Intereting Posts

Lebesgue measurability of nearly identical sets
Understanding the multiplication of fractions
What exactly is Laplace transform?
What does it mean when a function is finite?
Proving an entire function which misses a ball is constant
Some hints for “If a prime $p = n^2+5$, then $p\equiv 1\mod 10$ or $p\equiv 9\mod 10$”
Points in general position
Fourier Series on a 2-Torus
Does the ring of integers have the following property?
slightly tricky integral
Bounds on the size of these intersecting set families
Integrating a matrix
Counting non-isomorphic relations
Fourier transform of unit step?
Prove $ 1 + 2 + 4 + 8 + \dots = -1$

Each of the 25 cells in a five-by-five grid of squares is filled with a 0, 1, or 2 in such a way that the numbers written in neighboring cells differ from the number in that cell by 1. Two cells are considered neighbors if they share a side. How many different arrangements are possible?

- Probability: the average times to make all the balls the same color
- covariance of increasing functions
- Variance of the number of empty cells
- What are some open research problems in Stochastic Processes?
- Why can 2 uncorrelated random variables be dependent?
- Given K balls and N buckets what is the expected number of occupied buckets
- Counting some special derangements
- What is the problem in this solution to the Two Child Problem?
- Sum of stars and bars
- Can $18$ consecutive integers be separated into two groups,such that their product is equal?

Even and odd numbers must alternate, so we can checker the grid odd and even and put $1$s on the odd squares and arbitrary even numbers on the even squares.

If we make $12$ squares odd and $13$ even, that yields $2^{13}$ possibilities, and if we make $13$ squares odd and $12$ even, that yields another $2^{12}$ possibilities, for a total of $2^{12}+2^{13}=3\cdot2^{12}=12288$.

- Solve $ 1 + \dfrac{\sqrt{x+3}}{1+\sqrt{1-x}} = x + \dfrac{\sqrt{2x+2}}{1+\sqrt{2-2x}} $
- A classic exponential inequality: $x^y+y^x>1$
- $M$ be a finitely generated module over commutative unital ring $R$ , $N,P$ submodules , $P\subseteq N \subseteq M$ and $M\cong P$ , is $M\cong N$?
- For a trigonometric polynomial $P$, can $\lim \limits_{n \to \infty} P(n^2) = 0$ without $P(n^2) = 0$?
- Differentiable $f$ such that the set of translates of multiples of $f$ is a vector space of dimension two
- Evaluation of $\sum_{n=1}^\infty \frac{(-1)^{n-1}\eta(n)}{n} $ without using the Wallis Product
- Limit of Integral of Difference Quotients of Measurable/Bounded $f$ Being $0$ Implies $f$ is Constant
- Random variable independent of itself
- Singular $\simeq$ Cellular homology?
- Is there a primitive recursive function which gives the nth digit of $\pi$, despite the table-maker's dilemma?
- Let $\displaystyle f$ be differentiable, $\displaystyle f(x)=0$ for $|x| \geq 10 $ and $g(x)=\sum_{k \in \mathbb Z}f(x+k).$
- Evaluating $\int _0^{\frac{\pi }{2}}\:\frac{\sqrt{\sin^2\left(x\right)}}{\sqrt{\sin^2\left(x\right)}+\sqrt{\cos^2\left(x\right)}}dx$
- Taylor's Theorem with Peano's Form of Remainder
- Sequences of integers with lower density 0 and upper density 1.
- When a change of variable results in equal limits of integration