In recent years,there has been growing interest in the study of chiral active materials,which consist of building blocks that show active dynamics featuring chiral symmetry breaking,e.g.,particles that rotate in a com...In recent years,there has been growing interest in the study of chiral active materials,which consist of building blocks that show active dynamics featuring chiral symmetry breaking,e.g.,particles that rotate in a common direction.These materials exhibit fascinating phenomena such as odd viscosity,odd diffusivity,active turbulence in fluids,vivid dislocation dynamics or odd elasticity in crystals or elastic materials,and hyperuniform states.The systematic study of soft chiral active matter systems is relatively new,starting around 2017,but has already shown promising applications in robust cargo transport,segregation and mixing dynamics,or manipulation of metamaterials.In this review,we summarize recent experimental and theoretical advances in this field,highlighting the emergence of anti-symmetric and odd stresses and ensuring effects such as odd viscosity or topologically protected edge modes.We further discuss the underlying mechanisms and provide insights into the potential of chiral active matter for various applications.展开更多
The filtration structure of finite-dimensional special odd Hamilton superalgebras over a field of prime characteristic was studied. By determining ad-nilpotent dements in the even part, the natural filtration of speci...The filtration structure of finite-dimensional special odd Hamilton superalgebras over a field of prime characteristic was studied. By determining ad-nilpotent dements in the even part, the natural filtration of special odd Hamiltonian superalgebras is proved to be invariant. Using this result, the special odd Hamilton superalgebras is classified. Finally, the automorphism group of the restricted special odd Hamilton superalgebras is determined.展开更多
In this paper, the forced odd order neutral differential equations of the form are considered d n d t n[x(t)-R(t)x(t-τ)]+P(t)x(t-σ)=f(t),t≥t 0.A sufficient condition for the oscillation of all solutions is...In this paper, the forced odd order neutral differential equations of the form are considered d n d t n[x(t)-R(t)x(t-τ)]+P(t)x(t-σ)=f(t),t≥t 0.A sufficient condition for the oscillation of all solutions is obtained.展开更多
The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generali...The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .展开更多
This paper introduces the basic idea and provides the mathematical formulation of the delayed feedback control （DFC） methodology, which has been widely used in chaos control. Stability analysis including the well-kn...This paper introduces the basic idea and provides the mathematical formulation of the delayed feedback control （DFC） methodology, which has been widely used in chaos control. Stability analysis including the well-known odd number linfitation of the DFC is reviewed. Some new developments in characterizing the limitation of the DFC are presented. Various modified DFC methods, which are developed in order to overcome the odd number limitation, are also described. Finally, some open problems in this research field are discussed.展开更多
We propose a new generator of continuous distributions with at least four positive parameters called the Kumaraswamy-Odd Rayleigh-G family. Some special cases were presented. The plots of the Kumaraswamy Odd Rayleigh ...We propose a new generator of continuous distributions with at least four positive parameters called the Kumaraswamy-Odd Rayleigh-G family. Some special cases were presented. The plots of the Kumaraswamy Odd Rayleigh Log-Logistic (</span><i><span style="font-family:Verdana;">KORLL</span></i><span style="font-family:Verdana;">) distribution indicate that the distribution can take many shapes depending on the parameter values. The negative skewness and kurtosis indicates that the distribution has li</span><span style="font-family:Verdana;">ghter tails than the normal distribution. The Monte Carlo simulation results indicate that the estimated biases decrease when the sample size increases. Furthermore, </span></span><span style="font-family:Verdana;">the root mean squared error estimates decay</span><span style="font-family:""><span style="font-family:Verdana;"> towards zero as the sample size increases. This part shows the consistency of the maximum likelihood estimators. From the considered analytical measures, the new </span><i><span style="font-family:Verdana;">KORLL</span></i><span style="font-family:Verdana;"> provides </span></span><span style="font-family:Verdana;">the best fit to the analysed five real data sets indicating that this new model outclasses its competitors.展开更多
The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact...The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes. The present paper contains explicit additional and complementary details of the proof, insisting on the existence and the number of Goldbach’s representations of even positive integers as sums of pairs of primes.展开更多
We study the decays ofΛb→Λ(→pπ^(−))ℓ^(+)ℓ^(−)withℓ=(e,μ,τ).We examine the full angular distributions with polarizedΛb,where the T-odd observables are identified.We discuss the possible effects of new physics(N...We study the decays ofΛb→Λ(→pπ^(−))ℓ^(+)ℓ^(−)withℓ=(e,μ,τ).We examine the full angular distributions with polarizedΛb,where the T-odd observables are identified.We discuss the possible effects of new physics(NP)and find that the T-odd observables are sensitive to them as they vanish in the standard model.Special attention is given to the interference of(pseudo)scalar operators with(axial)vector operators in polarized Λ_(b)→Λ(→pπ^(−))τ^(+)τ^(−),which are studied for the first time.Their effects are proportional to the lepton masses and therefore may evade the constraint from Λ_(b)→Λ(→pπ^(−))μ^(+)μ^(−) at the LHCb naturally.AsΛ_(b)→Λ(→pπ^(−))τ^(+)τ^(−) is uncontaminated by the charmonia resonance,it provides a clean background to probe NP.In addition,we show that the experimental central value of K10 in Λ_(b)→Λ(→pπ^(−))μ^(+)μ^(−) at the LHCb can be explained by the NP case,which couples to the right-handed quarks and leptons.The polarization fraction of Λ_(b) at the LHCb is found to be consistent with zero regardless of the NP scenarios.展开更多
Research on the pairing phase transition in the odd-A nucleus ^(161)Dy is based on a sophisticated blend of the covariant density functional theory and the shell-model-like approach.It has been observed that variation...Research on the pairing phase transition in the odd-A nucleus ^(161)Dy is based on a sophisticated blend of the covariant density functional theory and the shell-model-like approach.It has been observed that variations in thermodynamic quantities at the critical temperature do not exclusively align with pairing phase transitions.The presence of an S-shaped heat capacity curve,often interpreted as an indicator of such transitions,does not offer a definitive confirmation.Additional factors,including the blocking effect,can modify the heat capacity curve and impede the transition process.The pairing phase transition in ^(161)Dy,which occurs approximately from 0.7 to 1.0 MeV,is unequivocally characterized as a first-order transition.Furthermore,the analysis of the impact of varying strengths of pairing correlations on these transitions reveals a nonlinear relationship,thereby adding complexity to the transition dynamics.展开更多
This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its general...This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. <em>Note that the second solution is very short and does not exceed one and a half pages</em>. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. <em>To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.</em>展开更多
A full planar tunable band pass resonator is introduced, which is constructed by using novel symmetric step impedance resonator （SIR） and hyperabrupt varactors for wide bandwidth tuning and size reduction. The equiv...A full planar tunable band pass resonator is introduced, which is constructed by using novel symmetric step impedance resonator （SIR） and hyperabrupt varactors for wide bandwidth tuning and size reduction. The equivalent circuit model of the proposed resonator is set up. Theoretical analysis based on transmission line as well as odd and even-mode method is completed. The attractiveness of the approach presented lies in its simplicity. Based on the detailed analysis, a 6 GHz to 10 GHz varactor tuned resonator is designed, fabricated, and measured. It shows wideband tuning ability of 37%. The experimental results of the resonator have a good agreement with the analysis results.展开更多
This article presents a novel method to prove that： let E be an AM-space and if dim E ≥ 3, then there does not exist any odd subtractive.isometric mapping from the unit sphere S（E） into S[L（Ω, μ）]. In particul...This article presents a novel method to prove that： let E be an AM-space and if dim E ≥ 3, then there does not exist any odd subtractive.isometric mapping from the unit sphere S（E） into S[L（Ω, μ）]. In particular, there does not exist any real linear isometry from E into L（Ω, μ）.展开更多
基金the National Natural Sience Foundation of China for supporting this project within the Research Fund for International Young Scientists(12350410368)financial support from the Natural Science Foundation of Guangdong Province(2024A1515011343)the Key Project of Guangdong Provincial Department of Education(2023ZDZX3021)
文摘In recent years,there has been growing interest in the study of chiral active materials,which consist of building blocks that show active dynamics featuring chiral symmetry breaking,e.g.,particles that rotate in a common direction.These materials exhibit fascinating phenomena such as odd viscosity,odd diffusivity,active turbulence in fluids,vivid dislocation dynamics or odd elasticity in crystals or elastic materials,and hyperuniform states.The systematic study of soft chiral active matter systems is relatively new,starting around 2017,but has already shown promising applications in robust cargo transport,segregation and mixing dynamics,or manipulation of metamaterials.In this review,we summarize recent experimental and theoretical advances in this field,highlighting the emergence of anti-symmetric and odd stresses and ensuring effects such as odd viscosity or topologically protected edge modes.We further discuss the underlying mechanisms and provide insights into the potential of chiral active matter for various applications.
文摘研究了一类不定方程求正整数解的问题.借助一个引理,推导并证明了不定方程x2-py2=z2(p为奇素数)正整数解的一般公式.不定方程x2-py2=z2(p为奇素数)满足(x,y)=1的一切正整数解可表示为x=12(a2+pb2),y=ab,z=12a2-pb2,这里a>0,b>0,a,b都是奇数,p a;或x=a2+pb2,y=2ab,z=a2-pb2,这里a>0,b>0,a,b一奇一偶,p a.
基金Sponsored by the Scientific Research Fund of Heilongjiang Provincial Education Department (11541109)the Science Foundation of Harbin Normal University (KM2007-11)
文摘The filtration structure of finite-dimensional special odd Hamilton superalgebras over a field of prime characteristic was studied. By determining ad-nilpotent dements in the even part, the natural filtration of special odd Hamiltonian superalgebras is proved to be invariant. Using this result, the special odd Hamilton superalgebras is classified. Finally, the automorphism group of the restricted special odd Hamilton superalgebras is determined.
文摘In this paper, the forced odd order neutral differential equations of the form are considered d n d t n[x(t)-R(t)x(t-τ)]+P(t)x(t-σ)=f(t),t≥t 0.A sufficient condition for the oscillation of all solutions is obtained.
文摘The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .
文摘This paper introduces the basic idea and provides the mathematical formulation of the delayed feedback control （DFC） methodology, which has been widely used in chaos control. Stability analysis including the well-known odd number linfitation of the DFC is reviewed. Some new developments in characterizing the limitation of the DFC are presented. Various modified DFC methods, which are developed in order to overcome the odd number limitation, are also described. Finally, some open problems in this research field are discussed.
文摘We propose a new generator of continuous distributions with at least four positive parameters called the Kumaraswamy-Odd Rayleigh-G family. Some special cases were presented. The plots of the Kumaraswamy Odd Rayleigh Log-Logistic (</span><i><span style="font-family:Verdana;">KORLL</span></i><span style="font-family:Verdana;">) distribution indicate that the distribution can take many shapes depending on the parameter values. The negative skewness and kurtosis indicates that the distribution has li</span><span style="font-family:Verdana;">ghter tails than the normal distribution. The Monte Carlo simulation results indicate that the estimated biases decrease when the sample size increases. Furthermore, </span></span><span style="font-family:Verdana;">the root mean squared error estimates decay</span><span style="font-family:""><span style="font-family:Verdana;"> towards zero as the sample size increases. This part shows the consistency of the maximum likelihood estimators. From the considered analytical measures, the new </span><i><span style="font-family:Verdana;">KORLL</span></i><span style="font-family:Verdana;"> provides </span></span><span style="font-family:Verdana;">the best fit to the analysed five real data sets indicating that this new model outclasses its competitors.
文摘The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes. The present paper contains explicit additional and complementary details of the proof, insisting on the existence and the number of Goldbach’s representations of even positive integers as sums of pairs of primes.
基金Supported in part by the National Key Research and Development Program of China (2020YFC2201501)the National Natural Science Foundation of China (NSFC) (12147103)。
文摘We study the decays ofΛb→Λ(→pπ^(−))ℓ^(+)ℓ^(−)withℓ=(e,μ,τ).We examine the full angular distributions with polarizedΛb,where the T-odd observables are identified.We discuss the possible effects of new physics(NP)and find that the T-odd observables are sensitive to them as they vanish in the standard model.Special attention is given to the interference of(pseudo)scalar operators with(axial)vector operators in polarized Λ_(b)→Λ(→pπ^(−))τ^(+)τ^(−),which are studied for the first time.Their effects are proportional to the lepton masses and therefore may evade the constraint from Λ_(b)→Λ(→pπ^(−))μ^(+)μ^(−) at the LHCb naturally.AsΛ_(b)→Λ(→pπ^(−))τ^(+)τ^(−) is uncontaminated by the charmonia resonance,it provides a clean background to probe NP.In addition,we show that the experimental central value of K10 in Λ_(b)→Λ(→pπ^(−))μ^(+)μ^(−) at the LHCb can be explained by the NP case,which couples to the right-handed quarks and leptons.The polarization fraction of Λ_(b) at the LHCb is found to be consistent with zero regardless of the NP scenarios.
基金Supported by the National Natural Science Foundation of China(11775099)the Jiangnan University Basic Research Program(JUSRP202406002)。
文摘Research on the pairing phase transition in the odd-A nucleus ^(161)Dy is based on a sophisticated blend of the covariant density functional theory and the shell-model-like approach.It has been observed that variations in thermodynamic quantities at the critical temperature do not exclusively align with pairing phase transitions.The presence of an S-shaped heat capacity curve,often interpreted as an indicator of such transitions,does not offer a definitive confirmation.Additional factors,including the blocking effect,can modify the heat capacity curve and impede the transition process.The pairing phase transition in ^(161)Dy,which occurs approximately from 0.7 to 1.0 MeV,is unequivocally characterized as a first-order transition.Furthermore,the analysis of the impact of varying strengths of pairing correlations on these transitions reveals a nonlinear relationship,thereby adding complexity to the transition dynamics.
文摘This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. <em>Note that the second solution is very short and does not exceed one and a half pages</em>. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. <em>To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.</em>
文摘A full planar tunable band pass resonator is introduced, which is constructed by using novel symmetric step impedance resonator （SIR） and hyperabrupt varactors for wide bandwidth tuning and size reduction. The equivalent circuit model of the proposed resonator is set up. Theoretical analysis based on transmission line as well as odd and even-mode method is completed. The attractiveness of the approach presented lies in its simplicity. Based on the detailed analysis, a 6 GHz to 10 GHz varactor tuned resonator is designed, fabricated, and measured. It shows wideband tuning ability of 37%. The experimental results of the resonator have a good agreement with the analysis results.
基金This study is supported by the National Natural Science Foundation of China (10571090)the Research Fund for the Doctoral Program of Higher Education (20060055010)
文摘This article presents a novel method to prove that： let E be an AM-space and if dim E ≥ 3, then there does not exist any odd subtractive.isometric mapping from the unit sphere S（E） into S[L（Ω, μ）]. In particular, there does not exist any real linear isometry from E into L（Ω, μ）.
