UNIQUE CONTINUATION AND PERSISTENCE PROPERTIES OF SOLUTIONS OF THE 2-COMPONENT DEGASPERIS-PROCESI EQUATIONS

In this article, the unique continuation and persistence properties of solutions of the 2-component Degasperis-Procesi equations are discussed. It is shown that strong solutions of the 2-component Degasperis-Procesi equations, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if they also decay exponentially at a later time.

Exploring and exploiting unique properties of the hippocampal dentate gyrus for post-stroke therapy:astrocytes link ischemic resistance with neurogenic potential

Acute cerebral ischemia can occur secondary to embolism, cardiac arrest, hemorrhage, traumatic brain injury, edema, vascular compression, or any physiologic condition resulting in low cardiac output state. Survivors of cerebral ischemic events frequently suffer from profound disability,

ON DOUBLING PROPERTIES ANDTHE UNIQUE CONTINUATION AT THEBOUNDARY IN DINI DOMAINS

Let u be a solution to second order elliptic equations in Dini domains, a direct and elementary proof of the doubling property for u 2 over balls centered at points in the domain is presented. Moreover, under the zero Dirichlet boundary condition, the unique continuation at the boundary for Dini domains has been proved.

Crystalline framework nanosheets as platforms for functional materials

The integration of organic and inorganic materials has been widely used in various applications to generate novel functional nanomaterials characterized by unique properties.Functional crystalline framework nanosheets and their synergistic effects have been studied recently for possessing the advantages of functional species as well as crystalline framework nanosheets.Hence,we have focused on the preparation methods and applications of functional crystalline framework nanosheets in this review.We introduced crystalline framework nanosheets and discussed the importance of integrating functional species with nanosheets to form functional crystalline framework nanosheets.Then,two aspects of the preparation methods of functional crystalline framework nanosheets were reviewed:in situ synthesis and post-synthesis modification.Subsequently,we discussed the properties of the crystalline framework nanosheets combined with various functional species and summarized their applications in catalysis,sensing,separation,and energy storage.Finally,we have shared our insights on the challenges of functional crystalline framework nanosheets,hoping to contribute to the knowledge base for optimizing the preparation methods,expanding categories,improving stability,and exploring potential applications.

Alternating Double Trinucleate[Yb_3(μ_3-OH)_2O]_2 Chain Bridged byμ_3-OH Mode in a 3D Yb^(3+)-CP and Its Near Infrared Luminescent Properties

A new 3D coordination polymer,namely,[Yb_3（μ_3-OH）_3（μ_2-OH）（DDLA）_（2.5）（BIPY）]（1,H_2DDLA = 2,2-dipyridyl-4,4-dicarboxylic acid,BIPY = 2,2？-bipyridine）,has been synthesized via solvothermal method. The complex crystallizes in triclinic,space group P1 with a = 7.7339（4）,b = 15.6699（8）,c = 17.2221（9） ？,α = 68.8630（10）,β = 88.1890（10）,γ = 88.1740（10）°,V = 1945.27（17） ？~3,Z = 2,D_c = 2.296 g·cm^3,μ（MoKα） = 1.075 mm^（-1）,F（000） = 1268,S = 1.068,R = 0.0263 and wR = 1.067 for 5176 observed reflections with I 〉 2σ（I）. Compound 1 contains a unique alternating double trinucleate {[Yb_3（μ_3-OH）_2OH]_2}_n chain,in which Yb^（3＋） is bridged through μ_3-OH and μ_2-OH two coordination modes. The alternating double trinucleate chains are further extended into a 3D structure through DDLA^（2-） ligands by different coordination modes. The photoluminescence spectra show 1 is highly sensitized by the characteristic emission of Yb^（3＋） in the NIR region.

Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials

In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.