Fermats ?ad?quare? – und kein Ende?

In his papers on the determination of maxima and minima and on the calculation of tangents Pierre Fermat uses two different Latin verbs, ?quare and ad?quare, which do not differ semantically but are used by him obviously in different meanings. While ?quabitur is used unambiguously in the sense of “is equal” the meaning of ad?quabitur is disputed by the experts since Tannery’s French translation (Œuvres complètes de Fermat, Vol. III, 1896). Herbert Breger (Arch. Hist. Exact Sci. 46, 193–219, (1994), p. 197 f), for instance, holds the view that Fermat used the word ad?quare in the sense of “to put equal” and adds: In a mathematical context, the only difference between “?quare” and “ad?quare” (if there is any) seems to be that the latter gives more stress on the fact that the equality is achieved. In contrast to this Michael Mahoney holds the thesis that ad?quare describes a counterfactual equality (Mahoney, M.S.: Fermat, Pierre de. In: Dictionary of Scientific Biography, vol. IV (1971), p. 569) or a pseudo-equality (Mahoney, M.S.: The Mathematical Career of Pierre de Fermat (1601–1665), (1973), p. 164), whatever that may mean. This viewpoint has been taken up again recently by Enrico Giusti (Ann. Fac. Sci. Toulouse, Math. (6), 18 fascicule spécial, 59–85 (2009)) in order to bring arguments to bear against Breger. In contrast to these (and other) authors, I show that Fermat makes a subtle logical distinction between the words ?quare and ad?quare. The same distinction is made by Nicolas Bourbaki introducing his ?théorie égalitaire?. Notwithstanding: both verbs stand for a ?relation d’égalité?. On this premiss, I describe—using six selected examples—that Fermat’s “method” may be justified right down to the last detail, even from the view of today’s mathematical knowledge.     

多讲?少讲?不讲?

近日读到著名经济学家张五常回忆其大学老师的一篇文章 ,引起我的一些思考 .文章说 ,老师让学生花了四个星期的时间自己去研究问题解决的办法 ,但学生们都找不到办法 .第五个星期 ,老师进入课室 ,“在五十分钟内 ,他如长江大河 ,滔滔不绝 ,使我听得呆了 .……”作者说 ,“假若没有经过那四个星期去想 ,我不会在五十分钟之内被老师说服的吧” .虽然故事讲的是大学教学 ,但是 ,我想我们的中学教学中 ,不也存在一个“讲”与“不讲” ,“多讲”与“少讲”的问题吗 ?在“满堂灌”的授课方式受到批评后 ,不少老师又都在培养学生能力的口号下祭起了…     

O?shore ?oating wind turbine and its dynamic problems?

Green energy sources and ocean wind power are plentiful in deep sea. More and more o?shore wind power plants are constructed in the deep water over hundred meters below the surface. While o?shore ?oating wind turbine system is working, wind turbine, ?oating foundation, and mooring system a?ect each other with wind, waves, and currents acting on them. Various o?shore ?oating wind turbine systems and the encoun-tered environmental loads are brie?y reviewed and discussed. It is di?cult and crucial to comprehensively analyze the aerodynamic-hydrodynamic-service system-structure un-der the coupling e?ect of o?shore ?oating wind turbine system. The environmental ?ow ?eld, structure scale, and rational applications of theories and approaches should be well considered in advance.     

Beitr?ge zur St?rungstheorie

Ohne ZusammenfassungHerrn Professor Dr. H.Kneser zum 60. Geburtstag gewidmet     

How Close is the Sample Covariance Matrix to?the?Actual Covariance Matrix?

Given a probability distribution in ? ⁿ with general (nonwhite) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent points. What is the optimal sample size N=N(n) that guarantees estimation with a fixed accuracy in the operator norm? Suppose that the distribution is supported in a centered Euclidean ball of radius $O(\sqrt{n})$ . We conjecture that the optimal sample size is N=O(n) for all distributions with finite fourth moment, and we prove this up to an iterated logarithmic factor. This problem is motivated by the optimal theorem of Rudelson (J. Funct. Anal. 164:60?C72, 1999), which states that N=O(nlog?n) for distributions with finite second moment, and a recent result of Adamczak et al. (J. Am. Math. Soc. 234:535?C561, 2010), which guarantees that N=O(n) for subexponential distributions.     

问题何在?

题:若复数z满足z~10=-1,则z~4-z~6-z~2 1的值是( )。0(A)-1;(B)1;(C)0;(D)以上都不对。此题见于一些书刊,其答案为(A) 解法一:(方程法或设值法) 设(1) (1)式两边同乘以z~4得 z~8-z~6-z~2 1=z~4t(2) 因为z~10=-1,(1)式分子,分母同乘以-z~6。并整理得 z~8-z~6 z~4-z~2=t(3) 由(2)得z~8-z~6-z~2=z~4t-1(4) (4)式代入(3)得 z~4t-1 z~4=t ∴(z~4-1)t=1-z~4 (5) 两边同除以z~4-1得:t=-1,故选(A)。解法二:(用数列求和解) ∵ z~10=-1, ∴原式的分子、分母同乘以-z~6,得: =-z~2 z~4-z~6 z~8 (看成首项,公比皆为-z~2的等比数列) 故选(A)。以上两种解法及原书刊给出的答案都是(A),可是若本题用满足z~10=-1的一值z=i(或-i)代入,得上面结果都错了,这就奇怪了,问题何在?     

Moir? Separations

The analysis of the moir? phenomenon by means of elementarygeometry which was begun by Firby (1984) and Firby & Stone(1984) is extended from first order to all integral and fractionalorders, and from consideration of the plane only to considerationof a general finite-dimensional space . The idea of the -curve, which forms the boundaryof the set in in which first-order interference might be observed, is extended to the idea of aboundary in for any given integral or fractional order of interference. Separating setsin , which form the boundary of regions in which two given integral or fractional ordersof interference might dominate relative to one another, areintroduced. The wavelength and contrast analysis previouslycarried out only for first-order interference in is extended to deal with all integral and fractionalorders of interference, and this is used to explain why highintegral orders, and all fractional orders, of interferenceare observed only in exceptional circumstances.     

A family of new smoothing functions and?a?nonmonotone smoothing Newton method for?the?nonlinear complementarity problems

In this paper, based on a p-norm with p being any fixed real number in the interval (1,+??), we introduce a family of new smoothing functions, which include the smoothing symmetric perturbed Fischer function as a special case. We also show that the functions have several favorable properties. Based on the new smoothing functions, we propose a nonmonotone smoothing Newton algorithm for solving nonlinear complementarity problems. The proposed algorithm only need to solve one linear system of equations. We show that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. Numerical experiments indicate that the method associated with a smaller p, for example p=1.1, usually has better numerical performance than the smoothing symmetric perturbed Fischer function, which exactly corresponds to p=2.     

The pexiderized Go?a?b-Schinzel functional equation

Let X be a linear space over a commutative field K. We characterize a general solution f,g,h,k:X→K of the pexiderized Go?a?b-Schinzel equation f(x+g(x)y)=h(x)k(y), as well as real continuous solutions of the equation.     

A Bernstein type theorem in ? × ? n

In this paper, as suitable application of the so-called Omori-Yau generalized maximum principle, we obtain a Bernstein type theorem concerning to complete hypersurfaces immersed with constant mean curvature in the product space ℝ × ℍ ⁿ . Furthermore, we treat the case that such hypersurfaces are vertical graphs.