In this two_part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODF...In this two_part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically_based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (Ⅰ), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N_dimensional (N_D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2_ and 3_D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point_group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work -Part (Ⅱ), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point_group symmetries are derived.展开更多
The explicit representations for tensorial Fourier expansion of 3_D crystal orientation distribution functions (CODFs) are established. In comparison with that the coefficients in the mth term of the Fourier expansion...The explicit representations for tensorial Fourier expansion of 3_D crystal orientation distribution functions (CODFs) are established. In comparison with that the coefficients in the mth term of the Fourier expansion of a 3_D ODF make up just a single irreducible mth_order tensor, the coefficients in the mth term of the Fourier expansion of a 3_D CODF constitute generally so many as 2m+1 irreducible mth_order tensors. Therefore, the restricted forms of tensorial Fourier expansions of 3_D CODFs imposed by various micro_ and macro_scopic symmetries are further established, and it is shown that in most cases of symmetry the restricted forms of tensorial Fourier expansions of 3_D CODFs contain remarkably reduced numbers of mth_order irreducible tensors than the number 2m+1 . These results are based on the restricted forms of irreducible tensors imposed by various point_group symmetries, which are also thoroughly investigated in the present part in both 2_ and 3_D spaces.展开更多
A general method was proposed to evaluate the distribution function of （C1｜C2 ） . Some examples were presented to validate the application of the method. Then the sufficient and necessary condition for that the dis...A general method was proposed to evaluate the distribution function of （C1｜C2 ） . Some examples were presented to validate the application of the method. Then the sufficient and necessary condition for that the distribution function of （ C1 ｜ C2 ） is uniform was proved.展开更多
This study investigates the form factors and impact parameter space parton distribution functions of the p meson derived from the generalized parton distributions within the Nambu-Jona-Lasinio model framework,employin...This study investigates the form factors and impact parameter space parton distribution functions of the p meson derived from the generalized parton distributions within the Nambu-Jona-Lasinio model framework,employing a proper time regularization scheme.We compare the charge G_(C),magnetic G_(M),and quadrupole G_(Q) form factors with latice data.The dressed form factors,G_(C)^(D) and G_(M)^(D),exhibit good agreement with latice results;however,G_(Q)^(D) is found to be harder than what is observed in latice calculations.The Rosenbluth cross section for elastic electron scattering on a spin-one particle can be expressed through the structure functions A(Q^(2))and B(Q^(2)).Additionally,the tensor polarization T_(20)(Q^(2),θ)can also be formulated in terms of these form factors.We analyze the structure functions A(Q2),B(Q2)and tensor polarization function T2o(Q2,);our findings quantitatively align with predicted values across various limits.In impact parameter space,we examine parton distribution functions along with their dependence on longitudinal momentum fraction x and impact parameter b+.The width distributions in impact parameter space reveal that the range of the charge distribution q_(C)(x,b_(⊥)^(2))is the most extensive.In contrast,the transverse magnetic radius falls within a moderate range,while the quadrupole distribution q_(Q)(x,b_(⊥)^(2))demonstrates the narrowest extent.展开更多
The recent measurement of the differential γ ＋ c-jet cross section, performed at the Tevatron collider in Run II by the D0 collaboration, is studied in a next-to-leading order（NLO） global QCD analysis to assess it...The recent measurement of the differential γ ＋ c-jet cross section, performed at the Tevatron collider in Run II by the D0 collaboration, is studied in a next-to-leading order（NLO） global QCD analysis to assess its impact on the proton parton distribution functions（PDFs）. We show that these data lead to a significant change in the gluon and charm quark distributions. We demonstrate also that there is an inconsistency between the new high precision HERA I＋II combined data and Tevatron measurement. Moreover, in this study we investigate the impact of older EMC measurements of charm structure function F_c^2 on the PDFs and compare the results with those from the analysis of Tevatron data. We show that both of them have the same impact on the PDFs, and thus can be recognized as the same evidence for the inefficiency of perturbative QCD in dealing with charm production in some kinematic regions.展开更多
A new and simple statistical approach is performed to calculate the parton distribution functions （PDFs） of the nucleon in terms of light-front kinematic variables.Analytic expressions of x-dependent PDFs are obtain...A new and simple statistical approach is performed to calculate the parton distribution functions （PDFs） of the nucleon in terms of light-front kinematic variables.Analytic expressions of x-dependent PDFs are obtained in the whole x region.And thereafter,we treat the temperature T as a parameter of the atomic number A to explain the nuclear EMC effect in the region x ∈ [0.2,0.7].We give the predictions of PDF ratios,and they are very different from those by other models,thus experiments aiming at measuring PDF ratios are suggested to provide a discrimination of different models.展开更多
The influence of the use of the generalized Hermite polynomial on the Hermite-based lattice Bofiz- mann （LB） construction approach, lattice sets, the thermal weights, moments and the equilibrium distribution functi...The influence of the use of the generalized Hermite polynomial on the Hermite-based lattice Bofiz- mann （LB） construction approach, lattice sets, the thermal weights, moments and the equilibrium distribution function （EDF） are addressed. A new moment system is proposed. The theoretical possibility to obtain a unique high-order Hermite-based singel relaxation time LB model capable to exactly inatch some first hydrodynamic inoments thermally i） on-Cartesian lattice, ii） with thermal weights in the EDF, iii） whilst the highest possible hydrodynamic moments that are exactly reatched are obtained with the shortest on-Cartesian lattiee sets with some fixed real-valued temperatures, is also analyzed.展开更多
Suppose that Z1 , Z2,’’’ Zn are independent normal random variables with common meanμ and variance σ2. Then S2 and have distribution andtn-1 distribution respectively. If the normal assumption fails, there will b...Suppose that Z1 , Z2,’’’ Zn are independent normal random variables with common meanμ and variance σ2. Then S2 and have distribution andtn-1 distribution respectively. If the normal assumption fails, there will be the remaindersof the distribution functions and density functions. This paper gives the direct expansions ofdistribution functions and density functions of S2 alld T up to o(n-1). They are more intuitiveand convenient than usual Edgeworth expansions.展开更多
Gas distribution function plays a crucial role in the description of gas flows at the mesoscopic scale.In the presence of non-equilibrium flow,the distribution function loses its rotational symmetricity,making the mat...Gas distribution function plays a crucial role in the description of gas flows at the mesoscopic scale.In the presence of non-equilibrium flow,the distribution function loses its rotational symmetricity,making the mathematical derivation difficult.From both the Chapman-Enskog expansion and the Hermite polynomial expansion(Grad’s method),we observe that the non-equilibrium effect is closely related to the peculiar velocity space(C).Based on this recognition,we propose a new methodology to construct the non-equilibrium distribution function from the perspective of polynomial expansion in the peculiar velocity space of molecules.The coefficients involved in the non-equilibrium distribution function can be exactly determined by the compatibility conditions and the moment relationships.This new framework allows constructing non-equilibrium distribution functions at any order of truncation,and the ones at the third and the fourth order have been presented in this paper for illustration purposes.Numerical validations demonstrate that the new method is more accurate than the Grad’s method at the same truncation error for describing non-equilibrium effects.Two-dimensional benchmark tests are performed to shed light on the applicability of the new method to practical engineering problems.展开更多
This work continues the studies on searching for plasma media with the inverse electron energy distribution function(EEDF)and providing recommendations for setting up subsequent experiments.The inverse EEDF is a distr...This work continues the studies on searching for plasma media with the inverse electron energy distribution function(EEDF)and providing recommendations for setting up subsequent experiments.The inverse EEDF is a distribution function that increases with an increase in energy at zero electron energy.The inverse EEDF plays a central role in the problem of negative conductivity.Based on the previously obtained criterion for the formation of an inverse EEDF in a spatially inhomogeneous plasma,a heuristic method is proposed that allows one to avoid resource-intensive calculations for spatially two-dimensional(2D)kinetic modeling on a large array of different glow discharges.It is shown that the conditions for EEDF inversion can be realized in two-chamber discharge structures due to violating the known Boltzmann distribution for electron density.The theoretical conclusions are validated by numerical modeling of lowpressure two-chamber inductively-coupled plasma(ICP)discharges in the COMSOL Multiphysics environment.As a result,areas of conditions with inverse EEDF were found for subsequent detailed kinetic analysis and experimental studies.展开更多
In recent years,many phase space distributions have been proposed,and one of the more independently interesting is the Bai distribution function(BDF).The BDF has been shown to interpolate between the instantaneous aut...In recent years,many phase space distributions have been proposed,and one of the more independently interesting is the Bai distribution function(BDF).The BDF has been shown to interpolate between the instantaneous auto-correlation function and the Wigner distribution function,and be applied in linear frequency modulated signal parameter estimation and optical partial coherence areas.Currently,the BDF is only defined for continuous signals.However,for both simulation and experimental purposes,the signals must be discrete.This necessitates the development of a BDF analysis workflow for discrete signals.In this work,we analyze the sampling requirements imposed by the BDF and demonstrate their correctness by comparing the continuous BDFs of continuous test signals with their numerically approximated counterparts.Our results permit more accurate simulations using BDFs,which will be useful in applying them to problems such as partial coherence.展开更多
Suppose {Xi, i≥1} and {Yi, i≥1} are two independent sequences with distribution functions FX(x) and FY(x), respectively. Zi is the combination of Xi and Yi with a probability pn for each i with 1≤i≤n. The extreme ...Suppose {Xi, i≥1} and {Yi, i≥1} are two independent sequences with distribution functions FX(x) and FY(x), respectively. Zi is the combination of Xi and Yi with a probability pn for each i with 1≤i≤n. The extreme value distribution ,n GZ(x) of this particular triangular array of the i.i.d. random variables Z1, , Z2, ,…, Zn n n ,nis discussed. We found a new form of the extreme value distribution ΛA(ρx)Λ(x)(0<ρ <1), which is not max-stable. It occurs if FX(x) and FY(x) belong to the same MDA(Λ). GZ(x) does not exist as mixture forms of the different types of extreme value distributions.展开更多
For performance optimization such as placement,interconnect synthesis,and routing, an efficient and accurate interconnect delay metric is critical,even in design tools development like design for yield （DFY） and des...For performance optimization such as placement,interconnect synthesis,and routing, an efficient and accurate interconnect delay metric is critical,even in design tools development like design for yield （DFY） and design for manufacture （DFM）. In the nanometer regime, the recently proposed delay models for RLC interconnects based on statistical probability density function （PDF）interpretation such as PRIMO,H-gamma,WED and RLD bridge the gap between accuracy and efficiency. However, these models always require table look-up when operating. In this paper, a novel delay model based on the Birnbaum-Saunders distribution （BSD） is presented. BSD can accomplish interconnect delay estimation fast and accurately without table look-up operations. Furthermore, it only needs the first two moments to match. Experimental results in 90nm technology show that BSD is robust, easy to implement,efficient,and accurate.展开更多
In this paper we discuss a step further some convergence and continuity problems of distribution function on R^i. We give the following results: (1)distribution function F(x_1,…,x_k) on R^k is continuous if and only ...In this paper we discuss a step further some convergence and continuity problems of distribution function on R^i. We give the following results: (1)distribution function F(x_1,…,x_k) on R^k is continuous if and only if all marginal distribution functions of F is continuous on R^1. (2)If limF_n(x_1,……,x_k)=F(x_1,…,x_k) and limF_n(x_1—0,…,x_k—0)=F(x_1—0,…,x_k—0) at all non-continuity points of F, then展开更多
In this paper,we calculate the scalar a_(0)(980)-meson leading-twist wave function by using the light-cone harmonic oscillator model(LCHO),where the model parameters are determined by fitting theξ-moments■of its lig...In this paper,we calculate the scalar a_(0)(980)-meson leading-twist wave function by using the light-cone harmonic oscillator model(LCHO),where the model parameters are determined by fitting theξ-moments■of its light-cone distribution amplitudes.Then,the a_(0)(980)-meson leading-twist light-cone distribution amplitudes with three different scalesζ=(1.0,2.0,5.2)Ge V are given.After constructing the relationship between the a_(0)(980)-meson leading-twist parton distribution functions/valence quark distribution function and its LCHO wave function,we exhibit the■(x,ζ)and■(x,ζ)with different scales.Furthermore,we also calculate the Mellin moments of the a_(0)(980)-meson’s valence quark distribution function■with n=(1,2,3),i.e.■=0.027,■=0.018 and■=0.013.Finally,the scale evolution for the ratio of the Mellin moments x■are presented.展开更多
Dislocation distribution functions of the edges of mode Ⅲ propagation crack subjected to three types of loads were studied by the methods of the theory of complex variable functions,by which,the problems researched w...Dislocation distribution functions of the edges of mode Ⅲ propagation crack subjected to three types of loads were studied by the methods of the theory of complex variable functions,by which,the problems researched were facilely transformed into Riemann-Hilbert problems and Keldysh-Sedov mixed boundary value problems. Analytical solutions of stresses,displacement and dynamic stress intensity factor were obtained by the measures of the theory of self-similar functions and corresponding differential and integral operation. In terms of the relationship between dislocation distribution functions and displacements,analytical solutions of dislocation distribution functions were obttained,and variational rules of dislocation distribution functions were depicted.展开更多
Stimulated Raman scattering(SRS)is one of the main instabilities affecting success of fusion ignition.Here,we study the relationship between Raman growth and Landau damping with various distribution functions combinin...Stimulated Raman scattering(SRS)is one of the main instabilities affecting success of fusion ignition.Here,we study the relationship between Raman growth and Landau damping with various distribution functions combining the analytic formulas and Vlasov simulations.The Landau damping obtained by Vlasov-Poisson simulation and Raman growth rate obtained by Vlasov-Maxwell simulation are anti-correlated,which is consistent with our theoretical analysis quantitatively.Maxwellian distribution,flattened distribution,and bi-Maxwellian distribution are studied in detail,which represent three typical stages of SRS.We also demonstrate the effects of plateau width,hot-electron fraction,hot-to-cold electron temperature ratio,and collisional damping on the Landau damping and growth rate.They gives us a deep understanding of SRS and possible ways to mitigate SRS through manipulating distribution functions to a high Landau damping regime.展开更多
The sequences {Zi , 1≤i≤n}, n≥1 have multi-nomial distribution among i.i.d. random variables {X1, , i≥1}, {X2, , ,n i i i≥1}, …, {Xm , i≥1}. The extreme value distribution GZ(x) of this particular triangular ar...The sequences {Zi , 1≤i≤n}, n≥1 have multi-nomial distribution among i.i.d. random variables {X1, , i≥1}, {X2, , ,n i i i≥1}, …, {Xm , i≥1}. The extreme value distribution GZ(x) of this particular triangular array of i.i.d. random variables Z1, , Z2, , …, ,i n n r ?1 Zn is discussed in this paper. We found a new type of not max-stable extreme value distributions, i) GZ (x) = ,n ∏Φα Ai(x)×Φαr (x); i i=1 r ?1 r?1 ii) GZ (x) = ∏Ψα Ai(x)×Ψαr (x); iii) GZ (x) = ∏Λ Ai(λix)×Λ(x), r≥2, 0<α1≤α2≤…≤αr and λi∈(0,1] for i, 1≤i≤r?1 which occur if i i=1 i=1 Fj, …, Fm belong to the same MDA.展开更多
The sequences {Zi,n, 1≤i≤n}, n≥1 are multi-nomial distribution among i.i.d, random variables {X1,i, i≥1}, {X2,i, i≥1 } {Xm,i, i≥1 }. The extreme value distribution Gz（x） of this particular triangular array of ...The sequences {Zi,n, 1≤i≤n}, n≥1 are multi-nomial distribution among i.i.d, random variables {X1,i, i≥1}, {X2,i, i≥1 } {Xm,i, i≥1 }. The extreme value distribution Gz（x） of this particular triangular array of i.i,d, random variables Z1,n, Z2 n,...,Zn,n is discussed. A new type of not max-stable extreme value distributions which are Fréchet mixture, Gumbel mixture and Weibull mixture has been found if Fj,…… Fm belong to the same MDA. Whether mixtures of different types of extreme value distributions exist or not and the more general case are discussed in this paper. We found that Gz（x） does not exist as mixture forms of the different types of extreme value distributions after we investigated all cases.展开更多
In order to quantify the seismic performance of buildings reasonably and effectively,the paper puts forward the method of earthquake loss evaluation of buildings based on story EDP-DV functions.Firstly,considering the...In order to quantify the seismic performance of buildings reasonably and effectively,the paper puts forward the method of earthquake loss evaluation of buildings based on story EDP-DV functions.Firstly,considering the randomness of seismic response parameters,the distribution functions of story EDPs at a given level of ground motion intensity can be achieved through Incremental Dynamic Analysis.Meanwhile,story EDP-DV functions which relate story response parameters(story EDPs)directly to economic losses(DVs)can be created beforehand by integrating component fragility functions and loss functions;they can be called directly without double computation for the same type of buildings.Lastly,the economic losses of a building at a given level of ground motion intensity can be achieved by combining the distribution functions of story EDPs and story EDP-DV functions.On this occasion,the proposed method omits the damage measure in PEER methodology and makes the story not component as a unit of account to evaluate the earthquake losses of a building reasonably and accurately.The numerical example shows that the proposed method is feasible and reasonable to quantify the seismic performance of buildings.展开更多
文摘In this two_part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically_based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (Ⅰ), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N_dimensional (N_D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2_ and 3_D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point_group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work -Part (Ⅱ), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point_group symmetries are derived.
文摘The explicit representations for tensorial Fourier expansion of 3_D crystal orientation distribution functions (CODFs) are established. In comparison with that the coefficients in the mth term of the Fourier expansion of a 3_D ODF make up just a single irreducible mth_order tensor, the coefficients in the mth term of the Fourier expansion of a 3_D CODF constitute generally so many as 2m+1 irreducible mth_order tensors. Therefore, the restricted forms of tensorial Fourier expansions of 3_D CODFs imposed by various micro_ and macro_scopic symmetries are further established, and it is shown that in most cases of symmetry the restricted forms of tensorial Fourier expansions of 3_D CODFs contain remarkably reduced numbers of mth_order irreducible tensors than the number 2m+1 . These results are based on the restricted forms of irreducible tensors imposed by various point_group symmetries, which are also thoroughly investigated in the present part in both 2_ and 3_D spaces.
文摘A general method was proposed to evaluate the distribution function of （C1｜C2 ） . Some examples were presented to validate the application of the method. Then the sufficient and necessary condition for that the distribution function of （ C1 ｜ C2 ） is uniform was proved.
基金Supported by the Scientific Research Foundation of Nanjing Institute of Technology(YKJ202352)the Natural Science Foundation of Jiangsu Province,China(BK20191472),and the China Postdoctoral Science Foundation(2022M721564)。
文摘This study investigates the form factors and impact parameter space parton distribution functions of the p meson derived from the generalized parton distributions within the Nambu-Jona-Lasinio model framework,employing a proper time regularization scheme.We compare the charge G_(C),magnetic G_(M),and quadrupole G_(Q) form factors with latice data.The dressed form factors,G_(C)^(D) and G_(M)^(D),exhibit good agreement with latice results;however,G_(Q)^(D) is found to be harder than what is observed in latice calculations.The Rosenbluth cross section for elastic electron scattering on a spin-one particle can be expressed through the structure functions A(Q^(2))and B(Q^(2)).Additionally,the tensor polarization T_(20)(Q^(2),θ)can also be formulated in terms of these form factors.We analyze the structure functions A(Q2),B(Q2)and tensor polarization function T2o(Q2,);our findings quantitatively align with predicted values across various limits.In impact parameter space,we examine parton distribution functions along with their dependence on longitudinal momentum fraction x and impact parameter b+.The width distributions in impact parameter space reveal that the range of the charge distribution q_(C)(x,b_(⊥)^(2))is the most extensive.In contrast,the transverse magnetic radius falls within a moderate range,while the quadrupole distribution q_(Q)(x,b_(⊥)^(2))demonstrates the narrowest extent.
文摘The recent measurement of the differential γ ＋ c-jet cross section, performed at the Tevatron collider in Run II by the D0 collaboration, is studied in a next-to-leading order（NLO） global QCD analysis to assess its impact on the proton parton distribution functions（PDFs）. We show that these data lead to a significant change in the gluon and charm quark distributions. We demonstrate also that there is an inconsistency between the new high precision HERA I＋II combined data and Tevatron measurement. Moreover, in this study we investigate the impact of older EMC measurements of charm structure function F_c^2 on the PDFs and compare the results with those from the analysis of Tevatron data. We show that both of them have the same impact on the PDFs, and thus can be recognized as the same evidence for the inefficiency of perturbative QCD in dealing with charm production in some kinematic regions.
基金Supported by National Natural Science Foundation of China (10721063,10975003)Hui-Chun Chin and Tsung-Dao Lee Chinese Undergraduate Research Endowment (Chun-Tsung Endowment) at Peking UniversityNational Fund for Fostering Talents of Basic Science (J0630311,J0730316)
文摘A new and simple statistical approach is performed to calculate the parton distribution functions （PDFs） of the nucleon in terms of light-front kinematic variables.Analytic expressions of x-dependent PDFs are obtained in the whole x region.And thereafter,we treat the temperature T as a parameter of the atomic number A to explain the nuclear EMC effect in the region x ∈ [0.2,0.7].We give the predictions of PDF ratios,and they are very different from those by other models,thus experiments aiming at measuring PDF ratios are suggested to provide a discrimination of different models.
文摘The influence of the use of the generalized Hermite polynomial on the Hermite-based lattice Bofiz- mann （LB） construction approach, lattice sets, the thermal weights, moments and the equilibrium distribution function （EDF） are addressed. A new moment system is proposed. The theoretical possibility to obtain a unique high-order Hermite-based singel relaxation time LB model capable to exactly inatch some first hydrodynamic inoments thermally i） on-Cartesian lattice, ii） with thermal weights in the EDF, iii） whilst the highest possible hydrodynamic moments that are exactly reatched are obtained with the shortest on-Cartesian lattiee sets with some fixed real-valued temperatures, is also analyzed.
文摘Suppose that Z1 , Z2,’’’ Zn are independent normal random variables with common meanμ and variance σ2. Then S2 and have distribution andtn-1 distribution respectively. If the normal assumption fails, there will be the remaindersof the distribution functions and density functions. This paper gives the direct expansions ofdistribution functions and density functions of S2 alld T up to o(n-1). They are more intuitiveand convenient than usual Edgeworth expansions.
基金supported by MOE Tier 1 project at National University of Singapore(No.A-0005235-01-00).
文摘Gas distribution function plays a crucial role in the description of gas flows at the mesoscopic scale.In the presence of non-equilibrium flow,the distribution function loses its rotational symmetricity,making the mathematical derivation difficult.From both the Chapman-Enskog expansion and the Hermite polynomial expansion(Grad’s method),we observe that the non-equilibrium effect is closely related to the peculiar velocity space(C).Based on this recognition,we propose a new methodology to construct the non-equilibrium distribution function from the perspective of polynomial expansion in the peculiar velocity space of molecules.The coefficients involved in the non-equilibrium distribution function can be exactly determined by the compatibility conditions and the moment relationships.This new framework allows constructing non-equilibrium distribution functions at any order of truncation,and the ones at the third and the fourth order have been presented in this paper for illustration purposes.Numerical validations demonstrate that the new method is more accurate than the Grad’s method at the same truncation error for describing non-equilibrium effects.Two-dimensional benchmark tests are performed to shed light on the applicability of the new method to practical engineering problems.
基金supported by the National Key R&D Program of China(No.2022YFE0204100)National Natural Science Foundation of China(Nos.12205067 and 12375199)the Fundamental Research Funds for the Central University(No.HIT.D?J.2023178)。
文摘This work continues the studies on searching for plasma media with the inverse electron energy distribution function(EEDF)and providing recommendations for setting up subsequent experiments.The inverse EEDF is a distribution function that increases with an increase in energy at zero electron energy.The inverse EEDF plays a central role in the problem of negative conductivity.Based on the previously obtained criterion for the formation of an inverse EEDF in a spatially inhomogeneous plasma,a heuristic method is proposed that allows one to avoid resource-intensive calculations for spatially two-dimensional(2D)kinetic modeling on a large array of different glow discharges.It is shown that the conditions for EEDF inversion can be realized in two-chamber discharge structures due to violating the known Boltzmann distribution for electron density.The theoretical conclusions are validated by numerical modeling of lowpressure two-chamber inductively-coupled plasma(ICP)discharges in the COMSOL Multiphysics environment.As a result,areas of conditions with inverse EEDF were found for subsequent detailed kinetic analysis and experimental studies.
基金the support of the University College Dublin through a John Sheridan Scholarship.Min Wan thanks 4TU.RECENTRE program(Award No.OA102070)the National Growth Fund programme PhotonDelta in The Netherlands.
文摘In recent years,many phase space distributions have been proposed,and one of the more independently interesting is the Bai distribution function(BDF).The BDF has been shown to interpolate between the instantaneous auto-correlation function and the Wigner distribution function,and be applied in linear frequency modulated signal parameter estimation and optical partial coherence areas.Currently,the BDF is only defined for continuous signals.However,for both simulation and experimental purposes,the signals must be discrete.This necessitates the development of a BDF analysis workflow for discrete signals.In this work,we analyze the sampling requirements imposed by the BDF and demonstrate their correctness by comparing the continuous BDFs of continuous test signals with their numerically approximated counterparts.Our results permit more accurate simulations using BDFs,which will be useful in applying them to problems such as partial coherence.
文摘Suppose {Xi, i≥1} and {Yi, i≥1} are two independent sequences with distribution functions FX(x) and FY(x), respectively. Zi is the combination of Xi and Yi with a probability pn for each i with 1≤i≤n. The extreme value distribution ,n GZ(x) of this particular triangular array of the i.i.d. random variables Z1, , Z2, ,…, Zn n n ,nis discussed. We found a new form of the extreme value distribution ΛA(ρx)Λ(x)(0<ρ <1), which is not max-stable. It occurs if FX(x) and FY(x) belong to the same MDA(Λ). GZ(x) does not exist as mixture forms of the different types of extreme value distributions.
文摘For performance optimization such as placement,interconnect synthesis,and routing, an efficient and accurate interconnect delay metric is critical,even in design tools development like design for yield （DFY） and design for manufacture （DFM）. In the nanometer regime, the recently proposed delay models for RLC interconnects based on statistical probability density function （PDF）interpretation such as PRIMO,H-gamma,WED and RLD bridge the gap between accuracy and efficiency. However, these models always require table look-up when operating. In this paper, a novel delay model based on the Birnbaum-Saunders distribution （BSD） is presented. BSD can accomplish interconnect delay estimation fast and accurately without table look-up operations. Furthermore, it only needs the first two moments to match. Experimental results in 90nm technology show that BSD is robust, easy to implement,efficient,and accurate.
文摘In this paper we discuss a step further some convergence and continuity problems of distribution function on R^i. We give the following results: (1)distribution function F(x_1,…,x_k) on R^k is continuous if and only if all marginal distribution functions of F is continuous on R^1. (2)If limF_n(x_1,……,x_k)=F(x_1,…,x_k) and limF_n(x_1—0,…,x_k—0)=F(x_1—0,…,x_k—0) at all non-continuity points of F, then
基金supported in part by the National Natural Science Foundation of China under Grant No.12265010,No.12265009the Project of Guizhou Provincial Department of Science and Technology under Grant No.ZK[2021]024the Project of Guizhou Provincial Department of Education under Grant No.KY[2021]030。
文摘In this paper,we calculate the scalar a_(0)(980)-meson leading-twist wave function by using the light-cone harmonic oscillator model(LCHO),where the model parameters are determined by fitting theξ-moments■of its light-cone distribution amplitudes.Then,the a_(0)(980)-meson leading-twist light-cone distribution amplitudes with three different scalesζ=(1.0,2.0,5.2)Ge V are given.After constructing the relationship between the a_(0)(980)-meson leading-twist parton distribution functions/valence quark distribution function and its LCHO wave function,we exhibit the■(x,ζ)and■(x,ζ)with different scales.Furthermore,we also calculate the Mellin moments of the a_(0)(980)-meson’s valence quark distribution function■with n=(1,2,3),i.e.■=0.027,■=0.018 and■=0.013.Finally,the scale evolution for the ratio of the Mellin moments x■are presented.
基金Sponsored by the Postdoctoral Science Fundation of China (Grant No. 200303337 )the National Natural Science Foundation of China (Grant No.30205035)
文摘Dislocation distribution functions of the edges of mode Ⅲ propagation crack subjected to three types of loads were studied by the methods of the theory of complex variable functions,by which,the problems researched were facilely transformed into Riemann-Hilbert problems and Keldysh-Sedov mixed boundary value problems. Analytical solutions of stresses,displacement and dynamic stress intensity factor were obtained by the measures of the theory of self-similar functions and corresponding differential and integral operation. In terms of the relationship between dislocation distribution functions and displacements,analytical solutions of dislocation distribution functions were obttained,and variational rules of dislocation distribution functions were depicted.
基金supported by the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDA25050700)the National Natural Science Foundation of China(Grant Nos.11805062,11875091 and 11975059)+1 种基金the Science Challenge Project(Grant No.TZ2016005)the Natural Science Foundation of Hunan Province,China(Grant No.2020JJ5029)。
文摘Stimulated Raman scattering(SRS)is one of the main instabilities affecting success of fusion ignition.Here,we study the relationship between Raman growth and Landau damping with various distribution functions combining the analytic formulas and Vlasov simulations.The Landau damping obtained by Vlasov-Poisson simulation and Raman growth rate obtained by Vlasov-Maxwell simulation are anti-correlated,which is consistent with our theoretical analysis quantitatively.Maxwellian distribution,flattened distribution,and bi-Maxwellian distribution are studied in detail,which represent three typical stages of SRS.We also demonstrate the effects of plateau width,hot-electron fraction,hot-to-cold electron temperature ratio,and collisional damping on the Landau damping and growth rate.They gives us a deep understanding of SRS and possible ways to mitigate SRS through manipulating distribution functions to a high Landau damping regime.
基金Project partially supported by the Swiss National Science Foundation
文摘The sequences {Zi , 1≤i≤n}, n≥1 have multi-nomial distribution among i.i.d. random variables {X1, , i≥1}, {X2, , ,n i i i≥1}, …, {Xm , i≥1}. The extreme value distribution GZ(x) of this particular triangular array of i.i.d. random variables Z1, , Z2, , …, ,i n n r ?1 Zn is discussed in this paper. We found a new type of not max-stable extreme value distributions, i) GZ (x) = ,n ∏Φα Ai(x)×Φαr (x); i i=1 r ?1 r?1 ii) GZ (x) = ∏Ψα Ai(x)×Ψαr (x); iii) GZ (x) = ∏Λ Ai(λix)×Λ(x), r≥2, 0<α1≤α2≤…≤αr and λi∈(0,1] for i, 1≤i≤r?1 which occur if i i=1 i=1 Fj, …, Fm belong to the same MDA.
基金Project partially supported by the National Natural Science Foundation of Switzerland
文摘The sequences {Zi,n, 1≤i≤n}, n≥1 are multi-nomial distribution among i.i.d, random variables {X1,i, i≥1}, {X2,i, i≥1 } {Xm,i, i≥1 }. The extreme value distribution Gz（x） of this particular triangular array of i.i,d, random variables Z1,n, Z2 n,...,Zn,n is discussed. A new type of not max-stable extreme value distributions which are Fréchet mixture, Gumbel mixture and Weibull mixture has been found if Fj,…… Fm belong to the same MDA. Whether mixtures of different types of extreme value distributions exist or not and the more general case are discussed in this paper. We found that Gz（x） does not exist as mixture forms of the different types of extreme value distributions after we investigated all cases.
文摘In order to quantify the seismic performance of buildings reasonably and effectively,the paper puts forward the method of earthquake loss evaluation of buildings based on story EDP-DV functions.Firstly,considering the randomness of seismic response parameters,the distribution functions of story EDPs at a given level of ground motion intensity can be achieved through Incremental Dynamic Analysis.Meanwhile,story EDP-DV functions which relate story response parameters(story EDPs)directly to economic losses(DVs)can be created beforehand by integrating component fragility functions and loss functions;they can be called directly without double computation for the same type of buildings.Lastly,the economic losses of a building at a given level of ground motion intensity can be achieved by combining the distribution functions of story EDPs and story EDP-DV functions.On this occasion,the proposed method omits the damage measure in PEER methodology and makes the story not component as a unit of account to evaluate the earthquake losses of a building reasonably and accurately.The numerical example shows that the proposed method is feasible and reasonable to quantify the seismic performance of buildings.
