数学专业英语论文
数学专业英语(5)
Mathematical English Dr. Xiaomin Zhang Email: zhangxiaomin@§2.5 Basic Concepts of Cartesian GeometryTEXT A The coordinate system of Cartesian geometryAs mentioned earlier, one of the applications of the integral is the calculation of area. Ordinarily we do not talk about area by itself, instead, we talk about the area of something. This means that we have certain objects (polygonal regions, circular regions, parabolic segments etc.) whose areas we wish to measure. If we hope to arrive at a treatment of area that will enable us to deal with many different kinds of objects, we must first find an effective way to describe these objects.The most primitive way of doing this is by drawing figures, as was done by the ancient Greeks. A much better way was suggested by Rena Descartes (1596-1650), who introduced the subject of analytic geometry (also known as Cartesian geometry). Descartes’ idea was torepresent geometric points by numbers. The procedure for points in a plane is this:Two perpendicular reference lines (called coordinate axes) are chosen, one horizontal (called the “x-axis”), the other vertical (the “y-axis”). Their point of intersection, denoted by O, is called the origin. On the x-axis a convenient point is chosen to the right of O and its distance form O is called the unit distance. Vertical distances along the y-axis are usually measured with the same unit distance, although sometimes it is convenient to use a different scale on the y-axis. Now each point in the plane (sometimes called the xy-plane) is assigned a pair of numbers, called its coordinates. These numbers tell us how to locate the point.Figure 2-5-1 illustrates some examples. The point with coordinates (3, 2) lies three units to the right of the y-axis and two units above the x-axis. The number 3 is called the x-coordinate of the point, 2 its y-coordinate. Points to the left of the y-axis have a negative x-coordinate; those belowthe x-axis have a negative y-coordinate. The x-coordinate of a point is sometimes called its abscissa and the y-coordinate is called its ordinate.When we write a pair of numbers such as (a, b) to represent a point, we agree that the abscissa or x-coordinate, a, is written first. For this reason, the pair (a, b) is often referred to as an ordered pair. It is clear that two ordered pairs (a, b) and (c, d) represent the same point if and only if we have a =c and b=d. Points (a, b) with both a and b positive are said to lie in the first quadrant, those with a<0 and b>0 are in the second quadrant; those with a<0 and b<0 are in the third quadrant; and those with a>0 and b<0 are in the fourth quadrant. Figure 2-5-1 shows one point in each quadrant.The procedure for points in space is similar. We take three mutually perpendicular lines in space intersecting at a point (the origin). These lines determine three mutually perpendicular planes, and each point inspace can be completely described by specifying, with appropriate regard for signs, its distances from these planes. We shall discuss three-dimensional Cartesian geometry in more detail later on; for the present we confine our attention to plane analytic geometry.NotationsRena Descartes (1596-1650)French scientific philosopher who developed a theory known as the mechanical philosophy. This philosophy was highly influential until superseded by Newton's methodology, and maintained, for example, that the universe was a plenum in which no vacuum could exist. Descartes was the first to make a graph, allowing a geometric interpretation of a mathematical function and giving his name to Cartesian coordinates (originated from Pappus’problem).Descartes believed that a system of knowledge should start from first principles and proceed mathematically to a series of deductions, reducing physics to mathematics. In Discours de la Méthode (1637), he advocated the systematic doubting of knowledge, believing as Plato that sense perception and reason deceive us and that man cannot have real knowledge of nature. The only thing that he believed he could be certain of was that he was doubting, leading to his famous phrase "Cogito ergo sum," (I think, therefore I am). From this one phrase, he derived the rest of philosophy, including the existence of God.Pappus’ problem The full enunciation of the problem is rather involved, but the most important case is to find the locus of a point such that the product of the perpendiculars on m given straight lines shall be in a constant ratio to the product of the perpendiculars on n other given straight lines. The ancients had solved this geometrically for the case m = 1, n = 1, and the case m = 1, n = 2. Pappus had further stated that, if m = n = 2, the locus is a conic, but he gave no proof; Descartes also failed to prove this by pure geometry, but he showed that the curve is represented by an equation of the second degree, that is, a conic; subsequently Newton gave an elegant solution of the problem by pure geometry.TEXT B Geometric figuresA geometric figure, such as a curve in the plane, is a collection of points satisfying one of more special conditions. By translating these conditions into expressions, involving the coordinates x and y, we obtain one or more equations which characterize the figure in question. For example, consider a circle of radius r with its center at the origin, as shown in Figure 2-5-2. Let P be an arbitrary point on this circle, and suppose P has coordinates (x, y). Then the line segment OP is the hypotenuse of a right triangle whose legs have lengths |x| and |y| and hence, by the theorem of Pythagoras,x2+y2=r2.The equation, called a Cartesian equation of the circle, is satisfied by all points (x, y) on the circle and by no others, so the equation completely characterizes the circle. This example illustrates how analytic geometry is used to reduce geometrical statements about points to analytical statements about real numbers.Throughout their historical development, calculus and analytic geometry have been intimately intertwined. New discoveries in one subject led to improvements in the other. The development of calculus and analytic geometry in this book is similar to the historical development, in that the two subjects are treated together. However, our primary purpose is to discuss calculus, Concepts from analytic geometry that are required for this purpose will be discussed as needed. Actually, only a few very elementary concepts of plane analytic geometry are required to understand the rudiments of calculus. A deeper study of analytic geometry is needed to extend the scope and applications of calculus, and this study will be carried out in later chapters using vector methods as well as the methods of calculus. Until then, all that is required form analytic geometry is a little familiarity with drawing graph of function.TEXT C Sets of points in the planeWe have already shown that there is a one-to-one correspondence between points in a plane and pairs of numbers(x, y). Certain sets of points in the plane may be of special interest. For example, we may wish to examine the set of points comprising the circumference of a certain circle, or the set of points constituting the interior of a certain triangle. One may wonder if such sets of point may be succinctly described in a compact mathematical notation.We may write{|(x, y)|y=2x} (1)to describe the set of ordered pairs (x, y), or corresponding points, such that the ordinate is equal to twice the abscissas. In effect, then, the vertical line in (1) is read “such that “. By “the graph of the set of ordered pairs” is meant the set of all points of the plane corresponding to the setof ordered pairs. The student will readily infer that the set of points constituting the graph lies on a straight line.Consider the set{(x, y)|y=x2}.Consistent with our previous interpretation, this symbol represents the set of ordered pairs (x, y) such that the ordinate is equal to the square of the abscissa. Here, the total graph comprises a simple recognizable geometrical figure, a curve known as a parabola.On the basis of these two examples, one may be tempted to believe that any arbitrarily drawn curve, which of course determines a set of points or ordered pairs, could be described succinctly by a simple equation. Unfortunately, this is not the case. For example, the broken line in figure 2-2-3 is one of such curves.Consider now the set{(x, y)|y>2x}to describe the set of points (x, y) whose ordinate is greater than twice its abscissa. In the case, our set of points constitutes not a curve, but a region of the coordinate plane.SUPPLEMENT Conic SectionThe conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone. For a plane perpendicular to the axis of the cone, a circle is produced. For a planethat is not perpendicular to the axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola. The curve produced by a plane intersecting both nappes is a hyperbola.Because of this simple geometric interpretation, the conic sections were studied by the Greeks long before their application to inverse square law orbits was known. Apollonius wrote the classic ancient work on the subject entitled On Conics. Kepler was the first to notice that planetary orbits were ellipses, and Newton was then able to derive the shape of orbits mathematically using calculus, under the assumption that gravitational force goes as the inverse square of distance. Depending on the energy of the orbiting body, orbit shapes that are any of the four types of conic sections are possible.A conic section may more formally be defined as the locus of a point P that moves in the plane of a fixed point F called the focus and a fixedline d called the conic section directrix (with F not on d) such that the ratio of the distance of P from F to its distance from d is a constant e called the eccentricity. If e=0, the conic is a circle, if 0<e<1, the conic is an ellipse, if e=1, the conic is a parabola, and if e>1, it is a hyperbola.A conic section with conic section directrix at x=0, focus at (p, 0), and eccentricity e>0 has Cartesian equationwhere p is called the focal parameter. Plugging in p givesfor an ellipse,for a parabola, andfor a hyperbola.The polar equation of a conic section with focal parameter p is given byFive points in a plane determine a conic, as do five tangent lines in a plane. This follows from the fact that a conic section is a quadratic curve, which has general formso dividing through by a to obtainleaves five constants. Five points, (x i, y i) for i=1, …, 5, therefore determine the constants uniquely. The geometric construction of a conic section from five points lying on it is called the Braikenridge-Maclaurin Construction. The explicit equation for this conic is given by the equationTwo conics that do not coincide or have an entire straight line in common cannot meet at more than four points. There is an infinite family of conics touching four lines. However, of the eleven regions into which plane division cuts the plane, only five can contain a conic section which is tangent to all four lines. Parabolas can occur in one region only (which also contains ellipses and one branch of hyperbolas), and the only closed region contains only ellipses.Let a polygon of 2n sides be inscribed in a given conic, with the sides of the polygon being termed alternately "odd" and "even" according to some definite convention. Then the n(n-2)points where an odd side meet a nonadjacent even side lie on a curve of order n-2. This fact includes Pascal’s Theorem as a special case.NotationsEllipse, parabola and hyperbolaAn ellipse is a curve that is the locus of all points in the plane the sum of whose distances r1and r2 from two fixed points F1and F2(the foci) separated by a distance of 2c is a given positive constant 2a.Problem Give similar definitions of parabola and hyperbola as above.A parabola is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus); a hyperbola is a conic section defined as the locus of all points in the plane the difference of whose distances from two fixed points (the foci) separated by a given distance is a given positive constant.Pascal’s Theorem (1640)The dual of Brianchon's theorem, discovered by B. Pascal in 1640 when he was just 16 years old. It states that, given a (not necessarily regular, or even convex) hexagon inscribed in a conic section, the three pairs of the continuations of opposite sides meet on a straight line, called the Pascal line.A very special case happens when the conic degenerates into two lines, however the theorem still holds although this particular case is usually called Pappus theorem. In 1847, Möbius published the following generalization of Pascal's theorem: if all intersectionpoints (except possibly one) of the lines prolonging two opposite sides of a (4n-2)-gon inscribed in a conic section are collinear, then the same is true for the remaining point.Brianchon's Theorem (1806)The dual of Pascal's theorem. It states that, given a hexagon circumscribed on a conic section, the lines joining opposite polygon vertices (polygon diagonals) meet in a single point.In 1847, Möbius gave a statement which generalizes Brianchon's theorem: if all lines (except possibly one) connecting two opposite vertices of a (4n-2)-gon circumscribed on a conic section meet in one point, then the same is true for the remaining line. Brianchon's theorem and Pascal's theorem are Example of dual geometric objects of Duality Principle.Duality Principle All the propositions in projective geometry occur in dual pairs which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words "point" and "line." The principle was enunciated by Gergonne (1826), A similar duality exists for reciprocation as first enunciated by Poncelet (1818).Problem 1In the following four sets, which one describes the set of points (x, y) whose ordinate is greater than twice its abscissa?A {(x, y)|y<2x}B {(x, y)|y>2x}C {(x, y)|x<2y}D {(x, y)|x<2y}Problem 2 A conic section is defined to be any of the curves of intersection of a plane with a cone, if the plane of the intersection is parallel to an element of the cone, the curve of intersection is called ____.A a circleB an ellipseC a parabolaD a hyperbola。
数学专业英语(Doc版).13
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(Doc版).Word4
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.。
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数学专业英语－The Normal DistributionWe shall begin by considering some simple continuously variable quantity like stature.We know this varies greatly from one individual to another ,and may al so expect to find certain average differences between people drawn from differ ent social classes or living in different geographical areas,etc.Let us suppose th at a socio-medical survey of a particular community has provided us with a re presentative sample of 117 males whose heights are distributed as shown in th e first and third columns of Table 1.Table 1.Distribution of stature in 117 malesWe shall assume that the original measurements were made as accurately as possible,but that they are given here only to the mearest 0.02 m (i.e.2 cm).Thu s the group labeled “1.66”contains all those men whose true measurements were between 1065 and 1067 m.One si biable to run into trouble if the exact methods of recording the measurements and grouping them are not specified ex actly.In the example just given the mid-point of the interval labeled”1.66”m. But suppose that the original readings were made only to the nearest 0.01 m (i.e. 1 cm )and then “rounded up “to the nearest multiple of 0.02 m.We sho uld then have “1.65”, which covers the range 1.645 to 1.655,included with “1.66”.The interval “1.66”would then contain all measurements lying betwe en 1.645 m and 1.665 m .for which the mid-point is 1.655 m. The difference of 5 mm from the supposed value of 1.66 m could lead to serious inaccuracy in certain types of investigation.A convenient visual way of presenting such data is shown in fig. 1, in which the area of e ach rectangle is ,on the scale used, equal to the observed proportion or percentage of indiv iduals whose height falls in the corresponding group.The total area covered by all the recta ngles therefore adds up to unity or 100per cent .This diagram is called a histogram.It is ea sily constructed when ,sa here ,all the groups are of the same width.It is also easily adapt ed to the case when the intervals are uneqal, provided we remember that the areas of the rec tangles must be proportional to the numbers of units concerned.If, for example, we wished to group togcther the entries for the three groups 1.80,1.82 and 1.84 m,totaling 7 individuals or 6 per cent of the total,then we should need a rectangle whose base covered 3working grou ps on the horizontal scale but whose height was only 2 units on the vertical scale shown in the diagram.In this way we can make allowance for unequal grouping intervals ,but it is usua lly less troublesome if we can manageto keep them all the same width.In some books histogram s are drawn so that the area of each rectangle is equal to the actual number (instend of the proportion) of individuals in the corresponding group.It is better, however, to use proport ions, sa different histograms can then be compared directly.----------------------------------------------The general appearance of the rectangles in Fig.1 is quite striking ,especially the tall hump in the centre and the rapidly falling tails on each side.There are certain minor irregularities in the pattern, and these would, in general ,be more ronounced if the size of the sample were smaller. Conversely, weth larger sa mples we usually find that the set of rectangles presents a more regular appearance. This suggests that if we had a very large number of measurements ,the ultimate shape of the picture for a suitably small wi dth of rectangle would be something very like a smooth curve,Such a curve could be regarded as represe nting the true ,theoretical or ideal distribution of heights in a very(or ,better,infinitely)large population of individuals.What sort of ideal curve can we expect ? There are seveala theoretical reasons for expecting the so-call ed Gaussiao or “normal “curve to turn up in practice;and it is an empirical fact that such a curve lften describes with sufficient accuracy the shape of histograms based on large numbers of obscrvations. More over,the normal curve is one of the easiest to handle theoretically,and it leads to types of statistical analy sis that can be carried out with a minimum amount of computation. Hence the central importance of this distribution in statistical work .The actual mathematical equation of the normal curve is where u is the mean or average value and is t he standard deviation, which is a measure of the concentration of frequency about the mean. More will b e said about and later .The ideal variable x may take any value from to .However ,some real measureme nts,like stature, may be essentially positive. But if small values are very rare ,the ideal normal curve ma y be a sufficiently close approximation. Those readers who are anxious to avoid as much algebraic mani pulation as possible can be reassured by the promise that no derect use will be made in this book of th e equation shown. Most of the practical numerical calculations to which it leads are fairly simple.Fig. 1 shows a normal curve, with its typocal symmetrical bell shape , fitted by suitable methods to the data embodied in the rectangles. This is not to say that the fitted curve is actually the t rue, ideal one t o which the histogram approxime.tes; it is merely the best approximation we can find.The mormal curve used above is the curve we have chosen to represent the frequency distribution of st ature for thr ideal or infinitely large population. This ideal poplation should be contrasted with the limite d sample of obsrever. Values that turns up on any occasion when we make actual measurements in the r eal world. In the survey mentioned above we had a sample of 117 men .If the community were sufficie ntly large for us to collect several samples of this size, we should find that few if any of the correspon ding histograms were exactly the same ,although they might all be taken as illustrating the underlying fre quency distribution. The differences between such histograms constitute what we call sampling variation, and this becomes more prominent at the size of sample decreases.VocabularySocio-medical survey 社会医疗调查表 visual 可见的。
数学专业英语(Doc版).21
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）。
数学专业英语(Doc版).19
数学专业英语－ProbabilityThe mathematics to which our youngsters are exposed at school is. With rare exceptions, based on the classical yes-or-no, right-or-wrong type of logic. It no rmally doesn’t include one word about probability as a mode of reasoning or as a basis for comparing several alternative conclusions. Geometry, for instance, is strictly devoted to the “if-then”type of reasoning and so to the notion (i dea) that any statement is either correct or incorrect.However, it has been remarked that life is an almost continuous experience o f having to draw conclusions from insufficient evidence, and this is what we h ave to do when we make the trivial decision as to whether or not to carry an umbrella when we leave home for work. This is what a great industry has to do when it decides whether or not to put $50000000 into a new plant abroad. In none of these case and indeed, in practically no other case that you can s uggest, can one proceed by saying:”I know that A, B, C, etc. are completely and reliably true, and therefore the inevitable conclusion is~~”For there is a nother mode of reasoning, which does not say: This statement is correct, and i ts opposite is completely false.”But which say: There are various alternative possibilities. No one of these is certainly correct and true, and no one certainl y incorrect and false. There are varying degrees of plausibility—of probability —for all these alternatives. I can help you understand how these plausibility’s compare; I can also tell you reliable my advice is.”This is the kind of logic, which is developed in the theory of probability. Th is theory deals with not two truth-values—correct or false—but with all the in intermediate truth values: almost certainly true, very probably true, possibly tr ue, unlikely, very unlikely, etc. Being a precise quantities theory, it does not u se phrases such as those just given, but calculates for any question under stud y the numerical probability that it is true. If the probability has the value of 1, the answer is an unqualified “yes”or certainty. If it is zero (0), the answer is an unqualified “no”i.e. it is false or impossible. If the probability is a h alf (0.5), then the chances are even that the question has an affirmative answer. If the probability is tenth (0.1), then the chances are only 1 in 10 that the a nswer is “yes.”It is a remarkable fact that one’s intuition is often not very good at csunati ng answers to probability problems. For ex ample, how many persons must the re are at least two persons in the room with the same birthday (born on the s ame day of the month)? Remembering that there are 356 separate birthdays po ssible, some persons estimate that there would have to be 50, or even 100, pe rsons in the room to make the odds better than even. The answer, in fact, is t hat the odds are better than eight to one that at least two will have the same birthday. Let us consider one more example: Everyone is interested in polls, w hich involve estimating the opinions of a large group (say all those who vote)by determining the opinions of a sample. In statistics the whole group in que stion is called the “universe”or “population”. Now suppose you want to c onsult a large enough sample to reflect the whole population with at least 98% precision (accuracy) in 99out of a hundred instances: how large does this ver y reliable sample have to be? If the population numbers 200 persons, then the sample must include 105 persons, or more than half the whole population. Bu t suppose the population consists of 10,000 persons, or 100,000 persons? In th e case of 10,000 persons, or 1000,000 person? In the case of 10,000 persons, a sample, to have the stated reliability, would have to consist of 213 persons: the sample increases by only 108 when the population increases by 9800. And if you add 90000 more to the population, so that it now numbers 100000, yo u have to add only 4 to the sample. The less credible this seems to you, the more strongly I make the point that it is better to depend on the theory of pr obability rather than on intuition.Although the subject started out (began) in the seventeenth century with game s of chance such as dice and cards, it soon became clear that it had important applications to other fields of activity. In the eighteenth century Laplace laid the foundations for a theory of errors, and Gauss later develop this into a real working tool for all experimenters and observers. Any measurement or set of measurement is necessarily is necessarily inexact; and it is a matter of the hig hest importance to know how to take a lot of necessarily discordant data, com bine them in the best possible way, and produce in addition some useful estim ate of the dependability of the results. Other more modern fields of application are: in life insurance; telephone traffic problems; information and communicati on theory; game theory, with applications to all forms of competition, includin g business international politics and war; modern statistical theories, both for th e efficient design of experiments and for the interpretation of the results of ex periments; decision theories, which aid us in making judgments; probability the ories for the process by which we learn, and many more.----Weaver, W.VocabularyProbability 概率论 permutation 置换Plausibility 似乎合理 binomial coefficient 二次式系数Affirmative 肯定的generating function 母函数Estimate 估计 even 事件Discordant 不一致的information and communication theoryCommunication theory 通讯理论信息与通讯论Decision theory 决策论 game theory 对策论，博弈论Notes1. Geometry, for example, is strictly devoted to the “if—then”type of reasoning and so to the noti on (idea) that any statement is either correct or incorrect.意思是：例如几何学就是严格地属于那种“如果，则”的推理类型，所以它也就属于那种对任何陈述要么是对的要么是不对的概念范围。
数学专业英语
学号：10901040201 姓名：曹菁 lg LinearA ebraNon-trivial linear combination. This is a contradiction and proves 1).Part 2) follows from 1) because dim(V)=n.Exercise Let A=0110⎛⎫ ⎪-⎝⎭()2R ∈. Find an invertible 2C C ∈ such that 1c AC - is diagonal.Show that C cannot be selected in 2R .Find the characteristic polynomial of A .Exercise Suppose V is a 3-dimensional vector space and :f V V → is an endomorphism with ()()3f CP x x λ=-.Show that ()f I λ- has characteristic polynomial and is thus anilpotent endomorphism. Show there is a basis for V so that the matrix representingis 001001λλλ⎛⎫ ⎪ ⎪ ⎪⎝⎭ ,001000λλλ⎛⎫ ⎪ ⎪ ⎪⎝⎭or.000000λλλ⎛⎫ ⎪ ⎪ ⎪⎝⎭. We could continue and finally give an ad hoc proof of the Jordan canonical form,but in this chapter we prefer to press on to inner product spaces.The Jordan form will be developed in Chapter 6 as of the general theory of finitely generated modules over Euclidean domains.The next section is included only as a convenient reference.This section should be just skimmed or omitted entirely.It is unnecessary for the rest of this chapter,and is not properly part of the flow of the chapter.The basic facts of Jordan form are summarized here simply for reference.The statement that a square matrix B over a field F is a Jordanblock means that F λ∃∈Such that B is a lower triangular matrix of the form 0101B λλλ⎛⎫⎪ ⎪⎪= ⎪ ⎪⎪⎝⎭.B gives a homomorphism :m m g F F →with ()m m g e e λ= and ()1i i i e e e λ+=+ for 1i m ≤<.Note that ()()mB CP x x m =-and so λ is the only eigenvalue of B ,and B satisfies its characteristic polynomial,i.e.,()0B CP B =.Definition A matrix n D F ∈ is in Jordan if ∃Jordan blocks i ni B F ∈ such that12000000000000000000000t B B D B ⎛⎫⎪⎪⎪= ⎪⎪ ⎪⎝⎭.Suppose D is of this form and i ni B F ∈has eigenvalue i λ.Then1..t n n n += and ()()()11..n ntD t CP x x x λλ=--.Note that a diagonal matrix is a special caseof Jordan form. D is a diagonal matrix iff each i n ,iff each Jordan block is 11⨯ a matrix. Fundamentals of StatisticsThe preceding chapter was mainly concerned with the theory of probability,including distribution theory. In practice researchers have to find methods to choose among distributions and to estimate distribution parameters from real data. The subject of sampling brings us now to the theory of statistics. Whereas probability assumes the distributions are known, statistics attempts to make, inferences from actual dataHere we sample from the distribution of a population, say the change in the Exchange rate,to make inferences about the population.The questions are ,what Is the best distribution for this random variable and what are the best parameters for This distribution ? Risk measurement, however, typically deals with large numbers of random variable. So, we also want to characterize the relationships between the risk factors to which the portfolio is exposed. For example, do we observe that movements in the yen/dollar rate are correlated with the dollar/euro rate? Another type of problem a to develop decision rule to rest some hypotheses, for instance Wherher the volatility remains stable over time.These examples illustrate two important problems in statistical inference, i.e, estimation and tests of hypotheses. With estimation, we wish to estimate the value of an unknown parameter from ample data. With tests of hypotheses, we wish to Verify a conjecture about the data.This examples reviews the fundamental tools of statistics theory for risk man- agers. Section 3.1 discusses s the sampling of real data and the constructionof returns. The problem of parameter estimation is presented in Section 3.2. Section 3.3 then turns to regression analysis's,summarizing important results as Well as common pitfalls in their interpretation. REAL DATATo start with an example, let us say that we observe movement in the daily Yen/dogar exchange rate and wish to characterize the distribution of tomorrow's Exchange rateThe risk manager ’s job is ro assess the range of potential gains and lasses on atrader's position . He or she observes a sequence a of past spot prices 0,1,...t S S S from which we have to infer the distribution of tomorrow's price, 1t S +. Measuring ReturnsThe truly random component in tomorrow ’s prices is not its level, but rather its Change relative to today's prices. We measure the relative rate of change in the spot price:()11/t t t t r S S S --=-Alternatively, we could construct the logarithm of the price ratio:[]1ln t t t R S S -=-which is equivalent to using continuous instead of discrete compounding. This is also()[]11ln 1/ln 1t t t t t R S S S r --⎡⎤=+-=+⎣⎦The return defined so far is the capital appreciation return, which ignores the income payment on the asset. Define the dividend or coupon as Dt.In the case of An exchange rate position, this is the interest payment in the foreign currency over the holding period. The rural mum av the asset is()11/TOT t t t t t r S D S S --=+-When the horizon is very short,the income return is typically very small compared to the capital appreciation return.the next question is whether the sequence of variables R can be viewed as independent observations. if so ,one could hypothesize,for instance,that the random variables are drawn from a normal distribution N(u,).we could then proceed to estimate u and --from the data and use this information to create a distribution for tomorrow;s spot price change.Independent observations have the very nice property that their joint distribution is the product of their marginal distribution ,which considerably simplifies the analysis.the obvious question is whether this assumption is a workable approximation. in fact,thee are good economic reasons to believe that rates of change on financial prices are close to independent .the hypothesis of efficient markets postulates that current prices convey all relevant information about the asset. if so,any change in the asset price must be due to news,or events which are by definition impossible to forecast(otherwise,it would not be news).this implies that changes in prices are unpredictable and,hence,satisfy our definition of independent random variables.conditional distribution of returns only on current prices, and not on the previous history of prices .if so,technical analysis must be a fruitless exercise. technical analysts try to forecast price movements from past price patterns.If in addition the distribution of return is constant over time, the variables are said to be independently and identically distributed(i.i.d.).so,we could consider that observations RT areindependent draws from the same distribution ()2,N u σ.Later, we will consider deviations from this basic model,distributions of financial returns typically display fat tails. also, variances are not constant and display some persistence; expected returns can also slightly vary over time. Time AggregationIt is often necessary to translate parameters over a given horizon to anther horizon . Forexample, we may have raw data for daily returns , from which we compute a daily volatility that we want to extend to a monthly volatility .Returns can be easily related across time when we use the log of the price ratio, because the log of a product is the sum of the logs of the individual terms.The two-day return, for example, can be decomposed as[]()[][]02211012210112ln //ln /ln /R S S S S S S S S R R =⨯=+=+This decomposition is only approximate if we use discrete returns, however. The expected return and variance are then ()()()020112E R E R E R =+ and()()()()020********,V R V R V R Cov R R =++.Assuming returns are uncorrelated and haveidentical distributions across days, we have ()()02012E R E R =and ()()02012V R V R =. Generalizing over T days, we can relate the moments of the T-day returns T R to those of the 1-day returns:1R()()1T E R E R T = ()()1T V R V R T =Expressed in terms of volatility, this yields the square root of time rule:()()1T SD R SD R T =KEY CONCEPTWhen successive returns are uncorrelated, the volatility increase as the horizon extends following the square root of time .More generally, the variance can be added up from different values across different periods . For instance, the variance over the next year can be computed as the average monthly variance over the first three months, multiplied by 3, plus the average variance over the last nine months, multiplied by 9 .This type of analysis is routinely used to construct a term structure of implied volatilities , which are derived from option data for different maturities .It should be emphasized that this holds only if returns have constant parameters across time and are uncorrelated . When there is non-zero correlation across days, the two-day variance is()()()()()()21111221V R V R V R V R V R ρρ=++=+Because we are considering correlations in the time series of the same variable ,P is called the autocorrelation coefficient ,or the serial autocorrelation coefficient .A positive value for P implies that a movement in one direction in one day is likely to be followed by another movement in the same direction the next day. A positive autocorrelation signals the existence of a trend. In this case, Equation(3.8)shows that the two-day variance is greater than the one obtained by the square root of time rule.A negative value for P implies that a movement in one direction in one day is likely to be followed by a movement in the other direction in one day is likely to be followed by a movement in the other direction the next day .So,prices tend to revert back to a mean value .A negative autocorrelation signals译文：非线性组合，这是一个矛盾和证明1)第2部分)遵循从1)因为V 的秩=n.例如：矩阵A=0110⎛⎫ ⎪-⎝⎭的秩是2，找到一个可逆，2C C ∈,比如1C AC -是对角线。
数学专业英语
integrable function the actual numerical value of the integral can only be estimated . This is usually done by approximating the integrand above and below by step function or by other simple functions whose integrals can be evaluated exactly , Then the comparison theorem is used to obtain corresponding approximations for the integral of the function in question.
姓名：张楚材
学号：20093970
From the definition of the integral , it is possible to deduce 从积分的定义中，可以推导出它的以下属性， 在 1.27
the following properties . Proofs are given in Section 1.27 THEOREM 1. LINEARITY WITH RESPECT TO THE
a k 1
k 1
a
定理 2.可加对集成。如果以下三个积分中的两个存在，
THEOREM 2. ADDITIVITY WITH PERSPECT TO INTERVAL OF INTEGRATION . If two of the following three integrals exist , the third also exist , and we have
数学专业英语(Doc版).9
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(吴炯圻)
New Words & Expressions:algebra 代数学geometrical 几何的algebraic 代数的identity 恒等式arithmetic 算术, 算术的measure 测量，测度axiom 公理numerical 数值的, 数字的conception 概念，观点operation 运算constant 常数postulate 公设logical deduction 逻辑推理proposition 命题division 除，除法subtraction 减，减法formula 公式term 项，术语trigonometry 三角学variable 变化的，变量2.1 数学、方程与比例Mathematics, Equation and Ratio4Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches.1－A What is mathematics数学来源于人类的社会实践，比如工农业生产，商业活动，军事行动和科学技术研究。
And in turn, mathematics serves the practice and plays a great role in all fields. No modern scientific and technological branches could be regularly developed without the application of mathematics.反过来，数学服务于实践，并在各个领域中起着非常重要的作用。
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1 数学专业英语论文 Security of Computer Network System ---------------------------------------------------------------------------------------
课程：数学专业英语 Security of Computer Network System Abstract: This paper discussed the secure and dependable problem about the computer network system. On some aspects: the importance of network security, basic theory, function, and the method of solving a 2
problem, etc. Good views for solving the problem are put forward. It strengthens people’s consciousness on network security.
Key words: Computer network Virtual private network Encryption techniques Firewall
Introduction: Along with the computer network technology development, the network security and the reliability have become the question of common interest by all users. The people all hoped their own network system can move reliably, not external intruder disturbance and destruction. Therefore solves the network security and the reliable problem carefully, is a guarantee the network normal operation’s premise and safeguard.
First, the importance of the network security. With the informationization developing fast today, the computer network obtained the widespread application, but along with the network information transmission capacity growing faster, some organizations and departments benefit the speedup with the service operation in the network, while, the data has also suffered to extent attack and destruction. The aggressor may intercept the information in the network, steals the user’s password, the database information; also may tamper with the database content, the forge user’s status, denies own signature. And what is more, the aggressor may delete the database content, the destroy node, releases computer virus and so on. This cause data security and own benefit have received the serious threat.
According to American FBI (US Federal Bureau of Investigation) investigation, the network security creates the economic loss surpasses 17,000,000,000 dollars every year.75% corporation report finance loss is because the computer system security problem creates. More than 50% safe threat come from inside. But only 59% loss could be possible estimate. In China, the economic loss amount in view of financial domain and the bank, negotiable securities computer system security problems creates has reached as high as several hundred million Yuan, also sometimes occurs in view of other profession network security threat.
Thus it can be seen, regardless of is the mean attack, or unconscious disoperation, will all be able to bring the inestimable loss to the system. Therefore, the computer network must have the enough strong security measure. Regardless of is in the local area network or in WAN, the network security measure should be Omni-directional in view of each kind of different threat and the vulnerability, so that it can guarantee the network information’s secrecy, the integrity and the usability. 3
Second, network security rationale. International Standardization Organization (ISO) once suggested the computer security the definition was: “The computer system must protect its hardware, the data not accidentally or reveals intentionally, the change and the destruction.” In order to help the computer user discrimination and the solution computer network security problem, the American Department of Defense announced “the orange peel book” (orange book, official name is “credible computer system standard appraisal criterion”), has carried on the stipulation to the multiuser computer system security rank division.
The orange peel book from low to high divides into the computer security four kinds of seven levels: D1, C1, C2, B1, B2, B3, A1.Above all,D1 level does not have the lowest safety margin rank, C1 and the C2 level has the lowest safety margin rank, B1 and the B2 level has the medium safekeeping of security ability rank, B3 and A1 belongs to the highest security rating.
In the network concrete design process, it should act according to each technology standard, the equipment type, the performance requirement as well as the funds which in the network overall plan proposed and so on, the overall evaluation determines one quite reasonably, the performance high network security rank, thus realization network security and reliability.
Third, the network security should have function. In order to adapt the information technology development well, the computer network application system must have following function: