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Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process
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作者 Chunlei Ruan Cengceng Dong +2 位作者 Kunfeng Liang Zhijun Liu Xinru Bao 《Computer Modeling in Engineering & Sciences》 SCIE EI 2024年第3期3033-3049,共17页
Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridyna... Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed. 展开更多
关键词 Population balance equation CRYSTALLIZATION peridynamic differential operator Euler’s first-order explicit method
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Efficient slope reliability and sensitivity analysis using quantile-based first-order second-moment method 被引量:1
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作者 Zhiyong Yang Chengchuan Yin +2 位作者 Xueyou Li Shuihua Jiang Dianqing Li 《Journal of Rock Mechanics and Geotechnical Engineering》 SCIE CSCD 2024年第10期4192-4203,共12页
This paper introduces a novel approach for parameter sensitivity evaluation and efficient slope reliability analysis based on quantile-based first-order second-moment method(QFOSM).The core principles of the QFOSM are... This paper introduces a novel approach for parameter sensitivity evaluation and efficient slope reliability analysis based on quantile-based first-order second-moment method(QFOSM).The core principles of the QFOSM are elucidated geometrically from the perspective of expanding ellipsoids.Based on this geometric interpretation,the QFOSM is further extended to estimate sensitivity indices and assess the significance of various uncertain parameters involved in the slope system.The proposed method has the advantage of computational simplicity,akin to the conventional first-order second-moment method(FOSM),while providing estimation accuracy close to that of the first-order reliability method(FORM).Its performance is demonstrated with a numerical example and three slope examples.The results show that the proposed method can efficiently estimate the slope reliability and simultaneously evaluate the sensitivity of the uncertain parameters.The proposed method does not involve complex optimization or iteration required by the FORM.It can provide a valuable complement to the existing approximate reliability analysis methods,offering rapid sensitivity evaluation and slope reliability analysis. 展开更多
关键词 Slope reliability Sensitivity analysis QUANTILE first-order second-moment method(FOSM) first-order reliability method(form)
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Comparative Studies between Picard’s and Taylor’s Methods of Numerical Solutions of First Ordinary Order Differential Equations Arising from Real-Life Problems
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作者 Khalid Abd Elrazig Awad Alla Elnour 《Journal of Applied Mathematics and Physics》 2024年第3期877-896,共20页
To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’... To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section. 展开更多
关键词 first-order differential Equations Picard Method Taylor Series Method Numerical Solutions Numerical Examples MATLAB Software
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Finite Element Approach for the Solution of First-Order Differential Equations
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作者 André Schmidt Horst R. Beyer +1 位作者 Matthias Hinze Evangelos N. Vandoros 《Journal of Applied Mathematics and Physics》 2020年第10期2072-2090,共19页
The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differe... The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differential equation into a system of algebraic equations by application of the method of weighted residuals in conjunction with a finite element ansatz. However, this procedure is restricted to even-ordered differential equations and leads to symmetric system matrices as a key property of the finite element method. This paper aims in a generalization of the finite element method towards the solution of first-order differential equations. This is achieved by an approach which replaces the first-order derivative by fractional powers of operators making use of the square root of a Sturm-Liouville operator. The resulting procedure incorporates a finite element formulation and leads to a symmetric but dense system matrix. Finally, the scheme is applied to the barometric equation where the results are compared with the analytical solution and other numerical approaches. It turns out that the resulting numerical scheme shows excellent convergence properties. 展开更多
关键词 Finite Element Method first-order differential Equations Fractional Powers of Operators
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Weak WT_2-class of differential forms and weakly A-harmonic tensors 被引量:3
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作者 GAO Hong-ya WANG Yan-yan 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2010年第3期359-366,共8页
In this paper, we give the definition of weak WT2-class of differential forms, and then obtain its weak reverse Holder inequality. As an application, we give an alternative proof of the higher integrability result of ... In this paper, we give the definition of weak WT2-class of differential forms, and then obtain its weak reverse Holder inequality. As an application, we give an alternative proof of the higher integrability result of weakly A-harmonic tensors due to B. Stroffolini. 展开更多
关键词 Weak WT2-class of differential forms weak reverse HSlder inequality weakly A-harmonic tensor higher integrability.
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General Integral Formulasof (0, q)(q 〉 0)Differential Forms on the Analytic Varieties 被引量:3
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作者 CHEN Shu-jin 《Chinese Quarterly Journal of Mathematics》 2017年第4期355-370,共16页
In this paper, firstly using different method and technique we derive the cor-responding integral representation formulas of (0, q)(q 〉 0) differential forms for the twotypes of the bounded domains in complex sub... In this paper, firstly using different method and technique we derive the cor-responding integral representation formulas of (0, q)(q 〉 0) differential forms for the twotypes of the bounded domains in complex submanifolds with codimension-m. Secondly weobtain the unified integral representation formulas of (0, q)(q 〉 0) differential forms for thegeneral bounded domain in complex submanifold with codimension-m, which include Hatzi-afratis formula, i.e. Koppelman type integral formula for the bounded domain with smoothboundary in analytic varieties. In particular, when m -- 0, we obtain the unified integralrepresentation formulas of (0, q)(q 〉 0) differential forms for general bounded domain in Cn,which are the generalization and the embodiment of Koppelman-Leray formula. 展开更多
关键词 Complex SUBMANIFOLD ANALYTIC VARIETIES UNIFIED formULA Extension Differ-ential form Integral representation
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Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations
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作者 XUE Xiaomin XU Juanjuan ZHANG Huanshui 《Journal of Systems Science & Complexity》 SCIE EI CSCD 2024年第1期230-252,共23页
This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discret... This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation. 展开更多
关键词 Discretization-then-continuousization method first-order hyperbolic partial differential equations forward and backward partial difference equations linear quadratic optimal control.
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Residues of Logarithmic Differential Forms in Complex Analysis and Geometry
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作者 A.G.Aleksandrov 《Analysis in Theory and Applications》 2014年第1期34-50,共17页
In the article, we discuss basic concepts of the residue theory of logarithmic and multi-logarithmic differential forms, and describe some aspects of the theory, de-veloped by the author in the past few years. In part... In the article, we discuss basic concepts of the residue theory of logarithmic and multi-logarithmic differential forms, and describe some aspects of the theory, de-veloped by the author in the past few years. In particular, we introduce the notion of logarithmic differential forms with the use of the classical de Rham lemma and give an explicit description of regular meromorphic differential forms in terms of residues of logarithmic or multi-logarithmic differential forms with respect to hypersurfaces, com-plete intersections or pure-dimensional Cohen-Macaulay spaces. Among other things, several useful applications are considered, which are related with the theory of holo-nomic D-modules, the theory of Hodge structures, the theory of residual currents and others. 展开更多
关键词 Logarithmic differential forms de Rham complex regular meromorphic forms holo-nomic D-modules Poincare lemma mixed Hodge structure residual currents.
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Strong/weak limits and strong/weak differentials of multi-variable indeterminate forms. I: Deduction and geometric illustration
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作者 宋光世 《Journal of Chongqing University》 CAS 2003年第2期104-108,共5页
Strong and weak limits as well as strong and weak differentials of multi-variable indeterminate forms are discussed based on the thorough exploration of differentiation to solve the strong and weak limits and differen... Strong and weak limits as well as strong and weak differentials of multi-variable indeterminate forms are discussed based on the thorough exploration of differentiation to solve the strong and weak limits and differentials of unitary indeterminate forms. The fruit of this work is going to be reported in three parts. The first part presents the standard analysis on this subject which supplements, systematizes and advances L. Hospital抯 principles on differential calculus by applying special ,general, and limit guaranteeing theories together with K(t) and XhK0 theories. The combination of theoretical analysis and geometric signification makes the derivation intuitional, visual and easy to perceive. 展开更多
关键词 strong limit weak limit strong differential weak differential indeterminate form multi-variable special form general form
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EXACT-VOLUME DIFFERENTIAL FORM DE RHAM COHOMOLOGY AND HAMILTON VARIATIONAL PRINCIPLE
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作者 王继春 《Acta Mathematica Scientia》 SCIE CSCD 1995年第4期415-421,共7页
In this paper, we suggested exact-volume differential form (for short:EVDF) and proved four theorems correlative with them: 1. existence theorem, 2. cohomology theorem,3. constant multiple theorem, and 4. equal gauge ... In this paper, we suggested exact-volume differential form (for short:EVDF) and proved four theorems correlative with them: 1. existence theorem, 2. cohomology theorem,3. constant multiple theorem, and 4. equal gauge theorem. And their application were discussed also. For examlpe, we deduced the particle dynamic equation of the special theory of relativity. At the same time we analyzed and contrasted cohomology theory with Hamilton's variational principle. The contrast showed the superiority of cohomology theory. Moreover,we gave a more complete classification list of differential forms. 展开更多
关键词 exact-volume differential form cohomology.
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An Extension of the Poincar’e Lemma of Differential Forms
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作者 Zhaoyang Tang Jianmin Zhu +1 位作者 Jianhua Huang Jin Li 《Applied Mathematics》 2013年第1期16-18,共3页
This paper is to extend the Poincar’e Lemma for differential forms in a bounded, convex domain [1] in Rn to a more general domain that, we call, is deformable to every point in itself. Then we extend the homotopy ope... This paper is to extend the Poincar’e Lemma for differential forms in a bounded, convex domain [1] in Rn to a more general domain that, we call, is deformable to every point in itself. Then we extend the homotopy operator T in [1] to the domain defromed to every point of itself. 展开更多
关键词 differential formS Poincar’e LEMMA Domain DEformED to EVERY Point
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STUDYING THE FOCAL VALUE OF ORDINARY DIFFERENTIAL EQUATIONS BY NORMAL FORM THEORY
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作者 张琪昌 梁以德 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1995年第9期891-900,共10页
We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation l... We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation language M ATHEMATICA,and extending the matrix representation method.This method can be used to calculate the focal value of any high order terms.This method has been verified by an example.The advantage of this method is simple and more readily applicable.the result is directly obtained by substitution. 展开更多
关键词 normal form ordinary differential equation focal value MATHEMATICA
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A NEW PROOF OF GAFFNEY’S INEQUALITY FOR DIFFERENTIAL FORMS ON MANIFOLDS-WITH-BOUNDARY:THE VARIATIONAL APPROACH à LA KOZONO-YANAGISAWA
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作者 Siran LI 《Acta Mathematica Scientia》 SCIE CSCD 2022年第4期1427-1452,共26页
Let(M,g_(0))be a compact Riemannian manifold-with-boundary.We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over M,via a variational approach a la Kozono-Yanag... Let(M,g_(0))be a compact Riemannian manifold-with-boundary.We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over M,via a variational approach a la Kozono-Yanagisawa[Lr-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains,Indiana Univ.Math.J.58(2009),1853-1920],combined with global computations based on the Bochner technique. 展开更多
关键词 Gaffney’s inequality differential form Sobolev spaces on manifolds Bochner technique variational approach
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Dynamics of Quantum State and Effective Hamiltonian with Vector Differential Form of Motion Method
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作者 Long Xiong Wei-Feng Zhuang Ming Gong 《Chinese Physics Letters》 SCIE EI CAS CSCD 2022年第7期36-40,共5页
Effective Hamiltonians in periodically driven systems have received widespread attention for realization of novel quantum phases, non-equilibrium phase transition, and Majorana mode. Recently, the study of effective H... Effective Hamiltonians in periodically driven systems have received widespread attention for realization of novel quantum phases, non-equilibrium phase transition, and Majorana mode. Recently, the study of effective Hamiltonian using various methods has gained great interest. We consider a vector differential equation of motion to derive the effective Hamiltonian for any periodically driven two-level system, and the dynamics of the spin vector are an evolution under the Bloch sphere. Here, we investigate the properties of this equation and show that a sudden change of the effective Hamiltonian is expected. Furthermore, we present several exact relations, whose expressions are independent of the different starting points. Moreover, we deduce the effective Hamiltonian from the high-frequency limit, which approximately equals the results in previous studies. Our results show that the vector differential equation of motion is not affected by a convergence problem, and thus, can be used to numerically investigate the effective models in any periodic modulating system. Finally, we anticipate that the proposed method can be applied to experimental platforms that require time-periodic modulation, such as ultracold atoms and optical lattices. 展开更多
关键词 Dynamics of Quantum State and Effective Hamiltonian with Vector differential form of Motion Method HAMILTONIAN VECTOR QUANTUM
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A UNIFORMLY CONVERGENT SECOND ORDER DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM
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作者 郭雯 林鹏程 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1989年第3期231-241,共11页
In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniform... In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme. 展开更多
关键词 exp A UNIformLY CONVERGENT SECOND ORDER DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ORDINARY differential EQUATION IN CONSERVATION form
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A HIGH ACCURACY DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF THE SECOND-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM
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作者 王国英 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1989年第5期465-470,共6页
In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the origi... In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3. 展开更多
关键词 A HIGH ACCURACY DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF THE SECOND-ORDER LINEAR ORDINARY differential EQUATION IN CONSERVATION form
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Research on Instruments of Formative Assessment for Differentiated Language Teaching 被引量:1
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作者 翟轶璠 《海外英语》 2012年第6X期283-284,288,共3页
Formative assessment is crucial to differentiated language teaching.To ensure the quality of differentiated teaching,some assessment instruments should be adopted during formative assessment,such as differentiated lea... Formative assessment is crucial to differentiated language teaching.To ensure the quality of differentiated teaching,some assessment instruments should be adopted during formative assessment,such as differentiated learning portfolios and group activity journal.The effective application of these instruments can also improve the students' autonomous language learning ability. 展开更多
关键词 formATIVE ASSESSMENT differentiATED INSTRUCTION in
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On the Conjecture of Gackstatter and Laine Concerning Algebraic Differential Equations
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作者 邱青 韩国平 《Chinese Quarterly Journal of Mathematics》 CSCD 2002年第2期16-21,共0页
In this paper we investigate the form of a class of complex differential equations withadmissible solution and obtain a main result.
关键词 admissible solution form of complex differential equations the value distribution
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Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method 被引量:3
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作者 K.DANESHJOU M.TALEBITOOTI R.TALEBITOOTI 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2013年第4期437-456,共20页
The generalized differential quadrature method (GDQM) is employed to con- sider the free vibration and critical speed of moderately thick rotating laminated compos- ite conical shells with different boundary conditi... The generalized differential quadrature method (GDQM) is employed to con- sider the free vibration and critical speed of moderately thick rotating laminated compos- ite conical shells with different boundary conditions developed from the first-order shear deformation theory (FSDT). The equations of motion are obtained applying Hamilton's concept, which contain the influence of the centrifugal force, the Coriolis acceleration, and the preliminary hoop stress. In addition, the axial load is applied to the conical shell as a ratio of the global critical buckling load. The governing partial differential equations are given in the expressions of five components of displacement related to the points ly- ing on the reference surface of the shell. Afterward, the governing differential equations are converted into a group of algebraic equations by using the GDQM. The outcomes are achieved considering the effects of stacking sequences, thickness of the shell, rotating velocities, half-vertex cone angle, and boundary conditions. Furthermore, the outcomes indicate that the rate of the convergence of frequencies is swift, and the numerical tech- nique is superior stable. Three comparisons between the selected outcomes and those of other research are accomplished, and excellent agreement is achieved. 展开更多
关键词 generalized differential quadrature method (GDQM) natural frequency rotating conical shell first-order shear deformation theory (FSDT) critical speed
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Study on the evolution of ore-formation fluids for Au-Sb ore deposits and the mechanism of Au-Sb paragenesis and differentiation in the southwestern part of Guizhou Province, China 被引量:10
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作者 WANG Zepeng XIA Yong +3 位作者 SONG Xieyan LIU Jianzhong YANG Chengfu YAN Baowen 《Chinese Journal Of Geochemistry》 EI CAS CSCD 2013年第1期56-68,共13页
Ore deposits (occurrences) of Au, As, Sb, Hg, etc. distributed in Southwest Guizhou constitute the important portion of the low-temperature metallogenic domain covering a large area in Southwest China, with the Carlin... Ore deposits (occurrences) of Au, As, Sb, Hg, etc. distributed in Southwest Guizhou constitute the important portion of the low-temperature metallogenic domain covering a large area in Southwest China, with the Carlin-type Au and Sb deposits being the most typical ones. In this paper the Au and Sb ore deposits are taken as the objects of study. Through the petrographic analysis, microthermomitric measurement and Raman spectrophic analysis of fluid inclusions in gangue minerals and research on the S and C isotopic compositions in the gold ore deposits we can reveal the sources of ore-forming materials and ore-forming fluids and the rules of ore fluid evolution. Ore deposits of Au, Sb, etc. are regionally classified as the products of ore fluid evolution, and their ore-forming materials and ore fluids were probably derived mainly from the deep interior of the Earth. Fluid inclusion studies have shown that the temperatures of Au mineralization are within the range of 170-361℃,the salinities are 0.35 wt%-8 wt% NaCl eq.; the temperatures of Sb mineralization are 129.4-214℃ and the salinities are 0.18 wt%- 3.23 wt% NaCl eq.; the ore-forming fluid temperatures and salinities tend to decrease progressively. In the early stage (Au metallogenic stage) the ore-forming fluids contained large amounts of volatile components such as CO2, CH4, N2 and H2S, belonging to the H2O-CO2-NaCl fluid system; in the late stage (Sb metallogenic stage) the ore-forming fluids belong to the Sb-bearing H2O-NaCl system. The primitive ore-forming fluids may have experienced at least two processes of immiscibility: (1) when early ore-bearing hydrothermal solutions passed through rock strata of larger porosity or fault broken zones, CO2, CH4, N2 would escape from them, followed by the release of pressure, resulting in pressure release and boiling of primitive homogenous fluids, thereafter giving rise to their phase separation, thus leading to Au unloading and mineralization; and (2) in the late stage (Sb metallogenic stage ) a large volume of meteoric water was involved in the ore-forming fluids, leading to fluid boiling as a result of their encounter, followed by the drop of fluid temperature. As a result, the dissolubility of Sb decreased so greatly that Sb was enriched and precipitated as ores. Due to differences in physic-chemical conditions between Au and Sb precipitates, Au and Sb were respectively precipitated in different structural positions, thus creating such a phenomenon of Au/Sb paragenesis and differentiation in space. 展开更多
关键词 成矿流体演化 中国西南地区 金锑矿床 分化 共生 贵州省 西南部 流体温度
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