Sheaves in Topology. A Taste of Topology. On Sums of Sixteen Biquadrates. The Four Pillars of Geometry. Mathematical Masterpieces. Further Chronicles by the Explorers. Reading, Writing and Proving. A Closer Look at Mathematics. The Fano Conference, September October 5, Groups St Andrews in Oxford. Proofs that Really Count. Set Theory. Boolean-Valued Models and Independence Proofs, third edition. Chaotic Dynamical Systems: An Introduction. Applied Quaternionic Analysis. Defining the Science of Stochastics. Artinian Modules over Group Rings.
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A Case Studies Approach. Introduction to Hodge Theory. This old problem is often referred to as Fermats problem for Torricelli, and the point F is often called the Fermat point of the trian- gle. Pierre Fermat gave the problem to Evangelista Torricelli , and Torricelli solved it in several ways. It is also known as Steiners problem. If ABC contains an angle of or more, the Fermat point is the vertex of this obtuse angle, so we will only consider triangles in which each angle measures less than There is a simple way to locate the Fermat point of such a triangle.
Con- struct equilateral triangles on the sides of triangle ABC; and join each ver- tex of triangle ABC to the exterior vertex of the opposite equilateral tri- angle. Those three lines intersect at the Fermat point F, as illustrated in Figure 6. Why does this work? The following proof [Niven, ] uses rotational isometry. This last sum will be a minimum when P and P 0 both lie on the line C 0 C because C 0 is the image of A under the rotation, its position does not depend on P. Since the choice of which side of the tri- angle to rotate was arbitrary, P must also lie on B 0 B and A0 A.
In what case does equality hold? This inequality is known in the literature e. Weitzenbock in Mathematische Zeitschrift. Many ana- lytical proofs of the inequality are known. There is a geometrical interpretation of this inequality that seems p to have been overlooked. That is, the sum of the areas Ka ; Kb , and Kc of the three equilateral triangles shaded gray in Figure 6. We now present a purely geometric proof [Alsina and Nelsen, ] of Weitzenbocks inequality using the Fermat point of the original triangle. As we saw in Section 6. When each of the angles of the triangle is smaller than , the point F is the point of intersection of the lines connecting the vertices A, B, and C to the vertices of equilateral triangles constructed outwardly on the sides of the triangle.
When one of the vertices of triangle ABC measures or more, then that vertex is the Fermat point. We first consider the case where each angle of the triangle is less than Let x; y, and z denote the lengths of the line segments joining the Fermat point F to the vertices, as illustrated in Figure 6. The same is true of the other triangles sharing the vertex F, and hence. When one angle say C measures or more, then, as illustrated in Figure 6. It follows from our proof that we have equality in Weitzenbocks inequal- ity if and only if x D y D z, so the three triangles with a common vertex at F are congruent and hence a D b D c, i.
The relationship in 6. The other two inequalities are established similarly, and hence from 6. When one angle say C measures or more, we have z D 0, x D b, and y D a. We refine the inequality in 6. Of all possible line segments passing through A with endpoints on the two circles, which is the longest? Those two squares and the one with area c 2 are the squares of the lengths of the sides of a right triangle, as seen in Figure 6.
We have equality if and only if sin C D 1 and cos C D 0, i. As a bonus, we obtain the law of cosines by simplifying the right-hand side of 6. The Pythagorean inequality and the law of cosines Application 6. Inequalities derived from equalities Finding bounds for a term in an equality is a natural method for deriving inequalities. To illustrate this, consider the following problem: If a; b, and c are the side lengths of a triangle, can a2 , b 2 , and c 2 be the side lengths of another triangle?
In general the answer is no. If ABC is a right triangle with hypotenuse c, then c 2 D a2 C b 2 , so a2 , b 2 , and c 2 do not determine a triangle. However, for acute triangles, the answer is yes, as a consequence of the Pythagorean inequality.
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As a second example, we use the law of cosines to prove Theorem 6. From the law of cosines, we have cosA D. In Applications 7. There is a geometric interpretation of the inequality in Theorem 6. In triangle ABC with circumradius R, let u; v, and w denote the distances from the circumcenter to the sides, as illustrated in Figure 6.
Hence we have proved Corollary 6. In Challenge 2. Using Figure 6.
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Hint: See Figure 4. Let Ki ; i D 1; 2; 3; 4; 5 denote the areas of the five regular pentagons constructed outwardly on the five sides of the given pentagon. Hint: Use Herons formula. Let a; b, and c be the side lengths of a triangle with the property that for any positive integer n, an , b n , and c n can also be the side lengths of a triangle. Prove that the original triangle is isosceles [Gelca and Andreescu, ].
CHAPTER 7 Employing non-isometric transformations In the two previous chapters, we examined how reflections and rotations isometric transformations may be used in a visual approach to inequalities. Non-isometric transformationstransformations that do not necessarily pre- serve lengthsconstitute an interesting class of mappings for proving some inequalities.
We consider three types of non-isometric transformations in this chapter: similarity of figures, measure-preserving transformations, and pro- jections. The first preserves shapes, but changes measures by a given factor or its powers while the others may change shapes of figures but preserve other properties. Prove that. Trigonometric solutions by Mordell and Barrow appeared in [Mordell and Barrow, ]; the proofs, however, were not elementary. In fact, no simple and elementary proof of what had become known as the Erdos-Mordell the- orem was known as late as [Steensholt, ].
Since then a variety of proofs have appeared, each in some sense simpler or more elementary than the preceding ones.
In Kazarinoff published a proof [Kazarinoff, ] based upon a theorem in Pappus of Alexandrias Mathematical Collection;. Proofs using area inequalities appeared in [Komornik, ] and [Dergiades, ]. Proofs employing Ptolemys the- orem appeared in [Avez, ] and [Lee, ]. The following visual proof appeared in [Alsina and Nelsen, b]. In Figure 7. Using similar figures, we construct a trapezoid in Figure 7. Our proof applies to acute triangles; when the triangle is obtuse, an analogous construction using similar triangles yields the same inequalities.
Lemma 7. For the triangle ABC in Figure 7. The dashed line segment in Figure 7. The other two inequalities are established analogously. We should note that the object in Figure 7. We now prove The Erdos-Mordell Theorem 7. If O is a point within a triangle ABC whose distances to the sides are u, v, and w and whose distances to the vertices are x, y, and z, then. From Lemma 7. But the AM-GM inequality insures that the coefficients of u; v, and w are each at least 2, from which the desired result follows. Note 1. The three inequalities in Lemma 7. This follows from the observation that the trapezoid in Figure 7.
The coefficients of u; v, and w in 7. Note 2. Many other inequalities relating x; y, and z to u; v, and w can be derived. In triangle ABC in Figure 7. In the proof of Lemma 4. As a consequence we have Theorem 7. Inequalities similar to 7. Examples [Op- penheim, ; Ehret, ] are the following: Corollary 7. From Figure 7. But K D. Applying 7. Now apply 7. Finally, 7. Application 7. Since r D x sin. Substituting cos A D 2 cos2. Many similar inequalities can be established, see Challenge 7.
Carnots theorem There are instances where the proof of an inequality yields as a special case the proof of an equality. Such is the case with Lemma 7. In that case we obtain a proof of the following theorem, due to Lazare Nicolas Marguerite Carnot Carnots Theorem 7. In any acute triangle, the sum of the distances from the circumcenter to the sides is equal to the sum of the inradius and circum- radius. As we observed in Note 1 following the proof of the Erdos-Mordell theorem, when O is the circumcenter the inequalities in Lemma 7.
The theorem can be extended to include obtuse triangles by using signed distances from O to the sides; see [Honsberger, ]. In any triangle, the distance from a vertex to a point on the opposite side is less than the longest side of the triangle. Duplicating the triangle to form the parallelogram in Figure 7.
Draw lines through P parallel to the sides of the triangle, partitioning it into three parallelograms and three triangles shaded gray in Figure 7. Hence, using Lemma 7. The Cauchy-Schwarz inequality Integral versions of this inequality were published by Victor Yacovlevich Bunyakovski in and Hermann Amandus Schwarz in Today the inequality is known either as the Cauchy-Schwarz inequality or the Cauchy-Bunyakovski- Schwarz inequality.
One of the simplest algebraic proofs of 7. This equality is easily established by algebra, and since. Diophantus used his identity to show that the product of any two integers that can be written as a sum of squares of integers is again a sum of integer squares. For a visual proof of Diophantus identity using non-isometric transformations, see [Nelsen, ]. We now present three geometric proofs and another algebraic p proof of 7. Proof 1 [Nelsen, a].
Since the area of a parallelogram with given sides is less than or equal to the area of a rectangle with the same given sides, the area of the rectangle in Figure 7. Proof 2 [Alsina, a]. In this proof see Figure 7. The third proof uses vector notation.
Let a D. This proof uses vectors and a projection, along with the Pythagorean comparison from Chapter 1. Proof 3. Proof 4. We conclude this section with another algebraic proof, showing that the Cauchy-Schwarz inequality for n numbers is a consequence of the AM-GM inequality for two numbers. Given, as above, a D. These vectors each have length 1. For each i between 1 and n, the AM-GM inequality yields. What are the largest and smallest values of f. From the Cauchy-Schwarz inequality, we have. Consequently, p the ex- treme values of f. First, note that Xn Xn Xn. The sample correlation coefficient In statistics, the sample correlation coefficient rxy is a measure of the strength and direction of a linear relationship between the x- and y-values in a set of n ordered pairs f.
Letting xN D. Inequalities and names Since there is ample evidence that Bunyakovskis work preceded that of Schwarz, calling 7. However, by modern standards, both Bun- yakovski and Schwarz should consider themselves fortunate to have their names associated with such an important result. It is rare today to receive much credit for merely finding a continuous analog of a discrete inequality or vice versa. The names of many inequalities are merely descriptive e. However, there is no rule governing that namingsometimes it is for the one who first discovered the inequality, but sometimes it is for the one who crafted its final form or the one responsible for an important application [Steele, ].
Aczels inequality We will use a visual argument to establish the n D 2 case, and then proceed to the general case. To establish 7. The Neuberg-Pedoe inequality. In , while doing some mathematics during an air raid on Southampton, the British geometer Dan Pedoe discovered the following re- markable two-triangle inequality: given two triangles with sides of length ai ; bi ; ci and area Ki ; i D 1; 2, we have. Later Pedoe learned that Jean Baptiste Joseph Neuberg had discovered the inequality in the 19th century, but had not proved that equality implies the similarity of the triangles.
An inequality of this sort is unusual in the sense that it involves two unrelated triangles. As noted by L. Carlitz [Carlitz, ], the Neuberg- Pedoe inequality is a consequence of Aczels inequalityp 2 p 7. What is the minimum value of ab C bc C ca? Second solution: Since. Which solution if either is correct? Hint: Use Carnots Theorem 7.
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Prove that the two inequal- ities are actually equivalent by proving the converse. In Corollary 6. CHAPTER 8 Employing graphs of functions Many simple properties of real-valued functions, such as boundedness, mono- tonicity, convexity, and the Lipschitz condition, can be expressed in terms of inequalities. Consequently there are visual representations of many of them, some of which are familiar.
In this chapter we introduce the idea of a moving frame to illustrate some of these properties, and then use them to establish additional inequalities. We also investigate the role played by the convexity or concavity of a function in establishing functional inequalities. We con- clude the chapter by examining inequalities in which areas under graphs of functions represent numbers.
A function f W S! Visually, this means that the entire graph of y D f. A moving frame is like a window, in this case with height M m units and some convenient width, such that, as the frame is moved horizontally with the opening always between the lines y D m and y D M , we see the graph inside the frame, never in the opaque regions shaded gray in Figure 8. T is nondecreasing if f satisfies the inequality f. Whenever one of these condi- tions holds throughout S, we say that f is monotonic or monotone.
A moving frame to show that a given function f is, say, nondecreasing, consists of a window formed by a pair of axes in which the second and fourth quadrants are opaque, and which moves along the curve with its origin on the graph of the function. The function is nondecreasing if one always sees the graph in the frame, here the first and third quadrants, as illustrated in Figure 8. Later in this chapter we will encounter a similar frame, transparent be- tween lines parallel to y D M x, for illustrating the Lipschitz condition.
Application 8. An inequality of L. Bankoff [Mitrinovic et al. Hence we have E3 n3. Now let f. Note that f. We have 2x sin. We have equality for triangular prisms with edges all the same length. The Cauchy-Schwarz inequality revisited In Section 7. Define a function f with domain. The graph of y D f. The statistical regression line In Application 7.
This line is called the least squares line, or the regression line of y on x. Our task is to find the numbers m and c in terms of the given x- and y-values that minimize Xn Q. It will be convenient to write, as we did in Application 7. To minimize Q m,c we will find the value of m which minimizes the first P term q. Now Xn Xn Xn Xn q. In terms of the correlation coefficient rxy from Application 7. Inequalities in convex polyhedra In , Leonhard Euler discovered the remarkable formula F C V E D 2, where F, V, and E are, respectively, the number of faces, vertices, and edges in a convex polyhedron.
Euler and this formula were honored in when Switzerland, the country of Eulers birth, issued a postage stamp commemorating the th anniversary of that event. See Figure 8. From these simple observations, many additional relationships both equal- ities and inequalities have been proved about F, V, and E [Euler, ; Fejes Toth, , ; Catalan, ; etc. In this section we will derive a few visually. Since the endpoints of the segment are. Thus we have proved. Theorem 8. For adults, h is fixed, so that BMI is propor- tional to weight which for many of us is not constant. The second inequality is equivalent to Leonardo da Vinci.
This definition captures the key idea of continuity by means of a rela- tionship between local inequalities, local in the sense that they hold in a window centered at. This can be visualized as illustrated in Fig- ure 8. We fix the point. The window cannot be viewed as a moving frame, because its shape may depend on the point. In Figure 8.
Figure 8. When such a frame exists with its width depending only on the height and not the coordinates of the center of the frame, we say that f is uniformly continuous on its domain. Then the entire graph of y D f. Satisfying a Lipschitz condition is a global property of the function, and it implies continuity. If the derivative of f is bounded, then f is Lipschitz. For example, f. We can illustrate the subadditivity of f. This graph is the same as the graph of f, but translated a units to the right and f. When f is subbadditive, the graph of y D f.
It is easy to show that all nonincreasing functions are subadditive, as are all linear functions f. The illustration in Figure 8. Superadditivity is the dual condition of subadditivity, i. In Challenge 8. Functions that are both subadditive and superadditive are called additive. Transforming the sides of a triangle Suppose a, b, and c are positive numbers that are the sides of a triangle i.
What properties of a function f insure that f. One answer is nondecreasing subadditive functions. For example, a, p p b, and c form a triangle. A function f whose domain is an interval I is convex if the set of points above its graph is convex, i. Thus a convex function satisfies the inequality. The inequality follows be- cause the line segment connecting the points. Convexity and concavity Thus the convexity of f implies that the chord connecting any two points on the graph of y D f.
If the inequality in 8. We define concavity from convexity: A function f whose domain is an in- terval I is concave if f is convex; or if the set C D f. Thus for concave functions, the inequality in 8.
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When the function is twice differentiable, convexity or concavity is easily established by examining the sign of the second derivative. A property of concave functions. Because f. The Kantorovich inequality. Our proof uses the convexity of f. Pn Proof. Let f. When the inequality in 8. In this case we say that f is midconvex on I.
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It can be shown [Roberts and Varberg, ] that continuous midconvex functions are convex, and so if f is continuous, verifying midconvexity suffices to show convexity. The midconvexity inequality in the preceding paragraph can be extended to three or more numbers. When f is strictly convex, equality in 8. The relationship between concave convex functions and subadditivity superadditivity is explored in Challenge 8.
Kantorovich J. Jensen Figure 8. Trigonometric inequalities Since the trigonometric functions are concave or convex on certain in- tervals, Jensens inequality is useful for establishing inequalities involving trigonometric functions of the angles of a triangle. Since the sine is concave on. The perimeter and circumradius of a triangle In triangle ABC, let a; b; c denote the sides, and R the circumradius.
Using 8. The Weitzenbock inequality revisited [Steele, ] In Section 6. We can now give a second proof using the convexity of the cosecant function on. We begin with the inequality in Lemma 2. In this section we examine a special case where one function is convex or concave and the other is a secant line or a tangent line.
Before considering some examples, we present a lemma and a theorem about. Tangent and secant lines The proofs are left as challenges. Lemma 8. Let f be differentiable and convex on an open interval I. Then i f 0 is nondecreasing on I, i. Analogous statements hold for concave functions, and the inequalities are strict for strictly convex and concave functions. Example 8. In Challenge 1.
The reciprocal. In Example 8. Since the sine function is concave on. If R denotes the circumradius, then as noted in Application 8. In this section we continue in that vein, using integrals to compute areas. The arithmetic mean-logarithmic mean-geometric mean in- equality. As mentioned in Example 8. Using integrals p In Figure 8. The area of each trapezoid simplifies to and one with.
The wheel of Theodorus The wheel of Theodorus of Cyrene circa 5th century BCE , or the square root spiral, is formed from right triangles, as illustrated in Figure 8. Start- ing with an isosceles right triangle, we construct right triangles whose legs are one unit and the hypotenuse p pof the preceding right triangle.
Thus the se- p quence of hypotenuses is 2; 3; : : : ; n; : : : : How does the spiral grow? Youngs inequality In this section we describe an inequality due to William Henry Young [Young, ] that relates the values of two integrals. Youngs result is so general that many interesting inequalities may be de- rived from it. For example, when '. Using integrals If we let '. Then by 8. A ladder is being carried down a hallway a feet wide.
At the end of the hallway is a 90 turn into another hallway b feet wide. What is the length of the longest ladder that can be carried horizontally around the corner? Ignore the width of the ladder. The situation is illustrated in Figure 8. The exercise can be solved immediately without calculus by using Holders inequality. The length of the ladder is L D. Bounded monotone sequences Then, using 8.
Often the ideas of this chapter concerning the representation of the values of an and M as slopes of line segments or areas of figures may be a convenient tool for establishing the required inequalities.
In order to show that 8. Hence the limit in 8. Challenges What are the geometric implications of this property for the graph of f? CHAPTER 9 Additional topics In this final chapter, we examine some methods for illustrating inequalities by combining two or more techniques from earlier chapters. We also give a brief introduction to the theory of majorization, which has proven to be remarkably effective in proving inequalities.
See, for example, Sections 4. We now revisit this metho- dology. Example 9. If a,pb, c are the sides of a right triangle with hypotenuse c, then c. Consider the square in Figure 9. Comparing the area of the large square topthe area of the four a by b rectangles yields.
Comparing the area of the square with side c to the area of the four gray triangles yields. Multiplying now yields c. Bellmans inequality By combining Minkowskis inequality see Application 8. Majorization This es- tablishes the first inequality below and the second follows from Minkowskis inequality 8. Application 9. Oppenheims inequality for two triangles An interesting result of A. Oppenheim is the following: let Ai Bi Ci. As remarked by Mitrinovic [Mitrinovic et al. Note the similarity to the connection between the Neuberg-Pedoe inequality Application 7.
Although the theory does not make extensive use of visual techniques it does have important applications in geometry. Olkin note in the Preface to their seminal work Inequali- ties: Theory of Majorization and its Applications [Marshall and Olkin, ]: Although they play a fundamental role in nearly all branches of mathe- matics, inequalities are usually obtained by ad hoc methods rather than as consequences of some underlying theory of inequalities.
For cer- tain kinds of inequalities, the notion of majorization leads to such a theory that is sometimes extremely useful and powerful for deriving inequalities. The theory of majorization requires advanced algebraic and functional machinery that is beyond the scope of this book, but we will present some basic definitions, examples and applications without proofs. Let x D. When discussing majorization, we usually write the components of a vector in nonincreasing order. Many inequalities relating three parameters angles, sides, altitudes, : : : of a triangle have succinct representations in terms of majorization.
In Figure 9. The triangles whose sides are dashed lines are equilateral triangles in a and isosceles right triangles in b. Functions that preserve majorization order are called Schur-convex. If S is a set of n-dimensional vectors, then F W S! Similarly, F W S! A classical result of I. For example, since the sine function is concave on. Flanders inequality Flanders inequality relates the product of the sines of the angles A; B; C of a triangle to the product of the magnitudes of the angles in radians :. Let g. Using elementary calculus, we can show that g is concave on.
The second re- quires that a; b; c be the sides of a triangle i. For example, the convexity of g. The first inequality in 9. The second inequality in 9. The isoperimetric inequality for triangles revisited In Challenge 4. Sporting Goods Categories. Masuk dengan. Login dengan Email. Daftar Disini. Produk dalam keranjang. Hapus semua. New of seller by.
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Product details:. To many, it is really two courses, the first linear algebra without proofs and the second with proofs. Recently, a third course has been added, computational linear algebra where the emphasis is on computer graphics and the generation of images. Nothing impresses potential math majors more than when they are told how many mathematicians work in the film industry. This book fits into the high end of the second course; it is almost exclusively proofs, although the author cannot resist putting in a few pages of computation.
It is not suitable for the traditional linear algebra course, not even for the last segment where proofs take over. In that niche, this book is excellent, Weintraub keeps the math flowing, appropriately directional and justified, and it is a rare occasion when he passes on including the detailed proof.
There are some times when a part of the proof is not included, but it is rare and generally inconsequential. If you have any current or potential need for understanding the theory of linear algebra, this is a book that you need to have on your easy access shelf. It plays an essential role in such widely differing fields as Galois theory, function spaces and homological algebra.
In this context linear algebra is about vector spaces and linear transformations, not about matrices. The natural and perhaps most enlightening approach to the canonical forms for linear transformations on finite-dimensional vector spaces, one of the main goals of this book, is via the basic structure theorems for modules over a principal ideal domain, but the author has not gone that far. The infinite-dimensional case is, however, treated. Thus, the book is for advanced students of pure mathematics and not for those requiring a textbook on numerically applicable linear algebra.
It is, moreover, written in a style to which a student of pure mathematics is fully accustomed. Although the book is for advanced students, it begins with the basics, but it does not deal with matrix operations or the solution of systems of linear equations. The chapter headings are: 1 Vector spaces and linear transformations, 2 Coordinates, 3 Determinants, 4 and 5 The structure of a linear transformation I and II, 6 Bilinear, sesquilinear and quadratic forms, 7 Real and complex inner product spaces, 8 Matrix groups as Lie groups, Appendix A: Polynomials.
Appendix B: Modules over principal ideal domains. This book can be warmly recommended to any student of pure mathematics requiring a precise and concise treatment of all the important and "well-known" results of linear algebra. The student will be grateful for the direct route that it follows and for the occasional explanations of the "right" way to understand the material.
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