For obtaining a formula for the local traveltime computation, we invoke the Fermat principle and a condition for limiting the position of the incident point.
为了得到局部走时计算公式，利用费马原理和关于入射点位置的限定条件。
In 2000, F. LUCA proved that Fermat number are anti-sociable numbers, and in 2005, M. H. LE proved all powers of 2 are anti-sociable numbers. We have used the method of M. H. LE to obtain some new results of the anti-sociable numbers. For every integer n containing prime divisors that are 1 mod 4, let p mod 4 be an arbitrary prime divisor of n. There is at least one anti-sociable number in n^2, p^2n^2, p^4n^2, and p^6n^2. Therefore we can prove that anti-sociable numbers have positive density in perfect square numbers. We also give a method to find the exact anti-sociable numbers.
LUCA证明了Fermat数都是孤立数；2005年，乐茂华教授证明了2的方幂都是孤立数，用乐茂华教授的方法给出孤立数的一些新的结果：对于任意含有4w+1型素因子的正整数n，设p为n的任意一个4w+1型素因子，则在n^2，p^2n^2，p^4n^2，p^6n^2里至少有一个是孤立数，因此可以证明孤立数在完全平方数里有正密度，另外也给出求解确定孤立数的方法。
Wavefront traveltimes on arbitrary positions in each cuboid cell can be expressed with linear interpolating function about wavefront traveltimes on grid points of the same cell.
在每个长方体单元内，首先把任意点上的波前传播时间用该单元网格节点上的已知波前传播时间的线性插值函数来表示，再根据Fermat原理，提出了确定三维声线路径的方法。
If you hope the certificate, when seeks for the Fermat`s theorem with ten year the primary proof defeat to be possible to come back, you definitely may prove, from the Maxwell time, these smatters to mathematics the human in starts with more and more complex mark cover mathematics thought insufficiency, hides constructs the natural law in the studio, gains God's favor.
如果你希望证明，当用十年的时间寻找费马大定理的初等证明的失败可回来，你完全可证明，自麦克斯韦时代起，那些对数学一知半解的人就在开始用越来越复杂的符号掩盖数学思维的不足，躲在书房里构造自然规律，获取上帝的宠幸。
The law of refraction, discovered first empirically by Willebrord SNELL, then subsequently derived formally by Descartes and Fermat, states that sin r = K sin i, where i is called the angle of incidence, the angle between the incident ray and the normal to the refracting surface.
折射的轨律，首先被 Willebrord SNELL 靠经验发现，接着后来正式被笛卡尔和 Fermat 认定，规定 sin r = K sin i ，这里 i 被称作入射角，入射光线和法线到折射平面之间的夹角。
Fermat saw K as`being proportional to the reciprocal of the velocity of propagation.
Fermat 认为 K 为与传播速度互为倒数的比例系数。
When propagating light|ray passes through two fixed points in the gradient index mediums and a cuspidal variable|point （at this point the trace equation of propagating light|ray is non|differentiable） at the borderline of two different GRIN mediums, the mathematical relationship, which the trace equation of propagating light|ray and borderline equation of two different GRIN mediums should satisfy, is derived by the Fermat`s principle and calculus of variation.
当光线通过梯度折射率介质中两个固定点和两个GRIN介质界线上的可动尖点（光线轨迹方程的不可微点）时，由费马原理和变分法运算推得传输光线的轨迹方程与两种不同GRIN介质的界线方程应满足的微分关系式。
In this paper we prove that it is sufficient to prove for Fermat`s last theorem using the complex hyperbolic functions in the hypercomplex variable theory.
摘要∶本文利用 P 阶复双曲函数，证明了只要证明就全部证明了费马大定理。
On one hand, the real-time imaging aglorithms are studied, which included the imaging methods using FFT and Fermat transformation which can further reduced the computation amount. On the other hand, for the separable two-dimensional imaging system, the lateral super-resolution imaging methods are focally studied in this paper.
继而，本文转入了成象的信号处理方法研究，一方面研究了实时成象问题，除了使用FFT实现二维脉冲压缩成象外，还研究了快速费马变换实现二维脉冲压缩的方法，进一步减少了成象运算量并提高了成象速度。
Based on Fermat's principle and light path function, it gets the spectral distribution of the incident rays, and it can get the ray tracing map through a matrix method.
在费马原理的基础上利用光程函数法得到入射光通过光栅后的光谱分布情况，再使用传输矩阵的计算方法构造出光谱的点列图。
A numerical ray-tracing algorithm is provided based on Fermat＇s principle.
并基于费马原理，提出了记录光路的光线追迹数值算法。
The proof of Fermat's Last Theorem was a major mathematical feat of the 20th century.
费马大定理的证明是20世纪数学的一个主要功绩。
L The proof of Fermat's Last Theorem was a major mathematical feat of the 20th century.
费马大定理的证明是20世纪数学的一个主要功绩。
It is such propositions that spur new ideas and inventiveness, just as Fermat's Last Theorem has done for centuries.
其实，正是这样的命题不断地激发着新的观念与新的创造，就像费马大定理几个世纪以来对数学所起的推动作用。
I'll solve one or more mathematics puzzles, such as Beal conjecture, and Fermat's last theorem etc.
后来，一个梦想伴随了我很多年，我将解决一个或更多的数学难题，如 Beal 猜想、费马大定理等。
The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem.
在数论中更深入的研究整数性质，通俗的成果例如费马大定理就是在数论中产生的。
Schur theorem was discovered by German mathematician Schur in 1916 when researching on the limited domain of Fermat's great theorem.
Schur定理是德国数学家Schur在1916 年研究有限域上的Fermat 大定理时发现的。
We are very possibly difficult the great philosophy which includes with the appropriate language summary Fermat`s last theorem's primary proof.
我们可能很难用合适的语言概括费马大定理的初等证明包括的伟大哲理。
Just how Fermat's ideas on coordinate geomitry evolved is not known.
Fermat的坐标几何究竟是怎样产生的，我们尚不知道。
The main features of these methods are that they aren't limited by the descriptions of the travel-time and raytracing of the seism wave. They set out directly from the principle of Huygens and the Fermat theory and then descript the features of the seism wave fields by adopting the equivalent fore-waves.
这些方法的主要特点在于不再局限于地震波的射线路径描述，而是直接从Huygens原理或者Fermat原理出发，采用等价波前描述地震波场的特征。