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数学专业英语－Mathematical DiscoveryTo give the flavor of Polya’s thinking and writing in a very beautiful but sub tle case , a case that involve a change in the conceptual mode , I shall quote at length from his Mathematical Discovery (vol.II , pp.54 ff):EXAMPLE I take the liberty a little experiment with the reader , I shall sta te a simple but not too commonplace theorem of geometry , and then I shall t ry to reconstruct the sequence of idoas that led to its proof . I shall proceed s lowly , very slowly , revealing one clue after the other , and revealing each g radually . I think that before I have finished the whole story , the reader will seize the main idea (unless there is some special hampering circumstance ) . B ut this main idea is rather unexpected , and so the reader may experience the pleasure of a little discovery .A.If three circles having the same radius pass through a point , the circle th rough their other three points of intersection also has the same radius .Fig.1 Three circles through one point.This is the theorem that we have to prove . The statement is short and clea r , but does not show the details distinctly enough . If we draw a figure (Fig .1) and introduce suitable notation , we arrive at the following more explicit restatement :B . Three circles k , l , m have the same radius r and pass through the sa me point O . Moreover , l and m intersect in the point A , m and k in B , k and l inC . Then the circle e through A , B , C has also the radiusFig .2 too crowded .Fig .1 exhibits the four circles k , l , m , and e and their four points of in tersection A, B , C , and O . The figure apt to be unsatisfactory , however , for it is not simple , and it is still incomplete ; something seems to be missin g ; we failed to take into account something essential , it seems .We are dialing with circles . What is a circle ? A circle is determined by c enter and radius ; all its points have the same distance , measured by the leng th of the radius , from the center . We failed to introduce the common radius r , and so we failed to take into account an essential part of the hypothesis . Let us , therefore , introduce the centers , K of k , L of l , and M of m . Where should we exhibit the radius r ? there seems to be no reason to treat a ny one of the three given circles k ; l , and m or any one of the three points of intersection A , B , and C better than the others . We are prompted to connect all three centers with all the points of intersection of the respective circl e ; K with B , C , and O , and so forth .The resulting figure (Fig . 2) is disconcertingly crowded . There are so many lines , straight and circular , that we have much trouble old－fashioned maga zines . The drawing is ambiguous on purpose ; it presents a certain figure if you look t it in the usual way , but if you turn it to a certain position and lo ok at it in a certain peculiar way , suddenly another figure flashes on you , s uggesting some more or less witty comment on the first . Can you recognize i n our puzzling figure , overladen with straight and circles , a second figure th at makes sense ?We may hit in a flash on the right figure hidden in our overladen drawing , or we may recognize it gradually . We may be led to it by the effort to sol ve the proposed problem , or by some secondary , unessential circumstance . For instance , when we are about to redraw our unsatisfactory figure , we ma y observe that the whole figure is determined by its rectilinear part (Fig . 3) .This observation seems to be significant . It certainly simplifies the geometri c picture , and it possibly improves the logical situation . It leads us to restate our theorem in the following form .C . If the nine segmentsKO , KC , KB ,LC , LO , LA ,MB , MA , MO ,are all equal to r , there exists a point E such that the three segmentsEA , EB , EC ,are also equal to r .Fig . 3 It reminds you －of what ?This statement directs our attention to Fig . 3 . This figure is attractive ; it reminds us of something familiar . (Of what ?)Of course , certain quadrilaterals in Fig .3 . such as OLAM have , by hypo thesis , four equal sided , they are rhombi , A rhombus I a familiar object ; having recognized it , we can “see “the figure better . (Of what does the whole figure remind us ?)Oppositc sides of a rhombus are parallel . Insisting on this remark , we reali ze that the 9 segments of Fig . 3 . are of three kinds ; segments of the same kind , such as AL , MO , and BK , are parallel to each other . (Of what d oes the figure remind us now ?)We should not forget the conclusion that we are required to attain . Let us a ssume that the conclusion is true . Introducing into the figure the center E or the circle e , and its three radii ending in A , B , and C , we obtain (suppos edly ) still more rhombi , still more parallel segments ; see Fig . 4 . (Of wha t does the whole figure remind us now ?)Of course , Fig . 4 . is the projection of the 12 edges of a parallele piped h aving the particularity that the projection of all edges are of equal length .Fig . 4 of course !Fig . 3 . is the projection of a “nontransparent “parallelepiped ; we see o nly 3 faces , 7 vertices , and 9 edges ; 3 faces , 1 vertex , and 3 edges are invisible in this figure . Fig . 3 is just a part of Fig . 4 . but this part define s the whole figure . If the parallelepiped and the direction of projection are so chosen that the projections of the 9 edges represented in Fig . 3 are all equa l to r (as they should be , by hypothesis ) , the projections of the 3 remainin g edges must be equal to r . These 3 lines of length r are issued from the pr ojection of the 8th, the invisible vertex , and this projection E is the center o f a circle passing through the points A , B , and C , the radius of which is r .Our theorem is proved , and proved by a surprising , artistic conception of a plane figure as the projection of a solid . (The proof uses notions of solid g eometry . I hope that this is not a treat wrong , but if so it is easily redresse d . Now that we can characterize the situation of the center E so simply , it i s easy to examine the lengths EA , EB , and EC independently of any solid geometry . Yet we shall not insist on this point here .)This is very beautiful , but one wonders . Is this the “light that breaks fo rth like the morning . “the flash in which desire is fulfilled ? Or is it merel y the wisdom of the Monday morning quarterback ? Do these ideas work out in the classroom ? Followups of attempts to reduce Polya’s program to practi cal pedagogics are difficult to interpret . There is more to teaching , apparentl y , than a good idea from a master .——From Mathematical ExperienceVocabularysubtle 巧妙的，精细的clue 线索，端倪hamper 束缚，妨碍disconcert 使混乱，使狼狈ambiguous 含糊的，双关的witty 多智的，有启发的rhombi 菱形（复数）rhombus 菱形parallelepiped 平行六面体projection 射影solid geometry 立体几何pedagogics 教育学，教授法commonplace 老生常谈；平凡的。
数学专业英语－Continuous Functions of One Real VariableThis lesson deals with the concept of continuity, one of the most important an d also one of the most fascinating ideas in all of mathematics. Before we give a preeise technical definition of continuity, we shall briefly discuss the concep t in an informal and intuitive way to give the reader a feeling for its meanin g.Roughly speaking the situation is this: Suppose a function f has the value f ( p )at a certain point p. Then f is said to be continuous at p if at every ne arby point x the function value f ( x )is close to f ( p ). Another way of pu tting it is as follows: If we let x move toward p, we want the corresponding f unction value f ( x )to become arbitrarily close to f ( p ), regardless of the manner in which x approaches p. We do not want sudden jumps in the values of a continuous function.Consider the graph of the function f defined by the equation f ( x ) = x –[ x ], where [ x ] denotes the greatest integer < x. At each integer we have what is known ad a jump discontinuity. For example, f ( 2 ) = 0 ,but as x a pproaches 2 from the left, f ( x )approaches the value 1, which is not equal to f ( 2 ).Therefore we have a discontinuity at 2. Note that f ( x )does appro ach f ( 2 )if we let x approach 2 from the right, but this by itself is not en ough to establish continuity at 2. In case like this, the function is called conti nuous from the right at 2 and discontinuous from the left at 2. Continuity at a point requires both continuity from the left and from the right.In the early development of calculus almost all functions that were dealt with were continuous and there was no real need at that time for a penetrating loo k into the exact meaning of continuity. It was not until late in the 18th centur y that discontinuous functions began appearing in connection with various kind s of physical problems. In particular, the work of J.B.J. Fourier(1758-1830) on the theory of heat forced mathematicians the early 19th century to examine m ore carefully the exact meaning of the word “continuity”.A satisfactory mathematical definition of continuity, expressed entirely in term s of properties of the real number system, was first formulated in 1821 by the French mathematician, Augustin-Louis Cauchy (1789-1857). His definition, whi ch is still used today, is most easily explained in terms of the limit concept to which we turn now.The definition of the limit of a function.Let f be a function defined in some open interval containing a point p, altho ugh we do not insist that f be defined at the point p itself. Let A be a real n umber.The equationf ( x ) = Ais read “The limit of f ( x ), as x approached p, is equal to A”, or “f ( x )approached A as x approached p.”It is also written without the limit symb ol, as follows:f ( x )→A as x →pThis symbolism is intended to convey the idea that we can make f ( x )as close to A as we please, provided we choose x sufficiently close to p.Our first task is to explain the meaning of these symbols entirely in terms of real numbers. We shall do this in two stages. First we introduce the concept of a neighborhood of a point, the we define limits in terms of neighborhoods.Definition of neighborhood of a point.Any open interval containing a point p as its midpoint is called a neighborho od of p.NOTATION. We denote neighborhoods by N ( p ), N1( p ), N2( p )etc. S ince a neighborhood N( p )is an open interval symmetric about p, it consists of all real x satisfying p-r < x < p+r for some r > 0. The positive number r is called the radius of the neighborhood. We designate N ( p )by N ( p; r )if we wish to specify its radius. The inequalities p-r < x < p+r are equiv alent to –r<x-p<r,and to ∣x-p∣< r. Thus N ( p; r )consists of all points x whose distance from p is less than r.In the next definition, we assume that A is a real number and that f is a fun ction defined on some neighborhood of a point p(except possibly at p) . Th e function may also be defined at p but this is irrelevant in the definition.Definition of limit of a function.The symbolismf ( x ) = A or [ f ( x )→A as x→p ]means that for every neighborhood N1( A )there is some neighborhood N2( p)such thatf ( x )∈N1( A ) whenever x ∈N2( p ) and x ≠p (*)The first thing to note about this definition is that it involves two neighborho ods,N1( A) andN2( p). The neighborhood N1( A)is specified first; it tells us how close we wish f ( x )to be to the limit A. The second neighborhood, N2( p ),tells u s how close x should be to p so that f ( x ) will be within the first neighbor hood N1( A). The essential part of the definition is that, for every N1( A),n o matter how small, there is some neighborhood N2(p)to satisfy (*). In gener al, the neighborhood N2( p)will depend on the choice of N1( A). A neighbo rhood N2( p )that works for one particular N1( A) will also work, of course, for every larger N1( A), but it may not be suitable for any smaller N1( A).The definition of limit can also be formulated in terms of the radii of the n eighborhoodsN1( A)and N2( p ). It is customary to denote the radius of N1( A) byεan d the radius of N2( p)by δ.The statement f ( x )∈N1( A ) is equivalent to the inequality ∣f ( x ) –A∣<ε,and the statement x ∈N1( A) ,x ≠p ,is equivalent to the inequalities 0<∣x-p∣<δ. Therefore, the definition of limit can also be expressed as follows:The symbol f ( x ) = A means that for everyε> 0, there is aδ> 0 such th at∣f ( x ) –A∣<εwhenever 0 <∣x –p∣<δ“One-sided”limits may be defined in a similar way. For example, if f ( x )→A as x→p through values greater than p, we say that A is right-hand limi t of f at p, and we indicate this by writingf ( x ) = AIn neighborhood terminology this means that for every neighborhood N1( A) ,t here is some neighborhood N2( p) such thatf ( x )∈N1( A) wheneve r x ∈N1( A) and x > pLeft-hand limits, denoted by writing x→p-, are similarly defined by restricti ng x to values less than p.If f has a limit A at p, then it also has a right-hand limit and a left-hand li mit at p, both of these being equal to A. But a function can have a right-hand limit at p different from the left-hand limit.The definition of continuity of a function.In the definition of limit we made no assertion about the behaviour of f at the point p itself. Moreover, even if f is defined at p, its value there need not b e equal to the limit A. However, if it happens that f is defined at p and if it also happens that f ( p ) = A, then we say the function f is continuous at p. In other words, we have the following definition.Definition of continuity of a function at a point.A function f is said to be continuous at a point p if( a ) f is defined at p, and ( b ) f ( x ) = f ( p )This definition can also be formulated in term of neighborhoods. A function f is continuous at p if for every neighborhood N1( f(p))there is a neighborhood N2(p)such thatf ( x ) ∈N1( f (p)) whenever x ∈N2( p).In theε-δterminology , where we specify the radii of the neighborhoods, the definition of continuity can be restated ad follows:Function f is continuous at p if for every ε> 0 ,there is aδ> 0 such that ∣f( x ) –f ( p )∣< εwhenever ∣x –p∣< δIn the rest of this lesson we shall list certain special properties of continuou s functions that are used quite frequently. Most of these properties appear obvi ous when interpreted geometrically ; consequently many people are inclined to accept them ad self-evident. However, it is important to realize that these state ments are no more self-evident than the definition of continuity itself, and ther efore they require proof if they are to be used with any degree of generality. The proofs of most of these properties make use of the least-upper bound axio m for the real number system.THEOREM 1. (Bolzano’s theorem) Let f be continuous at each point of a cl osed interval [a, b] and assume that f ( a )an f ( b )have opposite signs. T hen there is at least one c in the open interval (a ,b) such that f ( c )= 0.THEOREM 2. Sign-preserving property of continuous functions. Let f be conti nuious at c and suppose that f ( c )≠0. Then there is an interval (c-δ,c +δ) about c in which f has the same sign as f ( c ).THEOREM 3. Let f be continuous at each point of a closed interval [a, b]. Choose two arbitrary points x1<x2in [a, b] such that f ( x1 ) ≠f ( x2 ). T hen f takes every value between f ( x1) and f(x2)somewhere in the interval ( x1,x2 ).THEOREM 4. Boundedness theorem for continuous functions. Let f be contin uous on a closed interval [a, b]. Then f is bounded on [a, b]. That is , there is a number M > 0, such that∣f ( x )∣≤M for all x in [a, b].THEOREM 5. (extreme value theorem) Assume f is continuous on a closed i nterval [a, b]. Then there exist points c and d in [a, b] such that f ( c ) = su p f and f ( d ) = inf f .Note. This theorem shows that if f is continuous on [a, b], then sup f is its absolute maximum, and inf f is its absolute minimum.Vocabularycontinuity 连续性 assume 假定,取continuous 连续的 specify 指定, 详细说明continuous function 连续函数statement 陈述,语句intuitive 直观的 right-hand limit 右极限corresponding 对应的left-hand limit 左极限correspondence 对应 restrict 限制于graph 图形 assertion 断定approach 趋近,探索,入门consequently 因而,所以tend to 趋向 prove 证明regardless 不管,不顾 proof 证明discontinuous 不连续的 bound 限界jump discontinuity 限跳跃不连续least upper bound 上确界mathematician 科学家 greatest lower bound 下确界formulate 用公式表示,阐述boundedness 有界性limit 极限maximum 最大值Interval 区间 minimum 最小值open interval 开区间 extreme value 极值equation 方程extremum 极值neighborhood 邻域increasing function 增函数midpoint 中点decreasing function 减函数symmetric 对称的strict 严格的radius 半径(单数) uniformly continuous 一致连续radii 半径(复数) monotonic 单调的inequality 不等式monotonic function 单调函数equivalent 等价的Notes1. It wad not until late in the 18th century that discontinuous functions began appearing in connection with various kinds of physical problems.意思是:直到十八世纪末,不连续函数才开始出现于与物理学有关的各类问题中.这里It was not until …that译为“直到……才”2. The symbol f ( x ) = A means that for every ε> 0 ,there is a δ> 0, such that|f( x ) - A|<εwhenever 0 <|x –p |<δ注意此种句型.凡涉及极限的其它定义,如本课中定义函数在点P连续及往后出现的关于收敛的定义等,都有完全类似的句型,参看附录IV.有时句中there is可换为there exists; such that可换为satisfying; whenever换成if或for.3. Let…and assume (suppose)…Then…这一句型是定理叙述的一种最常见的形式;参看附录IV.一般而语文课Let假设条件的大前提,assume (suppo se)是小前提(即进一步的假设条件),而if是对具体而关键的条件的使用语.4. Approach在这里是“趋于”,“趋近”的意思,是及物动词.如:f ( x ) approaches A as x approaches p. Approach有时可代以tend to. 如f ( x )tends to A as x tends t o p.值得留意的是approach后不加to而tend之后应加to.5. as close to A as we please = arbitrarily close to A..ExerciseI. Fill in each blank with a suitable word to be chosen from the words given below:independent domain correspondenceassociates variable range(a) Let y = f ( x )be a function defined on [a, b]. Then(i) x is called the ____________variable.(ii) y is called the dependent ___________.(iii) The interval [a, b] is called the ___________ of the function.(b) In set terminology, the definition of a function may be given as follows:Given two sets X and Y, a function f : X →Y is a __________which ___________with each element of X one and only one element of Y.II. a) Which function, the exponential function or the logarithmic function, has the property that it satisfi es the functional equationf ( xy ) = f ( x ) + f ( v )b) Give the functional equation which will be satisfied by the function which you do not choose in (a).III. Let f be a real-valued function defined on a set S of real numbers. Then we have the following two definitions:i) f is said to be increasing on the set S if f ( x ) < f ( y )for every pair of points x and y with x < y.ii) f is said to have and absolute maximum on the set S if there is a point c in S such that f ( x ) < f ( c )for all x∈S.Now definea) a strictly increasing function;b) a monotonic function;c) the relative (or local ) minimum of f .IV. Translate theorems 1-3 into Chinese.V. Translate the following definition into English:定义:设E 是定义在实数集E上的函数,那么, 当且仅当对应于每一ε>0(ε不依赖于E上的点)存在一个正数δ使得当p和q属于E且|p –q| <δ时有|f ( p ) – f ( q )|<ε,则称f在E上一致连续.。
第一段翻译(2)：what is the exact value of the number pai?a mathematician made an experiment in order to find his own estimation of the number pai.in his experiment,he used an old bicycle wheel of diameter 63.7cm.he marked the point on the tire where the wheel was touching the ground and he rolled the wheel straight ahead by turning it 20 times.next,he measured the distance traveled by the wheel,which was 39.69 meters.he divided the number 3969 by 20*63.7 and obtained 3.115384615 as an approximation of the number pai.of course,this was just his estimate of the number pai and he was aware that it was not very accurate.数π的精确值是什么？一位数学家做了实验以便找到他自己对数π的估计。
在试验中，他用了一直径63.1厘米的旧自行车轮。
他在车轮接触地面的轮胎上做了标记，而且将车轮向前转动20次。
接下来，他测量了车轮经过的距离，是39.69米。
他用3969除20*63.7得到了数π的近似值3.115384615。
当然，这只是对数π的估计值，并且他也意识到不是很准确。
第二段翻译(5)：one of the first articles which we included in the "History Topics" section archive was on the history of pai.it is a very popular article and has prompted many to ask for a similar article about the number e.there is a great contrast between the historical developments of these two numbers and in many ways writing a history of e is a much harder task than writing one of pai.the number e is,compared to pai,a relative newcomer on the mathematical scene.我们包括在“历史专题”部分档案中的第一篇文章就是历史上的π，这是一篇很流行的文章，也促使许多人想了解下一些有关数e的类似文章。
数学专业英语（Doc版）.12数学专业英语－Linear ProgrammingLinear Programming is a relatively new branch of mathematics.The cornerstone of this exciting field was laid independently bu Leonid V. Kantorovich,a Russ ian mathematician,and by Tjalling C,Koopmans, a Yale economist,and George D. Dantzig,a Stanford mathematician. Kantorovich’s pioneering work was moti vated by a production-scheduling problem suggested by the Central Laboratory of the Len ingrad Plywood Trust in the late 1930’s. The development in the U nited States was influenced by the scientific need in World War II to solve lo gistic military problems, such as deploying aircraft and submarines at strategic positions and airlifting supplies and personnel.The following is a typical linear programming problem:A manufacturing company makes two types of television sets: one is black and white and the other is color. The company has resources to make at most 300 sets a week. It takes $180 to make a black and white set and $270 to make a color set. The company does not want to spend more than $64,800 a wee k to make television sets. If they make a profit of $170 per black and white set and $225 per color set, how many sets of each type should the company make to have a maximum profit?This problem is discussed in detail in Supplementary Reading Material Lesson 14.Since mathematical models in linear programming problems consist of linear in equalities, the next section is devoted to suchinequalities.Recall that the linear equation lx+my+n=0represents a straight line in a plane. Every solution (x,y) of the equation lx+my+n=0is a point on this line, and vice versa.An inequality that is obtained from the linear equation lx+my+n=0by replacin g the equality sign “=”by an inequality sign < (less than), ≤(less than or equal to), > (greater than), or ≥(greater than or equal to) is called a linear i nequality in two variables x and y. Thus lx+my+n≤0, lx+my+n≥0are all lin ear liequalities. A solution of a linear inequality is an ordered pair (x,y) of nu mbers x and y for which the inequality is true.EXAMPLE 1 Graph the solution set of the pair of inequalities SOLUTION Let A be the solution set of the inequality x+y-7≤0 and B be th at of the inequalit y x-3y +6 ≥0 .Then A∩B is the solution set of the given pair of inequalities. Set A is represented by the region shaded with horizontal lines and set B by the region shaded with vertical lines in Fig.1. Therefore thecrossed-hatched region represents the solution set of the given pair of inequali ties. Observe that the point of intersection (3.4) of the two lines is in the solu tion set.Generally speaking, linear programming problems consist of finding the maxim um value or minimum value of a linear function, called the objective function, subject to some linear conditions, called constraints. For example, we may wa nt to maximize the production or profit of a company or to maximize the num ber of airplanes that can land at or take off from an airport during peak hours; or we may want to minimize the cost of production or of transportation or to minimize grocery expenses while still meeting the recommended nutritional re quirements, all subject to certain restrictions. Linearprogramming is a very use ful tool that can effectively be applied to solve problems of this kind, as illust rated by the following example.EXAMPLE 2 Maximize the function f(x,y)=5x+7y subject to the constraintsx≥0 y≥0x+y-7≤02x-3y+6≥0SOLUTION First we find the set of all possible pairs(x,y) of numbers that s atisfy all four inequalities. Such a solution is called a feasible sulution of the problem. For example, (0,0) is a feasible solution since (0,0) satisf ies the giv en conditions; so are (1,2) and (4,3).Secondly, we want to pick the feasible solution for which the giv en function f (x,y) is a maximum or minimum (maximum in this case). S uch a feasible solution is called an optimal solution.Since the constraints x ≥0 and y ≥0 restrict us to the first quadrant, it follows from example 1 that the given constraints define the polygonal regi on bounded by the lines x=0, y=0,x+y-7=0, and 2x-3y+6=0, as shown in Fig.2.Fig.2.Observe that if there are no conditions on the values of x and y, then the f unction f can take on any desired value. But recall that our goal is to determi ne the largest value of f (x,y)=5x+7y where the values of x and y are restrict ed by the given constraints: that is, we must locate that point (x,y) in the pol ygonal region OABC at which the expression 5x+7y has the maximum possibl e value.With this in mind, let us consider the equation 5x+7y=C, where C is any n umber. This equation represents a family ofparallel lines. Several members of this family, corresponding to different values of C, are exhibited in Fig.3. Noti ce that as the line 5x+7y=C moves up through the polygonal region OABC, th e value of C increases steadily. It follows from the figure that the line 5x+7y =43 has a singular position in the family of lines 5x+7y=C. It is the line farth est from the origin that still passes through the set of feasible solutions. It yiel ds the largest value of C: 43.(Remember, we are not interested in what happen s outside the region OABC) Thus the largest value of the function f(x,y)=5x+7 y subject to the condition that the point (x,y) must belong to the region OAB C is 43; clearly this maximum value occurs at the point B(3,4).Fig.3.Consider the polygonal region OABC in Fig.3. This shaded region has the p roperty that the line segment PQ joining any two points P and Q in the regio n lies entirely within the region. Such a set of points in a plane is called a c onvex set. An interesting observation about example 2 is that the maximum va lue of the objective function f occurs at a corner point of the polygonal conve x set OABC, the point B(3,4).The following celebrated theorem indicates that it was not accidental.THEOREM (Fundamental theorem of linear programming) A linear objective function f defined over a polygonal convex set attains a maximum (or minim um) value at a corner point of the set.We now summarize the procedure for solving a linear programming problem:1.Graph the polygonal region determined by the constraints.2.Find the coordinates of the corner points of the polygon.3.Evaluate the objective function at the corner points.4.Identify the corner point at which the function has an optimal value.Vocabularylinear programming 线形规划 quadrant 象限objective function 目标函数 convex 凸的constraints 限制条件，约束条件 convex set 凸集feaseble solution 容许解，可行解corner point 偶角点optimal solution 最优解simplex method 单纯形法Notes1. A Yale economist, a Stanford mathematician 这里Yale Stanford 是指美国两间著名的私立大学：耶鲁大学和斯坦福大学，这两间大学分别位于康涅狄格州（Connecticut）和加里福尼亚州(California)2. subject to some lincar conditions 解作“在某些线形条件的限制下”。
专业英语论文（优秀）浅谈大学英语学习论文篇一一、前言开放30年来，我同一直处于一个与时俱进，开拓进取的年代，冈而，教育，作为一项关系到国家进步，人民发展和民族繁荣的大事业，也正在跟着时代的步伐做出适当的调整。
纵观以前，我国传统的教育一度以行为主义理论为核心指导理论，应用到外语教学中主去，就出现了外语教学结构皆以教师为主，学生为辅从而使我们的外语教学一直处于事倍功半的不良状态。
但随着建构主义理论在我国的日益流行，同内越来越多的外语学专家和教师才开始将注意力慢慢投向外语学习的真正主体一一学生。
此外，国家教育部也在2003年了启动大学英语教学的试点工作。
同家教育司亦于2023年颁布了《大学英语课程教学要求》其中提出新教学模式应能使学生选择适合自己需要的材料和方法进行学习，获得学习策略的指导逐步提高其自主学习的能力。
因此，在当前新的时代背景下，只有认真研究大学英语的自主学习模式才能真正实现我同大学英语教学的最终目的，培养出适应时代要求，能够真正为现代化建设贡献一份力的人才。
三自主学习模式一般来说自主学习模式是指学主在明确的学习目标要求之下，以及英在教师的正确指导下，根据自身认识结陶和需要，积极、主动、创造性地完成具本学习任务的一种教学方式。
自主学习具体模式有:1.问题式学习模式，2.锚式情境教学，3.协作学习，4.交互式教学。
这种教学模式有其实践的必要性小具本如下:传统外语教学法为我国的教育发展阳人才培养做出了巨大贡献，但目前已凋显滞后，存在着各种严重弊端。
自主学习教育模式有其优越性。
自主学习教育模式的提出主要出且于以下两苦:一是随着丰杜十会的迸步以人为本发展完善自我的价直取向成为社会发展的必然。
在教学理念上以学生为中心，在传授知识和技能拍同时特别注重语言的运用能力和自主学习能力的培养，在课程体系上也充分本现了分类教学的思想，注重个体差异，散学手段要求多样化，教学测评更重视学生的语言运用能力和过程性评估。
数学专业英语数学，作为一门自然科学，对于任何一个国家和民族来说都具有非常重要的意义。
作为一种研究物质与现象本质的学科，数学通过一系列公理和推理，进行精确的描述和分析，从而刻画出了丰富的数学结构和表现形式，成为自然科学中应用最广泛的基础学科之一。
为了更好地传承和发展数学学科，我国数学教育体系逐渐完善，大量优秀的数学教师和研究人员涌现出来，为数学事业不断贡献力量。
同时，我国的数学研究不仅在量上有了大幅度的提升，更在质上得到了越来越多的认可和尊重。
在数学研究领域，英语作为国际性的语言，具有不可替代的地位。
数学专业英语，是由数学领域专业人士使用的一种专门的术语和表达方式。
正如同其他学科一样，数学专业英语也有其独特的语言规范、术语和语法，并且赋予了数学的某些具体含义。
因此，掌握数学专业英语，不仅是进入国际学术圈的重要标志，同时也是增进数学研究深度和广度的关键所在。
为此，本文将从数学专业英语的基本要素、相关研究和未来发展趋势等方面进行探讨和解析。
一、数学专业英语的基本要素1. 语法语法是数学专业英语的基本要素之一，它是制定语言规范的重要基础。
英语语法主要包括名词、动词、形容词和副词等四个基本部分。
而在数学专业英语中，这些基本部分的构成都会涉及到特定的数学概念和符号。
此外，数学专业英语还包括数学符号、公式等。
例如，对于一个数学论文中的一个公式，“f(x)=ax+b”，这个公式中的"f(x)"，代表一个函数；"a"和"b"分别代表函数的系数；"x"则代表自变量。
在这个公式中，如果说我们把"f(x)"看作数学专业英语中的名词，那么"a"和"b"就是形容词，而"x"可以看作动词。
因此，理解数学专业英语中的语法结构和语法规则，对于正确地理解和应用数学专业英语极为重要。
学专业英语－StatisticsThe term statistics is used in either of two senses.In common parlance it is ge nerally employed synonymously with the word data.Thus someone may say tha t he has seen”statistics of industrial accidents in the United States.”It would be conducive to greater precision of meaning if we were not to use statistics i n this sense,but rather to say “data (or figures ) of industrial accidents in the United States.”“Statistics”also refers to the statistical principles and methods which have be en developed for handling numerical data and which form the subject matter o f this text.Statistical methods,or statistics, range form the most elementary descr iptive devices, which may be understood by anyone , to those extremely compl icated mathematical procedures which are comprehended by only the most expe rt theoreticians.It is the purpose of this volume not to enter into the highly ma thematical and theoretical aspects of the subject but rather to treat of its more elementary and more frequently used phases.Statistics may be defined as the collection, presentation, analysis, and interpreta tion of numerical data.The facts which are dealt with must be capable of num erical expression.We can make little use statistically of the information that dw ellings are built of brick, stone, wood, and other materials; however, if we are able to determine how many or what proportion of,dwellings are constructed of each type of material, we have numerical data suitable for statistical analysi s.Statistics should not be thought of as a subject correlative with physics, chemi stry, economics, and sociology. Statistics is not a science; it is a scientific met hod. The methods and procedures which we are about to examine constitute a useful and often indispensable tool for the research worker. Without an adequat e understanding of statistics, the investigator in the social sciences may frequen tly be like the blind man groping in a dark closet for a black cat that isn’t t here. The methods of statistics are useful in an ever---widening range of huma n activities, in any field of thought in which numerical data may be had.In defining statistics it was pointed out that the numerical data are collected, p resented, analyzed, and interpreted. Let us briefly examine each of these four p rocedures.COLLECTION Statistical data may be obtained from existing published or un published sources, such as government agencies, trade associations, research bur eaus, magazines, newspapers, individual research workers, and elsewhere. On th e other hand, the investigator may collect his own information, going perhaps f rom house to house or from firm to firm to obtain his data. The first-hand col lection of statistical data is one of the most difficult and important tasks whicha statistician must face. The soundness of his procedure determines in an ove rwhelming degree the usefulness of the data which he obtains.It should be emphasized, however, that the investigator who has experience an d good common sense is at a distinct advantage if original data must be colle cted. There is much which may be taught about this phase of statistics, but th ere is much more which can be learned only through experience. Although a p erson may never collect statistical data for his own use and may always use p ublished sources, it is essential that he have a working knowledge of the proce sses of collection and that he be able to evaluate the reliability of the data he proposes to use. Untrustworthy data do not constitute a satisfactory base upon which to rest a conclusion.It is to be regretted that many people have a tendency to accept statistical dat a without question. To them, any statement which is presented in numerical ter ms is correct and its authenticity is automatically established.PRESENTATION Either for one’s own use or for the use of others, the dat a must be presented in some suitable form. Usually the figures are arranged in tables or presented by graphic devices.ANALYSIS In the process of analysis, data must be classified into useful and logical categories. The possible categories must be considered when plans are made for collecting the data, and the data must be classified as they are tabu lated and before they can be shown graphically. Thus the process of analysis i s partially concurrent with collection and presentation.There are four important bases of classification of statistical data: (1) qualitativ e, (2) quantitative, (3) chronological, and (4) geographical, each of which will be examined in turn.Qualitative When, for example, employees are classified as union or non—uni on, we have a qualitative differentiation. The distinction is one of kind rather t han of amount. Individuals may be classified concerning marital status, as singl e, married, widowed, divorced, and separated. Farm operators may be classified as full owners, part owners, managers, and tenants. Natural rubber may be de signated as plantation or wild according to its source.Quantitative When items vary in respect to some measurable characteristics, a quantitative classification is appropriate. Families may be classified according t o the number of children. Manufacturing concerns may be classified according to the number of workers employed, and also according to the values of goods produced. Individuals may be classified according to the amount of income ta x paid.Chronological Chronological data or time series show figures concerning a par ticular phenomenon at various specified times. For example, the closing price o f a certain stock may be shown for each day over a period of months of year s; the birth rate in the United States may be listed for each of a number of y ears; production of coal may be shown monthly for a span of years. The anal ysis of time series, involving a consideration of trend, cyclical period (seasonal ), and irregular movements, will be discussed.In a certain sense, time series are somewhat akin to quantitative distributions i n that each succeeding year or month of a series is one year or one month fu rther removed from some earlier point of reference. However, periods of time —or, rather, the events occurring within these periods—differ qualitatively from each other also. The essential arrangement of the figures in a time sequence i s inherent in the nature of the data under consideration.Geographical The geographical distribution is essentially a type of qualitative distribution, but is generally considered as a distinct classification. When the p opulation is shown for each of the states in the United States, we have data which are classified geographically. Although there is a qualitative difference b etween any two states, the distinction that is being made is not so much of ki nd as of location.The presentation of classified data in tabular and graphic form is but one elem entary step in the analysis of statistical data. Many other processes are describ ed in the following passages of this book. Statistical investigation frequently en deavors to ascertain what is typical in a given situation. Hence all type of occ urrences must be considered, both the usual and the unusual.In forming an opinion, most individuals are apt to be unduly influenced by un usual occurrences and to disregard the ordinary happenings. In any sort or inve stigation, statistical or otherwise, the unusual cases must not exert undue influe nce. Many people are of the opinion that to break a mirror brings bad luck. H aving broken a mirror, a person is apt to be on the lookout for the unexpecte d”bad luck “and to attribute any untoward event to the breaking of the mir ror. If nothing happens after the mirror has been broken, there is nothing to re member and this result (perhaps the usual result )is disregarded. If bad luck oc curs, it is so unusual that it is remembered, and consequently the belief is rein forced. The scienticfic procedure would include all happenings following the br eaking of the mirror, and would compare the “resulting”bad luck to the am ount of bad luck occurring when a mirror has not been broken.Statistics, then, must include in its analysis all sorts of happenings. If we are studying the duration of cases of pncumonia, we may study what is typical by determining the average length and possibly also the divergence below and ab ove the average. When considering a time series showing steel—mill activity,we may give attention to the typical seasonal pattern of the series, to the gro wth factor( trend) present, and to the cyclical behaviour. Sometimes it is found that two sets of statistical data tend to be associated.Occasionlly a statistical investigation may be exhaustive and include all possibl e occurrences. More frequently, however, it is necessary to study a small grou p or sample. If we desire to study the expenditures of lawyers for life insuran ce, it would hardly be possible to include all lawyers in the United States. Re sort must be had to a sample;and it is essential that the sample be as nearly r epresentative as possible of the entire group, so that we may be able to make a reasonable inference as to the results to be expected for an entire populatio n. The problem of selecting a sample is discussed in the following chapter.Sometimes the statistician is faced with the task of forecasting. He may be req uired to prognosticate the sales of automobile tires a year hence, or to forecast the population some years in advance. Several years ago a student appeared i n summer session class of one of the writers. In a private talk he announced t hat he had come to the course for a single purpose: to get a formula which w ould enable him to forecast the price of cotton. It was important to him and h is employers to have some advance information on cotton prices, since the con cern purchased enormous quantities of cotton. Regrettably, the young man had to be disillusioned. To our knowledge, there are no magic formulae for forecas ting. This does not mean that forecasting is impossible; rather it means that fo recasting is a complicated process of which a formula is but a small part. And forecasting is uncertain and dangerous. To attempt to say what will happen in the future requires a thorough grasp of the subject to be forecast, up-to-the-m inute knowledge of developments in allied fields, and recognition of the limitat ions of any mechanica forecasting device.INTERPRETATION The final step in an investigation consists of interpreting the data which have been obtained. What are the conclusions growing out of t he analysis? What do the figures tell us that is new or that reinforces or casts doubt upon previous hypotheses? The results must be interpreted in the light of the limitations of the original material. Too exact conclusions must not be drawn from data which themselves are but approximations. It is essential, howe ver, that the investigator discover and clarify all the useful and applicable mea ning which is present in his data.VocabularyStatistics统计学in tables 列成表Statistical 统计的tabular列成表的Statistical data统计数据sample样本Statistical method统计方法 inference推理，推断Original data原始数据 reliance信赖Qualitative定性的forecasting预测Quantitative定量的 in common parlance按一般说法Chronological年代学的 conducive有帮助的Time series时间序列grope摸索Cyclical循环的 akin to类似Period周期apt to易于Periodic周期的 undue不适当的Prognosticate预测 sociology社会学Authenticity可信性，真实性phase相位；方面Synonymously同义的categories范畴，类型Correlative相互关系的，相依的 concurrent会合的，一致的，同时发生的Notes1. It is the purpose of this volume not to enter into the highly mathematical and theoretical a spects of the subject but rather to treat of its more elementary and more frequently used phases.意思是：本书的目的并不是要深入到这个论题的有关高深的数学与理论的方面，而是要讨论它的更为初等和更为常用的方面，not…but rather 意思是“不是…而是”，而rather than意思是“宁愿…而不”，两者意思相近但有差别（主要表现为强调哪方面的差别）。
如何提升数学的英语作文（中英文版）How to Enhance Your Mathematical Essay Writing in English?提升数学英语作文的几种策略：Firstly, expanding your mathematical vocabulary is crucial.Familiarize yourself with key terms and concepts in English, such as "algebra," "geometry," and "calculus." Understanding these terms will allow you to convey your ideas more effectively in your essay.首先，扩大你的数学词汇量至关重要。
熟悉英文中的关键术语和概念，比如“代数”，“几何”，“微积分”。
理解这些术语将使你能够更有效地在作文中表达你的观点。
Secondly, practice reading and analyzing mathematical essays or articles in English.This will help you grasp the structure and style commonly used in English mathematical writing, enabling you to craft a well-organized and coherent essay.其次，练习阅读和分析英文数学论文或文章。
这将帮助你掌握英文数学写作中常用的结构和风格，使你能够撰写出组织良好、条理清晰的作文。
Additionally, incorporating real-world examples into your essay can make it more engaging.Explain how mathematical concepts are applied in everyday life or various fields, using simple yet precise language.此外，将现实世界的例子融入你的作文可以使文章更加引人入胜。
Some Properties of Solutions of Periodic Second OrderLinear Differential Equations1. Introduction and main resultsIn this paper, we shall assume that the reader is familiar with the fundamental results and the stardard notations of the Nevanlinna's value distribution theory of meromorphic functions [12, 14, 16]. In addition, we will use the notation )(f σ，)(f μand )(f λto denote respectively the order of growth, the lower order of growth and the exponent of convergence of the zeros of a meromorphic function f ，)(f e σ（[see 8]），the e-type order of f(z), is defined to berf r T f r e ),(log lim )(+∞→=σ Similarly, )(f e λ，the e-type exponent of convergence of the zeros of meromorphic function f , is defined to berf r N f r e )/1,(log lim )(++∞→=λ We say that )(z f has regular order of growth if a meromorphic function )(z f satisfiesrf r T f r log ),(log lim )(+∞→=σ We consider the second order linear differential equation0=+''Af fWhere )()(z e B z A α=is a periodic entire function with period απω/2i =. The complex oscillation theory of (1.1) was first investigated by Bank and Laine [6]. Studies concerning (1.1) have een carried on and various oscillation theorems have been obtained [2{11, 13, 17{19]. When )(z A is rational in z e α，Bank and Laine [6] proved the following theoremTheorem A Let )()(z e B z A α=be a periodic entire function with period απω/2i = and rational in z e α.If )(ζB has poles of odd order at both ∞=ζ and 0=ζ, then for every solution )0)((≠z f of (1.1), +∞=)(f λBank [5] generalized this result: The above conclusion still holds if we just suppose that both ∞=ζ and 0=ζare poles of )(ζB , and at least one is of odd order. In addition, the stronger conclusion)()/1,(log r o f r N ≠+ (1.2)holds. When )(z A is transcendental in z e α, Gao [10] proved the following theoremTheorem B Let ∑=+=p j j j b g B 1)/1()(ζζζ,where )(t g is a transcendental entire function with 1)(<g σ, p is an odd positive integer and 0≠p b ，Let )()(z e B z A =.Then anynon-trivia solution f of (1.1) must have +∞=)(f λ. In fact, the stronger conclusion (1.2) holds.An example was given in [10] showing that Theorem B does not hold when )(g σis any positive integer. If the order 1)(>g σ , but is not a positive integer, what can we say? Chiang and Gao [8] obtained the following theoremsTheorem 1 Let )()(z e B z A α=,where )()/1()(21ζζζg g B +=,1g and 2g are entirefunctions with 2g transcendental and )(2g μnot equal to a positive integer or infinity, and 1g arbitrary. If Some properties of solutions of periodic second order linear differential equations )(z f and )2(i z f π+are two linearly independent solutions of (1.1), then+∞=)(f e λOr2)()(121≤+--g f e μλWe remark that the conclusion of Theorem 1 remains valid if we assume )(1g μis not equal to a positive integer or infinity, and 2g arbitrary and still assume )()/1()(21ζζζg g B +=,In the case when 1g is transcendental with its lower order not equal to an integer or infinity and 2g is arbitrary, we need only to consider )/1()()/1()(*21ηηηηg g B B +==in +∞<<η0,ζη/1<.Corollary 1 Let )()(z e B z A α=,where )()/1()(21ζζζg g B +=,1g and 2g areentire functions with 2g transcendental and)(2g μno more than 1/2, and 1g arbitrary. (a)If f is a non-trivial solution of (1.1) with +∞<)(f e λ,then )(z f and )2(i z f π+are linearly dependent. (b) If 1f and 2f are any two linearly independent solutions of (1.1),then +∞=)(21f f e λ.Theorem 2 Let )(ζg be a transcendental entire function and its lower order be no more than 1/2. Let )()(z e B z A =,where ∑=+=p j j j b g B 1)/1()(ζζζand p is an odd positive integer, then +∞=)(f λ for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.We remark that the above conclusion remains valid if∑=--+=pj j j b g B 1)()(ζζζWe note that Theorem 2 generalizes Theorem D when )(g σis a positive integer or infinity but 2/1)(≤g μ. Combining Theorem D with Theorem 2, we haveCorollary 2 Let )(ζg be a transcendental entire function. Let )()(z e B z A = where ∑=+=pj j j b g B 1)/1()(ζζζand p is an odd positive integer. Suppose that either (i) or (ii) below holds:(i) )(g σ is not a positive integer or infinity;(ii) 2/1)(≤g μ;then +∞=)(f λfor each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.2. Lemmas for the proofs of TheoremsLemma 1 ([7]) Suppose that 2≥k and that 20,.....-k A A are entire functions of period i π2,and that f is a non-trivial solution of 0)()()(20)(=+∑-=k i j j z y z A k ySuppose further that f satisfies )()/1,(log r o f r N =+; that 0A is non-constant and rationalin z e ，and that if 3≥k ，then 21,.....-k A A are constants. Then there exists an integer q with k q ≤≤1 such that )(z f and )2(i q z f π+are linearly dependent. The same conclusionholds if 0A is transcendental in z e ，and f satisfies )()/1,(log r o f r N =+，and if 3≥k ，thenas ∞→r through a set 1L of infinite measure, wehave )),((),(j j A r T o A r T =for 2,.....1-=k j .Lemma 2 ([10]) Let )()(z e B z A α=be a periodic entire function with period 12-=απωi and be transcendental in z e α, )(ζB is transcendental and analytic on +∞<<ζ0.If )(ζB has a pole of odd order at ∞=ζ or 0=ζ(including those which can be changed into this case by varying the period of )(z A and Eq . (1.1) has a solution 0)(≠z f which satisfies )()/1,(log r o f r N =+, then )(z f and )(ω+z f are linearly independent.3. Proofs of main resultsThe proof of main results are based on [8] and [15].Proof of Theorem 1 Let us assume +∞<)(f e λ.Since )(z f and )2(i z f π+are linearly independent, Lemma 1 implies that )(z f and )4(i z f π+must be linearly dependent. Let )2()()(i z f z f z E π+=,Then )(z E satisfies the differential equation222)()()(2))()(()(4z E c z E z E z E z E z A -''-'=, (2.1) Where 0≠c is the Wronskian of 1f and 2f (see [12, p. 5] or [1, p. 354]), and )()2(1z E c i z E =+πor some non-zero constant 1c .Clearly, E E /'and E E /''are both periodic functions with period i π2,while )(z A is periodic by definition. Hence (2.1) shows that 2)(z E is also periodic with period i π2.Thus we can find an analytic function )(ζΦin +∞<<ζ0,so that )()(2z e z E Φ=Substituting this expression into (2.1) yields ΦΦ''+ΦΦ'-ΦΦ'+Φ=-2222)(43)(4ζζζζc B (2.2) Since both )(ζB and )(ζΦare analytic in }{+∞<<=ζζ1:*C ,the Valiron theory [21, p. 15] gives their representations as)()()(ζζζζb R B n =，)()()(11ζφζζζR n =Φ， （2.3）where n ，1n are some integers, )(ζR and )(1ζR are functions that are analytic and non-vanishing on }{*∞⋃C ，)(ζb and )(ζφ are entire functions. Following the same arguments as used in [8], we have),(),()/1,(),(φρρφρφρS b T N T ++=， （2.4）where )),((),(φρφρT o S =.Furthermore, the following properties hold [8])}(),(max{)()()(222E E E E f eL eR e e e λλλλλ===,)()()(12φλλλ=Φ=E eR ,Where )(2E eR λ(resp, )(2E eL λ) is defined to berE r N R r )/1,(log lim 2++∞→(resp, r E r N R r )/1,(log lim 2++∞→), Some properties of solutions of periodic second order linear differential equationswhere )/1,(2E r N R (resp. )/1,(2E r N L denotes a counting function that only counts the zerosof 2)(z E in the right-half plane (resp. in the left-half plane), )(1Φλis the exponent of convergence of the zeros of Φ in *C , which is defined to beρρλρlog )/1,(log lim )(1Φ=Φ++∞→N Recall the condition +∞<)(f e λ,we obtain +∞<)(φλ.Now substituting (2.3) into (2.2) yields+'+'+-'+'++=-21112111112)(43)()()()()(4φφζζφφζζζφζζζζζR R n R R n R c b R n n)222)1((1111111112112φφφφζφφζφφζζζ''+''+'''+''+'+'+-R R R R R n R R n n n （2.5） Proof of Corollary 1 We can easily deduce Corollary 1 (a) from Theorem 1 .Proof of Corollary 1 (b). Suppose 1f and 2f are linearly independent and +∞<)(21f f e λ,then +∞<)(1f e λ,and+∞<)(2f e λ.We deduce from the conclusion of Corollary 1 (a) that)(z f j and )2(i z f j π+are linearly dependent, j = 1; 2. Let )()()(21z f z f z E =.Then we can find a non-zero constant 2c such that )()2(2z E c i z E =+π.Repeating the same arguments as used in Theorem 1 by using the fact that 2)(z E is also periodic, we obtain2)()(121≤+--g E e μλ,a contradiction since 2/1)(2≤g μ.Hence +∞=)(21f f e λ.Proof of Theorem 2 Suppose there exists a non-trivial solution f of (1.1) that satisfies )()/1,(log r o f r N =+. We deduce 0)(=f e λ, so )(z f and )2(i z f π+ are linearly dependent by Corollary 1 (a). However, Lemma 2 implies that )(z f and )2(i z f π+are linearly independent. This is a contradiction. Hence )()/1,(log r o f r N ≠+holds for each non-trivial solution f of (1.1). This completes the proof of Theorem 2.Acknowledgments The authors would like to thank the referees for helpful suggestions to improve this paper.References[1] ARSCOTT F M. Periodic Di®erential Equations [M]. The Macmillan Co., New York, 1964.[2] BAESCH A. On the explicit determination of certain solutions of periodic differentialequations of higher order [J]. Results Math., 1996, 29(1-2): 42{55.[3] BAESCH A, STEINMETZ N. Exceptional solutions of nth order periodic linear differentialequations [J].Complex Variables Theory Appl., 1997, 34(1-2): 7{17.[4] BANK S B. On the explicit determination of certain solutions of periodic differential equations[J]. Complex Variables Theory Appl., 1993, 23(1-2): 101{121.[5] BANK S B. Three results in the value-distribution theory of solutions of linear differentialequations [J].Kodai Math. J., 1986, 9(2): 225{240.[6] BANK S B, LAINE I. Representations of solutions of periodic second order linear differentialequations [J]. J. Reine Angew. Math., 1983, 344: 1{21.[7] BANK S B, LANGLEY J K. Oscillation theorems for higher order linear differential equationswith entire periodic coe±cients [J]. Comment. Math. Univ. St. Paul., 1992, 41(1): 65{85.[8] CHIANG Y M, GAO Shi'an. On a problem in complex oscillation theory of periodic secondorder lineardifferential equations and some related perturbation results [J]. Ann. Acad. Sci. Fenn. Math., 2002, 27(2):273{290.一些周期性的二阶线性微分方程解的方法1． 简介和主要成果在本文中，我们假设读者熟悉的函数的数值分布理论[12，14，16]的基本成果和数学符号。
数学毕业论文外文翻译数学毕业论文外文翻译数学作为一门精确的科学，不仅在理论研究中发挥着重要作用，也在实际应用中有着广泛的应用。
因此，对于数学专业的学生来说，毕业论文是一项重要的任务。
在撰写数学毕业论文时，往往需要参考大量的外文文献，以获取最新的研究成果和理论进展。
然而，由于外文文献的语言障碍，很多学生在翻译外文文献时遇到了困难。
本文将探讨数学毕业论文外文翻译的一些技巧和方法。
首先，对于数学毕业论文外文翻译，学生应该具备一定的英语基础。
毕竟，数学领域的外文文献往往使用专业术语和复杂的数学符号，如果没有足够的英语能力，很难准确理解和翻译。
因此，学生在大学期间应该注重英语的学习，提高自己的阅读和翻译能力。
其次，数学毕业论文外文翻译需要学生具备一定的数学知识。
毕竟，数学是一门高度抽象和逻辑性强的学科，对于非数学专业的学生来说，理解和翻译数学文献是一项挑战。
因此，学生在进行外文翻译时，需要事先对相关数学知识进行充分的学习和了解，以确保能够准确理解和翻译文献中的数学内容。
此外，数学毕业论文外文翻译需要学生具备良好的翻译技巧。
在进行翻译时，学生应该注重准确性和流畅性。
准确性是指翻译的内容要与原文保持一致，不能出现歧义或错误。
流畅性是指翻译的内容要符合中文的表达习惯和语法规则，使读者能够顺畅地理解翻译的内容。
为了提高翻译质量，学生可以参考一些翻译技巧，比如使用词典、查阅专业术语表、请教专业人士等。
最后，数学毕业论文外文翻译需要学生具备耐心和毅力。
由于数学文献的复杂性和难度，翻译过程可能会遇到各种困难和挑战。
学生需要保持耐心，逐句逐词地进行翻译，确保每个细节都准确无误。
同时，学生需要具备毅力，不断学习和提高自己的翻译能力，以应对更高难度的数学文献翻译任务。
总之，数学毕业论文外文翻译是一项具有挑战性的任务，需要学生具备一定的英语基础、数学知识和翻译技巧。
同时，学生还需要具备耐心和毅力，不断学习和提高自己的翻译能力。
学专业英语－How to Organize a paper (For Beginers)?The usual journal article is aimed at experts and near-experts, who are the peo ple most likely to read it. Your purpose should be say quickly what you have done is good, and why it works. Avoid lengthy summaries of known results, and minimize the preliminaries to the statements of your main results. There ar e many good ways of organizing a paper which can be learned by studying pa pers of the better expositors. The following suggestions describe a standard acc eptable style.Choose a title which helps the reader place in the body of mathematics. A use less title: Concerning some applications of a theorem of J. Doe. A. good title contains several well-known key words, e. g. Algebraic solutions of linear parti al differential equations. Make the title as informative as possible; but avoid re dundancy, and eschew the medieval practice of letting the title serve as an infl ated advertisement. A title of more than ten or twelve words is likely to be m iscopied, misquoted, distorted, and cursed.The first paragraph of the introduction should be comprehensible to any mathe matician, and it should pinpoint the location of the subject matter. The main p urpose of the introduction is to present a rough statement of the principal resul ts; include this statement as soon as it is feasible to do so, although it is som etimes well to set the stage with a preliminary paragraph. The remainder of th e introduction can discuss the connections with other results.It is sometimes useful to follow the introduction with a brief section that estab lishes notation and refers to standard sources for basic concepts and results. N ormally this section should be less than a page in length. Some authors weave this information unobtrusively into their introductions, avoiding thereby a dull section.The section following the introduction should contain the statement of one or more principal results. The rule that the statement of a theorem should precede its proof a triviality. A reader wants to know the objective of the paper, as well as the relevance of each section, as it is being read. In the case of a ma jor theorem whose proof is long, its statement can be followed by an outline of proof with references to subsequent sections for proofs of the various parts. Strive for proofs that are conceptual rather than computational. For an example of the difference, see A Mathematician’s Miscellany by J.E.Littlewood, in wh ich the contrast between barbaric and civilized proofs is beautifully and amusin gly portrayed. To achieve conceptual proofs, it is often helpful for an author t o adopt an initial attitude such as one would take when communicating mathe matics orally (as when walking with a friend). Decide how to state results wit h a minimum of symbols and how to express the ideas of the proof without computations. Then add to this framework the details needed to clinch the resul ts.Omit any computation which is routine (i.e. does not depend on unexpected tri cks). Merely indicate the starting point, describe the procedure, and state the o utcome.It is good research practice to analyze an argument by breaking it into a succe ssion of lemmas, each stated with maximum generality. It is usually bad practi ce to try to publish such an analysis, since it is likely to be long and unintere sting. The reader wants to see the path-not examine it with a microscope. A p art of the argument is worth isolating as a lemma if it is used at least twice l ater on.The rudiments of grammar are important. The few lines written on the blackbo ard during an hour’s lecture are augmented by spoken commentary, and aat t he end of the day they are washed away by a merciful janitor. Since the publ ished paper will forever speak for its author without benefit of the cleansing s ponge, careful attention to sentence structure is worthwhile. Each author must develop a suitable individual style; a few general suggestions are nevertheless a ppropriate.The barbarism called the dangling participle has recently become more prevalen t, but not less loathsome. “Differentiating both sides with respect to x, the eq uation becomes---”is wrong, because “the equation”cannot be the subject th at does the differentiation. Write instead “differentiating both sides with respec t to x, we get the equation---,”or “Differentiation of both sides with respect to x leads to the equation---”Although the notion has gained some currency, it is absurd to claim that infor mal “we”has no proper place in mathematical exposition. Strict formality is appropriate in the statement of a theorem, and casual chatting should indeed b e banished from those parts of a paper which will be printed in italics. But fif teen consecutive pages of formality are altogether foreign to the spirit of the t wentieth century, and nearly all authors who try to sustain an impersonal digni fied text of such length succeed merely in erecting elaborate monuments to slu msiness.A sentence of the form “if P,Q”can be understood. However “if P,Q,R,S,T”is not so good, even if it can be deduced from the context that the third co mma is the one that serves the role of “then.”The reader is looking at the paper to learn something, not with a desire for mental calisthenics.Vocabularypreliminary 序,小引(名)开端的,最初的(形) eschew 避免medieval 中古的，中世纪的inflated 夸张的comprehensible 可领悟的，可了解的pinpoint 准确指出（位置）weave 插入，嵌入unobtrusivcly 无妨碍地triviality 平凡琐事barbarism 野蛮，未开化portray 写真，描写clinch 使终结rudiment 初步，基础commentary 注解，说明janitor 看守房屋者sponge 海绵dangling participle 不连结分词prevalent 流行的，盛行loathsome 可恶地absurd 荒谬的banish 排除sustain 维持，继续slumsiness 粗俗，笨拙monument 纪念碑calisthenics 柔软体操，健美体操notes1. 本课文选自美国数学会出版的小册子A mamual for authors of mathematical paper的一节，本文对准备投寄英文稿件的读者值得一读。
数学专业英语－Historical introduction of CalculusThe Two Basic Concepts of CalculusThe remarkable progress that has been made in science and technology during the last century is due in large part to the development of mathematics. That branch of mathematics known as integral and differential calculus serves as a natural and powerful tool for attacking a variety of problems that arise in phys ics,engineering,chemistry,geology,biology, and other fields including,rather recentl y,some of the social sciences.To give the reader an idea of the many different types of problems that can b e treatedby the methods of calculus,we list here a few sample questions.With what speed should a rocket be fired upward so that it never returns to e arth? What is the radius of the smallest circular disk that can cover every isos celes triangle of a given perimeter Ｌ? What volume of material is removed fr om a solid sphere of radius 2 r if a hole of redius r is drilled through the ce nter? If a strain of bacteria grows at a rate proportional to the amount present and if the population doubles in one hour,by how much will it increase at th e end of two hours? If a ten-pound force stretches an elastic spring one inch,h ow much work is required to stretch the spring one foot?These examples,chosen from various fields,illustrate some of the technical quest ions that can be answered by more or less routine applications of calculus.Calculus is more than a technical tool－it is a collection of fascinating and ex eiting idea that have interested thinking men for centuries.These ideas have to do with speed,area,volume,rate of growth,continuity,tangent line,and otherconcept s from a varicty of fields.Calculus forces us to stop and think carefully about the meanings of these concepts. Another remarkable feature of the subject is it s unifying power.Most of these ideas can be formulated so that they revolve a round two rather specialized problems of a geometric nature.We turn now to a brief description of these problems.Consider a cruve C which lies above a horizontal base line such as that show n in Fig.1. We assume this curve has the property that every vertical line inter sects it once at most.The shaded portion of the figure consists of those pointe which lie below the curve C , above the horizontal base,and between two para llel vertical segments joining C to the base.The first fundamental problem of c alculus is this: To assign a number which measures the area of this shaded re gion.Consider next a line drawn tangent to the curve,as shown in Fig.1. The second fundamental problem may be stated as follows:To assign a number which me asures the steepness of this line.Basically,calculus has to do with the precise formulation and solution of these two special problems.It enables us to define the concepts of area and tangent l ine and to calculate the area of a given region or the steepness of a given an gent line. Integral calculus deals with the problem of area while differential cal culus deals with the problem of tangents.Historical BackgroundThe birth of integral calculus occurred more than 2000 years ago when the Gr eeks attempted to determine areas by a procees which they called the method of exhaustion.The essential ideas of this ,method are very simple and can be d escribed briefly as follows:Given a region whose area is to be determined,we i nscribe in it a polygonal region which approximates the given region and whos e area we can easily compute.Then we choose another polygonal region which gives a better approximation,and we continue the process,taking polygons with more and more sides in an attempt to exhaust the given region.The method is illustrated for a scmicircular region in Fig.2. It was used successfully by Arch imedes(287－212 B.C.) to find exact formulas for the area of a circle and a fe w other special figures.The development of the method of exhaustion beyond the point to which Ar chimcdcs carried it had to wait nearly eighteen centuries until the use of algeb raic symbols and techniques became a standard part of mathematics. The eleme ntary algebra that is familiar to most high-school students today was completel y unknown in Archimedes’time,and it would have been next to impossible to extend his method to any general class of regions without some convenient w ay of expressing rather lengthy calculations in a compact and simpolified form.A slow but revolutionary change in the development of mathematical notations began in the 16th century A.D. The cumbersome system of Roman numerals was gradually displaced by the Hindu-Arabic characters used today,the symbol s “+”and “-”were introduced for the forst time,and the advantages of the decimal notation began to be recognized.During this same period,the brilliant su ccesse of the Italian mathematicians Tartaglia,Cardano and Ferrari in finding al gebraic solutions of cubic and quadratic equations stimulated a great deal of ac tivity in mathematics and encouraged the growth and acceptance of a new and superior algebraic language. With the wide spread introduction of well-chosen algebraic symbols,interest was revived in the ancient method of exhaustion and a large number of fragmentary results were discovered in the 16 th century by such pioneers as Cavalieri, Toricelli, Roberval, Fermat, Pascal, and Wallis. Fig.2. The method of exhaustion applied to a semicircular region.Gradually the method of exhaustion was transformed into the subject now calle d integral calculus,a new and powerful discipline with a large variety of applic ations, not only to geometrical problems concerned with areas and volumes but also to jproblems in other sciences. This branch of mathematics, which retaine d some of the original features of the method of exhaustion,received its bigges t impetus in the 17 th century, largely due to the efforts of Isaac Newion (16 42—1727) and Gottfried Leibniz (1646—1716), and its development continued well into the 19 th century before the subject was put on a firm mathematical basis by such men as Augustin-Louis Cauchy (1789-1857) and Bernhard Riem ann (1826-1866).Further refinements and extensions of the theory are still being carried o ut in contemporary mathematicsVocabularygeology 地质学decimal 小数，十进小数biology 生物学discipline 学科social sciences 社会科学 contemporary 现代的disk (disc) 圆盘bacteria 细菌isosceles triangle 等腰三角形 elastic 弹性的perimeter 周长 impetus 动力volume 体积 proportional to 与…成比例center 中心 inscribe 内接steepness 斜度 solid sphere 实心球method of exhaustion 穷举法 refinement 精炼，提炼polygon 多边形，多角形 cumbersome 笨重的，麻烦的polygonal 多角形fragmentary 碎片的，不完全的approximation 近似，逼近 background 背景。
数学专业英语－Operations ResearchThe start of operations research took place in a military context in the United Kingdom during World War Ⅱ, and it was quickly taken up under the name operations research (OR) in the United States. After the war it evolved in co n nection with industrial organization, and its many techniques allowed for expan ding areas of application in the United States, the United Kingdom, and in oth er industrial countries. It is, however, not easy to give a precise definition of operations research, There are three different representative definitions.According to the classical definition, due to P. M. Morse and G. E. Kimball, operations research is a scientific method of providing executives with a quanti tative basis for decisions regarding operations under their control.The second definition, due to C. W. Churchman, R. L. Ackoff, and E. L. Arn off, is as follows: operations research in the most general sense can be charact erized as the application of scientific methods, techniques, and tools to the ope rations of systems so as to provide those in control with optimum solutions to problems.As the third definition we mention the suggestion due to S. Beer: operations r esearch is the attack of modern science on problems of likelihood (accepting mischance) that arise in the management and control of men, machines, materi als, and money in their natural environments. Its special technique is to invent a strategy of control by measuring, comparing, and predicting probable behavi our through a scientific model of a situation.These three definitions have several common features. In the first place, operati ons research serves executives by providing partial observations and advice whi ch they can use in judging a situation. Second, the applicability of operations r esearch is limited to areas where scientific methods can be successfully applied. This is the reason why operations research would not be considered to extend beyond only partial observation and advice. A fundamental requirement for a scientific approach is that it must have a mathematical model whose validity c an be tested by actual data, Third, any operation should satisfy three necessary conditions in order that it may be an object of scientific approach: (1) the op eration should be defined objectively; (2) the results, consequences, and effects of its application should be objectively measurable; (3) the operation should b e capable of repetition. Fourth , operations research should aim at finding a pr actical strategy. Although operations research is based on scientific methodolog y, it does not aim at establishing general scientific assertions that are valid for all situstions.These four points are essential to any operations research, and are implicit in each of the three aforementioned definitions.On the other hand these three definitions emphasize differently some specific f eatures of operations research, according to their historical positions. In compar ion with the first definition, the second makes clearer the place where operatio ns research is applied by pointing out that it is concerned with the operations of systems, and, instead of the vague mention of quantitative basis for decision s in the first definition, it states that operations research seeks optimum solutio ns, reflecting a stage where optimum solutions were sought by applications of mathematical programming techniques. In the third definition of operations rese arch the notion of system is defined explicitly, the notion of operation is defin ed to be its special technique, and the objectives of operations research are giv en. It is clearly asserted that operations research belongs to the methodology o f applied sciences. In operations research, operations and systems are dealt wit h in their intimate interconnection. The methodology of operations research ther efore relies on an overall approach for which interdisciplinary cooperation is in dispensable and in which the operations research team plays an important role. In applying the operations research approach to the circumstances with which we are concerned, we concentrate our interest on mutual relationships among i nput and output characteristics. A black-box method by which the interrelation between input and output can be clarified without entering the actual mechanis m of the transformation yielded by the system or by its subsystems plays a fu ndamental role in operations research. The following are major phases of an o perations research project: (1) formulating the problem; (2) constructing a math ematical model to represent the system under study and deriving a solution fro m the model; (3) testing the model and the solution derived form it; (4) the i mplementation stage of establishing controls over the solution and putting it to work. It is important to construct a model of information communication in c onnection. With a mathematical model of any problem in operations research. Process of aliocation, competition, queuing, inventory, and production appear fr equently in the mathematical models of operations research.——From Encyclopedic Dictionaryof MathematicsVocabularyOperations research (OR) 运筹interdiscipline 交叉学科Executive 行政人员 interdisciplinary cooperation 交叉学科的likelihood 似然合作scientific approach 科学方法black-box method 黑箱方法methodology 方法论 implementation stage 实现阶段aforementioned 前述的queue 排队mathematical programming 数学规划Notes1. Operations Research运筹学. 运筹学是第二次世界大战期间,为解决后勤供应问题而发展起来的一门学科,它运用最优化技术去解决管理和决策问题.2. According to the classical definition, due to P. M. Morse and G. E. Kimball, operations research i s a scientific method of providing executives with a quantitive basis for decisions regarding operations un der their control.意思是:根据P. M. Morse和G. E. Kimball提出的古典定义,运筹学是一种科学方法,它提供行政人员一种定量基础,以便他们对所控的操作进行决策,这里due to是”归功于”“由…提出”之意,providing…with…for…是”提供…给…用于…”之意.3. …instead of the vague mention of quantitative basis for decisions in the first definition, it states th at …by applications of mathematical programming techniques.意思是:代替第意个定义中对于决策的定量基础那种模糊的提法,它(第二个定义)阐明了运筹学用于寻求最优解,反映了运用数学规划方法求最优解的阶段.这里reflecting至句子结束一段,属独立分词结构,用以补充说明it states that…的句子.4. The methodology of operations research therefore relies on…the operations research team plays an important role.意思是:因而运筹学的方法论依赖于…一种全面的研究,对这种研究来说,各交叉学科的合作是不可避免的,而且,在这种研究中,运筹学小组起了重要的作用.注意:前后两个which都是approach的关系代词,很容易误认为第二个which 是cooperation的关系代词,虽然这在意思上说得过去,但从语法结构上却不然.5. A black-box method by which the interrelation between input and output …plays a fundamental role in operations research.意思是:黑箱方法不需引进由系统或它的子系统所产生的变换的确切机制而能阐明输入和输出的相互关系,这种方法在运筹学中起了重要的作用,注意这一句中的主语A black-box and method和谓语plays相隔甚远.ExerciseⅠ. Answer the following questions :1. What are the necessary conditions for operation to become an object of scientific approach?2. Point out the main points the 2nd and the 3rd definitions emphasize as compared with the first defi nition.Ⅱ. 1. Translate the third definitions of OR due to S. Beer.2. Translate the following sentences into Chinese ;ⅰ) It was G. Gantor who first introduced the concept of the set as object of mathematical study.ⅱ) The definition of probability due to Laplace provoked a great deal of argument when it was applie d;ⅲ) Nowadays, we usually adopted measure theoretic foundations of probability initiated by A. N. Kol omogorov.。
学专业英语－How to Write Mathematics?How to Write Mathematics?------ Honesty is the Best PolicyThe purpose of using good mathematical language is, of course, to make the u nderstanding of the subject easy for the reader, and perhaps even pleasant. The style should be good not in the sense of flashy brilliance, but good in the se nse of perfect unobtrusiveness. The purpose is to smooth the reader’s wanted, not pedantry; understanding, not fuss.The emphasis in the preceding paragraph, while perhaps necessary, might see m to point in an undesirable direction, and I hasten to correct a possible misin terpretation. While avoiding pedantry and fuss, I do not want to avoid rigor an d precision; I believe that these aims are reconcilable. I do not mean to advise a young author to be very so slightly but very very cleverly dishonest and to gloss over difficulties. Sometimes, for instance, there may be no better way t o get a result than a cumbersome computation. In that case it is the author’s duty to carry it out, in public; the he can do to alleviate it is to extend his s ympathy to the reader by some phrase such as “unfortunately the only known proof is the following cumbersome computation.”Here is the sort of the thing I mean by less than complete honesty. At a certa in point, having proudly proved a proposition P, you feel moved to say: “Not e, however, that p does not imply q”, and then, thinking that you’ve done a good expository job, go happily on to other things. Your motives may be per fectly pure, but the reader may feel cheated just the same. If he knew all abo ut the subject, he wouldn’t be reading you; for him the nonimplication is, qui te likely, unsupported. Is it obvious? (Say so.) Will a counterexample be suppl ied later? (Promise it now.) Is it a standard present purposes irrelevant part of the literature? (Give a reference.) Or, horrible dictum, do you merely mean th at you have tried to derive q from p, you failed, and you don’t in fact know whether p implies q? (Confess immediately.) any event: take the reader into y our confidence.There is nothing wrong with often derided “obvious”and “easy to see”, b ut there are certain minimal rules to their use. Surely when you wrote that so mething was obvious, you thought it was. When, a month, or two months, or six months later, you picked up the manuscript and re-read it, did you still thi nk that something was obvious? (A few months’ripening always improves ma nuscripts.) When you explained it to a friend, or to a seminar, was the someth ing at issue accepted as obvious? (Or did someone question it and subside, mu ttering, when you reassured him? Did your assurance demonstration or intimidation?) the obvious answers to these rhetorical questions are among the rules th at should control the use of “obvious”. There is the most frequent source o f mathematical error: make that the “obvious”is true.It should go without saying that you are not setting out to hide facts from the reader: you are writing to uncover them. What I am saying now is that you should not hide the status of your statements and your attitude toward them eit her. Whenever you tell him something, tell him where it stands: this has been proved, that hasn’t, this will be proved, that won’t. Emphasize the importan t and minimize the trivial. The reason saying that they are obvious is to put t hem in proper perspecti e for the uninitiated. Even if your saying so makes an occasional reader angry at you, a good purpose is served by your telling him how you view the matter. But, of course, you must obey the rules. Don’t le t the reader down; he wants to believe in you. Pretentiousness, bluff, and conc ealment may not get caught out immediately, but most readers will soon sense that there is something wrong, and they will blame neither the facts nor them selves, but quite properly, the author. Complete honesty makes for greatest clar ity.---------Paul R.Haqlmosvocabularyflashy 一闪的 counter-example 反例unobtrusiveness 谦虚dictum 断言；格言forestall 阻止，先下手deride嘲弄anticipate 预见 subside沉静pedantry 迂腐；卖弄学问 mutter出怨言，喃喃自语fuss 小题大做 intimidation威下reconcilable 使一致的 rhetorical合符修辞学的gloss 掩饰 pretentiousness自命不凡alleviate 减轻，缓和bluff 欺骗implication 包含，含意concealment隐匿notes1. 本课文选自美国数学学会出版的小册子How to write mathematics 中Paul R.Halmos. 的文章第9节2. The purpose is smooth the reader’way, to anticipates his difficulties and to forestall them. Clarit y is what’s wanted, not pedantry; understanding, not fuss.意思是：目的是为读者扫清阅读上的障碍，即预先设想读者会遇到什么困难，并力求避免出现这类困难。
数学专业英语－(a) How to define a mathematical term?数学术语的定义和数学定理的叙述，其基本格式可归纳为似“if …then …”的格式，其他的格式一般地说可视为这一格式的延伸或变形。
如果一定语短语或定语从句，以界定被定义的词，所得定义表面上看虽不是“If ……then ……”的句型，而实际上是用“定语部分”代替了“If ”句，因此我们可以把“定语部分”写成If 句，从而又回到“If ……then ……”的句型。
至于下面将要叙述的“Let …if …then ”，“Let and assume …, If …then …”等句型，其实质也是基本句型“If ……then ……”的延伸。
有时，在定义或定理中，需要附加说明某些成份，我们还可在“if …then …”句中插入如“where …”等的句子，加以延伸（见后面例子）。
总之，绝大部分（如果不是全部的话）数学术语的定义和定理的叙述均可采用本附录中各种格式之。
（a ）How to define a mathematical term?Something something The union of A and B is defined as the set of those elements which are in A, inBor in both.The mapping , ad-bc 0, is called a Mobius transformation.Something something(or adjective) The difference A-B is defined tobe the set of all elements of A which are notin B.A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.Real numbers which are greater than zero are said to be positive.3. We something to be something.We define the intersection of A and B to be the set of those elements common to both A and B. We call real numbers that are less than zero (to be) negative numbers.4. 如果在定义某一术语之前，需要事先交代某些东西（前提），可用如下形式：, then…) be an n-tuple of real numbers. Then the set of all such n-tuples is definthe Euclidean n-space R.Let d(x,y) denote the distance between two points x and y of a set A. Then the numberD=is called the diameter of A.5．如果被定义术语，需要满足某些条件，则可用如下形式：If…, then…If the number of rows of a matrix A equals the number of its columns, thenis called a square matrix.If a function f is differentiable at every point of a domain D, then it is said to be analytic in D.6.如果需要说明被定义术语应在什么前提下，满足什么条件，则可用下面形式：is calledis said to beLetSuppose…. If…then……Let f(z) be an analytic function defined on a domain D (前提条件). If for every pair or points , and in D with , we have f( ) f( ) (直接条件)，then f(z) is called a schlicht function or is said to be schlicht i n D.7. 如果被定义术语需要满足几个条件（大前提，小前提，直接条件）则可用如下形式：supposeassumeLet…and …. If…then…is called…Let D be a domain and suppose that f(z) is analytic in D. If for every pair of points and in D with , we have f( ) f( ), then f(z) is called a schlicht function.Notes:(a) 一种形式往往可写成另一种形式。
Differential Calculus Newton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insurmountable problems could be solved by more or less routine methods.The successful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch of calculus,differential calculus.
In this article, we give sufficient conditions for controllability of some partial neutral functional differential equations with infinite delay. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. The results are obtained using the integrated semigroups theory. An application is given to illustrate our abstract result. Key words Controllability; integrated semigroup; integral solution; infinity delay
1 Introduction In this article, we establish a result about controllability to the following class of partial neutral functional differential equations with infinite delay:
0,),()(0tx
xttFtCuADxtDxt
t (1)
where the state variable(.)xtakes values in a Banach space).,(Eand the control (.)u is given in 0),,,0(2TUTL,the Banach space of admissible control functions with U a Banach space. C
is a bounded linear operator from U into E, A : D(A) ⊆ E → E is a linear operator on E, B is the phase space of functions mapping (−∞, 0] into E, which will be specified later, D is a bounded linear operator from B into E defined by BDD,)0(0
0Dis a bounded linear operator from B into E and for each x : (−∞, T ] → E, T > 0, and t ∈ [0,
T ], xt represents, as usual, the mapping from (−∞, 0] into E defined by ]0,(),()(txxt
F is an E-valued nonlinear continuous mapping on.
The problem of controllability of linear and nonlinear systems represented by ODE in finit dimensional space was extensively studied. Many authors extended the controllability concept to infinite dimensional systems in Banach space with unbounded operators. Up to now, there are a lot of works on this topic, see, for example, [4, 7, 10, 21]. There are many systems that can be written as abstract neutral evolution equations with infinite delay to study [23]. In recent years, the theory of neutral functional differential equations with infinite delay in infinite dimension was developed and it is still a field of research (see, for instance, [2, 9, 14, 15] and the references therein). Meanwhile, the controllability problem of such systems was also discussed by many mathematicians, see, for example, [5, 8]. The objective of this article is to discuss the controllability for Eq. (1), where the linear part is supposed to be non-densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We shall assume conditions that assure global existence and give the sufficient conditions for controllability of some partial neutral functional differential equations with infinite delay. The results are obtained using the integrated semigroups theory and Banach fixed point theorem. Besides, we make use of the notion of integral solution and we do not use the analytic semigroups theory.
Treating equations with infinite delay such as Eq. (1), we need to introduce the phase space B. To avoid repetitions and understand the interesting properties of the phase space, suppose that ).,(BB is a (semi)normed abstract linear space of functions mapping (−∞, 0] into E, and
satisfies the following fundamental axioms that were first introduced in [13] and widely discussed in [16]. (A) There exist a positive constant H and functions K(.), M(.):,with K continuous and M locally bounded, such that, for any and 0a,if x : (−∞, σ + a] → E, Bx and (.)xis continuous on [σ, σ+a], then, for every t in [σ,
σ+a], the following conditions hold: (i) Bxt, (ii) BtxHtx)(,which is equivalent to BH)0(or everyB
(iii) BxtMsxtKxttsB)()(sup)( (A) For the function (.)xin (A), t → xt is a B-valued continuous function for t in [σ, σ + a]. (B) The space B is complete. Throughout this article, we also assume that the operator A satisfies the Hille-Yosida condition : (H1) There exist and ，such that )(),(A and MNnAInn,:)()(sup （2）
Let A0 be the part of operator A in )(AD defined by
)(,,)(:)()(000ADxforAxxAADAxADxAD
It is well known that )()(0ADADand the operator 0A generates a strongly continuous semigroup ))((00ttTon )(AD. Recall that [19] for all)(ADx and 0t,one has )()(000ADxdssTft and xtTxsdssTAt)(0)(00
