1
vote
0answers
9 views

Recursive Equation.

We have a recursive function: $$a_n = Aa_{n-1} + Ba_{n-2} $$ We assume that $ x ^ n = a_n $ From this equality quadratic equation we have two solutions $$ \alpha, \beta, \alpha \neq \beta $$ In that ...
0
votes
1answer
14 views

Confidence interval - No sample

"According to thorough measurements, the average waiting time at a restaurant is normally distributed with an average of $28$ minutes and a standard deviation of $5$ minutes. Calculate a $95\%$ ...
0
votes
0answers
3 views

Question about isotypical components

Consider $V=\bigotimes^3(\mathbb{C}^2)$ as a $\mathfrak{S}_3$ representation. One of its isotypical component is $S^3(\mathbb{C}^2)$, which is a linear subspace of symmetric tensors of ...
-1
votes
2answers
27 views

High school Math Team problems.

In triangle $\triangle ABC$, $AB=5,BC=6$ and $AC=7$. The circle with diameter $AB$ intersects $BC$ at $D$, ($D$ doesn't equal to $B$). Compute $BD$.
1
vote
1answer
12 views

A function which is not identically zero has positive integral for some ball.

Assume that $f$ is integrable on $\mathbb{R}^d$, and $f$ is not identically zero. The hint in my book is telling me that there exists some ball such that $\int_{B} |f| > 0$ Suppose $f= ...
0
votes
1answer
20 views

infimum and supremum finding

Find the $\sup$ and $\inf$ of $E=\{x\in\mathbb{R}\mid 1-\frac{1}{n} < x < 2-\frac{1}{n}, n \in \Bbb{N}\}$. Justify your answers I claim that $\sup E = 2$ and $\inf E = 0$. Let $(1-1/n,2-1/n]$, ...
0
votes
0answers
16 views

how much certain mathematician has been cited in recent articles

I want to know how much Galois work has been cited in recent articles. So I'm looking for a tool that can help me do that, more simply is there a tool to know how much certain mathematician has been ...
1
vote
1answer
6 views

How to solve transcendental hyperbolic equation

How can I solve the functional relation $$ e^{-af(x)}\cosh( f'(x) ) = bx $$ for $f(x)$? It would suffice to solve for $x>0$, $a>0$ and $b>0$.
0
votes
1answer
10 views

Is $Ape_1+Aqe_2$ where A (3x3) matrix, considered as a linear combination of $e_1,e_2$

$$\alpha=-8$$ Eigenvectors: $$e_1 = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} \text{ and } e_2 = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$$ What I did : (i) $x ∈ V \implies x$ of the ...
0
votes
0answers
12 views

Normal Distribution E(X^4)?

So i have the Normal Distribution f(z)=(1/sqrt(2pi))*e^((-z^2)/2) I know any E(Z^(any odd #)) makes you integrate an odd function thus giving an answer of zero (i.e. E(Z^1) and E(Z^3) both =0). And ...
0
votes
0answers
5 views

SPDE solution with Girsanov Theorem

can someone help me with this exercise? How can I show using the Girsanov Theorem that $u$, defined as: ...
0
votes
0answers
17 views

Is this the wrong way to find $\delta$?

Find $\delta$ such that $$|x^n - a^n| < \epsilon $$ whenever $$|x-a|<\delta$$ My thought is factoring $|x^n-a^n|$ so we have $|(x-a)|\cdot|x^{n-1}+x^{n-2}a+x^{n-3}a^2+...+xa^{n-2}+a^{n-1}|$ In ...
0
votes
0answers
8 views

Checking if transformation T(p(x)) is diagonalizable?

Say you have a transformation of $P_3$ to $P_3$ defined by, say, T(p(x)) = p'(x) + p''(x) + p'''(x). How would you determine if this is diagonalizable? Do I sub in a standard basis of {1,x,x^2,x^3} ...
0
votes
0answers
11 views

$\mathbb{E} e^{t S_n}=e^{C(t^2 \sigma^2 )}$

Let $S_n=X_1 + \cdots+ X_n$ be a sum of independent random variables such that each $X_i$ has mean zero, variance $\sigma_i ^2$ and lies in $[-1,1]$. Why does then $$\mathbb{E} e^{t S_n}=e^{C(t^2 ...
1
vote
0answers
5 views

Equality of regular language

I am showing that two regular expressions are equal. In one of intermediate steps, I thought of the equaility: $$(RS+R)^* = (RS)^*R(RS)^*R(RS)^*....R(RS)^* = R^*(RS)R^*(RS)...R^*$$ Is it true? I ...
1
vote
2answers
28 views

Why can we take the log of both sides?

I was watching a video that proves the "Log of a power" rule. I'm just having trouble understanding the loga(a^x) = x rule - which he uses in the proof And I don't get why you can log both sides. ...
1
vote
1answer
12 views

Generalisations of primes

I've read of (normal) primes, Gaussian primes and Eisenstein primes, which all uses different ways to define an integer to be a prime. For instance, $2$ factors into $1-i$ and $1+i$ for guassian ...
0
votes
2answers
29 views

Prove $\int_0^{2\pi}\frac{3a\sin^2\theta}{(1-a\cos \theta)^4}$ or $\int_0^{2\pi}\frac{\cos\theta}{(1-a\cos\theta)^3}=\frac{3a\pi}{(1-a^2)^{5/2}}$

While doing some mathematical modelling of planetary orbits I have come up with two definite integrals $D_1$ and $D_2$ which appear to produce the same result R when tested with various values of $a$ ...
1
vote
0answers
10 views

isotopy of homeomorphisms of a torus

Let's consider some homeomorphism of a torus which is isotopic to identity. Is it possible to construct an explicit isotopy?
1
vote
1answer
18 views

Lp spaces are nested but then why is 1/x square summable but not summable?

If $1\leq s<r<\infty$ and $f\in L^r$ then $f\in L^s$, so then why is $\frac{1}{x}$ not in $L^1$ but is in $L^2$ for the counting measure $c:\mathbb{N}\rightarrow \mathbb{R}$?
0
votes
1answer
5 views

Generalization of the birthday problem with convergence in distribution.

I have independent identically distributed random variables $Y_1, Y_2, ...$ that are uniformly distributed on the set $\lbrace1,2,...,n\rbrace$. I define $X^{(n)}=min\lbrace k:Y_k=Y_j$ for some $ j ...
2
votes
0answers
8 views

Geometric meaning of intersection multiplicities?

I am wondering about the geometric significance of the intersection multiplicity of two curves as defined in Hartshorne 5.4 (The length of $O_p/(f,g)$ is the intersection multiplicity of $Z(f)$ and ...
1
vote
1answer
13 views

Why Does This Summation Pattern Occur?

This is probably something stupid and for some reason I can't see it, but let's say you are taking the sum of the first $10^{i}$ squares. If $i=1$ we get $385$,$i=2$ we get $338350$, $i=3$ we get ...
0
votes
1answer
14 views

Normal tree with $\aleph_1$ nodes and no branch of cardinality $\aleph_1$

I am trying to understand how a normal tree with $\aleph_1$ node can fail to have a branch of cardinality $\aleph_1$. Consider the tree of height $\omega_1$ whose nodes are $\mathbb{Q}$-valued ...
1
vote
0answers
13 views

Elimination theory in Hartshorne

Does anyone know a good reference for elimination theory (Theorem 5.7A) mentioned in Hartshorne? The reference he gives is Van der Waerden modern algebra volume two, but it didn't feel locally ...
0
votes
0answers
10 views

Tangent to circle + bisecting an angle

Given a triangle ABC, and a circle centered at A such that B & C are outside the circle. How can I find a point Q on the circle such that QM is tangential to the circle, and bisects angle BQC? ...
0
votes
2answers
35 views

How many 7-digit ID numbers do not contain three consecutive sixes.

I have a homework question in a discrete mathematics class that asks me to determine how many 7-digit id numbers do not contain three consecutive sixes. It seems clear that I should approach this by ...
0
votes
1answer
10 views

$E[X^2]$ of the Beta Distribution

So I know the Beta distribution is $$f(x) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\cdot x^{a-1}\cdot(1-x)^{b-1}$$ I know the $E[X^r] = \dfrac{\Gamma(a+b)\Gamma(a+r)}{\Gamma(a)\Gamma(a+r+b)}$ And I ...
0
votes
0answers
6 views

Height of a Trapezoid given diagonals

The diagonals of a trapezoid have lengths 17 and 15, and the segment connecting the midpoints of the bases has a length of 4. The height of the trapezoid can be expressed as (x√y)/y Find x + y. I ...
0
votes
0answers
9 views

Game placing numbers in increasing order

Let $k\leq m\leq 100$ be positive integers. Aaron and Britney play a game on a $1\times m$ board, using $100$ paper pieces numbered from $1$ to $100$. The game has $k$ turns. In each turn, Aaron ...
1
vote
0answers
15 views

Statement about $(I-A)^{-1}$ matrices

Let $A \in \mathbb{R}^{n \times n}$ and let denote $I$ the $n \times n$ identitiy matrix. Theorem. If $(I-A)$ is invertible and $(I-A)^{-1}$ is a nonnegative matrix and there is a diagonal element in ...
1
vote
1answer
29 views

A tough integral and its generalization:

I happened to encounter an integral, a definite while I was walking the other day: $$ \int\limits_0^{\pi} \frac{ \sin ( 100 t ) }{\sin t } dt $$ I have tried the usual methods and nothing. I have ...
1
vote
2answers
23 views

How to get the radii for this volume integral?

I need to find the volume of the solid obtained by rotating the region bounded by the curves $y=x, y=0, x=2$, and $x=4$ around the line $x=1$ I know I need to integrate $\pi*((\text{outer radius})^2 ...
3
votes
3answers
57 views

A surprisingly resistant elementary numerical inequality

Let $a$ and $b$ be real numbers such that $a\geq 1$ and $b$ lies in the interval $[0,a-1]$. How can I then prove that $$(a-b)^{a-b}\geq a^{-b-a}\quad ?$$ This innocent looking inequality seems ...
0
votes
0answers
7 views

What is it called when we interpolate a point INTO a grid…

I suspect there is a terminological mish-mash going on in my understanding here: Consider a uniform 2D grid, where each $(x,y)$ value on this grid has a corresponding value. So, if I want to find ...
3
votes
1answer
26 views

Find $ \lim_{x\rightarrow 0}\frac{\tan^2 x+2x}{x+x^2} $ without L'Hôpital's Rule

I need to find $$\lim_{x\to 0}\frac{\tan^2 x+2x}{x+x^2}$$ This is what I did: Let ...
0
votes
0answers
7 views

Books for difficult questions on limits, continuity, differentiability of functions of two variables

i have done basic limits conti diff questions on functions of two variables but need some more difficult questions or question bank for these topics for my upcoming exam .
0
votes
0answers
5 views

Show that the function of 2 variables in greater or equal than 0 on specific set.

$D=\{(x,y) \in \mathbb{R^2} : \frac{x^2}{2} + y^2 \le 1 \} $ $ f : D \rightarrow \mathbb{R}$ is continuous and differentiable inside $Int(D)$ $ (x,y) \in \partial D \Rightarrow f(x,y) \ge 0$ $ ...
2
votes
1answer
18 views

Is there a simpler approach to this application of Dominated Convergence?

For a measure theory class, I'm trying to evaluate: $$\lim_{n\to\infty}\int^\infty_1\frac 1 {nx} e^{-x/n}\ \text d\lambda$$ Obviously I want to try and move the limit through the integral and ...
0
votes
1answer
17 views

$−8\sin 3x+5\cos 3x=4.3$ for $0<x<360.$

can you tell me please how to solve $−8\sin3x + 5\cos3x = 4.3$ for $0<x<360$? I find 6 solutions! and I don't know if they are the correct, although they seem to fit in the equation! thanks!
0
votes
1answer
33 views

If the inner product of Ax with x is 0 for all x, then A=0.

Given matrix $A\in M_{n}(\mathbb{C})$, if $\left<Ax,x\right>=0$ for all $x\in \mathbb{C^n}$, then $A=0_{n}$. (Here $\left<a,b\right> = b^{\ast}a$ where "*" is the conjugate transpose.) ...
0
votes
0answers
4 views

Gaussian Elimination Pseudocode to MATLAB script

So I'm having trouble understanding this pseudocode. It says that n is the size of the coefficient array, A(i,j) is the n-by-n coefficient array and l(i) is the index array l. I'm having trouble ...
0
votes
0answers
10 views

How to get state space equation for given matrix

state space equations (any realization) How to write state space equation for this matrix having two transfer function on it. [(s2 + 1)/(s2 + 5s + 6) (s + 2)/(s + 3)] I also need to check this ...
1
vote
0answers
28 views

what is the limit of f(n)/g(n)?

I am trying to solve this problem imagine that \begin{array}{lr} f(n) = 2^{\dfrac{1}{\sqrt{5}}\left[\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n} - \left(\dfrac{1 - \sqrt{5}}{2}\right)^{n}\right]} \\ ...
2
votes
0answers
33 views

Roots of $z^6+6z+10=0$

Find the number of roots of $z^6+6z+10=0$ in each quadrant. I want to use the argument principle. In the first quadrant, $f(z)=z^6+6z+10$ can be written as $f(R e^{i \theta})= R^6 e^{i6\theta}\{ ...
0
votes
0answers
16 views

Two proofs of cartesian product in topological space

Let $(X \times Y, \tau)$ be cartesian product of topological spaces $(X, \tau_X)$, $(Y, \tau_Y)$. Let $ A \subset X$, $ B \subset Y$. A) Prove that $\overline{A\times B}= \overline{A} \times ...
1
vote
1answer
25 views

Cashier has no change… catalan numbers.. probability question

I think this question uses catalan numbers.. but I don't know exactly how to answer it... its not homework or anything but I need to understand how to do it.. I tried drawing up likes for each 5r ...
0
votes
1answer
25 views

For every prime of the form 6x-1 are there comparable number of primes of the form 6x+1

All primes except $2$ and $3$ are of the form $6x-1$ and $6x+1$. For every prime of the form $6x-1$ are there comparable number of primes of the form $6x+1$ in the first $10000$ primes or is there an ...
0
votes
0answers
12 views

Clifford algebra - Gamma matrices

Let's say we have $\gamma^{a}$ matrices $(a=1,2,...,D)$. They satisfy the following condition $$\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\delta^{ab}I^{N\times N}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ ...
0
votes
2answers
17 views

almost every where property

I have question ,let $f$ is measurable and integrable i.e $f\in(\Omega,\mathbf{A}, \mathbb{R})$ and for all $A\in \mathbf{A}$ $\int_Af(\omega)d\mu=0$ show that $f=0$ almost every where. answer: I ...

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