In a dark, narrow alley, a function and a differential operator meet:
"Get out of my way  or I'll differentiate you till you're zero!"
"Try it  I'm e^x..."

Same alley, same function, but a different operator:
"Get out of my way  or I'll differentiate you till you're zero!"
"Try it  I'm e^x..."
"Too bad... I'm d/dy."

Q: What is the value of the contour integral around Western Europe?
A: Zero.
Q: Why?
A: Because all poles are in Eastern Europe!

Q: What is the difference between a Ph.D. in mathematics and a large pizza?
A: A large pizza can feed a family of four...

A newlywed husband is discouraged by his wife's obsession with mathematics. Afraid of being second fiddle to her profession, he finally confronts her: "Do you love math more than me?"
"Of course not, dear  I love you much more!"
Happy, although sceptical, he challenges her: "Well, then prove it!"
Pondering a bit, she responds: "Ok... Let epsilon be greater than zero..."

Top of Page
Analysis is full of elementary formulated problems which have a large significance for quite nonelementary subjects. We suggest to your attention some of them.
Let D_{1},…, D_{N} are discs on the plane. Their centers are c_{1},…, c_{N}. For each center count two numbers: m(c_{i})= the number of our discs conataining c_{i}, i=1,…,N; and M(c_{i}) =the number of discs in the sequence 2D_{1},…, 2D_{N} conatining c_{i}, i=1,…,N. Here 2D stands for the disc with the same center but the doubled radius.
Obviously 1 ≤ m_{i}≤ M_{i} ≤ N. Give an elementary prove that there exists an absolute constant A such that independently of the number N and independently of collection D_{1},…, D_{N} there exists 1≤j≤ N such that M_{j} ≤ A≤ m_{j}.

One is given the collection of N different positive integers: a_{1}, a_{2},…, a_{N}. Prove that whatever the collection of N1 positive integers 0<b_{1}<b_{2}<...<b_{N1}< a_{1}+a_{2}+...+a_{N} is given one can permute her/his given a_{i} in such a way that
a_{i_1}≠ b_{1},
a_{i_1}+a_{i_2}≠ b_{2},
…
a_{i_1}+a_{i_2}+…+a_{i_(N1)} ≠ b_{N1}.

Give an elementary prove of the following fact: let f, g, h be real valued, mesurable functions on the real line. Also they are 1periodic. Assume also that f< 1, g< 1, h< 1. We are interested in the following integral:
Obviously L is less or equal to 1, and 1 can be obviously attained. We add the property:
Prove that L then is at most ½.

Top of Page
