3
votes
2answers
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 ...
7
votes
3answers
300 views

Confused by textbook solution to trig problem

The following is part of a question and the solution from my textbook: Question Given that $2 \sin{2\theta} = \cos{2\theta}$, show that $\tan{2\theta} = 0.5$ Solution $2 \sin{2\theta} = ...
4
votes
1answer
142 views

Two questions on determinability (probability theory)

The following are some problems I encountered when self-learning GTM 261 "Probability and Stochastics". Definition (determinability) If $X$ and $Y$ are random variables taking values in ...
4
votes
1answer
128 views

Dodgy Reciprocal Issues in Complex Integration?

In studying early for qualification exams, I came across the following problem (UMass Amherst Graduate Qualifying Exams / Fall 2010 Complex Analysis Exam (see #10)): Let $D$ denote the open set ...
4
votes
1answer
235 views

What is the Pacman lemma?

I've seen "the Pacman lemma" mentioned in the context of reduction orderings on logical terms, but a Google search doesn't find a definition; what exactly is it? Closest a search found was a "Pacman ...
4
votes
2answers
907 views

Irreducible representations of a semidirect product

I have two finite groups. The irreducible representations of their product are given by tensor products of the irreducible of representations of the groups. Is there a way to build the irreducible ...
1
vote
1answer
76 views

Calculate the sides of a 2d polygon from the vertices?

So how would I go about calculating the sides of a 2d polygon? They're non-concave poly's, by the way.
0
votes
1answer
130 views

Trouble finding the right DTFT pair

i have a periodic signal $x[n] = \cos (\frac{2 \pi}{10}n)$. I found this DTFT pair: That's the only pair for a cosine function i found. But what is $\delta _{2\pi}$ ? Is it just a Dirac $\delta$ ...
8
votes
3answers
764 views

Differential forms

I failed to understand the definition of holomorphic $1$-form on Riemann surfaces. Can one explain it here? I saw two definitions in the books of Miranda and Farkas-Kra. Definition 1.: Suppose that ...
5
votes
1answer
712 views

On prime ideals in a polynomial ring over a PID (from Reid's _Commutative Algebra_)

More general version of this is in Reid's undergraduate commutative algebra. Prime ideals of $B[Y]$ where $B$ is a PID are as follows: $0$, $(f)$ for irreducible $f \in B[Y]$, and maximal ideals $m$. ...
6
votes
2answers
419 views

Derived sets and ordinals

Given a set $P$ of real numbers, its derived set is the set of all accumulation points - $a\in P^\prime$ if every open set containing $a$ also contains an infinite number of points from $P$ ...
0
votes
1answer
154 views

find skew lines on a cubic surface for a parametrization

I consider the hypersurface $Y = V(y(x^2+z^2)-x) \subset \mathbb{A}^3_k$ ($k = \mathbb{C}$) I've read that if you have two skew lines on a non-singular cubic surface $Y$, given by a polynomial of ...
3
votes
1answer
427 views

Practice Problems on the Elimination of Quantifiers

In a recent answer, JDH gave a remarkable proof that the integers are not definable in the structure $(\mathbb{Q},+,<)$ using Quantifier Elimination. Since I already have the old Enderton book, are ...
8
votes
0answers
286 views

Mixed Bessel Function integral $\int_{0}^{\infty} e^{- \lambda \left(\sqrt{(z+a)^2+b^2}+\sqrt{(z+c)^2+d^2}~\right)}\mathrm{d}z$

A tricky integral I have been working on, and probably doesn't have a solution in terms of known functions, is: $$\int_{0}^{\infty} e^{- \lambda ...
6
votes
1answer
1k views

Tensor Product of Vector Bundles

In Hatcher's book on Vector Bundles he states (page 14) that It is routine to verify that the tensor product operation for vector bundles over a fixed base space is commutative, associative, ...
8
votes
3answers
328 views

Two introductory linear algebra problems

I remember when I was in Moscow one of my homework questions was: Is there a $2\times 4$ matrix whose $2\times 2$ minors are: a) $(2,3,4,5,6,7)$ b) $(3,4,5,6,7,8)$ c) ...
14
votes
2answers
611 views

How to understand the Todd class?

I am reading the article "K-Theory and Elliptic Operators"(http://arxiv.org/abs/math/0504555), which is about Atiyah-Singer index theorem. In page 14 the article discussed the Thom isomorphism: ...
1
vote
1answer
91 views

questions about the paper: Affine quivers and canonical bases

I am reading the paper Affine quivers and canonical bases. I have a question on page 114 of the paper. In the proof of property (b), line 6 of page 114, why "for each $\gamma \neq 1$, $tr(\gamma, ...
0
votes
1answer
1k views

2D Epanechnikov Kernel

What is the equation for the $2D$ Epanechnikov Kernel? The following doesn't look right when I plot it. $$K(x) = \frac{3}{4} * \left(1 - \left(\left(\frac{x}{\sigma} \right)^2 + ...
1
vote
1answer
798 views

Distance of a point from a line passing through the intersection of lines

I'm given a point $P(1, 4)$ and its distance from a line passing through the intersection of the lines $x-2y+3=0$ and $x-y-5=0$ is 4 units. How do I find its equation? Here is how I tried to solve ...
1
vote
1answer
789 views

Proving regular expressions to be equivalent

I'm trying to prove that two regular expressions are equivalent. I mean prove in the rigorous sense of the word (i.e. this is a legit proof). The process is to show that R1 is a subset of R2, and ...
0
votes
1answer
164 views

What is the name of this function?

What is the name of the following function (if there is one)? $$f(x) = \begin{cases} x & \text{ if } -1 \leq x \leq 1\\ \frac 1x & \text{ if } x < -1 \text{ or } x >1\\ \end{cases} ...
1
vote
0answers
83 views

Lipschitz constant of the Laplace-Beltrami operator

I'm reading a paper on discrete differential geometry: Meyer et.al. They define the Laplace-Beltrami operator at a point $P$ by $$\vec{K}(p) = 2k_H(P)\vec{n(P)}$$ where $\vec{n}(p)$ is the normal ...
2
votes
1answer
169 views

Holomorphic Characteristic Function of a Random Variable

Let $X$ be a random variable with distribution $\mu _X$. Then, we define the characteristic function of $X$, $\phi _X$, by $$ \phi _X(t)\equiv \mathrm{E}\left[ e^{itX}\right] =\int _\mathbb{R} ...
6
votes
0answers
189 views

Continuous functions on [0,1] with f(0) = f(1) [duplicate]

Possible Duplicate: Horizontal chord of length $\frac{1}{2}$ in the graph of a continuous function. Consider the set $C$ of real continuous functions defined on $[0,1]$ such that $f(0) = ...
4
votes
3answers
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 ...
18
votes
7answers
631 views

Solving $\sqrt{x+5} = x - 1$

I'm currently learning about radicals and simplifying them, and I came across this problem on the internet and tried to solve it: $$\sqrt{x+5} = x - 1$$ So I used this logic: $$ \begin{align} ...
5
votes
4answers
3k views

Flip a coin until a head comes up. Why is “infinitely many tails” an event we need to consider?

Suppose we're considering the problem of flipping a coin until "heads" comes up. The sequences H, TH, TTH, TTTH, ... are all part of the sample space we need to consider. But what about the sequence ...
1
vote
1answer
183 views

Product of odd integers in Mod M

Does anyone know of any formular or algorithm (that runs sub-linear time) for computing the product of all odd integers in an interval in (mod M), or a similar product?
8
votes
4answers
2k views

Questions on atoms of a measure

In Kai Lai Chung's A course in probability theory, An atom of any probability measure $\mu$ on $(\mathbb{R}, \mathcal{B})$ is a singleton $\{x\}$ such that $\mu({x}) > 0$. In Wikipedia: ...
16
votes
3answers
584 views

Does a topology on a countable set always have a countable base?

I'm quite new to topology, and a homework question (which I solved without knowing the answer to this question) got me thinking: If $X$ is a countable set, and $\tau$ is a topology on it, does it ...
7
votes
1answer
136 views

Are there any strongly axiomatizable logics that are not compact?

I mean here a logic in the sense of a language and semantics. By strongly axiomatizable I mean strongly sound and strongly complete. So I'm basically asking if there is a particular deductive system ...
2
votes
1answer
2k views

Start learning A-Level further maths

I want to start learning what I need to know for A-Level further maths to take some of the "weight" off of year 12 and 13. What topics should I begin learning and what key ideas do I need to grasp and ...
3
votes
4answers
685 views

Compound angle formula confusion

I'm working through my book, on the section about compound angle formulae. I've been made aware of the identity $\sin(A + B) \equiv \sin A\cos B + \cos A\sin B$. Next task was to replace B with -B to ...
1
vote
2answers
343 views

What is a reference for the ( classical and well-known ) proof of Weyl's lemma?

What is a reference for the (classical and well-known) proof of Weyl's lemma that states: Let $U$ be an open subset of $R^n$. Then if $f\in L^1_{loc} (U)$ and if $\int_U ...
9
votes
2answers
892 views

$\Delta x$ in limit problem?

I was working on some limit homework and everything was going fine until I reached this problem: $$\lim_{\Delta x \to 0} \frac{2(x + \Delta x) - 2x}{\Delta x}.$$ I am understanding limits but the ...
1
vote
0answers
253 views

Vector Field/Differential Equation Correspondence

I have seen some examples (though I am currently looking for a good rigorous explanation and a source would be much appreciated) of taking a second order linear ODE and turning it into a linear system ...
2
votes
1answer
108 views

Support of a module and ideals

Let $R$ be a Noetherian ring , $M$ a finitely generated $R$-module and let $J$ be an ideal such that $Supp(M) \subset V(J)$ where $V(J) = \{P \in Spec(R) : P \supseteq J\}$. How to show there exists ...
14
votes
6answers
5k views

Zero probability and impossibility

I read a comment under this question: There are plenty of events that can occur that have zero probability. This reminds me that I have seen similar saying before elsewhere, and have never ...
3
votes
2answers
190 views

Fibonacci/Lucas Number Congruences

Is there a compendium of well-known (and elementary) Fibonacci/Lucas Number congruences? I've proven the following and would like to know if it is (a) trivial, (b) well-known, or (c) possibly new. $$ ...
5
votes
3answers
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 ...
14
votes
1answer
386 views

Fractional Part Double Summations

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 = ...
7
votes
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 + ...
4
votes
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? ...
2
votes
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 $X$?
1
vote
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 ...
2
votes
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 ...
4
votes
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 ...
2
votes
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 ...
1
vote
1answer
74 views

Terminology concerning Convergence of Fourier Series

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 ...

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