1
vote
1answer
44 views

Integral calculation - where the $i$ came from?

$$ \frac{e^{it(c-n)}}{i(c-n)} |_{-\pi}^\pi = \frac{e^{i\pi(c-n)} - e^{-i\pi(c-n)}}{i(c-n)} =\frac{2\sin(\pi(c-n))}{i(c-n)}$$ Correct answer is:$$\frac{2\color{red}{i}\sin(\pi(c-n))}{i(c-n)}$$ Why? :/...
3
votes
3answers
189 views

How would you prove this theory of computation problem?

I have trouble proving the following statement, I'm supposed to do it for our theory of computation course but since I've been trying for days I'm looking for a hint : What is the smallest value ...
1
vote
2answers
67 views

Probability that colored balls are separated

Say we throw $b$ blue balls and $r$ red balls uniformly into $n$ boxes. The probability that no box contains a red as well as a blue ball is then, by the inclusion exclusion principle: $$p = \sum_{k,\...
1
vote
1answer
65 views

Prove that every finite subgroup of SL2 is subgroup of one of the following groups

I need to prove that every finite subgroup of $SL_2(\mathbb{Q})$ is a subgroup of one of the following groups: $D_3, D_4, D_6$. Let $G$ is a finite subgroup of $SL_2(\mathbb{Q})$, $ g\in G$. I can ...
0
votes
1answer
38 views

Proving that something is a diagonal?

Suppose that D is a diagonal matrix with entries λ1, . . . , λn and that S is an invertible matrix. Suppose that each column of S is a multiple of a standard basis vector (i.e. an $~e_i$). Prove that $...
1
vote
0answers
102 views

Intuition for metric space completion

Is this intuition for the completion of an arbitrary metric space (X,$\rho$) correct? I am trying to understand Royden's argument in his Real Analysis book. Construct $\tilde X$ as union of two ...
-1
votes
2answers
273 views

partnership problems [closed]

A,B and C started a business by investing Rs 7,000, Rs 5,000 and Rs 3,000 respectively. If they earned a profit of Rs 9,000 , find the share of A ? Note: Rs = Indian Rupees
0
votes
1answer
646 views

Finding the set of all points equidistant between two planes

I'm trying to study for an upcoming exam in my math class and I came across an interesting question that I'm not entirely sure about. "Let $H_1$ be the plane $x + 2y − 2z = 1$ and $H_2$ the plane $y ...
-1
votes
1answer
37 views

Lie algebras with different bases

I am interesting to know that if a finite dimensional Lie algebra $L$ has two bases $\beta_1$ and $\beta_2$, how can we compare the cardinal of two sets $\{(x,y)\in \beta_1\times \beta_1~|~[x,y]=0\}$...
1
vote
0answers
36 views

Equivalent statements involving 'little o'

Let $A$ be a Banach algebra and $a\in A$. $\|z(z-a)^{-1}\|=1+o(\frac{1}{z})$ as $z\rightarrow +\infty$ iff $\|(z-a)^{-1}\|^{-1}=z+o(1)$ as $z\rightarrow +\infty$. i.e., $lim_{z\rightarrow +\infty}\|(...
1
vote
0answers
19 views

How to solve this degenerate parabolic euqation

I want to solve the following equation \begin{align*} \label{linear-prandtl-transform-3} \begin{cases} & \partial_t \hat{w}(t, x, y) + \tilde{u(t, y)}\partial_x \hat w(t, x, y) - 2 \partial_y \...
0
votes
2answers
324 views

Putnam 1990 A1 Induction Help

A1. $(150,9,1,0,0,0,0,0,1,1,6,33)$ Let $$T_0=2,\quad T_1=3,\quad T_2=6,$$ and for $n\ge3$, $$T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.$$ The first few terms are $$2,3,6,14,40,152,784,5168,...
5
votes
1answer
94 views

Marbles in a bag (Combinatorics)

There are 10 numbered from 1 to 10 marbles. The marbles are placed in an opaque bag and shuffled. One random marble is taken out, its number is written on a piece of paper, marble is then returned to ...
0
votes
0answers
19 views

Simple odds calculation

I'm stuck with a simple expression creation problem. I'd like to express odds by removing values from $100$ and arriving at a number. Every variable I use has value, that can either be $+10$ or $-10$, ...
3
votes
3answers
84 views

Evaluation of $\lim_{n\rightarrow \infty} \int_{0}^{1}\frac{1}{1+x^2+x^4+…+x^{2n}}dx\;,$ Where $n\in \mathbb{N.}$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty} \int_{0}^{1}\frac{1}{1+x^2+x^4+.........+x^{2n}}dx\;,$ where $n\in \mathbb{N.}$ $\bf{My\; Solution:}$ First we will simplify $\displaystyle 1+x^...
11
votes
2answers
167 views

Prove $x^5-2$ is irreducible over $\mathbb{Z}_{31}$

I am currently studying for my Algebra Qualifying Exam. I came across the following problem and am stuck on where to even begin: Prove that $x^5-2$ is irreducible over $\mathbb{Z}_{31}$. I have ...
0
votes
2answers
47 views

Proving a set is a subgroup

Let $G$ be an abelian group. Let $n$ be a fixed integer, and let $H = \{x \in G: x^n = e\}$. Prove that $H$ is a subgroup of $G$. Identity is given. Let $x$ and $y$ be in $H$. Since $H$ is abelian, ...
0
votes
2answers
69 views

Quadratic Equation Roots Prove

I have a question in my textbook from chapter of quadratic equations from exercise of sum of roots and product of roots that; Prove that the equation $$ a x^2 + b x + c = 0, \quad a > 0 $$ has ...
8
votes
2answers
987 views

How to tackle the depression and frustration regarding math? [closed]

I'm a high school student and it is my last year at school. I've always loved math from the core of my heart. Last year, I prepared a lot for the National olympiad. I cleared the regionals smoothly ...
2
votes
1answer
251 views

The order of a $k$-cycle in $S_n$ is $k$.

Here's what I have right now: The order of a $k$-cycle in $S_n$ is $k$. Proof. Let $\sigma$ represent the $k$-cycle $$\sigma=(x_1 \ x_2 \ \cdots \ x_k)$$with distinct elements $x_i$. Note that the ...
1
vote
3answers
105 views

Projection onto subspaces - point to line projection

In the following document about projection onto subspaces, the author is computing the transformation matrix to project a vector $b$ onto a line formed by vector $a$. Since the projected vector $p$...
1
vote
2answers
86 views

Integrate $ \int \frac{x^2}{x+1} dx $

How can I integrate by changing variable or by parts? $$ \int \frac{x^2}{x+1} dx $$ Thanks a lot
2
votes
1answer
89 views

Approximate a positive multivariate function with a sum of squares of polynomials?

I am constructing approximation to a multivariate function which I know is positive. My question is the following: Let $f(x)$ be a multivariate positive and continuous function. Can we approximate ...
0
votes
1answer
401 views

Odds of winning a two part drawing

There is a local drawing that involves being selected out of an estimated 6000 entries, and then correctly selecting 1 of 3 numbers in order to win. The numbers have are actually cards in a deck that ...
0
votes
1answer
69 views

Big-O complexity of iterating over every substring

What is the Big-O complexity of iterating over every possible non-empty substring of a string of length $N$? The simple way to do the iteration over string $S$ is: ...
1
vote
1answer
124 views

Showing Weierstrass Elliptic Function is meromorphic

Consider the Weierstrass Elliptic function, $$\rho(z) = \frac{1}{z^{2}} + \sum\bigg(\frac{1}{(z-m-nw)^{2}} - \frac{1}{(m+nw)^{2}}\bigg), $$ where $m,n \neq 0,0 $. How can one show that it is ...
0
votes
1answer
49 views

Can we make $(1/n)$ converge to any real $r$ w.r.t. a suitable metric ? and other related issues

For every $h \in \mathbb Z$ , I can construct a metric $d_h$ on $\mathbb R$ such that $\Big(\dfrac 1n \Big)$ converges to $h$ w.r.t. the metric $d_h$ , indeed I consider a function $f: \mathbb R \...
2
votes
1answer
76 views

A question about the definition of a stalk

In the definition of the stalk of a presheaf of abelian groups on a topological space, at a point, one uses the fact that the open sets containing that point form a poset, which is directed via ...
1
vote
0answers
42 views

If $ z_{0}=x_{0}+iy_{0}$ satisfy the equation $2\left|z_{0}\right|^2=r^2+2.$ Then $\left|\alpha \right| = $

Let a Complex no. $\alpha$ and $\displaystyle \frac{1}{\bar{\alpha}}$ lie on the Circles $\left|z-z_{0}\right|=r$ and $\left|z-z_{0}\right|=4r^2$ respectively. If $\displaystyle z_{0}=x_{0}+iy_{0}...
0
votes
2answers
151 views

A is a proper subset of B implies NOT(B subset of A) Proof

This is not homework. I'm just studying for my Discrete Mathematics course. I'd like to know how to prove the following using element wise proofs: Given a proper subset: $$A \subset B \iff (A \...
0
votes
1answer
58 views

Totally geodesic submanifold

I'm reading "Introduction to symplectic topology", D.McDuff, D.Salamonand and I have a problem with the exercise 1.26. According to the definition, a submanifold $L$ of a Riemannian manifold $(\mathbb ...
6
votes
4answers
97 views

How to prove this limit is $0$?

Let $f:[0,\infty )\rightarrow \mathbb{R}$ be a continuous function such that: $\forall x\ge 0\:,\:f\left(x\right)\ne 0$. $\lim _{x\to \infty }f\left(x\right)=L\:\in \mathbb{R}$. $\forall \...
0
votes
0answers
80 views

Is it correct to say that if $\lim\limits_{x \to a}f(x) = 0$ it is an Infinitesimal?

I think I'm misuderstanding something here, because to my understanding the definition of infinitesimal given in my textbook does not convey the same thing as in other sources. I've read the ...
1
vote
1answer
34 views

Do we have $proj_u(a) + proj_u(b) = proj_u(a+b)$?

Let $a, b, u$ be vectors in $\mathbb{R}^3$. For two vectors $r, u$ in $\mathbb{R}^3$, let $proj_u(r)$ be the projection of $r$ on the line of $u$ in $\mathbb{R}^3$. Do we have $proj_u(a) + proj_u(b) = ...
5
votes
2answers
3k views

Difference between sum and direct sum

What is the difference between sum of two vectors and direct sum of two vectors? My textbook is confusing about it. Any help would be appreciated. Thanks in advance!
1
vote
1answer
47 views

Is my proof for an invertible matrix correct?

I'm a bit confused in class over some rules for proofs, so I was wondering if this was the correct proof for the following question: Suppose that $A$ and $B$ are similar $n\times n$ matrices and ...
1
vote
0answers
33 views

Liapunov 's Direct Method $(S.L.Ross )$

Consider the Autonomous Non-Linear system $$ \frac{dx}{dt} = P(x,y),$$ $$ \frac{dy}{dt} = Q(x,y).$$ Assume that the given system has an isolated critical point $(0,0)$. Let $P \text{ and }Q$ ...
0
votes
2answers
62 views

Probability of a disintegration

The half-life of Uranium-$238$ is $5×10^9$ years. What is the probability than a uranium atom disintegrates in any one year? I think I have to use Poisson's law but I don't know how to apply it in ...
0
votes
1answer
51 views

Closure function of a matroid

I need some help to understand: If $M$ is matroid and e is an element in that matroid, what is the closure function of $M\setminus e$? And what is the closure function of $ M/e$? Any help would be ...
2
votes
3answers
110 views

$y''+y'^{2}+y=0$ equation solution

How would you solve this differential equation $y''+y'^{2}+y=0$? I can't apply the ansatz method (or more formally apply the characteristic polynomial method). Thanks
1
vote
2answers
33 views

Let $f: \Bbb R \to \Bbb R$ be such that $f^{-1} (a, \infty)$ and $f^{-1} (- \infty, b)$ are open for any $a,b \in \Bbb R$.

Let $f: \Bbb R \to \Bbb R$ be such that $f^{-1} (a, \infty)$ and $f^{-1} (- \infty, b)$ are open for any $a,b \in \Bbb R$. Show that $f$ is continuous. My Try: We first take an arbitary open subset $(...
0
votes
1answer
112 views

Problem in Gradient operator and Kronecker delta function

I have this expression $$\nabla_{i}\nabla_{j}\Big(\frac{1}{r}\Big)$$ Where $r$ is a distance. I tried this, but encountering manipulations of $\delta_{ij}$ with $\hat{r_i},\hat{r_{j}}$ and still ...
0
votes
1answer
51 views

Characteristic polynomial of recurrence relation $\lambda^4 + \lambda^3 - 9\lambda^2 + 11\lambda -4 = 0$

The characteristic polynomial of this recurrence relation is $$λ^4 + λ^3 - 9λ^2 + 11λ- 4 = 0$$ or $$(λ − 1)^3 \cdot (λ + 4) = 0.$$ So the solution is of the form $a_n = α({-4})^n + β~n^2 + γ~n + δ.$...
1
vote
3answers
43 views

Prove that f is continuous [closed]

Let F a set non-empty and closed. Give $x$∈$\mathbb{R}$ and let f(x)=inf{|x-y|, y∈F }.Prove that f is continuous and {$x$∈$\mathbb{R}$, f(x)=0} = F.
4
votes
1answer
35 views

For which real $x$ is this (monster) series convergent?

I'm practicing for an exam, and got to this example: $$\sum_{n=1}^{+\infty} \left (\frac{x^2n^2-2|x|^3n}{1+2xn^2} \right)^{7n}$$ I rearranged the expression to try to check for which $x$ the ...
2
votes
0answers
79 views

Hilbert style proof for $\Box A \vee \Box B \rightarrow \Box(A\vee B)$ in K.

I have to find a formal Hilbert style proof for $\Box A \vee \Box B \rightarrow \Box(A\vee B)$ on modal logic, K. I can use all classical propositional tautologies, Modus Ponens and Distribution axiom....
1
vote
2answers
435 views

Doing Michael Spivak's Exercises

I am doing Spivak's Calculus, and I find it EXTREMELY difficult. I usually ask questions here because I cannot do the problems on my own. How long should it take to do a Spivak problem? Is it ...
0
votes
3answers
54 views

Meaning of derivatives

I need to know the meaning of the higher order derivatives of a polynomial. Let make an example. Assume we have a polynomial of degree n. Then $$ f(x)=a_0+a_1x+a_2x^2+\ldots+a_nx^n $$ We know that ...
4
votes
2answers
798 views

Integration by parts using u-substitution and square root

I have to use the technique of integration by parts to evaluate the integrals. I'm having trouble with a particular problem: $$ \int x (\sqrt{x+2}) dx $$ I'm using u-substitution, but since I'm also ...
0
votes
1answer
22 views

Volume of Revolution $f(x) = x^2$

Suppose you are given $y = f(x)$ I want to use double integrals, instead of the traditional washers. Suppose even better, $f(x) = x^2$ Find the volume of $f(x) = x^2$, $x = 0$, $x = 4$, $y = 0$ &...

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