Prove that for all \(\displaystyle 0\le a\le b\) there exists a bounded set \(\displaystyle H\subset\R^p\) for which \(\displaystyle b(H)=a\) and \(\displaystyle k(H)=b\).

Difficulty: 2.

Interchange the order of integration.

\(\displaystyle \int_0^1\int_x^{2x} f(x,y)\dy\dx; \qquad \int_{-1}^1\int_{|x|}^{x^2+x+1} f(x,y)\dy\dx \)

Difficulty: 2.

Let \(\displaystyle H\subset\R^p\) be a bounded set. Determine whether the following statements are true or false.

(a) If \(\displaystyle k(H)=0\), then \(\displaystyle H\in\J\).

(b) If \(\displaystyle H\in\J\), then \(\displaystyle \partial H\in\J\).

(c) If \(\displaystyle \partial H\in\J\), then \(\displaystyle H\in\J\).

(d) If \(\displaystyle H\in\J\), then \(\displaystyle \INT H\in\J\).

(e) If \(\displaystyle H\in\J\), then \(\displaystyle \cl H\in\J\).

(f) If \(\displaystyle \INT H\in\J\) and \(\displaystyle \cl H\in\J\), then \(\displaystyle H\in\J\).

Difficulty: 3.

What is the moment of inertia for a cylinder of mass \(\displaystyle m\), radius \(\displaystyle r\), and height \(\displaystyle 2h\) about an axis that goes through its center but is orthogonal to its axis of symmetry?

Difficulty: 3.

\(\displaystyle \int_0^1\int_0^x y^2e^x \dy\dx = ? \)

Difficulty: 3.

Calculate the area of the set, defined with polar coordinates, by \(\displaystyle \beta-90^\circ\le\varphi\le90^\circ-\gamma\), \(\displaystyle \displaystyle0\le r\le \frac{m}{\cos\varphi}\).

Difficulty: 3.

\(\displaystyle \int_{\pi^2\le x^2+y^2\le4\pi^2}\sin(x^2+y^2)\dx\dy=? \)

Difficulty: 3.

Calculate the volume of \(\displaystyle \{(x,y,z)\in\R^3:x^2+y^2\le1\), \(\displaystyle |z|\le e^{\sqrt{x^2+y^2}}\}\).

Difficulty: 3.

What is the moment of inertia of a cone about its axis of rotation if it has homogeneous mass distribution with mass \(\displaystyle m\), its height is \(\displaystyle h\) and its base disc has radius \(\displaystyle r\).

Difficulty: 3.

Determine whether the following statements are true or false. Here \(\displaystyle f\) is a function from \(\displaystyle [a,b]\) to \(\displaystyle \R\).

(a) If \(\displaystyle f\) is monotonic, then \(\displaystyle f\) is of bounded variation.

(b) If \(\displaystyle f\) is continuous, then \(\displaystyle f\) is of bounded variation.

(c) If \(\displaystyle f\) is continuous and of bounded variation, then \(\displaystyle f\) is Lipschitz.

(d) If \(\displaystyle f\) is of bounded variation, then the interval \(\displaystyle [a,b]\) can be written as the union of countable many subintervals on each of which \(\displaystyle f\) is monotonic.

(e) If the \(\displaystyle \int_a^b\df\) Stieltjes-integral exists, then \(\displaystyle f\) is absolutely continuous.

(f) If \(\displaystyle f\) is absolutely continuous, then \(\displaystyle f\) is Riemann-integrable.

Difficulty: 4.

Let \(\displaystyle A\subset\R^p\), \(\displaystyle B\subset\R^q\) be bounded sets. True or false?

(a) \(\displaystyle k^{(p+q)}(A\times B) = k^{(p)}(A)\cdot k^{(q)}(B)\).

(b) \(\displaystyle b^{(p+q)}(A\times B) = b^{(p)}(A)\cdot b^{(q)}(B)\).

(c) If \(\displaystyle A\) and \(\displaystyle B\) are measurable, then \(\displaystyle A\times B\) is also measurable and \(\displaystyle t^{(p+q)}(A\times B) = t^{(p)}(A)\cdot t^{(q)}(B)\).

Difficulty: 4.

Is it true that if \(\displaystyle A\subset\R\) is measurable, then

\(\displaystyle \{(x,y) : \sqrt{x^2+y^2} \in A \} \subset \R^2 \)

is measurable?

Difficulty: 4.

The vertices of a triangle are \(\displaystyle A=(a,0)\), \(\displaystyle B=(b,0)\) and \(\displaystyle C=(0,m)\). For \(\displaystyle (x,y)\in[0,1]^2\) let

\(\displaystyle f(x,y) = (1-x)(1-y)\cdot A + x(1-y)\cdot B + y\cdot C. \)

Use this map and the theorem on measure transformation to determine the area of the triangle.

Difficulty: 4.

\(\displaystyle \int_0^1 \left( \int_{\sqrt{y}}^1 \sqrt{1+x^3} \, dx \right) \, dy=? \qquad \int_0^1 \left( \int_{y^{2/3}}^1 y\cos x^2 \, \, dx \right) \, dy=? \)

Difficulty: 4.

Give a function \(\displaystyle \varphi:[0,2]\to\R\) such that for any continuous function \(\displaystyle f:[0,1]\to\R\)

\(\displaystyle \int_0^1 \int_0^1 f(x^2+y^2) \dx \dy = \int_0^2 f \varphi. \)

Difficulty: 4.

\(\displaystyle \int_0^{\pi/2}\left(\int_x^{\pi/2}\frac{\sin y}{y}\;\dy\right)\dx=? \)

Difficulty: 4.

Prove that \(\displaystyle \sum\limits_{n=1}^\infty e^{-n^2x}\) is infinitely differentiable on \(\displaystyle (0,\infty)\).

Difficulty: 4.

\(\displaystyle \int_0^1 \sqrt{x} \left( \int_{x^{3/4}}^1 e^{y^3} \dy \right) \dx = ? \)

Difficulty: 4.

Let \(\displaystyle A,B\subset\R^p\) be disjoint bounded sets. Order the following numbers

\(\displaystyle k(A\cup B); \qquad b(A\cup B); \qquad k(A)+k(B); \qquad b(A)+b(B); \qquad k(A)+b(B); \qquad b(A)+k(B). \)

Difficulty: 5.

Let \(\displaystyle f:(0,1)\to\R\), \(\displaystyle f(x)=x\sin\log x\). Is this a function of bounded variation? Is it absolutely continuous?

Difficulty: 5.

Let \(\displaystyle H\subset\R^p\) be a bounded set. Are the following statements true or false?

(a) If \(\displaystyle \cl H\in\J\), then \(\displaystyle H\in\J\).

(b) If \(\displaystyle H\) is closed and \(\displaystyle H\in\J\), then \(\displaystyle \INT H\in\J\).

(c) If \(\displaystyle H\) is open and \(\displaystyle H\in\J\), then \(\displaystyle \cl H\in\J\).

(d) If \(\displaystyle k(\INT H)=b(\cl H)\), then \(\displaystyle H\in\J\).

(e) \(\displaystyle \partial H\in\J\).

Difficulty: 5.

Prove that if \(\displaystyle A\subset B\subset\R^p\) and \(\displaystyle B\) is Jordan-measurable, then

\(\displaystyle t(B) = k(A) + b(B\setminus A).\)

Difficulty: 5.

Show that a bounded set \(\displaystyle A\subset\R^p\) is measurable if and only if

\(\displaystyle k(B) = k(B\cap A)+k(B\setminus A)\)

for any set \(\displaystyle B\subset\R^p\).

Difficulty: 5.

Let \(\displaystyle A\subset [a,b]\) be Jordan-measurable. Connect the points of \(\displaystyle A\) to an arbitrary (but fixed) point of the plane. Show that the union of these line segments is Jordan-measurable in the plane. What is its ``area''?

Difficulty: 5.

Prove that if \(\displaystyle A,B\subset\R^p\) és \(\displaystyle \cl A\cap\cl B\) is of measure zero, then \(\displaystyle k(A\cup B)=k(A)+k(B)\).

Difficulty: 5.

Let \(\displaystyle A\subset\R^p\) be Jordan-measurable. Is it true that the set \(\displaystyle \displaystyle\bigcup_{a\in A}[0,a]\) is measurable?

Difficulty: 5.

Let \(\displaystyle f\) be bounded and non-negative on the measurable set \(\displaystyle A\). Prove that \(\displaystyle \int _A f=0\) implies that \(\displaystyle k(\{ x\in A: f(x)\ge a\})=0\) for all \(\displaystyle a>0\). Is the converse true?

Difficulty: 5.

Prove Steiner's theorem: if a rigid body has mass \(\displaystyle m\) and its moment of inertia about an axis \(\displaystyle l\) through its center of mass is \(\displaystyle I\), then the moment of inertia about an axis parallel to \(\displaystyle l\) and of distance \(\displaystyle r\) is \(\displaystyle I+mr^2\).

Difficulty: 5.

Show that Euler's Beta-function is infinitely differentiable and express its derivative as an integral.

Difficulty: 5.

Let \(\displaystyle A_1 ,\ldots , A_n\) be measurable sets in the unit cube whose measures add up to more than \(\displaystyle k\). Show that there is a point which is contained in at least \(\displaystyle k\) of these sets.

Difficulty: 6.

Prove that if \(\displaystyle m:\mathcal{J}\to\R\) is nonnegative, additive, translation-invariant and normed, then \(\displaystyle m=t\).

Difficulty: 6.

Prove that a bounded set \(\displaystyle A\subset\R^p\) is measurable if and only if

\(\displaystyle b(B) = b(B\cap A)+b(B\setminus A) \)

for any set \(\displaystyle B\subset\R^p\).

Difficulty: 6.

For any \(\displaystyle \varepsilon>0\) divide the \(\displaystyle n\)-dimensinal unit cube into an open and closed part in such a way that the inner Jordan measure of each is less than \(\displaystyle \varepsilon\).

Difficulty: 6.

Prove that a bounded set \(\displaystyle K\subset\R^n\) is Jordan-measurable if and only if it cuts all bounded open sets ``properly'' i.e. for all bounded open set \(\displaystyle X\subset\R^n\) one has \(\displaystyle b(X\cap K)+b(X\setminus K)=b(X)\).

Difficulty: 6.

Prove that a bounded set \(\displaystyle K\subset\R^n\) is Jordan-measurable if and only if it cuts all bounded closed sets ``properly'' i.e. for all bounded closed set \(\displaystyle X\subset\R^n\) one has \(\displaystyle k(X\cap K)+k(X\setminus K)=b(X)\).

Difficulty: 6.

Formulate a Weierstrass type criterion for improper Sieltjes-integrals.

Difficulty: 6.

Prove that if \(\displaystyle B_1,B_2,\ldots\subset\R^p\) are pairwise disjoint open balls then

\(\displaystyle b\bigg(\bigcup_{i=1}^\infty B_i\bigg) = \sum_{i=1}^\infty b(B_i). \)

Difficulty: 7.

Show that for any \(\displaystyle 0\le c\le d<\infty\) there exists a bounded, closed set with interior measure \(\displaystyle c\), and exterior measure \(\displaystyle d\).

Difficulty: 7.

Prove that if \(\displaystyle A\) is measurable with positive measure and \(\displaystyle f\) is integrable on \(\displaystyle A\), then there is at least one point where \(\displaystyle f\) is continuous.

Difficulty: 7.

Is it true that if \(\displaystyle f:[0,1]\times[0,1]\to\R\) is monotonic on every horizontal and vertical segments, then it is integrable?

Difficulty: 7.

Prove that if \(\displaystyle f>0\) on \(\displaystyle A\subset\R^n\) with positive Jordan-measure, then

Difficulty: 7.

Express the volume of the \(\displaystyle n\)-dimensional unit ball using Euler's \(\displaystyle \Gamma\)-function. What is the volume of the ``half-dimensional'' unit ball?

Difficulty: 7.

Prove that for \(\displaystyle s>0\) \(\displaystyle \Gamma(s)\cdot\Gamma''(s) > \big|\Gamma'(s)\big|^2\).

Difficulty: 7.

Formulate and prove the Dirichlet- and Abel-criterions for improper integrals.

Difficulty: 7.

Is \(\displaystyle \displaystyle f(t)=\int_1^t\int_1^t e^{xyt}\dx\dy\) (\(\displaystyle t>1\)) differentiable? What is its derivative?

Difficulty: 7.

Let \(\displaystyle f:\R^3\to\R\) be continuous, and \(\displaystyle G(r)=\int_{x^2+y^2\le r^2}f(x,y,r)\dx\dy\) (\(\displaystyle r>0\)).

(a) Show that \(\displaystyle G\) is continuous.

(b1) Show that if \(\displaystyle f\) continuously differentiable, then \(\displaystyle G\) is also continuously differentiable. What is \(\displaystyle G'\)?

(b2) Can the condition of continuous differentiablity be weakened?

Difficulty: 7.

Is \(\displaystyle \displaystyle f(t)=\int_1^t e^{x^2t}\dx\) differentiable? What is its derivative?

Difficulty: 7.

Let \(\displaystyle f:\R^2\to\R\) be continuous and \(\displaystyle \displaystyle G(x)=\int_{-x}^{x^2} f(x,y)\dy\).

(a) Prove that \(\displaystyle G\) is continuous.

(b1) Show that if \(\displaystyle f\) is continuously differentiable, then \(\displaystyle G\) is also continuously differentiable. What is \(\displaystyle G'\)?

(b2) Can the condition of continuously differentiability weakened?

Difficulty: 7.

Let \(\displaystyle f:[0,1]\to\R^2\) be a simple closed curve. Does it follow that its image has measure \(\displaystyle 0\)?

Difficulty: 8.

For all continuous functions \(\displaystyle f:\R\to\R\) let \(\displaystyle I_0f=f\) and for \(\displaystyle a\ge 0\) let \(\displaystyle I_af\) be the function for which

\(\displaystyle (I_a f)(x) = \int_0^x f(t) \frac{(x-y)^{a-1}}{\Gamma(a)} \dx. \)

Prove that (a) \(\displaystyle (I_1f)(x)=\int_0^xf\); (b) \(\displaystyle I_{a+b}=I_aI_b\).

Difficulty: 8.

Prove that if \(\displaystyle F_1\supset F_2\supset\ldots\) are bounded, closed sets and \(\displaystyle \bigcap\limits_{n=1}^\infty F_n\) is of measure zero, then \(\displaystyle {k(F_n)\to0}\).

Difficulty: 8.

Prove that Euler's Beta function is strictly convex.

Difficulty: 8. Hint is provided for this problem.

Is there a Peano-curve that is differentiable? (I.e. is there a surjective differentiable map \(\displaystyle [0,1]\to\R^2\)?)

Difficulty: 9.

Let \(\displaystyle \Gamma(s)=\int_0^\infty x^{s-1} e^{-x}\dx\) and \(\displaystyle B(s,u)=\int_0^1 x^{s-1} (1-x)^{u-1} \dx\) be Euler's Gamma- and Beta functions. Show that

\(\displaystyle B(s,u) = \frac{\Gamma(s)\Gamma(u)}{\Gamma(s+u)}. \)

Difficulty: 9.

Let \(\displaystyle B\) be Euler's Beta function. Prove that \(\displaystyle \log B\) is convex.

Difficulty: 9. Hint is provided for this problem.

For any \(\displaystyle H\subset\R^p\) bounded set let \(\displaystyle B(H)\) be (a) largest open ball in \(\displaystyle H\) if \(\displaystyle H\) has no interior then let \(\displaystyle B(H)=\emptyset\). Starting from an \(\displaystyle A_0\subset\R^p\) Jordan-measurable set let \(\displaystyle A_1=A\) and \(\displaystyle A_{n+1}=A_n\setminus B(A_n)\). Prove that \(\displaystyle \lim b(A_n)=0\).

Difficulty: 10.

We need a simulated random sequence of normal distribution, i.e. with density \(\displaystyle \displaystyle\varrho(x)=\frac1{\sqrt{2\pi}}e^{-x^2/2}\). Given a random-number generator that gives random numbers with uniform distribution in \(\displaystyle [0,1]\) how can one generate such a sequence.

Difficulty: 10. Hint is provided for this problem.

Let \(\displaystyle a\in\R\). \(\displaystyle \int_{-\infty}^\infty \frac{e^{-x^2/2}}{\sqrt{2\pi}}\cos(ax)\dx=?\)

Difficulty: 10.

According to Tauber's theorem if \(\displaystyle \displaystyle\lim_{r\to1-0} \sum_{n=0}^\infty a_nr^n=C\) exists and finite and moreover \(\displaystyle n a_n\to0\), then \(\displaystyle \displaystyle \sum_{n=0}^\infty a_n=C\).

(a) Formulate a Tauberian theorem for parametric integrals.

(b) Prove the Tauberian theorem for parametric integrals you formulated.

Difficulty: 10.

For \(\displaystyle x\in\R\) let \(\displaystyle \displaystyle I(x) = \int_{-\infty}^\infty \frac{e^{-t^2/2}}{\sqrt{2\pi}}\cos(xt)\dt\).

(a) Prove that \(\displaystyle I(x) \cdot I(y) = I\big(\sqrt{x^2+y^2}\big)\).

(b) Describe the behavior of \(\displaystyle I\) near \(\displaystyle 0\).

(c) \(\displaystyle I(x)=?\)

Difficulty: 10.