Inverse of a Function f（x）=y

For a function y= f（x） to have an inverse function, f must be one-to-one. Then for each x in its domian there is exactly one y in its range; furthermore, to each y in the range, there corresponds exactly one x in the domain. The correspondence from the range of f onto the domian of f is, therefore, also a function. It is this function that is the inverse of f.     

Puzzle and Solution:Diagonal String

“Suppose there were one nail in each of thefour walls of this room and in addition one nail onthe ceiling and one in the floor,Suppose furtherthat we had to tie strings between these nails.Ihave tWO colors of strings which I use—red andblue.Each connection between any two nails iseither with a red string or with a blue string. “All these strings make up many triangles,that is,any three nails can be considered apexesof a triangle formed by the strings connecting these three nails.My problem is to see if I could distribute the colors so that no triangle has all three sides the same color.”Said young Nicho—las. “This is rather complicated,”mused the mathematician. “It must involve calculations of permutations and combinations,and the like.I didn't think you knew that much algebra,Nick.”“I don’t,”replied young Nick。“but I can still do the problem.”“Well,all right,tell us how。”said the eld—er. “It is really very simple.”said young Nicho—las,“You only have to know enough to start rea-soning, “The answer is that there will be at least onetriangle all sides of which are the same color.Iwill show that it is impossible to avoid this. “Consider any one nail. Out from it theremust stretch five strings,one to each of the otherfive nails。No matter how you distribute the col—ors in these five strings,at least three of themmust be the same,since you have only two colors.For the sake of argument let assume that three ofthe strings are red. “Consider now the triangle formed betweenthree nails at which the three red strings haveterminated. “If we are to try to avoid a triangle with all three sides the same color,it follows thatthese three nails cannot all be joined to each other with one color.Putting this more simply,thetriangle formed by the three terminal nailsshould not be all blue.At least one of the stringsbetween the three nails must be red.But if so,we have completed a red triangle from the origi—nal nail.”     

ON AFFINE MINIMAL ROTATION SURFACES AND CONOID IN A~3

As we known,fon the minimal rotation surfaces and conoid in the 3-dim.Euclidean space E~3,we have the following classic theorems:Theorem A If a rotation surface is minimal,then it must be a plane orcatenoid.Theorem B If a conoid is minimal,then it must be a right helicoid.     

Periodic points and normal families concerning multiplicity

In 1992,Yang Lo posed the following problem:let F be a family of entire functions,let D be a domain in C,and let k 2 be a positive integer.If,for every f∈F,both f and its iteration f~khave no fixed points in D,is F normal in D?This problem was solved by Ess′en and Wu in 1998,and then solved for meromorphic functions by Chang and Fang in 2005.In this paper,we study the problem in which f and f~(k ) have fixed points.We give positive answers for holomorphic and meromorphic functions.(I)Let F be a family of holomorphic functions in a domain D and let k 2 be a positive integer.If,for each f∈F,all zeros of f(z)-z are multiple and f~khas at most k distinct fixed points in D,then F is normal in D.Examples show that the conditions"all zeros of f(z)-z are multiple"and"f~k having at most k distinct fixed points in D"are the best possible.(II)Let F be a family of meromorphic functions in a domain D,and let k 2 and l be two positive integers satisfying l 4 for k=2 and l 3 for k 3.If,for each f∈F,all zeros of f(z)-z have a multiplicity at least l and f~khas at most one fixed point in D,then F is normal in D.Examples show that the conditions"l 3for k 3"and"f~k having at most one fixed point in D"are the best possible.     

Read Carefully(2)——Finding the Meanings of Words and Symbols

If you do not know the meaning of a word, look it up in a dictionary. Some mathematics books have glossaries or indexes in the back that can help you locate words.Some terms have more than one meaning even in mathematics. The word divisor can mean the number divided by in a division problem. (In 12÷5=2.4, 5 is the divisor. ) But divisor also means a number that divides another with a zero remainder. For example, 7 is a divisor of 21. In this situation divisor has the same meaning as factor.     

Reflection Symmetry

Nearly a thousand years ago the Dine people,ancestors of today's Navajo in the Southwest United States,learned to weave on looms from the neighboring Pueblo Indians:If you fold a blanket along a horizontal line through its center,the top and bottom halves match.     

Trial and Error

Trial and error is a problem-solving strategy everyone uses at one time or another.In trial and error,you try ananswer.(The word trial comes from try.)If the answer is in error,you try something else.You keep trying untilyou get a correct answer.Trial and error is a particularly good strategy if a question has only a few possible answers.     

Irrational Numbers

The Greek philosopher Pythagoras(ca.580-ca.500 B.C.)believed that wholenumbers,including fractions,since they areratios of whole numbers,were the basis ofthe Universe.Thus,3/4 is the ratio of 3 to4.If you begin with 3 pies and divide them e-qually among 4 people,each person gets 3/4     

The Golden Rectangle

Fibonacci(Leonardo Pisano) was an outstanding mathematician and scholar whose interests lay particularly in solving practical problems.In Chapter 2,you came across the Fibonacci series.The series has a number of interesting properties.If we label each number(term) in the series using t1,     

ON A PROBLEM IN THE THEORY OF DETERMINANTS

<正> 1. In 1867 Sylvester published a paper in Philos. Mag. with perhaps the funniesttitle ever written in the mathematical literature. The various questions treated hereall led him to the problem to determine for which n-values one can construct a determinantof order n consisting exclusively from±1 and which is orthogonal in the sense that thecomposition of any two different rows is 0. If n is odd, then obviously no such determinantcan exist; Sylvester showed very easily that for n=2~k(k=1,2,…) there are such determinants. If n≥3, then it is easy to show that such a determinator can exist only inthe case n≡0 mod 4. For if it exists such a determinant with the elements a_(μν) then we have     

INFINITE ELEMENT METHOD FOR THE EXTERIOR PROBLEMS OF THE HELMHOLTZ EQUATIONS

1. IntroductionThe infinite element method has been successfully applied to some boundary valueproblems of partial differential equations, where the solutions possess corner singularpoints or the domains are exterior ones. If the equations are invariant under similaritytransformation the approaches have been given in [11] [131 for singular solutions, and in[12][15][181119] for the exterior problems. If the equations do not admit the above invariant property? one approach has been given in [14]…     

Normality and shared values concerning differential polynomials

Let F be a family of functions meromorphic in a domain D, let P be a polynomial with either deg P≥3 or deg P = 2 and P having only one distinct zero, and let b be a finite nonzero complex number. If, each pair of functions f and g in F, P (f)f and P (g)g share b in D, then F is normal in D.     

The Further Application Of Inductive Reasoning And Deductive Reasoning

Now we have a good understanding of inductive reasoning and deductive reasoning. In order to put them into practice, we should do some exercises. Practice Exercises Which reasoning process is shown in the following example? Explain your answer. 1. We examine the fingerprints of 1000 people. No two individuals in this group of people have identical fingerprints. We conclude that for all people, no two people have identical fingerprints.     

New Structures for Global Solutions of Two Dimensional Simplified Euler Equations

1 Introduction and Main Results Systems of two-dimensional （2D） hyperbolic conservation laws are more accurate modeling of many sophisticated physical phenomena. Although people have made great progress in one dimensional conservation laws.     

Being a Good Problem Solver（1）

When solving problems, You need not feel as if you are lost in a maze. Strategies can help you find solutions. What Is an Algorithm?you may have learned to solve any equation of the form x a = b for x. The method was to add -a to each side and then simplify. This method is an example of an algorithm. An algorithm is a sequence of steps that leads to a desired result. Not all algorithms are short. The algorithm called long division can involve many steps.     

Being a Good Problem Solver（2）

George Polya was a mathematician at Stanford University who was famous for his writing about solving problems. He once wrote:Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. Trying to solving problem, you have to observe and to imitate what other people do when solving problems and, finally, you learn to do problems by doing them.     

学英语（英文）

There are 9questions in total,presenting various different question types.While you attempt to resolve the problems,remember to be creative.During accomplishing these flexible mathematical exercises,you can inspire your mathematical thinking.1.What is the perimeter of a right triangle if the lengths of its two smallest sides are 15and 36?2.If a∶b∶c=6∶7∶11,what is the value of c-a?     

SOME NORMALITY CRITERIA OF MEROMORPHIC FUNCTIONS

Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, and f ＋ a（f^（k））^n≠b in D, then F is normal in D.     

Joins of 1-planar graphs

A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once.In this paper,we study 1-planar graph joins.We prove that the join G + H is 1-planar if and only if the pair [G,H] is subgraph-majorized by one of pairs [C3 ∪ C3,C3],[C4,C4],[C4,C3],[K2,1,1,P3] in the case when both elements of the graph join have at least three vertices.If one element has at most two vertices,then we give several necessary/sufficient conditions for the bigger element.     

学英语

There are 7 questions in total,presenting various different question types.While you attempt to resolve the problems,remember to be creative.During accomplishing these flexible mathematical exercises,you can inspire your mathematical thinking.1.If a-5=0,what is the value of a+5?