In this paper we study the boundedness and unboundedness of the solutions of the smooth and discontinuous(SD)oscillatorbegin{equation*}x''+f(x)x'+x-frac{x}{sqrt{x^{2}+alpha^{2}}}=p(t).end{equation*}Since f...In this paper we study the boundedness and unboundedness of the solutions of the smooth and discontinuous(SD)oscillatorbegin{equation*}x''+f(x)x'+x-frac{x}{sqrt{x^{2}+alpha^{2}}}=p(t).end{equation*}Since f(x)≠0,the system is non-Hamiltonian,so we have to introduce some reversibility assumptions to apply a suitable twist theorem,for reversible maps with small twist.Moreover,when the nonnegative parameterαdecreases to 0,the system becomes discontinuous.In this case,we need to introduce some suitable transformations to overcome the lack of regularity.We will prove that for any nonnegative parameterα,when p(t)is an odd periodic function satisfying∣∣∫2π0p(t)sintdift∣∣<4,all the solutions are bounded;when p(t)satisfies∣∣∫2π0p(t)sintdift∣∣>4,the SD oscillator has unbounded solutions,and when p(t)satisfies∣∣∫_(0)^(2π)p(t)sintdift∣∣≥4+|F|_(∞),all the solutions are unbounded.展开更多
Solid-state Na metal batteries(SSNBs),known for its low cost,high safety,and high energy density,hold a significant position in the next generation of rechargeable batteries.However,the urgent challenge of poor interf...Solid-state Na metal batteries(SSNBs),known for its low cost,high safety,and high energy density,hold a significant position in the next generation of rechargeable batteries.However,the urgent challenge of poor interfacial contact in solid-state electrolytes has hindered the commercialization of SSNBs.Driven by the concept of intimate electrode-electrolyte interface design,this study employs a combination of NaK alloy and carbon nanotubes to prepare a semi-solid NaK(NKC)anode.Unlike traditional Na anodes,the paintable paste-like NKC anode exhibits superior adhesion and interface compatibility with both current collectors and gel electrolytes,significantly enhancing the intimate contact of electrode-electrolyte interface.Additionally,the filling of SiO_(2)nanoparticles improves the wettability of NaK alloy on gel polymer electrolytes,further achieving a conformal interface contact.Consequently,the overpotential of the NKC symmetric cell is markedly lower than that of the Na symmetric cell when subjected to a long cycle of 300 h.The full cell coupled with Na_(3)V_(3)(PO_(4))_(2)cathodes had an initial discharge capacity of 106.8 mAh·g^(-1)with a capacity retention of 89.61%after 300 cycles,and a high discharge capacity of 88.1 mAh·g^(-1)even at a high rate of 10 C.The outstanding electrochemical performance highlights the promising application potential of the NKC electrode.展开更多
This paper studies the asymptotic behavior of solutions of the difference equation X（n＋1）=max{C1/Xn,C2/X（n-1）},n=0,1,…,where the parameters C1, C2 and the initial conditions x（-1）, xo are nonzero real numbers...This paper studies the asymptotic behavior of solutions of the difference equation X（n＋1）=max{C1/Xn,C2/X（n-1）},n=0,1,…,where the parameters C1, C2 and the initial conditions x（-1）, xo are nonzero real numbers. More precisely, it has been proved that： （1） if Ct 〈 0 and C2 〉 0, then every solution of the equation is eventually periodic; （2） if Ct 〈 0 and C2 〈 0, then every solution of the equation is unbounded when C1≠P C2 or is eventually periodic when C1 = C2.展开更多
基金supported by the Key Research Funds for the Universities of Henan Province(No.19A110018)the Foundation for Key Teachers of Henan Polytechnic University(No.2022XQG-09)。
文摘In this paper we study the boundedness and unboundedness of the solutions of the smooth and discontinuous(SD)oscillatorbegin{equation*}x''+f(x)x'+x-frac{x}{sqrt{x^{2}+alpha^{2}}}=p(t).end{equation*}Since f(x)≠0,the system is non-Hamiltonian,so we have to introduce some reversibility assumptions to apply a suitable twist theorem,for reversible maps with small twist.Moreover,when the nonnegative parameterαdecreases to 0,the system becomes discontinuous.In this case,we need to introduce some suitable transformations to overcome the lack of regularity.We will prove that for any nonnegative parameterα,when p(t)is an odd periodic function satisfying∣∣∫2π0p(t)sintdift∣∣<4,all the solutions are bounded;when p(t)satisfies∣∣∫2π0p(t)sintdift∣∣>4,the SD oscillator has unbounded solutions,and when p(t)satisfies∣∣∫_(0)^(2π)p(t)sintdift∣∣≥4+|F|_(∞),all the solutions are unbounded.
基金National Natural Science Foundation of China(52073253)。
文摘Solid-state Na metal batteries(SSNBs),known for its low cost,high safety,and high energy density,hold a significant position in the next generation of rechargeable batteries.However,the urgent challenge of poor interfacial contact in solid-state electrolytes has hindered the commercialization of SSNBs.Driven by the concept of intimate electrode-electrolyte interface design,this study employs a combination of NaK alloy and carbon nanotubes to prepare a semi-solid NaK(NKC)anode.Unlike traditional Na anodes,the paintable paste-like NKC anode exhibits superior adhesion and interface compatibility with both current collectors and gel electrolytes,significantly enhancing the intimate contact of electrode-electrolyte interface.Additionally,the filling of SiO_(2)nanoparticles improves the wettability of NaK alloy on gel polymer electrolytes,further achieving a conformal interface contact.Consequently,the overpotential of the NKC symmetric cell is markedly lower than that of the Na symmetric cell when subjected to a long cycle of 300 h.The full cell coupled with Na_(3)V_(3)(PO_(4))_(2)cathodes had an initial discharge capacity of 106.8 mAh·g^(-1)with a capacity retention of 89.61%after 300 cycles,and a high discharge capacity of 88.1 mAh·g^(-1)even at a high rate of 10 C.The outstanding electrochemical performance highlights the promising application potential of the NKC electrode.
文摘This paper studies the asymptotic behavior of solutions of the difference equation X（n＋1）=max{C1/Xn,C2/X（n-1）},n=0,1,…,where the parameters C1, C2 and the initial conditions x（-1）, xo are nonzero real numbers. More precisely, it has been proved that： （1） if Ct 〈 0 and C2 〉 0, then every solution of the equation is eventually periodic; （2） if Ct 〈 0 and C2 〈 0, then every solution of the equation is unbounded when C1≠P C2 or is eventually periodic when C1 = C2.
