2
votes
2answers
131 views

How do I solve the following recurrence?

Solve the recurrence $$X_n =\begin{cases} n & 0 \leq n < m\\ X_{n-m} + 1 & n \geq m.\end{cases}$$ So I've started with several base cases, but since the answer depends on $n$'s relation ...
0
votes
1answer
456 views

Pushout in Set example 2.6.2.2. in David Spivak's Book

Trying to understand Example 2.6.2.2. on page 50 of David I. Spivak's book "Category Theory for Scientists," Spivak gives $X$ as the interval $[0..1]=\{x\in\mathbb{R}|\,0\le x\le 1\}$, $Y$ as the ...
2
votes
1answer
104 views

If X is a countable KC-space, then every infinite $D\subset X$ contains an infinite subset with only a finite number of accumulation points (in X)

a topological space is called KC- space if every compact set is closed. a topological space is called US, if every convergent sequence has unique limit. generally, KC- space imply US - space. ...
5
votes
2answers
244 views

What is a noetherian category?

What is a noetherian category? I'm a little bit familiar with category theory, but I've no idea what this could be. Do you know what it is good for or examples?
1
vote
1answer
346 views

Probability and Statistics 2

John invites 12 friends to a dinner party, half of which are men. Exactly one man and one woman are bringing desserts. If one person from this group is selected at random, what is the probability that ...
2
votes
0answers
167 views

Integral of Bessel functions

Does anybody know if there is an analytical solution to the following integral of Bessel functions: $$\int J_m^*(kx) \, J_m(kx) \, x \,dx,$$ where $m$ is integer and the problem is that $k$ may be ...
6
votes
1answer
167 views

Prove: $\int_{-\infty}^{\infty} \frac{\tan^{-1}e^{-\pi x}}{\cosh\frac{3x\zeta(2)}{20}} dx = 10$

I want to solve the following integral: $$\int_{-\infty}^{\infty} \frac{\tan^{-1}e^{-\pi x}}{\cosh\frac{3x\zeta(2)}{20}} dx = 10$$ Have not tried it yet, but it may be tough. All I know is that ζ(2)...
1
vote
1answer
491 views

Diameter of a circle inscribing a regular pentagon

In his book, The Irrationals, Julian Havil presents a method used by classical Arab scholars for finding the diameter of a circle inscribing a regular pentagon - and then asks the reader to prove the ...
1
vote
4answers
2k views

Proof for the symmetric difference?

I want to prove that: a) $A \Delta B = \emptyset \Leftrightarrow A = B $ Prove for $\Leftarrow$ Then: $A \Delta B = (A$ \ $A) \cup (A $\ $A) = \emptyset$ Prove for $\Rightarrow$ Then(proof by ...
-1
votes
1answer
94 views

The smallest value of $|a|$ such that the lines $ x = a+m $, $y = -2 $ and $y = mx$ are concurrent

Question If the line $ x = a+m $, $y = -2 $ and $y = mx$ are concurrent, the least value of $|a|$ is (A) $\sqrt{2}$ (B) $2\sqrt{2}$ (C) $2\sqrt{3}$ (D) $3\sqrt{2}$ Solution Since the ...
5
votes
7answers
3k views

Self-teaching myself math from pre-calc and beyond.

Going to be starting grade 12 (pre-calculus) shortly and looking to get ahead. I would like to try some more rigorous stuff on my own and have a couple questions. Ideally I would like to be prepared ...
0
votes
1answer
38 views

Finding collinearity among variables

I have been reading about how (multi)-collinearity among predictor variables can be determined by looking at the condition number, or smallest eigenvalue, of the covariance matrix. My question is, if ...
5
votes
1answer
305 views

Hard integral that standard CAS get totally wrong

How to solve the following integral: $$\int_{-\infty }^{\infty }\exp \left ( i\left ( ax^3+bx^2 \right ) \right )dx$$ Standard CAS seem to get it totally wrong, see: http://www.walkingrandomly.com/?...
1
vote
2answers
77 views

Does the Pigeonhole principle apply in this problem?

I came accross this problem a while ago at school during a math contest. I dont remember the exact instruction (word for word) but it went something like : Randomed A and B, 2 natural integer $\in [0,...
6
votes
1answer
303 views

Why are there so many groups (up to isomorphism) of order $p^n$ for $n>2$, especially when compared to groups of similar sized order?

While bounds on the number of isomorphism classes of groups of order $p^n$ where $p$ is prime have been known for quite a while (such as the work of Higman$^{[1]}$ and Sims$^{[2]}$) which give us the ...
1
vote
0answers
69 views

Finding subgroups to special groups

Let $G$ and $H$ be groups. Is there any possibility to find all (normal) subgroups of $G\times H$ and $G*H$? I really hope that this task is easier, if $G$ and $H$ are cyclic groups. I tried to find ...
2
votes
1answer
258 views

Dummit Foote Exercise 13.3.15

A Field F is said to be formally real if $-1$ can not be written as sum of squares in F. let $f(x)$ be an irreducible polynomial in $F[x]$ of odd degree with $\alpha$ as a root. Now the Question is ...
1
vote
0answers
104 views

What do quadratic smoothing splines minimize?

Cubic smoothing splines minimize a combination of Interpolation cost and Smoothness (roughness) cost: $\qquad$ min Icost + $\lambda$ Scost where $\qquad$ Icost $\equiv \sum (Y_i - \mu(x_i))^2$ $\...
2
votes
3answers
105 views

Why $\int_{0}^{+\infty} \frac{\sin(x)}{x} \: dx = \int_{0}^{+\infty} \frac{\sin^{2}(x)}{x^{2}} \: dx$?

In an exercise, I saw the following equality : $$ \int_{0}^{+\infty} \frac{\sin(x)}{x} \: dx = \int_{0}^{+\infty} \frac{\sin^{2}(x)}{x^{2}} \: dx $$ At first, I was surprised by this equality. It is ...
0
votes
1answer
272 views

About zeros of vector fields in compact surfaces

I'm studying compact surfaces and in particular the relationship between zeros of vector fields defined on them and Euler characteristic of the surface herself. Let be $S$ a compact (smooth) surface ...
7
votes
5answers
635 views

What's the importance of the trig angle formulas?

What's the importance of the trig angle formulas, like the sum and difference formulas, the double angle formula, and the half angle formula? I understand that they help us calculate some trig ratios ...
1
vote
0answers
79 views

Lagrange multiplier on inequalities

Let's say we have an inequality, then we change it into a function so for example we have: $f(x,y) = x^2-xy-2$ Also we have the constraint $g(x,y) = x+y - 1 = 0$ Using Lagrange multiplier, I get ...
0
votes
3answers
410 views

Polynomials of roots. Sum and product of roots

If A and B are roots of the equation $px^2 +qx +r=0$ find in terms of $p$,$q$ and $r$ a) $$\frac{1}{A} + \frac{1}{B}$$ Hi, im new to this topic. This is not homework. im just curious of how to do ...
9
votes
1answer
370 views

A conjecture about vector space

Let $V$ be a $(r+1)$-dimensional vector space, and $p$ be a positive integer and $1\leq p\leq r-1$. Let $$X=\{v_1,\cdots,v_{2r+1-p}\}\subseteq V$$ be a finite set containing $(2r+1-p)$ different ...
3
votes
3answers
1k views

Convergence in measure of products

Let $\mu$ be a measure on $(X,\mathcal A)$ and let $f, f_1, f_2,\dots$ and $g, g_1, g_2,\dots $be real valued $\mathcal A$- measureable functions on $X$. Show that if $\mu$ is finite, $(f_n)$ ...
5
votes
5answers
5k views

What is the standard notation to represent the set of primes?

I have seen $\mathbb Z_p$. Are there others, perhaps $\mathbb N_p$? or the set of natural numbers where the totient of $ n $ equal $ n - 1 $ ? $$ \{n \in \mathbb N \mid n\ge2,\phi (n) = n-1\}$$
1
vote
1answer
71 views

Convergence in metric and in Borel measure

Suppose we have the space ($\mathbb{R}, B(\mathbb{R},\lambda$) and define e $h: \mathbb{R} \rightarrow \mathbb{R}$ by $h(t) = 1/(1+t^2)$. a) Prove that the formula $$d(f,g) = \int \frac{|f-g|}{1+|f-...
2
votes
4answers
115 views

Proof that $A \subseteq B \Leftrightarrow A \cup B = B$

I want to proof that: $A \subseteq B \Leftrightarrow A \cup B = B$ At the moment I have no idea on how to start. Please give me a hint. Thx in advance!
1
vote
1answer
208 views

Proof that $a\mid b \land b\mid c \Rightarrow a\mid c $

I am trying to prove: $a\mid b \land b\mid c \Rightarrow a\mid c $ $a\mid b$ means that a divides b if there is an integer k, that $b=k\cdot a$ Please give me a hint on how to start, because I ...
3
votes
2answers
343 views

Suppose $T^2$ is diagonalizable and $\ker{T}=\{0\}$, and every eigenvalue of $T^2$ is nonnegative. Show that $T$ is diagonalizable.

Suppose $T^2$ is diagonalizable and $\ker{T}=\{0\}$, and every eigenvalue of $T^2$ is nonnegative. Show that $T$ is diagonalizable. Of course $T$ is an operator on $V$. It seems to me that if I take ...
1
vote
1answer
188 views

Attempted exercise using Littlewood's theorem

This was an exercise to try to show we can use Littlewood's theorem$^1$ to prove that $$\lim_{N \to \infty}\frac{1}{N}\sum_{n=1}^{N} \frac{g(n)}{\log p_n} = 1 \hspace{30mm}(1)$$ If $\vartheta(p_k) \...
5
votes
3answers
368 views

How to find $\sum n^3$ if $\sum n^2$ is given

Problem : Find $\sum n^3$ if $\sum n^2 =2870$ Can we use the following method : $\sum n^2 = \frac{n(n+1)(2n+1)}{6}$ = 2870.. ( As sum of the square of first n natural number is $\frac{n(n+1)(2n+...
0
votes
2answers
240 views

Find the number of the ways of dividing $2p$ items into $2$ equal groups of $p$ each. Where the groups do not have distinct identity?

Find the number of the ways of dividing $2p$ items into $2$ equal groups of $p$ each, where the groups do not have distinct identity? Why is the solution to the above problem consisting of $\;\dfrac{...
8
votes
1answer
403 views

Examples of mathematical statements made with adjoint functors

I am wondering if it is possible to use the adjoint functors in topos theory for statements in analysis. Any examples would be warmly welcomed. Though I would prefer simpler, atomic, lemmas or ...
1
vote
0answers
96 views

Why the following observations regarding lattices hold?

The following is an excerpt of a recent paper on lattice cryptography: Let $n$ and $q$ be integers [...], and let $\beta > 0$ . Given a uniformly random matrix $A \in \mathbb Z^{n \times m}_q$ ...
1
vote
1answer
236 views

Convergence in metric and in measure

Let $\mu$ be a finite measure on $(X, A)$, with the semimetric $$ d(f,g) = \int \frac{|f-g|}{1+ |f-g|}d\mu$$ on all real-valued, A-measurable functions. Show that $$\lim_n d(f_n, f) = 0$$ holds iff ...
6
votes
2answers
1k views

Why is a projection matrix symmetric?

I am looking for an intuitive reason for a projection matrix of an orthogonal projection to be symmetric. The algebraic proof is straightforward yet somewhat unsatisfactory. Take for example another ...
4
votes
5answers
2k views

Probability of 2 Cards being adjacent

I read about a magic trick yesterday that relied on probability - I gave it a try a few times and it seemed to work, but I was wondering what the actual probability of success is. I understand basic ...
5
votes
2answers
138 views

Sequence and Series - If $a_n =\int^{\frac{\pi}{2}}_0 \frac{\sin^2nx}{\sin^2x}dx,$…

If $\displaystyle a_n =\int^{\frac{\pi}{2}}_0 \frac{\sin^2nx}{\sin^2x}dx, $ then find the value of $$\begin{vmatrix} a_1 & a_{51} & a_{101} \\ a_2 &a_{52} & a_{102}\\ a_3 & a_{53}...
3
votes
1answer
83 views

Cohomology calculation for maps to the 2-sphere.

Let $Y^3$ be a closed 3-manifold and $f\colon Y\to \operatorname{SO}(3)$, $g\colon Y\to S^2$ be smooth maps. Define $g'\colon Y\to S^2$ be the following composition: \begin{equation}Y\xrightarrow{...
6
votes
4answers
293 views

Alternating Recurrence relation $a_n = b_{n-1} + 5$ and $b_n = na_{n-1}$

I am racking my brain on solving the relation where: $$a_n = b_{n-1} + 5$$ $$b_n = na_{n-1}$$ where $a_0$ = $b_0$ = 1 I am trying to find the closed form for $a_n$. I have tried to shifting $b_n = ...
3
votes
0answers
109 views

free dimension of a module?

For any module $M$ there is defined the projective/injective/flat dimension, which is the length of the shortest projective/injective/flat resolution of $M$. Why isn't there defined a free dimension ...
0
votes
3answers
2k views

The definition of “span” and related theorem.

In wikipedia and the most of the linear algebra texts, The definition of the span is following as. "Given a vector space V over a field K, the span of a set S of vectors (not necessarily finite) is ...
7
votes
2answers
167 views

How much mathematicians rely on other results?

When mathematicians do research, do they think every detail so that they know exactly every step what it takes to achieve the result from axioms? Or is it possible to work as a researcher without ...
1
vote
0answers
139 views

Question On Modulo Notation

I understand what the Modulo/Modulus is and how it operates and everything, but a textbook is using some notation that I just can't quite grasp. Basically, we have the solution $(x,y) = (\pi($mod$(2\...
1
vote
1answer
227 views

How can the lyapunov exponents for the Mandelbrot Set be computed?

I am trying to find a way to calculate the Lyapunov exponents of the Mandelbrot set. There are some very nice diagrams that you can find on Flickr of a plot of the Lyapunov exponents of the Mandelbrot ...
3
votes
2answers
973 views

Every locally compact, second countable Hausdorff space has a countable basis of open sets with compact closure

Let $X$ be a locally compact, second countable Hausdorff space. I want to prove that it has a countable basis of opens with compact closure, and that this basis can be extracted as a subset of any ...
0
votes
2answers
60 views

Differentiable function (independent of three tetrameters)

Designate all triple of real numbers $A,B,C$ such that function $$f(x)= \begin{cases} \frac{e^{4x} + Ax + B}{x^2} \quad x \neq 0 \\ C \quad \quad \quad \quad x=0 \end{cases}$$ is differentiable in $...
3
votes
1answer
216 views

Can curvature be defined in Topos Theory?

I'm by no means an expert in either category theory or topos theory, but I'm trying to gain some perspective on traditional geometric ideas in this context. Topos theory claims that it is a geometric ...
7
votes
1answer
137 views

Introductory book on series

I'm looking for a good introductory book on sequences and series (mathematics). Do you have any advice?

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