Dierential geometry play a fundamental role in discussing partial dierential equations（PDEs） in mathematical physics. Recently discrete dierential geometry is an active mathematical terrain, which aims at the develo...Dierential geometry play a fundamental role in discussing partial dierential equations（PDEs） in mathematical physics. Recently discrete dierential geometry is an active mathematical terrain, which aims at the development and application of discrete equivalents of the geometric notions and methods of dierential geometry. In this paper, a discrete theory of exterior dierential calculus and the analogue of the Poincar′e lemma for dierential-dierence complex are proposed. They provide an intrinsic idea for developing the theory to discuss the integrability of dierence equations.展开更多
Using Nevanlinna theory of the value distribution of meromorphic functions, we investigate the problem of the existence of meromorphic solutions of some types of complex differential-difference equations and some prop...Using Nevanlinna theory of the value distribution of meromorphic functions, we investigate the problem of the existence of meromorphic solutions of some types of complex differential-difference equations and some properties of meromorphic solutions, and we ob- tain some results, which are the improvements and extensions of some results in references. Examples show that our results are precise.展开更多
In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex d...In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.展开更多
Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference...Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj（z）f1（λj1）（z＋cj）=R2（z,f2（z））,j=1∑nβj（z）f2（λj2）（z＋cj）=R1（z,f1（z））. where λij （j = 1, 2,…, n; i = 1, 2） are finite non-negative integers, and cj （j = 1, 2,… , n） are distinct, nonzero complex numbers, αj（z）, βj（z） （j = 1,2,… ,n） are small functions relative to fi（z） （i =1, 2） respectively, Ri（z, f（z）） （i = 1, 2） are rational in fi（z） （i =1, 2） with coefficients which are small functions of fi（z） （i = 1, 2） respectively.展开更多
In this paper,we mainly study the uniqueness of transcendental meromorphic solutions for a class of complex linear differential-difference equations.Specially,suppose that f(z)is a finite order transcendental meromorp...In this paper,we mainly study the uniqueness of transcendental meromorphic solutions for a class of complex linear differential-difference equations.Specially,suppose that f(z)is a finite order transcendental meromorphic solution of complex linear differential-difference equation:W_(1)(z)f'(z+1)+W_(2)(z)f(z)=W_(3)(z),where W_(1)(z),W_(2)(z),W_(3)(z) are nonzero meromorphic functions,with their orders of growth being less than one,such that W_(1)(z)+W_(2)(z)■0.If f(z) and a meromorphic function g(z) share 0,1,∞ CM,then either f(z)≡g(z) or f(z)+g(z)≡f(z)g(z) or f^(2)(z)(g(z)-1)^(2)+g^(2)(z)(f(z)-1)^(2)≡f(z)g(z)(f(z)g(z)-1) or there exists a polynomial φ(z)=az+b_(0) such that ■ where a(≠0),a_(0),b_(0) are constants with e^(a_(0))≠e^(b_(0)).展开更多
By using the Nevanlinna value distribution theory, we will mainly investigate the form of entire solutions with finite order on a type of system of differential-difference equations and a type of differential-differen...By using the Nevanlinna value distribution theory, we will mainly investigate the form of entire solutions with finite order on a type of system of differential-difference equations and a type of differential-difference equations, two interesting results are obtained. And it extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.展开更多
The main purpose of this paper is to study the growth of meromorphic solutions of complex linear differential-difference equations L（z, f） =n∑i=0m∑j=0Aij（z）f^（j）（z ＋ ci） = 0 or F（z）with entire or meromorp...The main purpose of this paper is to study the growth of meromorphic solutions of complex linear differential-difference equations L（z, f） =n∑i=0m∑j=0Aij（z）f^（j）（z ＋ ci） = 0 or F（z）with entire or meromorphic coefficients, and ci, i = 0,..., n being distinct complex numbers,where there is only one dominant coefficient.展开更多
Throughout this work,we explore the uniqueness properties of meromorphic functions concerning their interactions with complex differential-difference polynomial.Under the condition of finite order,we establish three d...Throughout this work,we explore the uniqueness properties of meromorphic functions concerning their interactions with complex differential-difference polynomial.Under the condition of finite order,we establish three distinct uniqueness results for a meromorphic function f associated with the differential-difference polynomial L_(η)^(n)f=Σ_(k=0)^(n)a_(k)f (z+k_(η))+a_(-1)f′.These results lead to a refined characterization of f (z)≡L_(η)^(n)f (z).Several illustrative examples are provided to demonstrate the sharpness and precision of the results obtained in this study.展开更多
In this paper,we mainly investigate the forms of entire solutions for certain Fermattype partial differential-difference equations in C^(2)by using Nevanlinna’s theory of several complex variables.
Three copper(Ⅱ),nickel and cadmium(Ⅱ)complexes,namely[Cu_(2)(μ-H2dbda)2(phen)2]·2H_(2)O(1),[Ni(μ-H2dbda)(μ-bpb)(H_(2)O)2]n(2),and[Cd(μ-H2dbda)(μ-bpa)]n(3),have been constructed hydrothermally using H4dbda(...Three copper(Ⅱ),nickel and cadmium(Ⅱ)complexes,namely[Cu_(2)(μ-H2dbda)2(phen)2]·2H_(2)O(1),[Ni(μ-H2dbda)(μ-bpb)(H_(2)O)2]n(2),and[Cd(μ-H2dbda)(μ-bpa)]n(3),have been constructed hydrothermally using H4dbda(4,4'-dihydroxy-[1,1'-biphenyl]-3,3'-dicarboxylic acid),phen(1,10-phenanthroline),bpb(1,4-bis(pyrid-4-yl)benzene),bpa(bis(4-pyridyl)amine),and copper,nickel and cadmium chlorides at 160℃.The products were isolated as stable crystalline solids and were characterized by IR spectra,elemental analyses,thermogravimetric analyses,and singlecrystal X-ray diffraction analyses.Single-crystal X-ray diffraction analyses revealed that three complexes crystallize in the monoclinic P21/n,tetragonal I42d,and orthorhombic P21212 space groups.The complexes exhibit molecular dimers(1)or 2D metal-organic networks(2 and 3).The catalytic performances in the Knoevenagel reaction of these complexes were investigated.Complex 1 exhibits an effective catalytic activity and excellent reusability as a heterogeneous catalyst in the Knoevenagel reaction at room temperature.CCDC:2463800,1;2463801,2;2463802,3.展开更多
Six new lanthanide complexes:[Ln(3,4-DEOBA)3(4,4'-DM-2,2'-bipy)]2·2C_(2)H_(5)OH,[Ln=Dy(1),Eu(2),Tb(3),Sm(4),Ho(5),Gd(6);3,4-DEOBA-=3,4-diethoxybenzoate,4,4'-DM-2,2'-bipy=4,4'-dimethyl-2,2'...Six new lanthanide complexes:[Ln(3,4-DEOBA)3(4,4'-DM-2,2'-bipy)]2·2C_(2)H_(5)OH,[Ln=Dy(1),Eu(2),Tb(3),Sm(4),Ho(5),Gd(6);3,4-DEOBA-=3,4-diethoxybenzoate,4,4'-DM-2,2'-bipy=4,4'-dimethyl-2,2'-bipyridine]were successfully synthesized by the volatilization of the solution at room temperature.The crystal structures of six complexes were determined by single-crystal X-ray diffraction technology.The results showed that the complexes all have a binuclear structure,and the structures contain free ethanol molecules.Moreover,the coordination number of the central metal of each structural unit is eight.Adjacent structural units interact with each other through hydrogen bonds and further expand to form 1D chain-like and 2D planar structures.After conducting a systematic study on the luminescence properties of complexes 1-4,their emission and excitation spectra were obtained.Experimental results indicated that the fluorescence lifetimes of complexes 2 and 3 were 0.807 and 0.845 ms,respectively.The emission spectral data of complexes 1-4 were imported into the CIE chromaticity coordinate system,and their corre sponding luminescent regions cover the yellow light,red light,green light,and orange-red light bands,respectively.Within the temperature range of 299.15-1300 K,the thermal decomposition processes of the six complexes were comprehensively analyzed by using TG-DSC/FTIR/MS technology.The hypothesis of the gradual loss of ligand groups during the decomposition process was verified by detecting the escaped gas,3D infrared spectroscopy,and ion fragment information detected by mass spectrometry.The specific decomposition path is as follows:firstly,free ethanol molecules and neutral ligands are removed,and finally,acidic ligands are released;the final product is the corresponding metal oxide.CCDC:2430420,1;2430422,2;2430419,3;2430424,4;2430421,5;2430423,6.展开更多
Three zinc(Ⅱ),nickel(Ⅱ),and cadmium(Ⅱ)complexes,namely[Zn(μ-Htpta)(py)_(2)]n(1),[Ni(H_(2)biim)2(H_(2)O)2][Ni(tpta)(H_(2)biim)2(H_(2)O)]2·3H_(2)O(2),and[Cd_(3)(μ4-tpta)2(μ-dpe)_(3)]_(n)(3),have been construc...Three zinc(Ⅱ),nickel(Ⅱ),and cadmium(Ⅱ)complexes,namely[Zn(μ-Htpta)(py)_(2)]n(1),[Ni(H_(2)biim)2(H_(2)O)2][Ni(tpta)(H_(2)biim)2(H_(2)O)]2·3H_(2)O(2),and[Cd_(3)(μ4-tpta)2(μ-dpe)_(3)]_(n)(3),have been constructed hydrothermally at 160℃ using H_(3)tpta([1,1':3',1″-terphenyl]-4,4',5'-tricarboxylic acid),py(pyridine),H_(2)biim(2,2'-biimidazole),dpe(1,2-di(4-pyridyl)ethylene),and zinc,nickel and cadmium chlorides,resulting in the formation of stable crystalline solids which were subsequently analyzed using infrared spectroscopy,element analysis,thermogravimetric analysis,as well as structural analyses conducted via single-crystal X-ray diffraction.The findings from these single-crystal Xray diffraction studies indicate that complexes 1-3 form crystals within the monoclinic system P2_(1)/c space group(1)or triclinic system P1 space group(2 and 3),and possess 1D,0D,and 3D structures,respectively.Complex 1 demonstrated substantial catalytic efficiency and excellent reusability as a heterogeneous catalyst in the reaction of Knoevenagel condensation under ambient temperature conditions.In addition,complex 1 also showcased notable anti-wear performance when used in polyalphaolefin synthetic lubricants.CCDC:2449810,1;2449811,2;2449812,3.展开更多
The complexes 1-4 of cyclobutanocucurbit[5]uril(CyB5Q[5])with Na^(+)/K^(+)have been synthesized and characterized by single-crystal X-ray diffraction.The results show that although the inorganic salts are used when th...The complexes 1-4 of cyclobutanocucurbit[5]uril(CyB5Q[5])with Na^(+)/K^(+)have been synthesized and characterized by single-crystal X-ray diffraction.The results show that although the inorganic salts are used when the cations are the same and the anions are different,in complex 1,Na^(+)closes one port of CyB5Q[5]through Na—O seven coordination bonds to form a molecular bowl;in complex 3,Na^(+)completely closes the two ports of CyB5Q[5]to form a molecular capsule with six Na—O coordination bonds;in complexes 2 and 4,the two ports of CyB5Q[5]are completely closed to form K—O coordinated molecular capsules,but the K^(+)of complex 2 is six-coordinated and that of complex 4 is eight-/nine-coordinated.and complex 4 are connected by three oxygen bridges to form a 1D molecular chain.CCDC:2457122,1;2457121,2;2457400,3;2457120,4.展开更多
Complex trimalleolar ankle fractures are a major orthopaedic challenge,with an incidence of 4.22 per 10000 person-years in the United States and an annual cost of 3.4 billion dollars.This review synthesizes current ev...Complex trimalleolar ankle fractures are a major orthopaedic challenge,with an incidence of 4.22 per 10000 person-years in the United States and an annual cost of 3.4 billion dollars.This review synthesizes current evidence on diagnostic protocols and management strategies,highlighting optimal approaches and emerging trends.Initial care emphasizes soft tissue assessment,often guided by the Tscherne classification,and fracture classification systems.External fixation may be required in open injuries,while early open reduction and internal fixation within six days is linked to improved outcomes.Minimally invasive techniques for the lateral malleolus,including intramedullary nailing and locking plates,are effective,while medial malleolus fractures are commonly managed with screw fixation or tension-band wiring.Posterior malleolus fragments involving more than 25%of the articular surface usually warrant fixation.Alternatives to syndesmotic screws,such as cortical buttons or high-strength sutures,reduce the need for secondary procedures.Arthroscopic-assisted open reduction and internal fixation benefits younger,active patients by enabling concurrent management of intra-articular and ligamentous injuries.Postoperative care prioritizes early weight-bearing and validated functional scores.Despite advances,complications remain common,and further research is needed to refine surgical strategies and improve outcomes.展开更多
The Jacobi elliptic function expansion method is extended to derive the explicit periodic wave solutions for nonlinear differential-difference equations. Three well-known examples are chosen to illustrate the applicat...The Jacobi elliptic function expansion method is extended to derive the explicit periodic wave solutions for nonlinear differential-difference equations. Three well-known examples are chosen to illustrate the application of the Jacobi elliptic function expansion method. As a result, three types of periodic wave solutions including Jacobi elliptic sine function, Jacobi elliptic cosine function and the third elliptic function solutions are obtained. It is shown that the shock wave solutions and solitary wave solutions can be obtained at their limit condition.展开更多
In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. By a fractional co...In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established.展开更多
This article is concerned with the Hirota direct method for studying novel multisoliton solutions of the discrete KdV equation. First the Hirota method was introduced, then the novel multisoliton solutions were obtain...This article is concerned with the Hirota direct method for studying novel multisoliton solutions of the discrete KdV equation. First the Hirota method was introduced, then the novel multisoliton solutions were obtained. Simultaneously the figures of the novel one-soliton solution and two-soliton solution were given and the singularity of the novel multisoliton solutions was discussed. Finally it was pointed out that the multisoliton solutions with sigularity can only be called soliton-like solutions. Key words differential-difference KdV equation - Hirota method - multisoliton-like solutions MSC 2000 35Q51 Project supported by the National Natural Science Foundation of China(Grant No. 19571052)展开更多
In this letter, the Clarkson-Kruskal direct method is extended to similarity reduce some differentialdifference equations. As examples, the differential-difference KZ equation and KP equation are considered.
A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations...A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, diseretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.展开更多
In this article, the interior layer for a second order nonlinear singularly perturbed differential-difference equation is considered. Using the methods of boundary function and fractional steps, we construct the formu...In this article, the interior layer for a second order nonlinear singularly perturbed differential-difference equation is considered. Using the methods of boundary function and fractional steps, we construct the formula of asymptotic expansion and point out that the boundary layer at t = 0 has a great influence upon the interior layer at t = a. At the same time, on the basis of differential inequality techniques, the existence of the smooth solution and the uniform validity of the asymptotic expansion are proved. Finally, an example is given to demonstrate the effectiveness of our result. The result of this article is new and it complements the previously known ones.展开更多
文摘Dierential geometry play a fundamental role in discussing partial dierential equations（PDEs） in mathematical physics. Recently discrete dierential geometry is an active mathematical terrain, which aims at the development and application of discrete equivalents of the geometric notions and methods of dierential geometry. In this paper, a discrete theory of exterior dierential calculus and the analogue of the Poincar′e lemma for dierential-dierence complex are proposed. They provide an intrinsic idea for developing the theory to discuss the integrability of dierence equations.
基金supported by the National Natural Science Foundation of China(11171013)supported by the Fundamental Research Funds for the Central Universitiesthe Research Funds of Renmin University of China(16XNH117)
文摘Using Nevanlinna theory of the value distribution of meromorphic functions, we investigate the problem of the existence of meromorphic solutions of some types of complex differential-difference equations and some properties of meromorphic solutions, and we ob- tain some results, which are the improvements and extensions of some results in references. Examples show that our results are precise.
文摘In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.
基金supported by the National Natural Science Foundation of China(10471067)NSF of Guangdong Province(04010474)
文摘Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj（z）f1（λj1）（z＋cj）=R2（z,f2（z））,j=1∑nβj（z）f2（λj2）（z＋cj）=R1（z,f1（z））. where λij （j = 1, 2,…, n; i = 1, 2） are finite non-negative integers, and cj （j = 1, 2,… , n） are distinct, nonzero complex numbers, αj（z）, βj（z） （j = 1,2,… ,n） are small functions relative to fi（z） （i =1, 2） respectively, Ri（z, f（z）） （i = 1, 2） are rational in fi（z） （i =1, 2） with coefficients which are small functions of fi（z） （i = 1, 2） respectively.
基金Supported by the National Natural Science Foundation of China (Grant No. 12001211)the Natural Science Foundation of Fujian Province,China (Grant No. 2021J01651)。
文摘In this paper,we mainly study the uniqueness of transcendental meromorphic solutions for a class of complex linear differential-difference equations.Specially,suppose that f(z)is a finite order transcendental meromorphic solution of complex linear differential-difference equation:W_(1)(z)f'(z+1)+W_(2)(z)f(z)=W_(3)(z),where W_(1)(z),W_(2)(z),W_(3)(z) are nonzero meromorphic functions,with their orders of growth being less than one,such that W_(1)(z)+W_(2)(z)■0.If f(z) and a meromorphic function g(z) share 0,1,∞ CM,then either f(z)≡g(z) or f(z)+g(z)≡f(z)g(z) or f^(2)(z)(g(z)-1)^(2)+g^(2)(z)(f(z)-1)^(2)≡f(z)g(z)(f(z)g(z)-1) or there exists a polynomial φ(z)=az+b_(0) such that ■ where a(≠0),a_(0),b_(0) are constants with e^(a_(0))≠e^(b_(0)).
文摘By using the Nevanlinna value distribution theory, we will mainly investigate the form of entire solutions with finite order on a type of system of differential-difference equations and a type of differential-difference equations, two interesting results are obtained. And it extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1130123311171119)+1 种基金the Natural Science Foundation of Jiangxi Province(Grant No.20132BAB211002)the Youth Science Foundation of Education Bureau of Jiangxi Province(Grant No.GJJ14271)
文摘The main purpose of this paper is to study the growth of meromorphic solutions of complex linear differential-difference equations L（z, f） =n∑i=0m∑j=0Aij（z）f^（j）（z ＋ ci） = 0 or F（z）with entire or meromorphic coefficients, and ci, i = 0,..., n being distinct complex numbers,where there is only one dominant coefficient.
基金Supported by the National Natural Science Foundation of China (Grant No.12161074)the Talent Introduction Research Foundation of Suqian University (Grant No.106-CK00042/028)+1 种基金Suqian Sci&Tech Program (Grant No.M202206)Sponsored by Qing Lan Project of Jiangsu Province and Suqian Talent Xiongying Plan of Suqian。
文摘Throughout this work,we explore the uniqueness properties of meromorphic functions concerning their interactions with complex differential-difference polynomial.Under the condition of finite order,we establish three distinct uniqueness results for a meromorphic function f associated with the differential-difference polynomial L_(η)^(n)f=Σ_(k=0)^(n)a_(k)f (z+k_(η))+a_(-1)f′.These results lead to a refined characterization of f (z)≡L_(η)^(n)f (z).Several illustrative examples are provided to demonstrate the sharpness and precision of the results obtained in this study.
基金Supported by the National Natural Science Foundation of China(Grant No.11971344).
文摘In this paper,we mainly investigate the forms of entire solutions for certain Fermattype partial differential-difference equations in C^(2)by using Nevanlinna’s theory of several complex variables.
文摘Three copper(Ⅱ),nickel and cadmium(Ⅱ)complexes,namely[Cu_(2)(μ-H2dbda)2(phen)2]·2H_(2)O(1),[Ni(μ-H2dbda)(μ-bpb)(H_(2)O)2]n(2),and[Cd(μ-H2dbda)(μ-bpa)]n(3),have been constructed hydrothermally using H4dbda(4,4'-dihydroxy-[1,1'-biphenyl]-3,3'-dicarboxylic acid),phen(1,10-phenanthroline),bpb(1,4-bis(pyrid-4-yl)benzene),bpa(bis(4-pyridyl)amine),and copper,nickel and cadmium chlorides at 160℃.The products were isolated as stable crystalline solids and were characterized by IR spectra,elemental analyses,thermogravimetric analyses,and singlecrystal X-ray diffraction analyses.Single-crystal X-ray diffraction analyses revealed that three complexes crystallize in the monoclinic P21/n,tetragonal I42d,and orthorhombic P21212 space groups.The complexes exhibit molecular dimers(1)or 2D metal-organic networks(2 and 3).The catalytic performances in the Knoevenagel reaction of these complexes were investigated.Complex 1 exhibits an effective catalytic activity and excellent reusability as a heterogeneous catalyst in the Knoevenagel reaction at room temperature.CCDC:2463800,1;2463801,2;2463802,3.
文摘Six new lanthanide complexes:[Ln(3,4-DEOBA)3(4,4'-DM-2,2'-bipy)]2·2C_(2)H_(5)OH,[Ln=Dy(1),Eu(2),Tb(3),Sm(4),Ho(5),Gd(6);3,4-DEOBA-=3,4-diethoxybenzoate,4,4'-DM-2,2'-bipy=4,4'-dimethyl-2,2'-bipyridine]were successfully synthesized by the volatilization of the solution at room temperature.The crystal structures of six complexes were determined by single-crystal X-ray diffraction technology.The results showed that the complexes all have a binuclear structure,and the structures contain free ethanol molecules.Moreover,the coordination number of the central metal of each structural unit is eight.Adjacent structural units interact with each other through hydrogen bonds and further expand to form 1D chain-like and 2D planar structures.After conducting a systematic study on the luminescence properties of complexes 1-4,their emission and excitation spectra were obtained.Experimental results indicated that the fluorescence lifetimes of complexes 2 and 3 were 0.807 and 0.845 ms,respectively.The emission spectral data of complexes 1-4 were imported into the CIE chromaticity coordinate system,and their corre sponding luminescent regions cover the yellow light,red light,green light,and orange-red light bands,respectively.Within the temperature range of 299.15-1300 K,the thermal decomposition processes of the six complexes were comprehensively analyzed by using TG-DSC/FTIR/MS technology.The hypothesis of the gradual loss of ligand groups during the decomposition process was verified by detecting the escaped gas,3D infrared spectroscopy,and ion fragment information detected by mass spectrometry.The specific decomposition path is as follows:firstly,free ethanol molecules and neutral ligands are removed,and finally,acidic ligands are released;the final product is the corresponding metal oxide.CCDC:2430420,1;2430422,2;2430419,3;2430424,4;2430421,5;2430423,6.
文摘Three zinc(Ⅱ),nickel(Ⅱ),and cadmium(Ⅱ)complexes,namely[Zn(μ-Htpta)(py)_(2)]n(1),[Ni(H_(2)biim)2(H_(2)O)2][Ni(tpta)(H_(2)biim)2(H_(2)O)]2·3H_(2)O(2),and[Cd_(3)(μ4-tpta)2(μ-dpe)_(3)]_(n)(3),have been constructed hydrothermally at 160℃ using H_(3)tpta([1,1':3',1″-terphenyl]-4,4',5'-tricarboxylic acid),py(pyridine),H_(2)biim(2,2'-biimidazole),dpe(1,2-di(4-pyridyl)ethylene),and zinc,nickel and cadmium chlorides,resulting in the formation of stable crystalline solids which were subsequently analyzed using infrared spectroscopy,element analysis,thermogravimetric analysis,as well as structural analyses conducted via single-crystal X-ray diffraction.The findings from these single-crystal Xray diffraction studies indicate that complexes 1-3 form crystals within the monoclinic system P2_(1)/c space group(1)or triclinic system P1 space group(2 and 3),and possess 1D,0D,and 3D structures,respectively.Complex 1 demonstrated substantial catalytic efficiency and excellent reusability as a heterogeneous catalyst in the reaction of Knoevenagel condensation under ambient temperature conditions.In addition,complex 1 also showcased notable anti-wear performance when used in polyalphaolefin synthetic lubricants.CCDC:2449810,1;2449811,2;2449812,3.
文摘The complexes 1-4 of cyclobutanocucurbit[5]uril(CyB5Q[5])with Na^(+)/K^(+)have been synthesized and characterized by single-crystal X-ray diffraction.The results show that although the inorganic salts are used when the cations are the same and the anions are different,in complex 1,Na^(+)closes one port of CyB5Q[5]through Na—O seven coordination bonds to form a molecular bowl;in complex 3,Na^(+)completely closes the two ports of CyB5Q[5]to form a molecular capsule with six Na—O coordination bonds;in complexes 2 and 4,the two ports of CyB5Q[5]are completely closed to form K—O coordinated molecular capsules,but the K^(+)of complex 2 is six-coordinated and that of complex 4 is eight-/nine-coordinated.and complex 4 are connected by three oxygen bridges to form a 1D molecular chain.CCDC:2457122,1;2457121,2;2457400,3;2457120,4.
文摘Complex trimalleolar ankle fractures are a major orthopaedic challenge,with an incidence of 4.22 per 10000 person-years in the United States and an annual cost of 3.4 billion dollars.This review synthesizes current evidence on diagnostic protocols and management strategies,highlighting optimal approaches and emerging trends.Initial care emphasizes soft tissue assessment,often guided by the Tscherne classification,and fracture classification systems.External fixation may be required in open injuries,while early open reduction and internal fixation within six days is linked to improved outcomes.Minimally invasive techniques for the lateral malleolus,including intramedullary nailing and locking plates,are effective,while medial malleolus fractures are commonly managed with screw fixation or tension-band wiring.Posterior malleolus fragments involving more than 25%of the articular surface usually warrant fixation.Alternatives to syndesmotic screws,such as cortical buttons or high-strength sutures,reduce the need for secondary procedures.Arthroscopic-assisted open reduction and internal fixation benefits younger,active patients by enabling concurrent management of intra-articular and ligamentous injuries.Postoperative care prioritizes early weight-bearing and validated functional scores.Despite advances,complications remain common,and further research is needed to refine surgical strategies and improve outcomes.
基金the State Key Programme of Basic Research of China under，高等学校博士学科点专项科研项目
文摘The Jacobi elliptic function expansion method is extended to derive the explicit periodic wave solutions for nonlinear differential-difference equations. Three well-known examples are chosen to illustrate the application of the Jacobi elliptic function expansion method. As a result, three types of periodic wave solutions including Jacobi elliptic sine function, Jacobi elliptic cosine function and the third elliptic function solutions are obtained. It is shown that the shock wave solutions and solitary wave solutions can be obtained at their limit condition.
文摘In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established.
文摘This article is concerned with the Hirota direct method for studying novel multisoliton solutions of the discrete KdV equation. First the Hirota method was introduced, then the novel multisoliton solutions were obtained. Simultaneously the figures of the novel one-soliton solution and two-soliton solution were given and the singularity of the novel multisoliton solutions was discussed. Finally it was pointed out that the multisoliton solutions with sigularity can only be called soliton-like solutions. Key words differential-difference KdV equation - Hirota method - multisoliton-like solutions MSC 2000 35Q51 Project supported by the National Natural Science Foundation of China(Grant No. 19571052)
文摘In this letter, the Clarkson-Kruskal direct method is extended to similarity reduce some differentialdifference equations. As examples, the differential-difference KZ equation and KP equation are considered.
基金supported by National Natural Science Foundation of China and the Natural Science Foundation of Shandong Province
文摘A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, diseretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.
基金Supported by the National Natural Science Funds (11071075)the Natural Science Foundation of Shanghai(10ZR1409200)+1 种基金the National Laboratory of Biomacromolecules,Institute of Biophysics,Chinese Academy of Sciencesthe E-Institutes of Shanghai Municipal Education Commissions(E03004)
文摘In this article, the interior layer for a second order nonlinear singularly perturbed differential-difference equation is considered. Using the methods of boundary function and fractional steps, we construct the formula of asymptotic expansion and point out that the boundary layer at t = 0 has a great influence upon the interior layer at t = a. At the same time, on the basis of differential inequality techniques, the existence of the smooth solution and the uniform validity of the asymptotic expansion are proved. Finally, an example is given to demonstrate the effectiveness of our result. The result of this article is new and it complements the previously known ones.
