In this paper,we define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better...In this paper,we define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better estimation for permuting tri-homomorphisms and permuting tri-derivations in unital C*-algebras and Banach algebras by the vector-valued alternative fixed point theorem.展开更多
The authors establish a kind of inequalities for nonnegative submartingales which depend on two functions ψ and ψ, and obtain the equivalent conditions for Φ and ψ such that this kind of inequalities holds. In t...The authors establish a kind of inequalities for nonnegative submartingales which depend on two functions ψ and ψ, and obtain the equivalent conditions for Φ and ψ such that this kind of inequalities holds. In the caseΦ=ψ ε△2, it is proved that this necessary and sufficient condition is equivalent to qΦ >1.展开更多
The synthesis of functionalized rubber copolymers is a topic of great research interest.In this study,we present a novel approach for the direct construction ofα-functionalized 3,4-polyisoprene through polymerization...The synthesis of functionalized rubber copolymers is a topic of great research interest.In this study,we present a novel approach for the direct construction ofα-functionalized 3,4-polyisoprene through polymerization of polar monomers and isoprene monomer.Theα-functionalized 3,4-polyisoprene was successfully synthesized via in situ sequential polymerization using the iron-based catalytic system(Fe(acac)_(3)/IITP/AliBu_(3)),exhibiting high activity and resistance to polar monomers without requiring protection of polar groups.The structure ofα-functionalized 3,4-polyisoprene was confirmed by proton nuclear magnetic resonance spectroscopy(^(1)H-NMR)and two-dimensional diffusion-ordered spectroscopy(2D DOSY)spectra analysis.The introduction of polar groups,particularly hydroxyl groups,enhanced the hydrophilicity of the copolymer.This was evidenced by a decrease in the water contact angle from 106.9°to 96.4°with increasing hydroxyl content in the copolymer.展开更多
In this paper,the authors prove that the parameterized area integralμ_(Ω,S)^(ρ)and the parameterized Littlewood-Paley g_(δ)^(*)-functionμ_(Ω,δ)^(*,ρ)are bounded on two-weight grand homogeneous variable Herz-Mo...In this paper,the authors prove that the parameterized area integralμ_(Ω,S)^(ρ)and the parameterized Littlewood-Paley g_(δ)^(*)-functionμ_(Ω,δ)^(*,ρ)are bounded on two-weight grand homogeneous variable Herz-Morrey spaces MK_(p),θ,q(·))^(α(·),λ)(ω_(1),ω_(2)),where θ>0,λ∈(2,∞),q(·)∈B(R^(n)),α(·)∈L^(∞)(R^(n)),ω_(1)∈A_(p_(ω_(1)))for p_(ω_(1))∈[1,∞]and ω_(2) is a weight.Furthermore,the authors prove that the commutators[b,μ_(Ω,S)^(ρ)]which is formed by b∈BMO(R^(n))and the μ_(Ω,S)^(ρ),and the[b,μ_(Ω,δ)^(*,ρ)]generated by b∈BMO(R^(n))and theμ_(Ω,δ)^(*,ρ)are bounded on MK_(p),θ,q(·))^(α(·),λ)(ω_(1),ω_(2)),respectively.展开更多
About the accurate value of ψ （ x ） = Γ′ （ x ） / Γ （ x）, we know only ψ （1）and ψ （1/ 2）.In this paper, by calculating the accurate values of some series expansion of a function and applying some skills...About the accurate value of ψ （ x ） = Γ′ （ x ） / Γ （ x）, we know only ψ （1）and ψ （1/ 2）.In this paper, by calculating the accurate values of some series expansion of a function and applying some skills of complex analysis, we prove that the expression of the ψ-function admits a finite expression in elementary function for rational number z. As an application, some examples are given.展开更多
Cubic equations of state(EOSs) are simple and easy at calculation. One way of improving the accuracy of a cubic EOS is through the modification of temperature-dependent energy parameter by using alpha-function.The ind...Cubic equations of state(EOSs) are simple and easy at calculation. One way of improving the accuracy of a cubic EOS is through the modification of temperature-dependent energy parameter by using alpha-function.The industrial applications of natural gas are very wide and as a result, prediction of thermodynamic properties and phase behavior of natural gas is an important part of design for such processes. In this work we develop a newα-function for the Peng-Robinson(PR) EOS with the parameters optimized especially for natural gas components.The parameters are generalized as a linear function of acentric factor. The results are compared to the predictions from original PR EOS and other α-functions in literature. It is shown that the new α-function presents a good accuracy with the average deviation of 1.42% for natural gas components.展开更多
Azulene, one of representative nonbenzenoid aromatic hydrocarbons, exhibits unique molecular structure and distinctive physical and chemical properties. Herein, azulenoisoindigo(Az II), an azulene-based isoindigo anal...Azulene, one of representative nonbenzenoid aromatic hydrocarbons, exhibits unique molecular structure and distinctive physical and chemical properties. Herein, azulenoisoindigo(Az II), an azulene-based isoindigo analogue, is designed and synthesized, which has a twisted molecular backbone and R/Sisomers in single crystals. Interestingly, Az II shows the characteristics of both isoindigo and azulene,such as reversible redox behavior and reversible proton responsiveness. UV-vis-NIR,^(1)H NMR and electron paramagnetic resonance(EPR) measurements were carried out to get insights into the possible mechanism of the proton-responsive property of Az II. The results demonstrated that only one azulenyl moiety of molecule of Az II was protonated and deprotonated, and the protonated Az II can be further oxidized to form azulenium cation radicals.展开更多
Humic acid(HA)was carbonized at 300,400 and 500℃ and then functionalized with 1 wt%–12 wt%Fe(Ⅲ)respectively[CHA300/400/500-Fe(Ⅲ)].Adsorption of such Fe(III)-functionalized carbonized HA as adsorbents to aqueous te...Humic acid(HA)was carbonized at 300,400 and 500℃ and then functionalized with 1 wt%–12 wt%Fe(Ⅲ)respectively[CHA300/400/500-Fe(Ⅲ)].Adsorption of such Fe(III)-functionalized carbonized HA as adsorbents to aqueous tetracycline(TC:25 mg·L^-1)was studied.The adsorption equilibrium time for CHA400-Fe(Ⅲ)to TC was 6 h faster and the adsorption removal efficiency(Re)was two times higher than that of HA/CHA.The adsorption Reof CHA400-Fe(Ⅲ)loaded 10%iron[CHA400-(10%)Fe(Ⅲ)]to TC could reach 99.8%at 8 h and still kept80.6%after 8 cycles.The adsorption kinetics were well fitted to the pseudo-second-order equation and the adsorption isotherms could be well delineated via Langmuir equations(R^2N 0.99),indicating that the homogeneous chemical adsorption of TC occurred on the adsorbents.The main adsorption mechanisms of TC were complexation Fe(III)and hydrophobic distribution.Electropositive and electronegative repulsion between TC and CHA400-(10%)Fe(Ⅲ)at lowly p H(2)and highly p H(8–10)respectively,leaded to the relatively low adsorption capacity and more notable influence of ion concentration.When the p H was between 4 and 8,TC mainly existed in neutral molecules(TCH2),so the influence of ion concentration was not obvious.The dynamic adsorption results showed that the CHA400-(10%)Fe(Ⅲ)could continuously treat about 2.4 L TC(27 mg·L^-1)wastewater with the effluent concentration as low as 0.068 mg·L^-1.Our study suggested a broad application prospect of a new,effective,lowcost and environment-friendly adsorbent CHA400-(10%)Fe(Ⅲ)for treatment of low-concentration TC polluted wastewater.展开更多
To better understand the role of the-NH_(2)group in adsorption process of phenolic wastewaters,NH_(2)-functionalized MIL-53(Al)composites with activated carbon(NH_(2)-M(Al)@(B)AC)were prepared.The results showed that ...To better understand the role of the-NH_(2)group in adsorption process of phenolic wastewaters,NH_(2)-functionalized MIL-53(Al)composites with activated carbon(NH_(2)-M(Al)@(B)AC)were prepared.The results showed that the-NH_(2)group could increase the mesopore volume for composites,which promotes mass transfer and full utilization of active sites,because hierarchical mesopore structure makes the adsorbent easier to enter the internal adsorption sites.Furthermore,the introduction of the-NH_(2)group can improve the adsorption capacity,decrease the activation energy,and enhance the interaction between the adsorbent and p-nitrophenol,demonstrating that the-NH_(2)group plays a crucial role in the adsorption of p-nitrophenol.The density functional theory calculation results show that the H-bond interaction between the-NH_(2)group in the adsorbent and the-NO_(2)in the p-nitrophenol(adsorption energy of -35.5 kJ·mol^(-1)),and base-acid interaction between the primary-NH_(2)group in the adsorbent and the acidic-OH group in the p-nitrophenol(adsorption energy of -27.3 kJ·mol^(-1))are predominant mechanisms for adsorption in terms of the NH_(2)-functionalized adsorbent.Both NH_(2)-functionalized M(Al)@AC and M(Al)@BAC composites exhibited higher p-nitrophenol adsorption capacity than corresponding nonfunctionalized composites.Among the composites,the NH_(2)-M(Al)@BAC had the highest p-nitrophenol adsorption capacity of 474 mg·g^(-1).展开更多
To prove RH, studying <span style="white-space:nowrap;">ζ and using pure analysis method likely are two kinds of the incorrect guide. Actually, a unique hope may study Riemann function <img src=&qu...To prove RH, studying <span style="white-space:nowrap;">ζ and using pure analysis method likely are two kinds of the incorrect guide. Actually, a unique hope may study Riemann function <img src="Edit_b4e53620-7ae2-4a2b-aee0-351c62aef8cd.png" width="250" height="20" alt="" /> by geometric analysis, which has the symmetry: <span style="white-space:nowrap;">v = 0 if <span style="white-space:nowrap;">β = 0, and <img src="Edit_8c67c5d7-c1d4-4cad-8792-78e4bd172ebd.png" width="150" height="28" alt="" /> Assume that |u| is single peak in each root-interval <img src="Edit_a91df253-2965-4b03-8033-54aba2e23036.png" width="85" height="27" alt="" /> of u for any fixed <span style="white-space:nowrap;">β <span style="white-space:nowrap;">∈ (0,1/2], using the slope u<sub>t </sub>of the single peak, we prove that v has opposite signs at two end-points of I<sub>j</sub>, there surely is an inner point so that v = 0, so {|u|,|v|/<span style="white-space:nowrap;">β}form a local peak-valley structure, and have positive lower bound <img src="Edit_04798c0f-8e21-4a3a-ae12-0e28b01ee348.png" width="167" height="22" alt="" />in I<sub>j</sub>. Because each t must lie in some <em style="white-space:normal;">I<sub style="white-space:normal;">j</sub> , then ||<span style="white-space:nowrap;">ξ|| > 0 is valid for any t. In this way, the summation process of <span style="white-space:nowrap;">ξ is avoided. We have proved the main theorem: Assume that u (t, <span style="white-space:nowrap;">β) is single peak, then RH is valid for any <img src="Edit_ed8521a3-63b1-417f-a3b0-a3c790bae519.png" width="140" height="19" alt="" />. If using the equivalence of Lagarias (1999), the assumption of single peak can be canceled. Therefore our new thinking is that we have found the local peak-valley structure of <span style="white-space:nowrap;">ξ, which may be the geometry structure expected by Bombieri (2000), and proposed a basic framework of proving RH by geometric analysis.展开更多
The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-fun...The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-function, Appell function F3 and a general class of poly- nomials. The results obtained provide unification and extension of the results given by Saxena et al. [13], Srivastava and Grag [17], Srivastava et al. [20], and etc. The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus. On account of the general nature of the Saigo-Maeda operators, R-function, and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notably H-function, /-function, Mittag-Leffier function, generalized Wright hypergeometric function, generalized Bessel-Maitland function follow as special cases of our main findings.展开更多
The functional equation f(z)^n+g(z)^n=1 can be interpreted as the Fermat-type equations over function field.In this paper,by using Nevanlinna theory of meromorphic functions,we investigate the existence of meromorphic...The functional equation f(z)^n+g(z)^n=1 can be interpreted as the Fermat-type equations over function field.In this paper,by using Nevanlinna theory of meromorphic functions,we investigate the existence of meromorphic solutions of hyper-order strictly less than 1 to the Fermat-type functional equation(a0f(z)+a1f(z+c))^(3)+(b0f(z)+b1f(z+c))3=e^(αz+β),where a0,a1,b0,b1,α,β,c are complex constants and c≠0.展开更多
基金partially supported by the Natural Sciences and Engineering Research Council of Canada(2019-03907)。
文摘In this paper,we define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better estimation for permuting tri-homomorphisms and permuting tri-derivations in unital C*-algebras and Banach algebras by the vector-valued alternative fixed point theorem.
文摘The authors establish a kind of inequalities for nonnegative submartingales which depend on two functions ψ and ψ, and obtain the equivalent conditions for Φ and ψ such that this kind of inequalities holds. In the caseΦ=ψ ε△2, it is proved that this necessary and sufficient condition is equivalent to qΦ >1.
基金financially supported by the National Key R&D Program of China(No.2022YFB3704701)the National Key R&D Program of China(No.2022YFC2603502)+1 种基金Natural Science Foundation of Shandong Province(No.ZR2022QE271)the financial support from the Taishan Scholars Program(No.tsqn202211165)。
文摘The synthesis of functionalized rubber copolymers is a topic of great research interest.In this study,we present a novel approach for the direct construction ofα-functionalized 3,4-polyisoprene through polymerization of polar monomers and isoprene monomer.Theα-functionalized 3,4-polyisoprene was successfully synthesized via in situ sequential polymerization using the iron-based catalytic system(Fe(acac)_(3)/IITP/AliBu_(3)),exhibiting high activity and resistance to polar monomers without requiring protection of polar groups.The structure ofα-functionalized 3,4-polyisoprene was confirmed by proton nuclear magnetic resonance spectroscopy(^(1)H-NMR)and two-dimensional diffusion-ordered spectroscopy(2D DOSY)spectra analysis.The introduction of polar groups,particularly hydroxyl groups,enhanced the hydrophilicity of the copolymer.This was evidenced by a decrease in the water contact angle from 106.9°to 96.4°with increasing hydroxyl content in the copolymer.
基金Supported by the National Natural Science Foundation of China(Grant No.12201500)。
文摘In this paper,the authors prove that the parameterized area integralμ_(Ω,S)^(ρ)and the parameterized Littlewood-Paley g_(δ)^(*)-functionμ_(Ω,δ)^(*,ρ)are bounded on two-weight grand homogeneous variable Herz-Morrey spaces MK_(p),θ,q(·))^(α(·),λ)(ω_(1),ω_(2)),where θ>0,λ∈(2,∞),q(·)∈B(R^(n)),α(·)∈L^(∞)(R^(n)),ω_(1)∈A_(p_(ω_(1)))for p_(ω_(1))∈[1,∞]and ω_(2) is a weight.Furthermore,the authors prove that the commutators[b,μ_(Ω,S)^(ρ)]which is formed by b∈BMO(R^(n))and the μ_(Ω,S)^(ρ),and the[b,μ_(Ω,δ)^(*,ρ)]generated by b∈BMO(R^(n))and theμ_(Ω,δ)^(*,ρ)are bounded on MK_(p),θ,q(·))^(α(·),λ)(ω_(1),ω_(2)),respectively.
基金Supported by the National Natural Science Foundation of China (10926128)
文摘About the accurate value of ψ （ x ） = Γ′ （ x ） / Γ （ x）, we know only ψ （1）and ψ （1/ 2）.In this paper, by calculating the accurate values of some series expansion of a function and applying some skills of complex analysis, we prove that the expression of the ψ-function admits a finite expression in elementary function for rational number z. As an application, some examples are given.
文摘Cubic equations of state(EOSs) are simple and easy at calculation. One way of improving the accuracy of a cubic EOS is through the modification of temperature-dependent energy parameter by using alpha-function.The industrial applications of natural gas are very wide and as a result, prediction of thermodynamic properties and phase behavior of natural gas is an important part of design for such processes. In this work we develop a newα-function for the Peng-Robinson(PR) EOS with the parameters optimized especially for natural gas components.The parameters are generalized as a linear function of acentric factor. The results are compared to the predictions from original PR EOS and other α-functions in literature. It is shown that the new α-function presents a good accuracy with the average deviation of 1.42% for natural gas components.
基金financially supported by the National Natural Science Foundation of China (Nos. 21790362, 22075310 and21522209)the “Strategic Priority Research Program of Chinese Academy of Sciences”(No. XDB12010100)+1 种基金the Science and Technology Commission of Shanghai Municipality (Nos. 19XD1424700 and18JC1410600)SIOC. An early preprint of this work appeared on Chem Rxiv [47]。
文摘Azulene, one of representative nonbenzenoid aromatic hydrocarbons, exhibits unique molecular structure and distinctive physical and chemical properties. Herein, azulenoisoindigo(Az II), an azulene-based isoindigo analogue, is designed and synthesized, which has a twisted molecular backbone and R/Sisomers in single crystals. Interestingly, Az II shows the characteristics of both isoindigo and azulene,such as reversible redox behavior and reversible proton responsiveness. UV-vis-NIR,^(1)H NMR and electron paramagnetic resonance(EPR) measurements were carried out to get insights into the possible mechanism of the proton-responsive property of Az II. The results demonstrated that only one azulenyl moiety of molecule of Az II was protonated and deprotonated, and the protonated Az II can be further oxidized to form azulenium cation radicals.
基金financially supported by the National Natural Science Foundation of China(No.51641209)。
文摘Humic acid(HA)was carbonized at 300,400 and 500℃ and then functionalized with 1 wt%–12 wt%Fe(Ⅲ)respectively[CHA300/400/500-Fe(Ⅲ)].Adsorption of such Fe(III)-functionalized carbonized HA as adsorbents to aqueous tetracycline(TC:25 mg·L^-1)was studied.The adsorption equilibrium time for CHA400-Fe(Ⅲ)to TC was 6 h faster and the adsorption removal efficiency(Re)was two times higher than that of HA/CHA.The adsorption Reof CHA400-Fe(Ⅲ)loaded 10%iron[CHA400-(10%)Fe(Ⅲ)]to TC could reach 99.8%at 8 h and still kept80.6%after 8 cycles.The adsorption kinetics were well fitted to the pseudo-second-order equation and the adsorption isotherms could be well delineated via Langmuir equations(R^2N 0.99),indicating that the homogeneous chemical adsorption of TC occurred on the adsorbents.The main adsorption mechanisms of TC were complexation Fe(III)and hydrophobic distribution.Electropositive and electronegative repulsion between TC and CHA400-(10%)Fe(Ⅲ)at lowly p H(2)and highly p H(8–10)respectively,leaded to the relatively low adsorption capacity and more notable influence of ion concentration.When the p H was between 4 and 8,TC mainly existed in neutral molecules(TCH2),so the influence of ion concentration was not obvious.The dynamic adsorption results showed that the CHA400-(10%)Fe(Ⅲ)could continuously treat about 2.4 L TC(27 mg·L^-1)wastewater with the effluent concentration as low as 0.068 mg·L^-1.Our study suggested a broad application prospect of a new,effective,lowcost and environment-friendly adsorbent CHA400-(10%)Fe(Ⅲ)for treatment of low-concentration TC polluted wastewater.
基金supported by the National Natural Science Foundation of China(22008134)。
文摘To better understand the role of the-NH_(2)group in adsorption process of phenolic wastewaters,NH_(2)-functionalized MIL-53(Al)composites with activated carbon(NH_(2)-M(Al)@(B)AC)were prepared.The results showed that the-NH_(2)group could increase the mesopore volume for composites,which promotes mass transfer and full utilization of active sites,because hierarchical mesopore structure makes the adsorbent easier to enter the internal adsorption sites.Furthermore,the introduction of the-NH_(2)group can improve the adsorption capacity,decrease the activation energy,and enhance the interaction between the adsorbent and p-nitrophenol,demonstrating that the-NH_(2)group plays a crucial role in the adsorption of p-nitrophenol.The density functional theory calculation results show that the H-bond interaction between the-NH_(2)group in the adsorbent and the-NO_(2)in the p-nitrophenol(adsorption energy of -35.5 kJ·mol^(-1)),and base-acid interaction between the primary-NH_(2)group in the adsorbent and the acidic-OH group in the p-nitrophenol(adsorption energy of -27.3 kJ·mol^(-1))are predominant mechanisms for adsorption in terms of the NH_(2)-functionalized adsorbent.Both NH_(2)-functionalized M(Al)@AC and M(Al)@BAC composites exhibited higher p-nitrophenol adsorption capacity than corresponding nonfunctionalized composites.Among the composites,the NH_(2)-M(Al)@BAC had the highest p-nitrophenol adsorption capacity of 474 mg·g^(-1).
文摘To prove RH, studying <span style="white-space:nowrap;">ζ and using pure analysis method likely are two kinds of the incorrect guide. Actually, a unique hope may study Riemann function <img src="Edit_b4e53620-7ae2-4a2b-aee0-351c62aef8cd.png" width="250" height="20" alt="" /> by geometric analysis, which has the symmetry: <span style="white-space:nowrap;">v = 0 if <span style="white-space:nowrap;">β = 0, and <img src="Edit_8c67c5d7-c1d4-4cad-8792-78e4bd172ebd.png" width="150" height="28" alt="" /> Assume that |u| is single peak in each root-interval <img src="Edit_a91df253-2965-4b03-8033-54aba2e23036.png" width="85" height="27" alt="" /> of u for any fixed <span style="white-space:nowrap;">β <span style="white-space:nowrap;">∈ (0,1/2], using the slope u<sub>t </sub>of the single peak, we prove that v has opposite signs at two end-points of I<sub>j</sub>, there surely is an inner point so that v = 0, so {|u|,|v|/<span style="white-space:nowrap;">β}form a local peak-valley structure, and have positive lower bound <img src="Edit_04798c0f-8e21-4a3a-ae12-0e28b01ee348.png" width="167" height="22" alt="" />in I<sub>j</sub>. Because each t must lie in some <em style="white-space:normal;">I<sub style="white-space:normal;">j</sub> , then ||<span style="white-space:nowrap;">ξ|| > 0 is valid for any t. In this way, the summation process of <span style="white-space:nowrap;">ξ is avoided. We have proved the main theorem: Assume that u (t, <span style="white-space:nowrap;">β) is single peak, then RH is valid for any <img src="Edit_ed8521a3-63b1-417f-a3b0-a3c790bae519.png" width="140" height="19" alt="" />. If using the equivalence of Lagarias (1999), the assumption of single peak can be canceled. Therefore our new thinking is that we have found the local peak-valley structure of <span style="white-space:nowrap;">ξ, which may be the geometry structure expected by Bombieri (2000), and proposed a basic framework of proving RH by geometric analysis.
基金NBHM Department of Atomic Energy,Government of India,Mumbai for the finanicai assistance under PDF sanction no.2/40(37)/2014/R&D-II/14131
文摘The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-function, Appell function F3 and a general class of poly- nomials. The results obtained provide unification and extension of the results given by Saxena et al. [13], Srivastava and Grag [17], Srivastava et al. [20], and etc. The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus. On account of the general nature of the Saigo-Maeda operators, R-function, and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notably H-function, /-function, Mittag-Leffier function, generalized Wright hypergeometric function, generalized Bessel-Maitland function follow as special cases of our main findings.
基金Supported by the National Natural Science Foundation of China(Grant No.11971344)。
文摘The functional equation f(z)^n+g(z)^n=1 can be interpreted as the Fermat-type equations over function field.In this paper,by using Nevanlinna theory of meromorphic functions,we investigate the existence of meromorphic solutions of hyper-order strictly less than 1 to the Fermat-type functional equation(a0f(z)+a1f(z+c))^(3)+(b0f(z)+b1f(z+c))3=e^(αz+β),where a0,a1,b0,b1,α,β,c are complex constants and c≠0.
