spin systems, and prove that for all small enough p (depending on the dimension) and any q>1, the FK‐dynamics exhibits the cutoff phenomenon at with a window size , where λ∞ is the large n limit of the spectral gap of the process. Our proof extends the information percolation framework of Lubetzky and Sly to the RCM and also relies on the arguments of Blanca and Sinclair who proved a sharp mixing time bound for the planar version. A key aspect of our proof is the analysis of the effect of a sequence of dependent (across time) Bernoulli percolations extracted from the graphical construction of the dynamics, on how information propagates. 相似文献
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In this work, a vanillin complex is immobilized onto MCM-41 and characterized by FT-IR, X-ray diffraction, scanning electron microscopy, energy dispersive spectroscopy, thermogravimetric analysis, and BET techniques. This supported Schiff base complex was found to be an efficient and recoverable catalyst for the chemoselective oxidation of sulfides into sulfoxides and thiols into their corresponding disulfides (using hydrogen peroxide as a green oxidant) and also a suitable catalyst for the preparation of 2,3-dihydroquinazolin-4(1H)-one derivatives in water at 90°C. Using this protocol, we show that a variety of disulfides, sulfoxides, and 2,3-dihydroquinazolin-4(1H)-one derivatives can be synthesized in green conditions. The catalyst can be recovered and recycled for further reactions without appreciable loss of catalytic performance. 相似文献
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ABSTRACT Multicolour emissive carbon dots (CDs) are widely investigated by virtue of their merits on fluorescent properties. Method on heteroatom doping assisted with various solvents has been proved efficient in achieving multiple-colour-emissive CDs, especially long-wavelength emission. Herein, a synthesis of multicolour-emissive CDs by controlled surface function is reported. By tuning the thermal-pyrolysis temperature and molar ratio of reactants, optimal emission of the resulted CDs gradually shifts from blue to yellow light with the assistance of different solvents. According to the emissive relationship dependent on excitation, fluorescence lifetimes, and FT-IR of these CDs, the different surface states participated with S and N elements on the surface of carbogenic core govern fluorescent colours of the CDs. In terms of the applications, blue CDs (B-CDs) exhibits high sensitivity for ion detections of Ag+ and Fe3+, which is further illustrated to have different quenching mechanisms each other because that these ions have the affinity interaction with different surface groups of the CDs. Moreover, blue and yellow CDs solutions are mixed with PVP water solution to fabricate white-light CDs/PVP film, which exhibits stable fluorescence with a CIE coordinate of (0.32, 0.33) and endows these CDs as potentially fluorescent nanomaterial in the solid state lighting field. 相似文献
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A. Othman Almatroud Amina-Aicha Khennaoui Adel Ouannas Giuseppe Grassi M. Mossa Al-sawalha Ahlem Gasri 《Entropy (Basel, Switzerland)》2020,22(12)
This article proposes a new fractional-order discrete-time chaotic system, without equilibria, included two quadratic nonlinearities terms. The dynamics of this system were experimentally investigated via bifurcation diagrams and largest Lyapunov exponent. Besides, some chaotic tests such as the 0–1 test and approximate entropy (ApEn) were included to detect the performance of our numerical results. Furthermore, a valid control method of stabilization is introduced to regulate the proposed system in such a way as to force all its states to adaptively tend toward the equilibrium point at zero. All theoretical findings in this work have been verified numerically using MATLAB software package. 相似文献