## The last effort

Here you can find the problems for the take-home exam. The rules of the game are outlined in the preamble, I copy them here for your convenience.

The following problems are suggested for the home exam, to be submitted no later than by

March 8, 2012. Almost each problem consists of several subproblems, arranged in a specific order. This order is not accidental and should be considered as an implicit hint: solutions of subsequent problems are based on the preceding ones. Please take care to avoid the words “obvious”, “clearly” etc., use as few “plain” words as possible and instead write the intermediate assertions in a closed and precise form using the quantifiers and standard set theoretic notations.The problems have different complexity: some are easier, some require additional ideas, but none of them is “computational”: if your solutions involves too many identical transformations and/or other computations, have a second look, whether you indeed answer the question that was asked, or something different.

To get the full score 100, it is

not necessaryto solveallproblems and answer all questions: the grade will be awarded based on your demonstrated understanding of mathematics and not on your familiarity with some theorems.Don’t forget to consult the lecture notes: sometimes you may find useful hints or examples there.

For your convenience Dima will soon post the Hebrew translation of these problems.

Don’t hesitate to ask questions in the comment field: we’ll try to answer them to the extent permissible for an independent home assignment 😉

## Good luck!

UPD (Feb 06, 2012, 8:30 am) A small correction of Problem 9 made (sign corrected + more accurate wording).

Question:

In problem 5 : Does “sumset A+B” refer to the Minkowski sum of the two sets?

Comment by Daniel Shallom — Tuesday, February 7, 2012 @ 10:30 |

Yes, the “sumset” always means the “set of sums” and often called the Minkowski sum.

Comment by Sergei Yakovenko — Wednesday, February 8, 2012 @ 10:17 |

In problem 9 (1) : Perhaps you meant “with the coefficient exp(-2 Pi b)” instead of exp(-2 Pi a) ?

Comment by Daniel Shallom — Thursday, February 9, 2012 @ 9:52 |

Yes you are right of course. The similarity factor should be .

Comment by dmitrybat — Thursday, February 9, 2012 @ 10:36 |

In problem 7 (5) How should I understand the “plus one” in D U (D+1) ? in what sense “plus one” when considering a set in R2 ?

Comment by Daniel Shallom — Thursday, February 9, 2012 @ 9:59 |

D+1 is a shift of D by 1 unit in the direction of, say, positive X axis.

Comment by dmitrybat — Thursday, February 9, 2012 @ 10:46 |

I couldn’t find a precise definition of injective, surjective and bijective maps in the lecture notes. I have something from my own notes but it is not very clear. Can you please tell me where to look or give a definition here?

Comment by Daniel Shallom — Thursday, February 9, 2012 @ 10:04 |

Injective means 1-1

Surjective means onto

Bijective is both 1-1 and onto

Comment by dmitrybat — Thursday, February 9, 2012 @ 10:31 |

In problem 3 (3) Do we need to prove that conjucate of a product equals product of conjugaes?

It can be done easily both by explicit expressions and by some logical sentences (in English) but is it necessary ?

Comment by Daniel Shallom — Thursday, February 9, 2012 @ 10:18 |

Since the conjugation operation was just defined, no a-priori knowledge about this operation is available and so every fact needs to be proved.

Comment by dmitrybat — Thursday, February 9, 2012 @ 10:41 |

This is all very clear now. Thanks.

Comment by Daniel Shallom — Friday, February 10, 2012 @ 11:18 |

I think It’s best to inform everyone of the correction in problem 9

Comment by Daniel Shallom — Friday, February 10, 2012 @ 11:20 |

שלום,

בשאלה 8 , מה הכוונה ב – ” איי גג ” ?

תודה.

Comment by אורן — Tuesday, February 14, 2012 @ 9:52 |

means the closed segment, [-1,1].

Comment by Anonymous — Tuesday, February 14, 2012 @ 10:00 |

האם בשאלה 8 הכוונה גם לפונקציה במשתנה אחד

אשמח להבין מי עונה לי

Comment by אורן — Tuesday, February 14, 2012 @ 10:22 |

Of course: this segment is a subset of the real line, .

–Sergei.

(The previous response was also mine 😉

Comment by Anonymous — Tuesday, February 14, 2012 @ 10:52 |

I have a question about problem 3 (2) :” Show that division by a non-invertible number is impossible”

Do we need to do it for the example we show or generally?

(I think it should be just for the example because we only develop the tools to do it generally in the following sub-problems)

Comment by Daniel Shallom — Tuesday, February 14, 2012 @ 5:58 |

The formulation is indeed awkward. Even if is non-invertible, some numbers still can be divisible by , e.g., 😉

Comment by Sergei Yakovenko — Wednesday, February 15, 2012 @ 4:21 |

So should we just give an example of a non invertible z , and skip the second sentence :”Show that…” ?

Comment by Daniel Shallom — Wednesday, February 15, 2012 @ 11:48 |

Yes.

Dima

Comment by Anonymous — Monday, February 20, 2012 @ 2:39 |

Just making sure: in problem 3(7) you mean:”any number that is not zero”

Comment by Daniel Shallom — Thursday, February 16, 2012 @ 9:55 |

Obviously 😉 Daniel, your attention to the details is really very commendable! יישר כח

Comment by Sergei Yakovenko — Thursday, February 16, 2012 @ 11:01 |

Still the remainder can be zero if a=5n, for example z=5+sqrt(5) has norm 20 and 20mod5=0

Comment by Daniel Shallom — Thursday, February 16, 2012 @ 11:37 |

I believe the correct formulation should be: “either 0 or “. Sergei?

Comment by dmitrybat — Friday, February 17, 2012 @ 4:17 |

So the formulation of subsection 8 should also change , right?

Comment by Daniel Shallom — Saturday, February 18, 2012 @ 11:30

No, (8) stays as is.

Comment by dmitrybat — Monday, February 20, 2012 @ 2:48

Yes, thanks, Dima.

Comment by Sergei Yakovenko — Monday, February 20, 2012 @ 4:06

Another question: In problem 9(2) I assume we have to prove it by explicitly writing the arc length integral . The similarity shown in 9(1) is not enough (because similarity properties are not formally proven) . Is my assumption correct?

Comment by Daniel Shallom — Thursday, February 16, 2012 @ 2:39 |

Yes.

Comment by dmitrybat — Friday, February 17, 2012 @ 4:19 |

May I use polar coordinates at problem 7? it seems easier to do the algebra in some sub-problems.

Comment by Daniel Shallom — Friday, February 17, 2012 @ 1:07 |

Of course, you may use whatever you like as long as it is correct and justified.

Comment by dmitrybat — Friday, February 17, 2012 @ 4:19 |

In problem 7 : If I find a bijective function – it is both injective and surjective so can I give the same answer for all three kinds or do I have to find an injective map that is not sujective?

Comment by Daniel Shallom — Friday, February 17, 2012 @ 11:08 |

You can use the continuous bijective map for all the three categories.

Comment by dmitrybat — Friday, February 17, 2012 @ 4:20 |

Comments 11 & 13 are still without conclusive reply

Comment by Daniel Shallom — Sunday, February 19, 2012 @ 9:59 |

Dear Sergei and Dima. In problem 2(2) perhaps you mean “infinitely many segments” instead of “finitely many segments” ?

Comment by Daniel Shallom — Wednesday, February 29, 2012 @ 1:32 |

No, the original formulation is correct. Note that if is a cover of , then usually contains some elements which are not in .

Dima

Comment by Anonymous — Wednesday, February 29, 2012 @ 9:17 |

Yes, of course. My mistake. The number might be different for different epsilons but it will always be finite.

Comment by Daniel Shallom — Friday, March 2, 2012 @ 1:09

האם בשאלה 9 חזקת האקספוננט לא צריכה להיות חיובית , כלומר

+b*pai*phi

?

Comment by אורן — Thursday, March 1, 2012 @ 10:07 |

Think about (a) which direction of the polar angle is positive, (b) what happens with the polar radius when increases.

Comment by Sergei Yakovenko — Thursday, March 1, 2012 @ 11:43 |

בשאלה 9 , אני לא מבין מה ההבדל בין סעיף 1 לסעיף 2.

אשמח אם תחדדו ההבדל.

Comment by אורן — Thursday, March 1, 2012 @ 10:14 |

They are indeed very close, yet you have to explain how the lengths of similar curves (defined by the corresponding integrals) are related to each other.

Comment by Sergei Yakovenko — Thursday, March 1, 2012 @ 11:44 |

האם בשאלה 4 ניתן להוכיח המתבקש ע”י סרטוט – כלומר דיאגרמת ון?

Comment by אורן — Thursday, March 1, 2012 @ 3:59 |

האם בשאלה 5.3 הקבוצה

A

היא אותה קבוצה

A

מסעיף 5.1

?

Comment by אורן — Friday, March 2, 2012 @ 8:08 |

Yes, in both cases is stands for an arbitrary subset of finite diameter.

Comment by Sergei Yakovenko — Sunday, March 4, 2012 @ 9:54 |

בשאלה 3 סעיפים 7,8 נראה כי האיברים

a,b

לא יכולים להיות אפס.(כל אחד מהם בנפרד)

האם ייתכן ובשאלה זו יש להתחשב במספר אפס כ-לא שייך לקבוצת השלמים ?

(בתחילת השאלה מצוין כי איברים אלו שייכים לשלמים ואפס מוגדר כמספר שלם)

Comment by אורן — Wednesday, March 7, 2012 @ 12:03 |

סעיף 7 – הכוונה שאיבר שהוא לא 0

– תיקון ניסוח סעיף זה – ראה הערה מס’ 13 לעיל

סעיף 8 – מופיע במפורש התנאי כי האיבר אינו אפס

Comment by dmitrybat — Wednesday, March 7, 2012 @ 1:01 |