The purpose is only to share and spread the awareness of availability of this second master piece on Euclid. Thanks to Clay Math Organization for serving students world wide, and thanks to the generous Mr and Mrs Clayton. I hope my math olympiad students will enjoy this and enrich themselves mathematically.
The purpose to share this here is to spread the awareness of availability of such a masterpiece by ClayMath organization, thanks of course to Mr and Mrs Clayton also.
Development of mathematics resembles a fast revolution of a wheel: sprinkles of water are flying in all directions. Fashion — it is the stream that leaves the main trajectory in the tangential direction. These streams of epigone works attract most attention, and they constitute the main mass, but they inevitably disappear after a while because they parted with the wheel. To remain on the wheel, one must apply the effort in the direction perpendicular to the main stream.
V I Arnold, translated from “Arnold in His Own words,” interview with the mathematician originally published in Kvant Magazine, 1990, and republished in the Notices of the American Mathematical Society, 2012.
- I am lying
- The only statement on this whiteboard is false.
Whereas Russell s paradox in set theory is mathematical.
Prove that the empty set is unique.
“What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement.”
— Bertrand Russell, “The Study of Mathematics” (1902)
An example from daily English language usage : “In this or the next block, you will find a taxi cab” and “the child just born is just male or female”.
Problem 1: Prove that
Problem 2: When does equality hold in the following theorem? ? Hint: Re-examine the proof of the theorem, the answer is not “when x and y are linearly dependent.”
Problem 3: Prove that . When does inequality hold?
Problem 4: Prove that ?
Problem 5: The quantity is called the distance between x and y. Prove and interpret geometrically the “triangle inequality” : .
Problem 6: Let functions f and g be integrable on .
(a) Prove that . Hint: Consider separately the cases for some and for all .
(b) If equality holds, must for some ? What if f and g are continuous?
(c) Show that the following theorem is a special case of (a) above: , equality holds if and only if x and y are linearly dependent.
Problem 7: A linear transformation is norm preserving if amd inner product preserving if
(a) Prove that T is norm preserving if and only if T is inner product preserving.
(b) Prove that such a linear transformation T is and is of the same sort.
If are non-zero, the angle between x and y, denoted is defined as , which makes sense by the following theorem :
The linear transformation T is angle preserving if T is 1-1, and for we have
(a) Prove that if T is norm preserving, then T is angle preserving.
(b) If there is a basis of and numbers such that , prove that T is angle preserving if and only if all are equal.
(c) What are all angle preserving ?
Problem 9: If , let have the matrix .
Show that T is angle preserving and if , then
Problem 10: If is a linear transformation, show that there is a number M such that for . Hint: Estimate in terms of and the entries in the matrix of T.
Problem 11: If and , show that and . Note that and denote points in .
Problem 12: Let denote the dual space of the vector space . If , define by . Define by . Show that T is a 1-1 linear transformation and conclude that every is for a unique .
Problem 13: If , then x and y are called perpendicular (or orthogonal) if . If x and y are perpendicular, prove that .