The conceptions of theorems, laws and corollaries of hydrology were put forward. Combining with hydrology practice, several theo- rems, laws as well as corollaries of hydrology were summarized. The study provided some...The conceptions of theorems, laws and corollaries of hydrology were put forward. Combining with hydrology practice, several theo- rems, laws as well as corollaries of hydrology were summarized. The study provided some references for accelerating the development of hydrology theory in these aspects and promoting the improvement of its production technology.展开更多
A universal symplectic structure for a Newtonian system including nonconservative cases can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics. In this paper the symplectic geometry...A universal symplectic structure for a Newtonian system including nonconservative cases can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics. In this paper the symplectic geometry structure of Birkhoffian system is discussed, then the symplecticity of Birkhoffian phase flow is presented. Based on these properties we give a way to construct symplectic schemes for Birkhoffian systems by using the generating function method.展开更多
The conservation laws of the Levi equation are presented.Two types of symmetry of the Levi equationhierarchy are deduced.Further it is proved that these symmetries construct an infinite-dimensional Lie algebra.
Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc., are reduced to an integrable ODE expressed by u" +p(u)(u')^2...Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc., are reduced to an integrable ODE expressed by u" +p(u)(u')^2 + q(u) = 0 whose generai solution can be given. Furthermore, combining complete discrimination system for polynomiai, the classifications of all single travelling wave solutions to these equations are obtained. The equation u"+p(u)(u')^2+q(u) = 0 includes the equation (u')^2 = f(u) as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.展开更多
New types of exact solutions of the (N + 1)-dimensional φ^4-model are studied in detail. Some types of interaction solutions such as the periodic-periodic interaction waves and the periodic-solitary wave interacti...New types of exact solutions of the (N + 1)-dimensional φ^4-model are studied in detail. Some types of interaction solutions such as the periodic-periodic interaction waves and the periodic-solitary wave interaction solutions are found.展开更多
In this paper,by improving some procedure of extended tanh-function method,some new exact solutions to the integrable Broer-Kaup equations in(2 + 1)-dimensional spaces are obtained,which include soliton-like solutions...In this paper,by improving some procedure of extended tanh-function method,some new exact solutions to the integrable Broer-Kaup equations in(2 + 1)-dimensional spaces are obtained,which include soliton-like solutions,solitary wave solutions,trigonometric function solutions,and rational solutions.展开更多
Variable separation approach is introduced to solve the (2+1)-dimensional KdV equation. A series of variable separation solutions is derived with arbitrary functions in system. We present a new soliton excitation m...Variable separation approach is introduced to solve the (2+1)-dimensional KdV equation. A series of variable separation solutions is derived with arbitrary functions in system. We present a new soliton excitation model (24). Based on this excitation, new soliton structures such as the multi-lump soliton and periodic soliton are revealed by selecting the arbitrary function appropriately.展开更多
Using the modified extended tanh-function method,explicit and exact traveling wave solutions for the(2+1)-dimensional higher-order Broer-Kaup(HBK)system,comprising new soliton-like and period-form solutions,are obtained.
By deriving a Korteweg-de Vries (KdV) equation for particular dusty plasma, we find the existing regions of solitary waves. The regions of both rarefactive and compressive solitary waves have also been given in this...By deriving a Korteweg-de Vries (KdV) equation for particular dusty plasma, we find the existing regions of solitary waves. The regions of both rarefactive and compressive solitary waves have also been given in this paper for a two-temperature ions and a hot adiabatic dusty fluid, where the dust charges can vary.展开更多
The experimental values of 2059 β-decay half-lives are systematically analyzed and investigated. We have found that they are in satisfactory agreement with Benford's law, which states that the frequency of occurrenc...The experimental values of 2059 β-decay half-lives are systematically analyzed and investigated. We have found that they are in satisfactory agreement with Benford's law, which states that the frequency of occurrence of each figure, 1-9, as the first significant digit in a surprisingly large number of different data sets follows a logarithmic distribution favoring the smaller ones. Benford's logarithmic distribution of β-deeay half-lives can be explained in terms of Neweomb's justification of Benford's law and empirical exponential law of β-decay half-lives. Moreover, we test the calculated values of 6721 β-decay half-lives with the aid of Benford's law. This indicates that Benford's law is useful for theoretical physicists to test their methods for calculating β-decay half-lives.展开更多
A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic w...A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions, and other exact excitations like polynomial solutions, exponential solutions, and rational solutions, etc., are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and a generalized cubic nonlinear Klein-Gordon equation.展开更多
For any finite-dimensional semisimple Lie algebra g, a Z+-graded vertex algebra is construsted on the vacuum representation Vk(g[θ]of g[θ]),which is a one-dimentionM central extension of 8-invariant subspace on t...For any finite-dimensional semisimple Lie algebra g, a Z+-graded vertex algebra is construsted on the vacuum representation Vk(g[θ]of g[θ]),which is a one-dimentionM central extension of 8-invariant subspace on the loop algebra Lg=g C((t^1/p)).展开更多
Using ab initio plane-wave pseudopotential density functional theory method, we have studied the structural and thermodynamic properties of the wurtzite boron nitride (w-BN). Through the quasi-harmonic Debye model, ...Using ab initio plane-wave pseudopotential density functional theory method, we have studied the structural and thermodynamic properties of the wurtzite boron nitride (w-BN). Through the quasi-harmonic Debye model, in which the phononic effects are considered, the dependencies of the normalized lattice parameters a/ao and c/co, the axial ratio c/a, and the normalized primitive cell volume V/Vo on pressure P and temperature T are investigated. Furthermore, the Debye temperature θ, the variation of thermal expansion α, as well as the heat capacity Cv as functions of P and T are studied systematically.展开更多
The present study was conducted to evaluate minimally invasive tube cystostomy technique in goats and calves suffering from obstructive urolithiasis having intact and ruptured urinary bladder. This clinical study was ...The present study was conducted to evaluate minimally invasive tube cystostomy technique in goats and calves suffering from obstructive urolithiasis having intact and ruptured urinary bladder. This clinical study was carried out on 84 clinical cases (61 goats and 23 calves) based on the history, clinical signs and physical examination along with common laboratory techniques at the Chittagong Veterinary and Animal Sciences University during January 2015-December 2016. Physical parameters like heart rate, respiratory rate, rectal temperature and dehydration status of animals were noted and corrected abnormality before surgery. Some cases had slightly higher haemoglobin (Hb), packed cell volume (PCV) and uric acid and epithelial casts in urine. The study revealed that abnormal calcium-phosphorus ratio was predisposing the animals to urolithiasis. It was found that young ruminants (3-5 months) were most commonly affected in both species. All affected animals were male in this study, in which calves were not castrated but in goats 95.08% animals were castrated. Rupture of bladder was more common in calves as compared to goats. Postoperatively all cases were administered with broad-spectrum antibiotic, anti-inflammatory agent and caliculolytic agents like ammonium chloride. Tube cystostomy with Foley’s catheter was the most satisfactory technique for the management of obstructive urolithiasis in small ruminants. Postoperative complications were recorded only in four animals and remaining 80 animals had a normal recovery from tube cystostomy. Tube cystostomy is a simple, inexpensive and very effective procedure for the treatment of urolithiasis in ruminants.展开更多
Spaces of equivalence modulo a relation of congruence are constructed on field solutions to establish a theory of the universe that includes the theory QFT (Quantum Field theory), the SUSY (Super-symmetry theory) ...Spaces of equivalence modulo a relation of congruence are constructed on field solutions to establish a theory of the universe that includes the theory QFT (Quantum Field theory), the SUSY (Super-symmetry theory) and HST (heterotic string theory) using the sheaves correspondence of differential operators of the field equations and sheaves of coherent D - Modules [1]. The above mentioned correspondence use a Zuckerman functor that is a factor of the universal functor of derived sheaves of Harish-Chandra to the Langlands geometrical program in mirror symmetry [2, 3]. The obtained development includes complexes of D - modules of infinite dimension, generalizing for this way, the BRST-cohomology in this context. With it, the class of the integrable systems is extended in mathematical physics and the possibility of obtaining a general theory of integral transforms for the space - time (integral operator cohomology [4]), and with it the measurement of many of their observables [5]. Also tends a bridge to complete a classification of the differential operators for the different field equations using on the base of Verma modules that are classification spaces of SO(l, n + 1), where elements of the Lie algebra al(1, n + 1), are differential operators, of the equations in mathematical physics [1]. The cosmological problem that exists is to reduce the number of field equations that are resoluble under the same gauge field (Verma modules) and to extend the gauge solutions to other fields using the topological groups symmetries that define their interactions. This extension can be given by a global Langlands correspondence between the Hecke sheaves category on an adequate moduli stack and the holomorphic L G - bundles category with a special connection (Deligne connection). The corresponding D - modules may be viewed as sheaves of conformal blocks (or co-invariants) (images under a version of the Penrose transform [1, 6]) naturally arising in the framework of conformal field theory.展开更多
Based on the experimental results of the references [1] and [2], i. e., the contact resistance increases due to the contaminants formed on contact surface with the operating cycles,the physical model and math model of...Based on the experimental results of the references [1] and [2], i. e., the contact resistance increases due to the contaminants formed on contact surface with the operating cycles,the physical model and math model of contact failure on small current contacts under mechanical actions are proposed through deep theoretical analysis in this paper.展开更多
It is exciting that the gravitational wave has been confirmed, according to the announcement of LIGO. Perhaps, this is the time for physicists to fix the Einstein equation for the gravitational wave and the nonexisten...It is exciting that the gravitational wave has been confirmed, according to the announcement of LIGO. Perhaps, this is the time for physicists to fix the Einstein equation for the gravitational wave and the nonexistence of the dynamic solution. These two problems are inextricably related. As a first step, theorists should improve their pure mathematics beyond Einstein on non-linear mathematics and related physical considerations. Then, theoretically we must first show that the gravitational waves necessarily exist that Einstein was not certain. We find that the existence of gravitational waves is due to that the photons must have a gravitational wave component. Next, we must rectify the Einstein equation that has no gravitational wave solution which Einstein has recognized, and no dynamic solution that Einstein failed to see. However, it is very questionable that the measured gravitational waves are due to the black holes that can be definitely valid due to the long distance of the sources. Moreover, since the repulsive gravitation can also generate a gravitational wave, the problem of gravitational wave is actually far more complicated than we have known. A useful feature of the gravitational wave based on repulsive gravitation is that it can be easily generated on earth. Thus this can be a useful tool for communication because it can penetrate any medium.展开更多
Our ability to arrive at knowledge by chains of judgment is constitutive of our rationality, likewise our ability to discern the self-evidence of logical and arithmetical laws. To count an activity as "thinking about...Our ability to arrive at knowledge by chains of judgment is constitutive of our rationality, likewise our ability to discern the self-evidence of logical and arithmetical laws. To count an activity as "thinking about the physical world" is to hold it assessable in the light of the laws of physics; whereas to count an activity as "thinking at all" is to hold it assessable in the light of the laws of logic. Thus, the kind of generality that distinguishes logic from the special sciences is a generality in the applicability of the norms it provides. Logical laws are more general than laws of the special sciences because they prescribe universally the way in which one ought to think, if one is to think all. Logicism is usually understood to be the thesis that all, or at least large parts of, mathematics can be reduced to logic. This thesis has two sides: (1) all mathematical concepts can be defined in terms of basic logical concepts; (2) all mathematical theorems can be deduced from basic logical truths. According to logicism all terms, including all mathematical terms, are to be given a definite meaning within the basic system. This paper aims at a comparative analysis of the contributions of Frege and Russell to the development of modem logic by reviewing in some detail their essential features and derivations. Without making any pretensions to proffering a definitive resolution of any puzzles, the discussion will, however, raise some fundamental questions, and offer a critical evaluation of the putative success or failure of the logicist programmes of Frege and Russell.展开更多
U(3)-O(4) transitional description of diatomic molecules in the U(4) vibron model is studied by usingthe algebraic Bethe ansatz, in which the O(4) limit is a special case of the theory. Vibrational band-heads of somet...U(3)-O(4) transitional description of diatomic molecules in the U(4) vibron model is studied by usingthe algebraic Bethe ansatz, in which the O(4) limit is a special case of the theory. Vibrational band-heads of sometypical diatomic molecules are fitted by both transitional theory and the O(4) limit within the same framework. Theresults show that there are evident deviations from the O(4) limit in description of vibrational spectra of some diatomicmolecules.展开更多
文摘The conceptions of theorems, laws and corollaries of hydrology were put forward. Combining with hydrology practice, several theo- rems, laws as well as corollaries of hydrology were summarized. The study provided some references for accelerating the development of hydrology theory in these aspects and promoting the improvement of its production technology.
基金The project supported by the Special Funds for State Key Basic Research Projects under Grant No.G1999,032800
文摘A universal symplectic structure for a Newtonian system including nonconservative cases can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics. In this paper the symplectic geometry structure of Birkhoffian system is discussed, then the symplecticity of Birkhoffian phase flow is presented. Based on these properties we give a way to construct symplectic schemes for Birkhoffian systems by using the generating function method.
基金National Natural Science Foundation of China under Grant Nos.10871165 and 10671121
文摘The conservation laws of the Levi equation are presented.Two types of symmetry of the Levi equationhierarchy are deduced.Further it is proved that these symmetries construct an infinite-dimensional Lie algebra.
文摘Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc., are reduced to an integrable ODE expressed by u" +p(u)(u')^2 + q(u) = 0 whose generai solution can be given. Furthermore, combining complete discrimination system for polynomiai, the classifications of all single travelling wave solutions to these equations are obtained. The equation u"+p(u)(u')^2+q(u) = 0 includes the equation (u')^2 = f(u) as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.
基金The project supported by National Natural Science Foundations of China under Grant Nos. 90203001, 10475055, and 90503006
文摘New types of exact solutions of the (N + 1)-dimensional φ^4-model are studied in detail. Some types of interaction solutions such as the periodic-periodic interaction waves and the periodic-solitary wave interaction solutions are found.
文摘In this paper,by improving some procedure of extended tanh-function method,some new exact solutions to the integrable Broer-Kaup equations in(2 + 1)-dimensional spaces are obtained,which include soliton-like solutions,solitary wave solutions,trigonometric function solutions,and rational solutions.
基金The author would like to thank Profs. Jie-Fang Zhang and Chun-Long Zheng for helpful discussions.
文摘Variable separation approach is introduced to solve the (2+1)-dimensional KdV equation. A series of variable separation solutions is derived with arbitrary functions in system. We present a new soliton excitation model (24). Based on this excitation, new soliton structures such as the multi-lump soliton and periodic soliton are revealed by selecting the arbitrary function appropriately.
文摘Using the modified extended tanh-function method,explicit and exact traveling wave solutions for the(2+1)-dimensional higher-order Broer-Kaup(HBK)system,comprising new soliton-like and period-form solutions,are obtained.
基金National Natural Science Foundation of China under Crant No.10575082Natural Science Foundation of Gansu Province under Grant No.3ZS061-A25-014Natural Science Foundation of Northwest Normal University under Grant No.NWNUKJCXGC-03-17
文摘By deriving a Korteweg-de Vries (KdV) equation for particular dusty plasma, we find the existing regions of solitary waves. The regions of both rarefactive and compressive solitary waves have also been given in this paper for a two-temperature ions and a hot adiabatic dusty fluid, where the dust charges can vary.
基金supported by the National Natural Science Foundation of China under Grant Nos. 10675090, 10535010, and 10775068the National Fund for Forstering Talents of Basic Science under Grant No. J0630316+2 种基金the 973 State Key Basic Research and Development Program of China under Grant No. 2007CB815004the CAS Knowledge Innovation Project under Grant No. KJCX2-SW-N02the Research Fund of Doctoral Points under Grant No. 20070284016
文摘The experimental values of 2059 β-decay half-lives are systematically analyzed and investigated. We have found that they are in satisfactory agreement with Benford's law, which states that the frequency of occurrence of each figure, 1-9, as the first significant digit in a surprisingly large number of different data sets follows a logarithmic distribution favoring the smaller ones. Benford's logarithmic distribution of β-deeay half-lives can be explained in terms of Neweomb's justification of Benford's law and empirical exponential law of β-decay half-lives. Moreover, we test the calculated values of 6721 β-decay half-lives with the aid of Benford's law. This indicates that Benford's law is useful for theoretical physicists to test their methods for calculating β-decay half-lives.
文摘A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions, and other exact excitations like polynomial solutions, exponential solutions, and rational solutions, etc., are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and a generalized cubic nonlinear Klein-Gordon equation.
基金National Key Basic Research Project of China under Grant Nos.2004CB318000 and 2006CB805905National Natural Science Foundation of China under Grant No.10471034+1 种基金the Outstanding Youth Fund of Henan Province under Grant No.0512000100Innovation Fund of Colleges and Universities in Henan Province
文摘For any finite-dimensional semisimple Lie algebra g, a Z+-graded vertex algebra is construsted on the vacuum representation Vk(g[θ]of g[θ]),which is a one-dimentionM central extension of 8-invariant subspace on the loop algebra Lg=g C((t^1/p)).
基金The project supported by National Natural Science Foundation of China under Grant No. 10576020
文摘Using ab initio plane-wave pseudopotential density functional theory method, we have studied the structural and thermodynamic properties of the wurtzite boron nitride (w-BN). Through the quasi-harmonic Debye model, in which the phononic effects are considered, the dependencies of the normalized lattice parameters a/ao and c/co, the axial ratio c/a, and the normalized primitive cell volume V/Vo on pressure P and temperature T are investigated. Furthermore, the Debye temperature θ, the variation of thermal expansion α, as well as the heat capacity Cv as functions of P and T are studied systematically.
文摘The present study was conducted to evaluate minimally invasive tube cystostomy technique in goats and calves suffering from obstructive urolithiasis having intact and ruptured urinary bladder. This clinical study was carried out on 84 clinical cases (61 goats and 23 calves) based on the history, clinical signs and physical examination along with common laboratory techniques at the Chittagong Veterinary and Animal Sciences University during January 2015-December 2016. Physical parameters like heart rate, respiratory rate, rectal temperature and dehydration status of animals were noted and corrected abnormality before surgery. Some cases had slightly higher haemoglobin (Hb), packed cell volume (PCV) and uric acid and epithelial casts in urine. The study revealed that abnormal calcium-phosphorus ratio was predisposing the animals to urolithiasis. It was found that young ruminants (3-5 months) were most commonly affected in both species. All affected animals were male in this study, in which calves were not castrated but in goats 95.08% animals were castrated. Rupture of bladder was more common in calves as compared to goats. Postoperatively all cases were administered with broad-spectrum antibiotic, anti-inflammatory agent and caliculolytic agents like ammonium chloride. Tube cystostomy with Foley’s catheter was the most satisfactory technique for the management of obstructive urolithiasis in small ruminants. Postoperative complications were recorded only in four animals and remaining 80 animals had a normal recovery from tube cystostomy. Tube cystostomy is a simple, inexpensive and very effective procedure for the treatment of urolithiasis in ruminants.
文摘Spaces of equivalence modulo a relation of congruence are constructed on field solutions to establish a theory of the universe that includes the theory QFT (Quantum Field theory), the SUSY (Super-symmetry theory) and HST (heterotic string theory) using the sheaves correspondence of differential operators of the field equations and sheaves of coherent D - Modules [1]. The above mentioned correspondence use a Zuckerman functor that is a factor of the universal functor of derived sheaves of Harish-Chandra to the Langlands geometrical program in mirror symmetry [2, 3]. The obtained development includes complexes of D - modules of infinite dimension, generalizing for this way, the BRST-cohomology in this context. With it, the class of the integrable systems is extended in mathematical physics and the possibility of obtaining a general theory of integral transforms for the space - time (integral operator cohomology [4]), and with it the measurement of many of their observables [5]. Also tends a bridge to complete a classification of the differential operators for the different field equations using on the base of Verma modules that are classification spaces of SO(l, n + 1), where elements of the Lie algebra al(1, n + 1), are differential operators, of the equations in mathematical physics [1]. The cosmological problem that exists is to reduce the number of field equations that are resoluble under the same gauge field (Verma modules) and to extend the gauge solutions to other fields using the topological groups symmetries that define their interactions. This extension can be given by a global Langlands correspondence between the Hecke sheaves category on an adequate moduli stack and the holomorphic L G - bundles category with a special connection (Deligne connection). The corresponding D - modules may be viewed as sheaves of conformal blocks (or co-invariants) (images under a version of the Penrose transform [1, 6]) naturally arising in the framework of conformal field theory.
文摘Based on the experimental results of the references [1] and [2], i. e., the contact resistance increases due to the contaminants formed on contact surface with the operating cycles,the physical model and math model of contact failure on small current contacts under mechanical actions are proposed through deep theoretical analysis in this paper.
文摘It is exciting that the gravitational wave has been confirmed, according to the announcement of LIGO. Perhaps, this is the time for physicists to fix the Einstein equation for the gravitational wave and the nonexistence of the dynamic solution. These two problems are inextricably related. As a first step, theorists should improve their pure mathematics beyond Einstein on non-linear mathematics and related physical considerations. Then, theoretically we must first show that the gravitational waves necessarily exist that Einstein was not certain. We find that the existence of gravitational waves is due to that the photons must have a gravitational wave component. Next, we must rectify the Einstein equation that has no gravitational wave solution which Einstein has recognized, and no dynamic solution that Einstein failed to see. However, it is very questionable that the measured gravitational waves are due to the black holes that can be definitely valid due to the long distance of the sources. Moreover, since the repulsive gravitation can also generate a gravitational wave, the problem of gravitational wave is actually far more complicated than we have known. A useful feature of the gravitational wave based on repulsive gravitation is that it can be easily generated on earth. Thus this can be a useful tool for communication because it can penetrate any medium.
文摘Our ability to arrive at knowledge by chains of judgment is constitutive of our rationality, likewise our ability to discern the self-evidence of logical and arithmetical laws. To count an activity as "thinking about the physical world" is to hold it assessable in the light of the laws of physics; whereas to count an activity as "thinking at all" is to hold it assessable in the light of the laws of logic. Thus, the kind of generality that distinguishes logic from the special sciences is a generality in the applicability of the norms it provides. Logical laws are more general than laws of the special sciences because they prescribe universally the way in which one ought to think, if one is to think all. Logicism is usually understood to be the thesis that all, or at least large parts of, mathematics can be reduced to logic. This thesis has two sides: (1) all mathematical concepts can be defined in terms of basic logical concepts; (2) all mathematical theorems can be deduced from basic logical truths. According to logicism all terms, including all mathematical terms, are to be given a definite meaning within the basic system. This paper aims at a comparative analysis of the contributions of Frege and Russell to the development of modem logic by reviewing in some detail their essential features and derivations. Without making any pretensions to proffering a definitive resolution of any puzzles, the discussion will, however, raise some fundamental questions, and offer a critical evaluation of the putative success or failure of the logicist programmes of Frege and Russell.
基金The project supported by National Natural Science foundation of China under Grant No.10175031the Natural Science Foundation of Liaoning Province of China under Grant No.2001101053
文摘U(3)-O(4) transitional description of diatomic molecules in the U(4) vibron model is studied by usingthe algebraic Bethe ansatz, in which the O(4) limit is a special case of the theory. Vibrational band-heads of sometypical diatomic molecules are fitted by both transitional theory and the O(4) limit within the same framework. Theresults show that there are evident deviations from the O(4) limit in description of vibrational spectra of some diatomicmolecules.