关于丢番图方程ax~m-by~n=4(英文)

1 Main Results The Diophantine equation ax~m-by~n=4,m>1,n>1 (1) is the popularization of many other diophantine equations,where,a,b,x,y∈Z_(>0),In     

Number of Negative Entries in A2≤0

Let A be a real matrix or a sign pattern of order n. N-(A) denotes the number of negative entries in A. In 1972 R DeMarr and A Steger conjectured: If A is a real matrix of order n such that A2≤0, then N_(A2)≤(n-1)2 1. Now the conjecture is proved to be true when A is reducible or a matrix of order n≤3 and some sufficient conditions for N_(A2)≤(n-1)2 1 are given. It is also proved that N_(A2)≤n2-4n 5 when A is a reducible combinatorially symmetric sign pattern such that A2≤0, and the extreme sign patterns are characterized.     

Number of Negative Entries in A2≤0

Let A be a real matrix or a sign pattern of order n. N_ (A) denotes the number of negative entries in A. In 1972 R DeMarr and A Steger conjectured: If A is a real matrix of order n such that A^2≤0, then N- (A^2)≤( n - 1)^2 1. Now the conjecture is proved to be true when A is reducible or a matrix of order n≤3 and some sufficient conditions for N- (A^2)≤(n - 1)^2 1 are given. It is also proved that N_ (A^2)≤n^2 -4n 5 when A is a reducible combinatorially symmetric sign pattern such that A^2≤0, and the extreme sign patterns are characterized.     

关于指数丢番图方程的Hugh Edgar问题

p~m=q~n 2 m>1, n>1 (1)where p and q are primes, has the solution 3~3=5~2 2.Hall asked a question how many solutions else are there? Later, Hugh Edgar asked another question how many solutions(m,n) does the Diophantine equation p~m-q~n=2~h;m>1,n≥1,(2) have for primes p, q and integer h? In the references at el., Hall's and Hugh Edgar's problems were partly solved. In this paper, we have proved the following results: THEOREM 1. If p=qt~2±4 or q=pt~2±4, then the equation p~m-q~n=4 has no integer solution in which m>1, n>1. THEOREM 2. If(p,q)=(3,5), (5,3), (5,7), (7,5)(mod 8), then Eq.(2) has no integer solution in which h≥2, except that 3~2-5~1=2~2, 5~3-11~2=2~2 and     

丢番图方程X^2^p—Dy^2=1与费马商Qp（m）

0 The Diophantine equation X~(2p)-Dy~2=1Let D be a positive integer which is square free,and p be a prime.In 1966,Ljunggren showed that if p=2 and D=q is a prime,then the Diophantine equationx~(2p)-Dy~2=1(1)has only positive integer solutions(q,x,y)=(5,3,4),(29,99,1820).In 1979,KoChao and Sun Qi showed that if p=2 and D=2q,then Eq.(1)has no positive inte-     

费马大定理：Abel猜想的一个证明

1 IntroductionIn 1823 Abel made a conjecture in a particular case of Fermat's last theorem:If n>2and x,y and z are nonzero integers such thatx~n+y~n=z~n(1)then none of x,y or z can be a prime power(see P.Ribenboim[1],p.25).It is clear that we may assume,without loss of generality,that(x,y)=(x,z)=(y,z)=1and 0相似文献   

The Incidence Coloring Number of Complete k-Fartite Graph

The concept of the incidence chromatic number of a graph was introduced by Brualdi and Massey. They conjectured that every graph G can be incidence colored with △(G) 2 colors. In this paper, the trueness of this conjecture for complete k-partite graph was proved, and the incidence chromatic number of complete k-partite graphs was calculated.     

关于丢番图方程ax~2+by~2=p~z

0 Introduction Let p be a prime, and let a, b∈Z_(>0), a>b>1 and (a, b)=1. If the equation X~2+abY~2=p~z, (X,Y)=1, Z>0 (1) has an integer solution (X,Y, Z), then there exists a unique integer solution (X_p, Y_p, Z_p,)which satisfies X_p>0, Y_p>0 and Z_p 0) satisfy the equation ax~2+by~2=2~z,z>2 (2) and x|~*a, y|~*b, where symbol x|~*a means that a is divided exactly by each prime factor of x.     

Quality evaluation of layerlike backfilling and flow pattern of backfill slurry in stope

Stability condition and quality evaluation formula of layerlike backfilling roof,Q≥C,where Q denotes is quality index depending on allowable compressive or tensile strength and integrity of backfilling,and C is the technical index depending on mining method and backfilling technology,were inferred according to simply supported beam theorem.Technical treatment measures for instable backfilling roof,including optimum of appropriate filling materials and dosage for excellent flow property and reduction of backfill cost.It is proved that slope equation of backfill slurry in a stope to be filled is y=hexp[x2/(2σ)2)],where h is height of cone and σ2 is mean square,and that optimum drainage point of backfill slurry can be determined by the equation and sizes of stope.Case study indicates that the results can give a theoretical support for quality evaluation and control of layerlike backfilling.     

Isothermal Oxidation Kinetics of Artificial Magnetite Pellets

In order to establish the kinetics of oxidation of artificial magnetite pellets, we comprehensively studied kinetics of the oxidation of artificial magnetite pellets from low temperature to high temperature using chemical analysis. The results show that when the oxidation temperature is below 1 073 K(800 ℃), the reaction is controlled by the step of internal diffusion, and the model function is 23 G(a) =1-3(1-x) ~(2/3)+2(1-x)(α, reaction degree). When the temperature is above 1 073 K(800 ℃), the reaction mechanism was chemical reaction, and the model function is 13 G(a) =1-(1-x)~(1/3). The apparent activation energy for the oxidation of artificial magnetite pellets was also determined, which was 8.90 kJ/mol for the low temperature and 67.79 kJ/mol for the high temperature. Based on the derived kinetic equation for the oxidation of artificial magnetite pellets, the calculated value is consistent with the experimental data. Compared with that of nature magnetite pellets, the apparent activation energy is decreased obviously, which indicates that the artificial magnetite pellets are oxidized more easily than nature magnetite pellets.     

Efficient Computation of Scattering from Targets with Negative Impedance Surface

Current surface integral equations used for computing scattering from targets with negative impedance boundary condition (IBC) are not efficient. A modified surface dual integral equation (M-SDIE) for targets with negative IBC is presented. A pure imaginary number is used to balance the formulations. It is proved that the M-SDIE is accurate and efficient with three numerical examples. The first numerical example shows that the M-SDIE is accurate compared with Mie. The second example shows that the presented SIE is efficient. In the third example, a missile head is selected to present the computing power of the M-SDIE. All the examples show that the M-SDIE is an efficient algorithm for negative IBC.     

Rate Equation of Small Particle-Air Bubble Attachment in Turbulent Flow of Flotation Cells

A rate equation of small particle-air bubble attachment in the turbulent now of flotation cells has beenderived. The equation, integrating both the collision probability and adhesion probability together, represents theprobability of attachment between particle and bubble in the turbulent flow. "Capture efficiency" f(a) is introducedinto the rate equation to reflect the influence of energy hairier on the attachment rate. Three typical situations of particle-bubble interaction in flotation process have been discussed. For a completely hydrophobic particle-bubble system,f(a) = 1. This means that all collision leads to attachment. Whereas for hydrophilic particle-bubble systems, .f(a) =0. Thus no adhesion of particle on bubble occurs at all. In real notation circumstances, however, there always existsa certain energy barrier between the particle and the bubble. Therefore, f(a) = 0～1. In such cases, not all collisionsresult in particle-bubble attachment.     

Josephson第一方程推导

1 Introduction In 1962, Josephson studied the physical phenomenon of two superconductors seperated by a thin layer of potential hill. He predicted from the point of theory that when Volt of two sides of junction is zero, there may be a superconducting current. He also thought that the critical current has a maximum J_c and regarded it due to the tunnel current of Cooper's pairs. A half year later,Anderson and Rowell proved by means of experiments that superconducting current exists in the superconducting-tunnel junction when v =0. During 1964-1965, Josephson proved again from the point of theory that the tunnel current is a sinusoidal function of     

An efficient method for train–track–substructure dynamic interaction analysis by implicit-explicit integration and multi-time-step solution

In this work, a method is put forward to obtain the dynamic solution efficiently and accurately for a largescale train–track–substructure(TTS) system. It is called implicit-explicit integration and multi-time-step solution method(abbreviated as m I-n E-MTS method). The TTS system is divided into train–track subsystem and substructure subsystem. Considering that the root cause of low efficiency of obtaining TTS solution lies in solving the algebraic equation of the substructures, the high-efficie...     

Determination of paclitaxel in southern yew tree by HPLC

Paclitaxel in southern yew tree was quantitatively determined by high performance liquid chromatography (HPLC) with ODS-C18 column. A mixture of CH3OH-H2O-CH3COOH(volume ratio: 55 :44 :1) is used as mobile phase and UV detection is carried out at 227 nm, and the column temperature is 20℃. The results show that there is a good linear felationship between the area of paclitaxel and the concentration of the sample in the range 50-500 mg/L for paclitaxel. The corresponding regression equation is Y= 13 021. 7 + 1. 01 × 106X, r=0. 9990. The average recovery is 95. 3% and the relative standard deviation is 2. 08%.     

Explicit formulation of the J_2-integral in anisotropic materials and its application in microcrack shielding problems

The explicit formulation of the J2-integral in anisotropic bodies and its application in microcrack shielding problems are discussed. With analytical treatments and numerical examinations, it is proved that there is a redistribution law for the remote J-integral in a discrete model of microcrack shielding problems, i.e. the projected conservation law of the Jk-vector. In this law, the J2-integral which was disregarded by Herrmann (1981) is proved to be of the same significance as the J1-integral. It is also concluded that the two energy dissipative processes due to the mi crocrack damage, i. e. the reduction in the effective moduli and the release of residual stresses, can be described by using the dissipation of the remote J-integral spreading across the microcrack damage zone.     

RESEARCH ON THE WATER-RESISTANCE OF MAGNESIUM OXYCHLORIDE CEMENT——I:THE STABILITY OF THE REACTION PRODUCTS OF MAGNESIUM OXYCHLORIDE CEMENT IN WATER

In this paper.the change of the crystalline phasesin hardened magnesium oxychloride cement(MOC)paste inwater was analyzed by XRD.It was developed that the reactionproducts 5 phase or 3 phase of MOC are instable in water andcan be changed into Mg(OH)_2 by the action of water,whichcauses the content of 5 phase or 3 phase to be less and less,thecontent of Mg(OH)_2 to be more and more and the strength to bethe lower the lower,after hardended MOC paste was immersedin water.The change of 5 phase and 3 phase into Mg(OH)_2 isnot a dissolve process,but a hydrolysis process.The hydrolysisproducts of 5 phase and 3 phase are Mg(OH)_2 precipitationand soluble Cl~-,Mg~+ ions and H_2O.The hydrolysis is sponta-neous thermodlynamically and its chemical kinatic equation is C=C_oe~(-kt) Thus,it is suggested that only by enhancing the stabilityof 5 phase or 3 phase in water and preventing 5 phase or 3phase from the hydrolyzing can the water resistance of MOC beimproved well.     

One-dimensional tensile constitutive equation cannot be directly generalized to deal with two-dimensional bulging mechanical problems

Superplastic forming has been extensively applied to manufacture parts and components with complex shapes or high-precisions. However, superplastic formation is in multi-stress state. In a long time, uniaxial tensile constitutive equation has been directly generalized to deal with multi-stress state. Whether so doing is feasible or not needs to be proved in theory. This paper first summarizes the establishing processes of superplastic tensile and bulging constitutive equation with variable m, and, using the analytical expressions of equivalent stress a and equivalent strain rate ε of free bulge based on the fundamentals of continuum medium plastic mechanics, derives the analytical expressions of optimum loading rules for superplastic free bulge. By comparing the quantitative results on typical superplastic alloy ZnAI22, it is shown that one-dimensional tensile constitutive equations cannot be directly generalized to deal with two-dimensional bulging quantitative mechanical problems; only superplastic b     

关于丢番图方程1 p~a=2~bq~c 2~dp~eq~f

It is well known that Diophantine equation is as follows:1 p~a=2~bq~c 2~dp~eq~f (1) where a, b, c, d, e, f are non-negative integcrs. In 1983, L. J. Alex and L. L. Foster (Rocky Mr. J. Math., 13(1983),     

The New Method for CNC Trajectory Machining Motion Planning Based on Reference Time

In order to make motion planning fitting practice,many characteristic of CNC trajectory motion are discussed, such as the geometric function,the motion and the time.It is found that the relation between orbit function and motional parame- ter,so the differential equation about the trajectory motion be set-up by the goal of trajectory motion.The actual motion process is defined as reference time to link planning and practice.Present a new movement planning method based on self-defining time.At rest state,the differential simultaneous equation can be calculated according geometric characteristic analysis,it can be get that simple function consisted of coordinate and reference time variants.At motive state,dynamic parameter can be worked out accord- ing practical value of reference time,It is proved by experiment and simulation that it is a good way to control geometry and motion comprehensively,to reduce computation times and to increase the ability of environmental adaptation for path planning