# All Questions

244 views

### What is the formal definition of $d$, or $\partial$, in differation and integration

This might sound a bit like a silly question, but i'm a second year math student, and so far i've encountered $d$ or $\partial$ in many cases ofcourse (mostly in calculus :)). Those letters or symbols ...
300 views

233 views

### Qualms about the axioms of probability

Let S be a set, and let $2^S$ be the power set of S. From my current understanding (which is very limited, especially since I haven't seen any measure theoretic approaches to probability), the axioms ...
631 views

567 views

### Algebraic Closure of Puiseux Series

Using the construction $R_N = K[t^\frac1N]$ $L_N = Quot(R_N)$ and $P = \bigcup_{N\in \mathbb{N}} L_N$ one automatically gets that the puiseux series are a field. Nevertheless they are also an ...
In attempt to deepen my understanding of Dedekind sums, I've proven the following identity $$\sum_{i = 0}^{t} \sum_{j = 0}^{b(t-i)} \left \lbrace c \left( t - i - \frac{j}{b} \right) \right \rbrace = ... 5answers 164 views ### Distance to Cross a City Diagonally If I had to cross from the southwest corner of a city to the northeast corner of a rectangular city and I could do so by helicopter, the distance would be \sqrt{x^2 + y^2}, which is less than x + ... 4answers 382 views ### Conditional probability Given the events A, B the conditional probability of A supposing that B happened is:$$P(A | B)=\frac{P(A\cap B )}{P(B)}$$Can we write that for the Events A,B,C, the following is true? ... 1answer 91 views ### Dimension problem Let f \colon \mathbb{C}^5 \rightarrow \mathbb{C}^7 a linear function, f(2 i e_1 + e_3) = f(e_2) and \mathbb{C}^7=X \oplus Im(f). What dimension has X? 1answer 178 views ### Probability of Falling leaves I was walking past a tree when I thought about a problem which I've been trying to solve. It states that "If there are 20 leaves on a tree and all the leaves fall on the floor, find the probability ... 2answers 232 views ### morphism of the local rings correspond to what kind of maps between varieties To a regular(or polynomial) map f: X \to Y between affine varieties we associate its pullback f^\ast: K[Y] \to K[X] and it holds that f is an isomorphism iff f^\ast is an isomorphism. Now if ... 1answer 988 views ### Finding the distance between two gears I have the following problem: In my class, we did a majorly complicated method to figure this out but I think there is a better way to do this... Here is the exact problem: A belt fits snugly ... 3answers 172 views ### Sum of coefficients of an orthogonal matrix Let (a_{ij})_{1 \le i,j \le n} be a real orthogonal matrix. Show that$$\left| \sum_{1 \le i,j \le n} a_{ij}\right| \le n. Naively applying the Cauchy-Schwarz inequality only gives ...
Let $f\in L^1(\mathbb{T})$, and $\sum_{n}a_{n}e^{int}$ its Fourier series. Fix a $t_{0}\in \mathbb{T}$. Suppose $\sum_{n}a_{n}e^{int}$ converges at $t_{0}$. But if it is still possible that ...