-2
votes
2answers
29 views

Solving Integral that includes radical expression

I need to solve this integral analytically. I used many methods but I can’t solve it. Please help me. Thank you $$\int\sqrt{x^4-c}\ dx$$ http://i62.tinypic.com/15heux1.png
0
votes
0answers
13 views

Path-components of the general linear group using only elementary algebra

Let $E(c)$ be an elementary matrix of the type to add $c$ times a row to another row when applied to another matrix on the left (with $c$ in some off-diagonal position $(i, j)$), and, with the usual ...
1
vote
1answer
16 views

Span the space of all polynomials of highest order?

I have 2 question in my homework I am not sure my answer. Can someone help me to solve it? Let $v_1 = t^2 + 2t + 1$ and $v_2 = t^2 + 2$ , where $t$ is a real number. Determine whether $v_1 , v_2$ ...
0
votes
1answer
30 views

Mentioning Professors in math PhD applications

When applying to grad school, is it a good idea to mention professors you want to work with? For example, "I am applying to this program because University X has leading experts in the field Y, such ...
0
votes
3answers
17 views

Greatest area using a string with the length of $l$

Suppose we have a string with length of $l$ what is the shape that has highest area? In other words,with a constant perimeter of $l$ what is the shape with the highest area? P.S:My own speculation ...
0
votes
0answers
8 views

Chain rule version for partiel derivative?

Non-math student here so go easy on me. How do we calculate a partial derivative in terms of $x$ when dealing with a multivariable composite function, and what 'chain rule version', if any, could one ...
2
votes
0answers
15 views

Finding elements in a set

Definitions You have a set $S$ with $n$ elements. Within these $n$ elements, we denote $2$ as "special numbers". Given a subset $T\subset S_n$ we say that the subset $T$ is "up" if it contains both ...
0
votes
0answers
10 views

Galois group of the field of all constructible complex numbers

I am trying to understand infinite galois theory by self-made examples. First example I am struggling with is the field of all constructible (by compass and straightedge) complex numbers - let's call ...
1
vote
2answers
19 views

$\Bbb R^n \times (0,\infty)$ what does this mean?

Just began to read about PDEs. There are a whole list of notations which I don't understand and the book isn't expecting a reader who is as inexperienced as me. Also don't understand what this ...
2
votes
2answers
28 views

What is an example of a homomorphism of rings that doesn't preserve gcd's?

Given a commutative ring $R$, we say that $x$ is a gcd of $(y,z)$ iff the following conditions hold: $x \mid y,z$ For all $x' \in R$, if $x' \mid y,z$, then $x' \mid x$. This gives a ternary ...
0
votes
0answers
6 views

T invariant subspace complement

Let $T$ be a linear operator on a linear space $V$ of finite dimension. How to prove that $\operatorname{ran} T$ has a complement T-invariant (i.e. a T-invariant subspace $W$ of $V$ such that ...
1
vote
1answer
21 views

Proving divisibility of $\sum\limits_{r=1}^{p-1} {r^{p^n}}$ by p.

Let $p>2$ be an odd number and let $n$ be a positive integer. Prove that $p$ divides $${\sum\limits_{r=1}^{p-1}{r^{p^n}}}$$ My Proof: From multinomial expansion, we know that $${(1 + 2 + 3 + ... + ...
0
votes
0answers
23 views

Number of m letter pairs with a vowel in the alphabet

The number of ways to choose m unique pairs of letters from the alphabet is: $$ \frac{26!}{(26-2m)! m! 2^m} $$ Which gives 325 single pairs, 44850 double pairs, 3453450 triple pairs... If I want to ...
0
votes
1answer
21 views

Closed Form Expression of sum with binomial coefficient

I have the following equation which is making me problems. $$A_{n} = \sum_{k=0}^{n} \binom{n-k}{k}(-1)^{k}$$ where $n\in\mathbb{N}$. The task is to find a closed form expression for $A_{n}$. I have ...
0
votes
2answers
38 views

show that $\mathbb{Z}$ is totally disconnected

Show that $(\mathbb{Z},d)$ is totally disconnected (where $d$ is the metric induced by the Euclidean metric on $\mathbb{R}$). I think that to prove this I should use contradiction but not really sure ...
7
votes
3answers
133 views

For what functions is $y'' = y$?

What functions $y = f(x)$ have the property that $f(x) = f''(x)$, i.e. what functions have the same integral and derivitive? I could think of $ce^x$ and $ce^{-x}$ (where $c$ is a constant), but are ...
1
vote
0answers
8 views

Othonormal basis for $L^2$ space on square

Can one describe an orthonormal basis for $L^2(\gamma,ds)$ where $\gamma$ is the square with vertices at $(1,1),(-1,1),(1,-1),(-1,-1)$ and $ds$ is the arc length. To be more precise can we express ...
1
vote
5answers
61 views

Why $\frac{x}{\sqrt{x+1}-1}$ can be written as $\sqrt{x+1}+1$?

I've evaluated the first formula on $W|A$ and it says that $\sqrt{x+1}+1$ is an alternate form to the first expression. I just don't see how it's possible. The first thing I imagined was to write: ...
0
votes
0answers
6 views

Transition rates and probabilities of a continuous markov chain

A certain type of component has two states: 0 = OFF and 1 = OPERATING. In state 0 , the process remains there an exponential amount of time with rate $ \alpha$, and then moves to state 1. The ...
0
votes
0answers
8 views

How do I find the rate of change in area of rotating loop?

Say we have a rectangle of side lengths $a$ and and $b$ that is rotating with one of its sides $a$ about a fixed axis with angular speed $v$. Consider a line $l$ a distance $x$ from the axis of the ...
2
votes
2answers
50 views

Prove that the following is a Field

Let $p$ be a prime Let $n$ be an element in $Z_{p}^*$ where $n\not=\pm1$. Define a ring structure on $F = Z_p \times Z_p$. We define the addition by $$ (a_1,b_1) + (a_2,b_2) = (a_1 + a_2, b_1 + ...
1
vote
0answers
13 views

Examples of Non-Markov process with continuous time and finite set of states.

What is the best real world examples of non-Markov process with continuous time, but with finite set of states?
2
votes
0answers
13 views

Positive definiteness of a matrix whose entries are continuous

Let $A$ be a $n \times n$ real matrix whose entries $a_{i,j}(x)$ are continuous functions from $\mathbb{R^n} \to \mathbb{R}^n$. Suppose that $A$ is positive definite at $x = x^\star$. Then, is it ...
1
vote
0answers
7 views

Does every Young diagram have a unique minimal major index?

Given a Young diagram, $Y_\rho$, corresponding to an irreducible complex representation $\rho$ of the symmetric group $S_n$, we can associate a set of major indices $\{ ...
3
votes
1answer
42 views

Trigonometric tough limit

I am still struggling with this one. Can't figure it out... $$\lim_{x\to 0} \frac{1-\cos x}{x^2}$$ I tried $\cos x = \sqrt{1 - \sin^2{x}}$
0
votes
0answers
6 views

Cramér Lundberg Risk Model - exponential distribution of claim sizes

I am studying the classical ruin model, which express the insurance company free surplus at time $t$ as $C_t=u+ct-\sum_{i=1}^{N(t)}Y_i $ where: $ct$ is the premium income up to time t $u$ is the ...
2
votes
1answer
19 views

Binomial sum identity

Everyone knows that $\sum{n \choose 2k}=\sum{n \choose 2k+1}$ My question is follwing What is the clear form of $\sum_{k=0} (-1)^k{n \choose 2k}$ and $\sum_{k=0} (-1)^k{n \choose 2k+1}$ For ...
2
votes
1answer
18 views

How to find cubic residues $\bmod p$ using WolframAlpha?

How to find cubic residues $\bmod p$ using WolframAlpha? Just type in "quadratic residues modulo p" and you're done, but typing in "cubic residues modulo p" does nothing. Logically, "x^3 ...
1
vote
0answers
16 views

Calculate the right Riemann sum to approximate the area of the region bounded by $f(x) = 25 - x^2$ on the interval $[-5, 5]$.

I'm attempting to calculate the right Riemann sum and approximate the area of the region bounded by $f(x) = 25 - x^2$ on the interval $[-5, 5] = [a, b]$. $$\sum_{k = 1}^{n}{f(a + k\Delta x)}\Delta ...
0
votes
1answer
10 views

An extension of a corollary to Fuglede's theorem

Fuglede's theorem states that if $T,N\in B(H)$ for some Hilbert space $H$ and $N$ is normal and $TN = NT$, then $TN^* = N^*T$. A corollary to this theorem is that if $M,N \in B(H)$ are normal and ...
0
votes
2answers
21 views

Find the radius of convergence of $\sum_{n=0}^{\infty} \frac{\ln(n+1)} {n!} (z+2)^n$

I am having a little difficulty with this. I need to find the radius of convergence of this problem: $$\sum_{n=0}^{\infty} \frac{\ln(n+1)} {n!} (z+2)^n$$ Using the root test I have ...
0
votes
1answer
17 views

Integrating a second order non homogeneous ODE

I took an exam and the teacher didn't solve this problem during the correction. I need to solve $$y''(x)-y(x)=\sin (e^x)$$I was able to find the solution to the homogeneous equation ...
2
votes
1answer
23 views

Inequality which is true for almost every n

In my assignment I have to prove the following statement: Let $l$ be a natrual number. Prove that for almost every $n$ the following inequality is true: $n\lt\sqrt{n ^ 2 + l}\lt n+1$ I chose to ...
1
vote
1answer
15 views

Real Canonical Form

Question: Let us consider the quadratic form $q: R^3 -> R$, $$q(x,y,z) = x^2+25y^2+10xy+2xz$$Find the corresponding symmetric bilinear form $f$ and a basis $B$ such that $[f]_B$ has the real ...
9
votes
3answers
32 views

Shared eigenvectors between $A$ and $A^k$

$\newcommand\la{\lambda}$ Thanks to the spectral mapping theorem, we know that if $\la_1,\ldots,\la_n$ are the eigenvalues of a $n\times n$ complex matrix $A$, then $\la_1^k,\ldots,\la_n^k$ are the ...
3
votes
0answers
19 views

Find out in how many ways the people left the floor.

Seven people enter a lift. The lift stops at three (unspecified) floors. At each of the three floors, no one enters the lift, but at least one person leaves the lift. After the three floor stops, the ...
3
votes
1answer
32 views

$x^p -x-c$ is irreducible over a field of characteristic p if it has no root in the field

Let $c$ be an element of a field $F$ of characteristic $p$ (prime). Then how to show that $x^p -x-c$ is irreducible over $F$ if it has no root in $F$. I was trying using contradiction and by ...
1
vote
0answers
16 views

Rotate basis to align with vector

I have a coordinate system with the Basis $B=(e_x, e_y, e_z)$ and two vectors $r$ and $a$. Now, I want to rotate the basis so that the $e_x$ unit vector points in the direction of the vector ...
1
vote
1answer
13 views

How to find a basis for the null space

$A$ is a $3x3$ matrix of rank $2$. The system of equations $Ax = [3\,\, 5\,\, 7]^T$ has infinitely many solutions, including $x = [1\,\,2\,\,3]^T$ and $x = [4\,\,4\,\,4]^T$. A basis for the null ...
4
votes
3answers
41 views

Evaluating $\int_{\sqrt{2}}^{\sqrt{5}} \frac{x^3}{\sqrt{x^2-1}} dx$ by substitution

$$\int_{\sqrt{2}}^{\sqrt{5}} \frac{x^3}{\sqrt{x^2-1}} dx$$ $u^2 = x^2 - 1$ I have worked out that $dx = du$ and that $u = x - 1$ so, $\int\frac{x^3}{u} du$ - but I'm stuck at this stage. Any ...
3
votes
0answers
41 views

Understanding proof of fundamental theorem of algebra

So this is the proof I have: If $p(z)$ is a non-constant polynomial, then there exists a $z \in \Bbb Z$ such that $p(z) = 0$. Let $p(z) = z^n + a_{n-1}z^{n-1} +a_{n-2} z^{n-2} + ... + a_0$ ...
2
votes
0answers
15 views

Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix?

I have a real symmetric $3\times3$ matrix $\mathbf{M}(\mathbf{r}$) which depends on $\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field $e_i(\mathbf{r})$ over ...
-3
votes
0answers
23 views

Statistics Random Variable [on hold]

Please help me with this question for stats
1
vote
1answer
19 views

Uncertain about which probability method to use for the problem

Suppose I want to catch a bus (which runs every 10 minutes on average). What is the probability that: 1). You will wait for at least fifteen minutes before the bus arrives, and then, 2). 3 buses ...
1
vote
0answers
6 views

Selfadjoint Operators: Sesquilinear Form (III)

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ and an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Denote for shorthand: ...
0
votes
1answer
19 views

Relation between finite Abelian Groups and traces?

I have recently read Kronecker's 1870 paper on finite Abelian groups, on the definition of abstract group and so on. It turns out that such definition is literally taken over (being probably unaware ...
2
votes
0answers
24 views

Adjoints functors in scheme theory

What's useful information available to us, when we state that : If $ f: X \to Y $ a morphism of schemes, and if $ \mathcal{F} $ denotes $ \mathcal{ O }_X $ - modules, and $ \mathcal { G} $ denotes $ ...
2
votes
1answer
18 views

Conditional expectation on Bi variate probability distribution

Two balls are drawn simultaneously from an urn containing 2$0$ balls numbered $1,2,3...20$. let $X$ be the number on one of the drawn ball and $Y$ be the number on the other. Find $E(XY)$. I try to ...
1
vote
0answers
26 views
1
vote
2answers
20 views

Kernel of a matrix with a 0 column

I'm trying to find the $$ ker\begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 4 \\ \end{bmatrix} $$ I would usually write the linearly ...

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