Uniform Formula Research Articles

KdV hierarchy with time-dependent coefficients: Lax integrability, bilinear Bäcklund transformation and soliton solutions

Starting from the linear problems, we first derive a new Lax integrable Korteweg-de Vries (KdV) hierarchy with time-dependent coefficients. Then a bilinear Bäcklund transformation (BT) of the variable-coefficient KdV (vcKdV) equation contained in the KdV hierarchy is given. Based on the given bilinear BT, one-soliton solution, two-soliton solution and three-soliton solution are obtained. From these obtained soliton solutions, a uniform formula of explicit n-soliton solutions of the vcKdV equation is summarized. It is graphically shown that the dynamical evolutions of such soliton solutions with time-dependent functions of the KdV hierarchy possess time-varying speeds in the process of propagations.

Products of Borel fixed ideals of maximal minors

We study a large family of products of Borel fixed ideals of maximal minors. We compute their initial ideals and primary decompositions, and show that they have linear free resolutions. The main tools are an extension of straightening law and a very uniform primary decomposition formula. We study also the homological properties of associated multi-Rees algebra which are shown to be Cohen–Macaulay, Koszul and defined by a Gröbner basis of quadrics.

Asymptotic results for ruin probability in a two-dimensional risk model with stochastic investment returns

This paper considers a two-dimensional time-dependent risk model with stochastic investment returns. In the model, an insurer operates two lines of insurance businesses sharing a common claim number process and can invest its surplus into some risky assets. The claim number process is assumed to be a renewal counting process and the investment return is modeled by a geometric Lévy process. Furthermore, claim sizes of the two insurance businesses and their common inter-arrival times correspondingly follow a three-dimensional Sarmanov distribution. When claim sizes of the two lines of insurance businesses are heavy tailed, we establish some uniform asymptotic formulas for the ruin probability of the model over certain time horizon. Also, we show the accuracy of these asymptotic estimates for the ruin probability under the risk model by numerical studies.

Phase Transitions for Scale-Free SAT Formulas

Recently, a number of non-uniform random satisfiability models have been proposed that are closer to practical satisfiability problems in some characteristics. In contrast to uniform random Boolean formulas, scale-free formulas have a variable occurrence distribution that follows a power law. It has been conjectured that such a distribution is a more accurate model for some industrial instances than the uniform random model. Though it seems that there is already an awareness of a threshold phenomenon in such models, there is still a complete picture lacking. In contrast to the uniform model, the critical density threshold does not lie at a single point, but instead exhibits a functional dependency on the power-law exponent. For scale-free formulas with clauses of length k=2, we give a lower bound on the phase transition threshold as a function of the scaling parameter. We also perform computational studies that suggest our bound is tight and investigate the critical density for formulas with higher clause lengths. Similar to the uniform model, on formulas with k>=3, we find that the phase transition regime corresponds to a set of formulas that are difficult to solve by backtracking search.

A cluster-formula composition design approach based on the local short-range order in solid solution structure

The composition design is of importance for developing high-performance complex alloys and is also the primary step to realize a new mode for material development via theoretical prediction and experimental verification, in comparison with the traditional experience-oriented experiments. Traditional alloy design approaches, including Hume-Rothery rule, electron theories, equivalent method, computer simulation, etc., are first reviewed from the viewpoints of their theoretical basis and applicability to limitations. Almost all the traditional alloys are based on solid solution structures, in which the typical characteristic is the chemical short-range order (CSRO) of the solute distribution. We propose a cluster-plus-glue-atom model for stable solid solutions in light of CSRO. A cluster-formula composition design approach is presented for developing the multi-component high-performance alloys. The cluster-plus-glue-atom model classifies the solid solution structure into two parts, i.e., the cluster part and the glue atom part, where the clusters are centered by solute atoms, showing the strong interactions of clusters with the solvent base and the weak interactions of clusters with solute atoms. The clusters are the nearest-neighbor polyhedrons, being cuboctahedron with a coordination number of 12 (CN12) in FCC structure and rhombic dodecahedron with a CN14 in BCC structure, respectively. Then a uniform cluster-formula of[CN12/14 cluster](glue atom)x is achieved from the cluster model. Its wide applications in different multi-component alloy systems confirm its universality as a simple and accurate tool for multiple-component complex alloy composition design. Such alloy systems include corrosion-resistant Cu alloys, high-performance Ni-base superalloys, high-strength maraging stainless steels, Ti/Zr alloys with low Young's modulus, high-entropy alloys, amorphous metallic glasses, quasicrystals, etc.. The specific alloy design steps are incarnated in the up-Ti alloys with low Young's modulus. Firstly, the necessary alloying elements are chosen according to the service requirements (BCC stability and low Young's modulus). Secondly, the local cluster unit to present CSRO and the corresponding cluster formula of[(Mo, Sn)-(Ti, Zr)14](Nb, Ta)x are built, in which the occupations of the alloying elements in the cluster formula are determined by the enthalpy of mixing H between them with the base Ti. Thirdly, these designed alloys are verified experimentally, and the lowest Young's modulus appears at the up-[(Mo0.5Sn0.5)-(Ti13Zr1)]Nb1. Finally, a new Mo equivalent formula under the guidance of phase diagram features is proposed to characterize the structural stability of Ti alloy. Thus all the Ti alloy compositions with different structural types can be expressed with a uniform cluster formula, in which the structural types of alloys are determined by the Mo equivalent.

Modeling plastic deformation effect on the hysteresis loops of ferromagnetic materials based on modified Jiles-Atherton model

Plastic deformation is one of the most important features that affect the hysteresis magnetic properties of steels, because it changes the dislocation density and affects domain-wall movement and pinning. In order to model the effect of plastic deformation on the magnetic properties, the prevailing Jiles-Atherton (J-A) theory is extensively used. However, the J-A models in a series of papers published by Jiles et al. are not completely consistent. As a result, there exists no uniform formula of magneto-plastic model established by different researchers, based on different J-A models, and various versions given by different mathematic expressions of magneto-plastic model often create difficulty in discriminating the accuracies and effectivenesses of the analyzed results. Therefore, it is necessary to establish an accurate and reasonable magneto-plastic model. In this paper, on the basis of magnetization mechanism of ferrimagnet and plastic deformation model, the effects of plastic deformation on the magnetic characteristic parameters adopted in magneto-plastic model, such as dislocation density, pinning coefficient and scaling constant, are analyzed and the relationship between them is first established. Then, by contrasting the fitting formula of the anhysteretic magnetization curve, the energy conservation equation and the effective magnetic field equation established by different researchers, several queries are proposed, and the irrationality and inaccuracy of the existing magneto-plastic model are elucidated, such as the mixing of anhysteresis magnetization and magnetization, the unreasonably regarding the irreversible magnetization energy as actual total magnetization energy. Thus, the energy conservation equation, the effective magnetic field equation and the anhysteretic magnetization equation are modified, and the differential expression of the magneto-plastic model is re-derived finally. Comparing the results calculated by the existing magneto-plastic models with the experimental results, it is seen indeed that a more sharp change of magnetization appears at small plastic deformation, then, the values of magnetization decrease more slowly with the increase of plastic deformation than those from the models respectively proposed by Li Jian-Wei, Leng Jian-Cheng and Wang Zheng-Dao; the saturation magnetization and residual magnetization decrease with the increase of plastic deformation, the coercive force is increased oppositely and the trend to reach the saturation magnetization becomes gentler, which is more exactly consonant with experiment observation than that calculated by the Sablik's model; additionally, the hysteresis loops of the plastically deformed carbon-steel samples calculated by the modified magneto-plastic model are also in better agreement with the experimental results than those from the existing models. Consequently, the modification is effective, and the modified magneto-plastic model is more accurate to simulate the plastic deformation effect on the magnetic property of ferromagnetic material.

A Multi-Directional Magnetic Treatment Approach to Improve the Dimensional Stability of Bearing Rings

Comparing with one directional magnetic treatment, a multidirectional magnetic treatment is applied to reduce residual stress in thin-walled bearing rings in order to improve the dimensional stability of bearing rings. The magnetic field was applied in both the axial and the radial directions of the thin walled bearing rings respectively. Six months of repeated testing results demonstrated that the dimension of bearing rings after magnetic treatment had no significant change. Experimental results shows both the major axis(the longer axis of an ellipse)and the minor axis(the shorter axis of an ellipse)appear shorten tendency and the dimension of bearing rings tends to be stable. Thin-walled uniform radial pressure deformation formulas are applied to calculate the reduced stress, and the maximum reduction is up to 45.97MPa. Research shows the multi-directional magnetic treatment can improve the dimensional stability of bearing rings and used in practical engineering.

Tail behavior for the sum of two correlated classes of discounted aggregate claims in a time-dependent risk model

ABSTRACTConsider a continuous-time risk model with two correlated classes of insurance business and a constant force of interest. Suppose that the correlation comes from a common shock, and that the claim sizes and inter-arrival times correspondingly form a sequence of random pairs, with each pair obeying a dependence structure. By assuming that the claim sizes are heavy tailed, a uniform tail asymptotic formula for the sum of the two correlated classes of discounted aggregate claims is obtained.

Effective width method for distortional buckling design of cold-formed lipped channel sections

The aim of this paper is to improve the effective width method for distortional buckling strength of cold-formed steel lipped channel members using the energy method and the large deflection theory based on nonlinear post-buckling strength analysis of the partially stiffened element. The accuracy of the proposed method was verified using the finite element method. Comparison on post-distortional buckling strength is conducted between the energy method and the effective width method in the Chinese code Technical code of cold-formed thin-walled steel structures (GB50018-2002). The compared results show that the energy method is very efficient and the effective width method formula in Chinese code can be used to calculate the post-distortional buckling strength. Then, The half-wave length, elastic buckling stress of distortional buckling and the corresponding stability coefficient of partially stiffened elements are developed based on the energy method. With comparison on the calculated results of elastic buckling stress using the proposed method and the Finite Strip Method, suitability and good precision of the developed method are illuminated. Finally, a uniform formula for the stability coefficient of partially stiffened elements considering both local and distortional buckling effect is established. Comparison with experimental results and calculated results predicted with North America specification indicates that the proposed method is reasonable and has a high accuracy and reliability for the distortional buckling load carrying capacities of cold-formed lipped channel members.

Basic geometry and aberration characteristics of conicoidal conformal domes.

This paper investigated the geometry and aberration characteristics of conicoidal conformal domes. First, on the basis of previous research, we got the expression that was suitable for describing the external surface of the conicoidal conformal dome. Based on the theory of differential geometry, this paper first proved that the Dupin index line of a quadric surface was an ellipsoid and the radius of curvature had extreme values in the meridian plane and sagittal plane. Then the uniform formulas of curvature which were suitable for ellipsoid, paraboloid, and hyperboloid were deduced in the meridian plane and sagittal plane, respectively. Meanwhile, the angle between the axis of imaging systems and the surface normal was calculated. With the help of computers, the plots of curvature differences and the angle in the case of different edge slopes, fineness ratios, and the locations of the rotational center were obtained. Finally, we analyzed the Zernike polynomial coefficients of Z4, Z5, and Z8, which represent defocus, astigmatism and coma, respectively for the model established in CODE V. The research indicates that the dynamic ranges of defocus, astigmatism, and coma increase with the growing of edge slopes and fineness ratios, but have little change with the variation of the rotational center positions. Moreover, the curves of Z5 and Z8 have turning points, and the curves of curvature differences and angle difference are only similar to the curves of Z5 and Z8 when the look angle changes after the turning point. For the look angle changing from zero to the turning point, the curves of Z5 and Z8 change rapidly. This is mainly caused by the significant variations of the symmetry of the conformal dome participating in imaging. Therefore, the aberrations with small scanning angles should be given more attention when designing the conformal systems.

Confined collapse of unbonded multi-layer pipe subjected to external pressure

If a leakage exists on the outer sheath of an unbonded multi-layer pipe, external hydrostatic pressure will be applied on the surface of the internal sheath inducing the possibility of a confined collapse. Also, when the pipe bore is depressurized at a fast rate, the pressure remaining in the pipe wall can cause the similar collapse of the internal sheath. The present paper uses a flexible pipe with multi-layer structure to examine the mechanical response of the internal cylinder confined by outer cylinders under external pressure. The previous related researches mainly focus on the rigid or almost rigid confinement conditions. However, this study emphasizes on structural elastic stability in terms of the different confinement levels and D/t ratios of the internal cylinder. The initial investigation is numerical and employs two-dimensional models to obtain the nonlinear pressure-deformation equilibrium paths using a Riks algorithm. The results are fitted as uniform formulas for the prediction to the critical pressure and corresponding collapse modes. The simplified analytical model followed is developed through the principle of virtual work to compare with the fitting formulas. Experimental tests of an unbonded flexible pipe with simple construction are also conducted to validate and agree with the theoretical estimations.

Estimation of final hydrogen temperature from refueling parameters

Compressed hydrogen storage is currently widely used in fuel cell vehicles due to its simplicity in tank structure and refueling process. For safety reason, the final gas temperature in the hydrogen tank during vehicle refueling must be maintained under a certain limit, e.g., 85 °C. Many experiments have been performed to find the relations between the final gas temperature in the hydrogen tank and refueling conditions. The analytical solution of the hydrogen temperature in the tank can be obtained from the simplified thermodynamic model of a compressed hydrogen storage tank, and it serves as function formula to fit experimental temperatures. From the analytical solution, the final hydrogen temperature can be expressed as a weighted average form of initial temperature, inflow temperature and ambient temperature inspired by the rule of mixtures. The weighted factors are related to other refueling parameters, such as initial mass, initial pressure, refueling time, refueling mass rate, average pressure ramp rate (APRR), final mass, final pressure, etc. The function formula coming from the analytical solution of the thermodynamic model is more meaningful physically and more efficient mathematically in fitting experimental temperatures. The simple uniform formula, inspired by the concept of the rule of mixture and its weighted factors obtained from the analytical solution of lumped parameter thermodynamics model, is representatively used to fit the experimental and simulated results in publication. Estimation of final hydrogen temperature from refueling parameters based on the rule of mixtures is simple and practical for controlling the maximum temperature and for ensuring hydrogen safety during fast filling process.

A uniform asymptotic formula for the second moment of primitive L-functions on the critical line

We prove an asymptotic formula for the second moment of primitive L-functions of even weight and prime power level. The error term is estimated uniformly in all parameters: level, weight, shift, and twist.

Uniform formula for the riemann solutions of a scalar combustion model

In this paper, we construct a uniform formula for the Riemann solutions of the simplified Chapman-Jouguet model. Firstly, we define a new functional, and then, we obtain that the Riemann solutions can be expressed by the maximum value point of this functional, while Riemann solutions may contain some of strong detonation waves, Chapman-Jouguet detonation waves and contact discontinuities. Finally, Chapman-Jouguet deflagration waves are also discussed.

Uniform asymptotic formula for the Aharonov Bohm wavefield

A uniform asymptotic formula for the Aharonov–Bohm wavefield (that of a plane quantum wave scattered by a thin straight solenoid) far away from the solenoid is obtained in a direct way. Actually quite good accuracy is achieved even down to one wavelength away. The error is numerically of order radius^(−3/2) for all values of polar angle, including directly forwards. Several previous formulas, uniform and otherwise, for the far field limit exist in the literature. All contain the essential ingredient: the Fresnel integral (complex error function), but ordinarily the error in these formulas is of order radius^(−1/2) in the forwards direction where the Fresnel contribution is most important.

Initial research on the theory and application of urban water demand field

With the rapid development and progress and the urbanization of human society, the demand for water of human activity is increasing. In the development of human society, with the climate changing and human activities increasing, water shortages have become increasingly prominent. The amount of water in some areas cannot be guaranteed, as well as a sharp deterioration in water quality. With the acceleration of urbanization, urban water demand is increasing. The water resources in the natural water cycle which are supplied for the city are not enough. Thus, in various urban water supply and water diversion projects, water is transported to the city, with the purpose of supplying water for the city production and life. Water cycle has also evolved from the original natural water cycle to dualistic nature-social water cycle. The dualistic water cycle theory indicates a new direction for the development of hydrology and water resources disciplines. So far there is still no uniform mathematical formula that can accurately describe the dualistic water cycle. In this study, we try to explore the mathematical and physical foundation of dualistic water cycle. We focused on the cities and did the in-depth analysis of the interference and driving principle of the socio-economic water demand on the water cycle. By analogy in physics Maxwell equations, the urban water demand field-the concept of “water field” was preliminary defined. According to the concept of electric charge, we define the “water charge” as the water requirement intensity. In the progressive development of the city, the city is in the analogy of a point electric charge in electric field. The electric field is formed by the point electric charge, while the water demand field is formed by the urban water demand. Point charge electric field can be superimposed over each other, so the electric field strength is also superimposed. Also, the water field strength of each cities which are close can be superimposed, and the city group will attract the water resources more far from their location, for example, the South-North water conversion project. And the formulas for calculating the city “water field” intensity were proposed. We also defined the constant in the formulas. This study took 22 major cities in the Haihe River Basin as an example, the water demands of every city are consider the city “water charge”. And the Haihe River Basin water field strength was quantitative described by applying the water field strength calculation formulas, and the trends in water flow were analyzed. The results showed that the water field strength formulas can describe the driving force and the mobility trend of urban water resources. Through the preliminary application, with the distribution of water field, the water cycle of social trends and the direction of water flow in the Haihe River Basin can be more accurately and effectively described. Meanwhile, using the field strength calculation formulas, the water field strength of city can be calculated, which represents the attracting strength of the city for the water resources around the city.

An improved algorithm for numerical calculation of seismic response spectra

The information of seismic response spectra is key to many problems concerned with aseismic structure and is also helpful for earthquake disaster relief if it is generated in time when earthquake happens. While current numerical calculation methods suffer from poor precision, especially in frequency band near Nyquist frequency, we present a set of improved parameters for precision improvement. It is shown that precision of displacement and velocity response spectra are both further improved compared to current numerical algorithms. A uniform fitting formula is given for computing these parameters for damping ratio range of 0.01–0.9, quite convenient for practical application.

Bilinearization and new multisoliton solutions for the (4+1)-dimensional Fokas equation

The (4 + 1)-dimensional Fokas equation is derived in the process of extending the integrable Kadomtsev–Petviashvili and Davey–Stewartson equations to higher-dimensional nonlinear wave equations. This equation is under investigation in this paper. Hirota’s bilinear method is, for the first time, used to solve such a higher-dimensional equation. In order to bilinearize the Fokas equation, some appropriate transformations are adopted. As a result, single-soliton solution, double-soliton solution and three-soliton solution are obtained. A new uniform formula of n-soliton solution is derived from this. It is shown that the transformations adopted in this work play a key role in converting the Fokas equation into Hirota’s bilinear form.

Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients

Abstract In this paper, Hirota’s bilinear method is extended to a new modified Kortweg–de Vries (mKdV) hierarchy with time-dependent coefficients. To begin with, we give a bilinear form of the mKdV hierarchy. Based on the bilinear form, we then obtain one-soliton, two-soliton and three-soliton solutions of the mKdV hierarchy. Finally, a uniform formula for the explicit N-soliton solution of the mKdV hierarchy is summarized. It is graphically shown that the obtained soliton solutions with time-dependent functions possess time-varying velocities in the process of propagation.

Studies of the spin Hamiltonian parameters and defect structures for Ag2+ in NaF and CsCdF3 crystals

The spin Hamiltonian parameters (g factors g//, g⊥, hyperfine structure constants A//, A⊥ and superhyperfine parameters Az′, Ax′ and Ay′) and defect structures for Ag2+ in NaF and CsCdF3 crystals are theoretically studied using the improved perturbation formulas of these quantities for a 4d9 ion in a tetragonally elongated octahedron. The contributions from both the crystal-field and charge transfer mechanisms are taken into account, and the relevant model parameters are quantitatively obtained from the cluster approach in a consistent way. The impurity centers are found to undergo the relative tetragonal elongations of about 9.4% and 8.2% for Ag2+ in NaF and CsCdF3, respectively, along the C4 axis due to the Jahn–Teller effect. By employing the few adjustable parameters, the calculated spin Hamiltonian parameters based on the above uniform formulas and the local tetragonal elongation distortions agree well with the experimental data. Despite dominant ionicity of the hosts, the charge transfer contributions are actually important to the spin Hamiltonian parameters (e.g., about 20% for the g-shifts) due to strong covalency of impurity Ag2+.