2
votes
2answers
3k views

Dual graph: Simple example

If I have a graph consisting of 2 disjoint triangles, which are connected by an additional edge, then I have difficulties understanding how its dual graph looks like.
0
votes
0answers
87 views

Non zero components in DFT

I wanted to do a simple example of DFT computation using the following python code (numpy + scipy). I am posting here because I am sure my problem is more related to my comprehension of the DFT ...
1
vote
1answer
127 views

How to work with Connections

I am currently reading a book which deals with complex manifolds. Since I am fairly new to the topic I don't know exactly the meaning of the followinig: Suppose we have a holomorphic vector bundle ...
9
votes
2answers
244 views

Relationship between different L-functions

What's the relationship between between Artin $L$-functions and Dirichlet or Hecke $L$-functions if $L/K$ is an abelian extension? I've been told that one can interpret the Artin $L$-functions as ...
0
votes
3answers
104 views

Finding the norm of $x \in \mathbb{R}^2$ if the unit ball is defined in a specific way

I need to find the norm of $x \in \mathbb{R}^2$ if the unit ball is defined by this inequality: $B=(\begin{pmatrix} x_1\\ x_2 \end{pmatrix}: -a_1\leq x_1\leq a_1, -a_2\leq x_2\leq a_2 ) $. What ...
2
votes
2answers
365 views

Injection from the set of countable ordinals $\Omega$ into $\mathbb{R}$

I'm reading through this and I'd like to define an injective function from the set of countable ordinals $\Omega$ into $\mathbb{R}$ using transfinite induction (or maybe transfinite recursion?). ...
6
votes
1answer
279 views

Calculating $\prod (\omega^j - \omega^k)$ where $\omega^n=1$.

Let $1, \omega, \dots, \omega^{n-1}$ be the roots of the equation $z^n-1=0$, so that the roots form a regular $n$-gon in the complex plane. I would like to calculate $$ \prod_{j \ne k} (\omega^j - ...
3
votes
0answers
68 views

Are PL maps determined by PL-paths?

Let $X,Y$ be polyhedra. If $f\colon X\rightarrow Y$ is a PL map and $g\colon I\rightarrow X$ is a PL map (where $I$ denotes the interval), then $f\circ g$ is a PL map. Is the converse true? i.e. ...
1
vote
1answer
735 views

Solving modular inequalities/constraint solving

A few of my current programming problems boil down to solving inequalities over modular domains and possibility could benifit from knowledge of efficient maths/algorithms rather than brute force ...
1
vote
2answers
140 views

Difference Equation

$y_{n+3} − 3 y_{n+1} + 2 y_n = (−2)^n$ I get the solution to be $y_n = A(-2)^n + Bn + C + \frac{1}{9}n(-2)^{n-1}$ but wolfram alpha gets $y(n) = c_1 (-2)^n+c_2+c_3 n+\frac{1}{27} 2^{n-1} e^{i \pi n} ...
2
votes
1answer
200 views

A question from Titchmarsh's Riemann Zeta Function textbook.

I have one query, concerning the newest edition of this monograph. At page 7, section 1.2, at the bottom of the page, it's written that: " It is easily seen that $\zeta(s)=2$ for $s=\alpha$, where ...
4
votes
5answers
838 views

Mathematical induction equation involving a sum of binomial coefficients

I have a problem with a mathematical equation. I don’t find the given solution. This is the equation: $\sum\limits_{k=2}^{n-1} {k \choose 2} = {n \choose 3} $ I should show with induction that the ...
6
votes
1answer
13k views

Simplest way to calculate the intersect area of two rectangles

I have a problem where I have TWO NON-rotated rectangles (given as two point tuples {x1 x2 y1 y2}) and I like to calculate their intersect area. I have seen more general answers to this question, ...
1
vote
1answer
501 views

Symmetric-decreasing rearrangement of a function

I'm studying section 3.3 of Analysis by Lieb and Loss, about symmetric-decreasing rearrangement of functions. Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define ...
4
votes
1answer
144 views

Necessary and sufficient conditions for $||f||_p = ||f||_q$ with $p \neq q$

Let $0 < p < q \leq \infty$ and suppose $ E\subset \mathbb{R}^N$ with $m(E)=1$ (where $m$ is the Lebesgue measure). I am asked to find necessary and sufficient conditions for: $$ ( ...
3
votes
1answer
495 views

Complexification of Tangent Bundle

I am currently reading a book where the author says that the tangent and cotangent bundles $TM$ and $T^*M$ of a manifold $M$ are complexified. I am not familiar with Complex Manifolds so looked it ...
6
votes
1answer
232 views

Construction of the global $\mathbf{Proj}$

I have many questions about the very abstract concept of global $\mathbf{Proj}$. I am following Hartshorne's book Algebraic Geometry, where this concept is on II.7, page 160. Let $(X, ...
2
votes
2answers
1k views

Generalization of variance to random vectors

Let $X$ be a random variable. Then its variance (dispersion) is defined as $D(X)=E((X-E(X))^2)$. As I understand it, this is supposed to be a measure of how far off from the average we should expect ...
4
votes
2answers
114 views

Commutator of a particular group product

Let $S \leq G$ and $N \lhd G$ two subgroups of $G$. Is the commutator $[SN,SN]$ equal to $[S,S]N$ ? The first is generated by commutators $[ax,by]$ (where $a,b\in S$ and $x,y \in N$), and each ...
1
vote
1answer
33 views

Range mapping process name

Does the process of mapping a random range, for example, [4; 55] to [0; 1] have a name? Maybe it's called normalization?
1
vote
2answers
126 views

Continuous Functionals and Norms

In Luenberger Optimization book, pg. 40 upper semicontinuity for a functional is defined as "if given $\epsilon > 0$ there is a $\delta > 0$ s.t. $f(x) - f(x_0) < \epsilon$ for $||x-x_0|| ...
1
vote
1answer
362 views

Proof: Matrix exponential maps from tangent space to Matrix (Lie) group

Let us assume we have a definition of the tangent space (e.g. as in Proof: Tangent space of the general linear group is the set of all squared matrices). Furthermore, we already verified that the ...
4
votes
1answer
102 views

Why does every finite subgroup of $\mathrm{Aut}(F_n)$ acts on a graph of Euler characteristic $n-1$?

My question is the following: In a paper I read that: Any finite subgroup of $\mathrm{Aut}(F_n)$ can be realised as agroup of baspoint-preserving isometries of a graph of Euler characteristic $1-n$. ...
3
votes
2answers
134 views

Solution of a differential equation that would be a generalized mean?

I am trying to solve this differential equation on which I've been stuck for several days now. $$\frac{d X}{d t}=\frac{\int_{-\infty}^{\infty}\frac{\partial f}{\partial t}\frac{\partial f}{\partial ...
0
votes
2answers
120 views

Limit and Series [duplicate]

Possible Duplicate: Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$ Compute: \begin{align*} \lim_{n\to+\infty} ...
0
votes
1answer
113 views

On Tangent vectors as jets & submanifolds

Here is my second question on understanding jets better: For a smooth manifold $M$ the set of jets $J^1_0(\mathbb{R},M)$ is the same as the Tangent bundle $TM$. This implies that any equivalence ...
1
vote
1answer
74 views

How to interpret this vector multiplication?

I have to solve the following problem (exact wording): Given is a Matrix $A_{m \times n} \in \mathbb{R}^{m \times n}$ and $r = \mathrm{rank}(A)$. The goal is to find the vectors ...
0
votes
1answer
100 views

Exponential growth on discrete quantities

I'm familiar with exponential growth, however I'm not sure how deal with situations where my quantities are discrete and rounding errors come into play. To be concrete, say I got $N$ items arranged ...
0
votes
1answer
341 views

If $\operatorname{Re}^{2}(x)=-1$, what is $x$?

$i=\sqrt{-1}$ $\operatorname{Re}(z)+i\cdot\operatorname{Im}(z)=z$ If $\operatorname{Re}^{2}(x)=-1$, what is $x$? $x$ cannot be defined in complex number as $(a+ib)$. { $a$ and $b$ are real numbers ...
3
votes
2answers
98 views

Convergence of the next series

I'm trying to determine the convergence of this series: $$\sum \limits_{n=1}^\infty\left(\frac12·\frac34·\frac56·...\frac{2n-3}{2n-2}·\frac{2n-1}{2n}\right)^a$$ I've tried using ...
-3
votes
1answer
130 views

Need a paper in mathematics relates to computer [closed]

I am studying master in computer engineering. For my advance Mathematics I need a paper in following titles. Linear Algebra, probability, random variables, stochastic variable, Que Theory. The paper ...
1
vote
1answer
104 views

Generating Functions and the Negative Binom

I'm reading from 3 different sets of notes on generating functions and having a little trouble integrating their approaches. First, I'm used to working with the following definitions: ...
1
vote
2answers
170 views

On the definition of jets

I have some problems with the definition of jets and it would be great if someone could help me here: In many books it is written, that the $r-th$ order jet $j^r_xf$ of a smooth function $f:M ...
-1
votes
1answer
366 views

how many elements are in A? (sets)

Five applicants (Jim, Don, Mary, Sue, and Nancy) are available for two identical jobs. A supervisor selects two applicants to fill these jobs. Let A denote the set of selections containing at least ...
0
votes
1answer
181 views

Manipulating a product term inside an integral

I have an expression of the form $$P(x) = \int_0^\infty \prod_{i=0}^{n} e^{-G(a, x_i)}\,\mathrm{d}a $$ and I was wondering if there was any way that I could swap the order of the product and the ...
0
votes
1answer
40 views

Sequence of complex $L_1$ functions with $L_1$-divergent limit of the fourier operator

In a task I should first show that $\mathcal{F}(e^{-\frac{x^2}{2 k}})=\sqrt{k} e^{-\frac{k \xi ^2}{2}}$ if $\text{Re}(k)>0$. Then they say that one may conclude that there is a sequence $(f_k)_{k ...
1
vote
3answers
1k views

Binary expansion Unique

I am trying to show that how the binary expansion of a given positive integer is unique. According to this link, http://www.math.fsu.edu/~pkirby/mad2104/SlideShow/s5_3.pdf, All I see is that I can ...
1
vote
1answer
71 views

Which ODEs guarantee that their solutions don't go through $x(t)=0$?

For $\ x(0)\equiv x_0>0\ $ and a system governed by $$\dot x(t)=-k\ x(t),$$ I find that $$x(t)>0\ \ \ \forall\ t.$$ (Because the solution is $x(t)=e^{-kt}x_0$.) For which $f$ and $$\dot ...
5
votes
1answer
570 views

Fast $L^{1}$ Convergence implies almost uniform convergence

$\sum_{n \in \mathbb{N}} ||f_{n}-f||_{1} < \infty$ implies $f_{n}$ converges almost uniformly to $f$, how to show this? EDIT: Egorov's theorem is available. I have been able to show pointwise ...
1
vote
1answer
178 views

Counting the number of Strings.

I encountered this question in a coding competition. The question: Given a string, calculate the number of permutations of that string such that no two identical letters lie adjacent to each ...
3
votes
1answer
86 views

Co-ordinate axes: What does the $e$ in ${\hat e}_x$ stand for?

In vector analysis for $\mathbb{R^3}$ we write standard basis vectors in various forms like $\{\hat{x}, \hat{y}, \hat{z} \}$, $\{ \hat{\imath}, \hat{\jmath}, \hat{k}\}$, $\{ {\hat e}_x, {\hat e}_y, ...
1
vote
0answers
56 views

Question on measure [duplicate]

Possible Duplicate: Existence of a measure Let $(X,\mathfrak{B})$ be a measurable space. Could someone give me the steps to show that if $\mu$ and $\nu$ are measures on $\mathfrak{B}$ and ...
1
vote
4answers
232 views

How can I show that this family of curves may be described by this differential equation?

I have a homework problem in which I wish to show that the family of curves given by $x^2 + y^2 = c x$ where $c$ is an abitrary constant may be described by the differential equation $\frac{dy}{dx} ...
2
votes
1answer
477 views

Derivation of the Riccati Differential Equation

I am attempting to derive the Riccati Equation for linear-quadratic control. The original equation is: $-\partial V/\partial t = \min_{u(t)} \{x^TQx + u^TRu + \partial V^T/\partial x(Ax + Bu) \}$ $x ...
4
votes
3answers
1k views

Do you need real analysis to understand complex analysis?

I'm debating whether I should take a course, in complex analysis (using Bak as a text). I've already taken Munkres level topology and "very light" real analysis (proving the basic theorems about ...
1
vote
1answer
55 views

arrangements of a platoon

Ten men from a platoon are arranged in two rows. Each row has the men arranged by increasing height from left to right, and every man in the back row must be taller than the man in front of him. In ...
3
votes
3answers
3k views

Mathematical Induction and “the product of odd numbers is odd”

I am extremely poor at proofs and logical manipulation so I am stuck on a lot of these questions especially induction. The question below I have been stuck at for a little over 1 hour and I can't ...
2
votes
1answer
115 views

Calculating the percent of a person's life that a certain time takes up?

A while ago I read an article on Cracked.com about why you wouldn't want to be immortal. Among many other reasons was that time would speed up for you after so many years and it would make every ...
5
votes
1answer
163 views

When does a biregular graph for the free product 2∗(2×2) have a 4 cycle?

I'd like to understand a graph theoretic property in terms of group theory. I have some boring graphs, and some neat graphs, all created from groups, but I don't know how to tell a boring group from ...
3
votes
1answer
104 views

What was done in this equation involving the fundamental theorem of calculus?

All I know is that it uses the fundamental theorem of calculus. $$\large\frac{d}{dx}\int_{x^2}^{\sin x} e^{xt^2}dt = e^{x\;\sin^2 x}\cos x - e^{x^5}2x+\int_{x^2}^{\sin x} t^2e^{xt^2}dt$$

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