Intereting Posts

Diophantine equation $a^2+b^2=c^2+d^2$
Separated scheme
Integral points on a circle
Number of ways of distributing balls into boxes
Modules over commutative rings
List all cosets of H and K
The Fibonacci sum $\sum_{n=0}^\infty \frac{1}{F_{2^n}}$ generalized
Limit of an expression
Is this determinant always non-negative?
Video lectures on Group Theory
Is there an analogue of the jordan normal form of an nilpotent linear transform of a polynomial ring?
Why $e^x$ is always greater than $x^e$?
Determinant of matrix with trigonometric functions
Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$
If $A$ is a complex matrix of size $n$ of finite order then is $A$ diagonalizable ?

The unit step function $I$ is defined by

$$

I(x)=

\begin{cases}0,\quad x \le 0, \\

1,\quad x>0.

\end{cases}

$$

Let $f$ be continuous on $[a,b]$ and suppose $c_n\geq 0$ for $n=1, 2, 3,\ldots$ and $\sum_n c_n$ is convergent. Let $\alpha=\sum_{n=1}^{N} c_n I(x-s_n)$ where ${s_n}$ is a sequence of distinct points in $(a,b)$. Then

$$

\int_{a}^{b}fd\alpha=\sum_{i=1}^{N}c_n f(s_n).

$$

I can’t understand why the last equation holds. Where does $f(s_n)$ comes from?

- Integer parts of multiples of irrationals
- if $A$ has Lebesgue outer measure $0$ then so does $B=\left\{x^2: x\in A \right\}$
- A limit about $a_1=1,a_{n+1}=a_n+$
- How to prove $\sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6$?
- Integral $\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$
- Proving that a convex function is Lipschitz

- When can we use Fubini's Theorem?
- A function and its Fourier transform cannot both be compactly supported
- What is the easiest known expression for inverse of Laplace transform?
- Fourier-Series of a part-wise defined function?
- Does $\zeta(3)$ have a connection with $\pi$?
- Type of singularity of $\log z$ at $z=0$
- Riemann-Stieltjes integral of unbounded function
- recurrence relation $f(n)=5f(n/2)-6f(n/4) + n$
- Show uncountable set of real numbers has a point of accumulation
- Differentiation under the integral sign for Lebesgue integrable derivative

Remember the definition of the Riemann-Stieltjes integral. In the present example, the integrator $\alpha$ is constant between two consecutive $s_n$, hence two consecutive points from a partition of $[a,b]$ only contribute to the corresponding Riemann-Stieltjes sum if there is a $s_n$ between them, i. e. if $\alpha$ has a jump in between them. Their contribution is exactly the sum of all jumps happening. If the partition is fine enough, at most one jump happens between two neighboring points of the partition.

$$

\begin{eqnarray}

\int_{a}^{b}f(x)d\alpha(x)&=&\int_{a}^{b}f(x)d\left(\sum_{n=1}^{N} c_n I(x-s_n)\right)\\

&=&\sum_{n=1}^{N}c_n\int_{a}^{b}f(x)d\left( I(x-s_n)\right)\\

&=&\sum_{n=1}^{N}c_n\int_{a}^{b}f(x)\frac{d\left( I(x-s_n)\right)}{dx}dx\\

&=&\sum_{n=1}^{N}c_n\int_{a}^{b}f(x)\delta(x-s_n)dx\\

&=&\sum_{i=1}^{N}c_n f(s_n)

\end{eqnarray}

$$

- Model existence theorem in set theory
- Is there a change of variables that allows one to calculate $\int_0^\pi \frac{1}{4-3\cos^2 x}\, \mathrm dx$ avoiding improperties?
- An overring of a polynomial ring, noetherian or not?
- How can we show that $(I-A)$ is invertible?
- Generating function for the number of positive integer solutions to the equation: $4a+2b+c+3d=n$
- Hensel lift of a monic polynomial over $F_{2}$ in $Z_{8}$
- Why infinite cardinalities are not “dense”?
- proving an invloved combinatorial identity
- Torsion free abelian groups $G,H$ such that $k \cong k$ (as rings) for any field $k$
- Automorphisms of the unit disc
- What is Modern Mathematics? Is this an exact concept with a clear meaning?
- Prove that the function is continuous and differentiable (not as easy as it sounds imo)
- Is the restriction $f$ of $F\in H^{-1}(\Omega,\mathbb R^d)$ to $C_c^\infty(\Omega,\mathbb R^d)$ a distribution?
- How to find continued fraction of pi
- How are the integral parts of $(9 + 4\sqrt{5})^n$ and $(9 − 4\sqrt{5})^n$ related to the parity of $n$?