0
votes
0answers
4 views

Real Analysis. Bounds, Infrimum and Supremum

Given that S={x|4x^(2) > x^(3) +x} (1)Determine whether S is bounded (2)Determine their Supremum and Infrimum.
0
votes
0answers
3 views

L1 norm differentiablility

I am trying to understand the Least Absolute Deviation algorithm, which basically is min l1-norm(z) subject to z=Ax-b I want to understand how is the l1-norm ...
0
votes
1answer
15 views

Prove that there is no permutation $\gamma$ such that $\gamma (1 2) \gamma^{-1} = (1 2 3)$

I need to prove that there is no $\gamma$ such that: $$\gamma (1 2) \gamma^{-1} = (1 2 3)$$ First of all, I'll try to write $\gamma$ in a generic way: $$\gamma = (a b c) \implies \gamma^{-1} = ...
2
votes
2answers
23 views

How to evaluate $\lim\limits_{n \to\infty} \left|\frac{(n+1)^n x^{(n+1)}}{n^n}\right|$?

I'm trying to find the radius of convergence for the series $\sum_{n=0}^{\infty} \frac{n^n}{n!}x^n$ and have used Wolfram Alpha to find that it is $|x| < \frac{1}{e}$ and am trying to show that ...
-4
votes
2answers
15 views

fermat's little theorem prove

Prove that the third number of fermat's is prime? any help with the prove ? I meant prove that $257$ is prime
-1
votes
0answers
15 views

Circular table problem

I've looked other questions that might help solve my problem, but haven't found any people who've used my method to solve it. The problem goes like this: Suppose there are 7 men and 5 women, and they ...
2
votes
2answers
23 views

What is the probability of getting a 3 or higher on a six sided die, if I reroll after failing the first time?

Just as the question says... What is the probability of rolling a 3 or higher on a six sided die, if I reroll the die a second time when I fail the first time?
0
votes
1answer
16 views

Trig substitution for integral of z/(x^2+z^2)?

So I have an integral $\int_1^4\int_y^4\int_0^z\frac{z}{x^2+z^2}dxdzdy$ but I can't figure out what trig substitution to use on the first step. When I try $z=cos$ and $x=sin$, I end up with $\int\cos$ ...
2
votes
0answers
24 views

What is uniform continuity geometrically

I just wonder if someone can give me clear picture because I do understand the technicalities but I still don't have a clear picture of the difference in the geometry between a uniform continuous ...
1
vote
1answer
40 views

Can we prove that matrix multiplication by its inverse is commutative?

We know that $AA^{-1} = I$ and $A^{-1}A = I$, but is there a proof for the commutative property here? Or is this just the definition of invertibility?
0
votes
0answers
9 views

Closed form solution (formula) for possible events

Let's have 100 time units and 4 possible events A1, A2, B1, B2 that might occur within the 100 units. A1 always occurs before A2, B1 always occurs before B2, t1 < t2 < t3 < t4. There are 2 ...
0
votes
0answers
17 views

Finding the volume of a solid region

I'm trying to find the volume of the solid region inside the sphere $x^2+y^2+z^2=4$, and the upper nappe of the cone $z^2=3x^2+3y^2$ (I only have to set up the triple integral itself, not evaluate ...
-1
votes
2answers
51 views

Solve $x,y\in \mathbb{Z}$

Solve for $x,y\in \mathbb{Z}$ $$x^{6}=y^{2}+53$$ I tried but I couldn't complete
0
votes
0answers
9 views

A combinatorial proof by tesselation of the plane.

Some days ago the following problem was posed in the site: given a set of $N$ points in the plane such that for each pair of points $p,q$ we have $\lVert p-q\rVert >1$, prove there is a subset of ...
0
votes
0answers
4 views

Optimization problem involving semidefinite matrix variable that is constrained to be a tensor product

I would like to solve the following optimization problem. With scalar $R$ and nine (mutually orthogonal) $9$-dimensional column vectors $\vec v_i$ all given ($\vec v_i\!'$ is the row vector Hermitian ...
1
vote
2answers
15 views

Under what conditions can a general 2-form be written as a wedge product of two 1-form

Assume we have a 2-form $\omega \in \Lambda^2\mathbb{R}^n$. It is usually stated one can write $$\omega = \alpha \wedge \beta,$$ with $\alpha, \beta \in \Lambda^1\mathbb{R}^n$ only for $n < 4$. How ...
0
votes
0answers
19 views

Difference between line integrals in complex analysis and real analysis,

The formula in complex analysis is $$\int f(\gamma(t))*(\gamma'(t)dt$$ and the formula in the real variable setting, for a gradient field, is: $$\int F.dr$$ $$=\int fxdx + fydy + fzdz,$$ where ...
1
vote
0answers
26 views

What is the number of set partitions of $\{1,1,2,2,3,3\}$?

It must be less than $B_6$ (where $B_6$ is the Bell number of $6$) since the elements are "duplicated". I would most appreciate a generating function that gives the number of set partitions of ...
-1
votes
1answer
21 views

Calculating limit using exponent rules or something else that I didn't think about, please help!

I need to calculate the following limit $$\lim_{n \to \infty}\frac{3\cdot2^n - 2\cdot3^n}{ 5\cdot2^n - 6\cdot3^n}.$$ Any way, back to our topic, according to my book and wolframalpha, the answer is ...
2
votes
2answers
63 views

Does $\frac{x+y}{2}>\frac{a+b}{2}$ hold?

$a$ and $b$ are two real positive numbers. Given that $x=\sqrt{ab}$ and $y=\sqrt{\frac{a^2+b^2}{2}}$, which one has a higher value, $\frac{x+y}{2}$ or $\frac{a+b}{2}$? We know that ...
0
votes
0answers
22 views

Every cycle is a composition of simple cycles

In a directed multigraph: Every cycle (closed walk) is a composition of simple cycles, right? Moreover, every finite path is a composition of simple paths, right? What is the simplest proof of ...
1
vote
1answer
28 views

Counting number of bijections

The question is: Let $S = \{a,b,c,d\}$ and let $X : = \{f\colon S \to S | f \, \, \text{is bijective and } f(x) \ne x \, \, \text{for each}\, \, x \in S \}$. What is $|X|$? Is there a simple ...
0
votes
0answers
30 views

Is there a closed-form approximation to a band-limited sawtooth?

A partial Fourier Series with no coefficients is equal to the closed form expression: $${A \over n} \sum_{k=1}^n \cos(k\theta) = {A \over 2n} \left\{{\sin([2n + 1]\theta/2) \over \sin(\theta/2)} - ...
0
votes
0answers
6 views

Comparison of Cramer Rao bound - deduction and conceptual question

The CRB gives the variance of the estimation error of the estimates and a lower value is preferred. I have computed the cramer rao bound (CRB) of the estimates of the coefficients $\mathbf{h^T}$ for ...
0
votes
0answers
15 views

Modified arithmetical functions from a modified Möbius function

In this post when $n>1$ we assume that $n=\prod_{k=1}^{\omega(n)}p_{n}^{e_{k}}$ its factorization in prime powers, where $\omega (n)$ is the number of distinct primes. It is well known the ...
0
votes
0answers
23 views

Finding the Zero's of a Second Derivative to Determine Points of Inflection

I have been told to graph the function $y=\cos \left(x^2\right)$ for -2π ≤ x ≤ 2π. I have determined key features of the graph but need help when it comes to determining the points of inflection for ...
1
vote
1answer
17 views

Finding the work from $(0,0)\to(1,1)$ of $\vec F(x,y)=xy^2\hat i+yx^2\hat j$

I need to find the work from $(0,0)\to(1,0)\to(1,1)$ of the following vector field:$\vec F(x,y)=xy^2\hat i+yx^2\hat j$ My attempt: $$\oint_{c}\vec F d\vec r=\int_{(0,0)\to (1,0)}\bigg(xy^2\; dx ...
0
votes
0answers
26 views

Probability Quesiton Help

I am working on some exercise questions from probability. I am stuck at this question. Can somebody help me to solve this. I would really appreciate. In a certain country, 35 percent of people are ...
1
vote
2answers
41 views

What is a real world example of “zero work” done by a conservative vector field?

I have only a high school physics background, so when I study the later parts of multivariable calculus, e.g., Greens, Gauss, and Stokes' theorems, there are some topics that I only know the ...
2
votes
0answers
23 views

proof of chinese remainder theorem $x=a_1M_1y_1+…+a_nM_ny_n$?

I can't understand the proof of Chinese Remainder Theorem let $x ≡ a_1 (\text{mod }m_1 ),$ $x ≡ a_2 (\text{mod }m_2 ),$ · · · $x ≡ a_n (\text{mod }m_n )$ such that $m_1,m_2,...,m_n$ are relatively ...
-2
votes
1answer
32 views

How to find triple integral of the following question?

I've been trying to solve that question and over and over again, I get answer of: Where as the online integral solver gives an answer of: I am really confused that If I am correct or the online ...
0
votes
0answers
15 views

Notation for separating out factors of a number

I have an integer (let's call it $n$), and I want to define it as the product of two values: one that's a pure power of two, and another that is odd. Obviously, these two values are unique for a ...
3
votes
0answers
9 views

Help on designing a dynamical system

I would like to build a four-dimensional dynamical system that has the following behavior: Here, $x_1, x_2, x_3$ and $x_4$ are the four dimensions, and each axis has a fixed point that should be a ...
0
votes
0answers
9 views

Picking Marbles and Estimating the Expected Cost

This is a general version of a game I play. Suppose there is a bag of marbles that consists of 4 different marbles : Silver ...
0
votes
4answers
19 views

Superior limit of a certain sequence

Let $(x_n)$ be the sequence: $$\{1, 2, 1 + \frac12, 2 + \frac12, 1 + \frac13, 2 + \frac13, \ldots \}$$ I (think I) understand why $\lim \inf x_n = 1$, but I'm not sure why $\lim \sup x_n =2$. Any ...
0
votes
1answer
12 views

How to interpret multiple critical points (from Lagrange multipliers) that all give a maximum value

If I have 6 critical points, 3 of which give the same maximum possible value of a function f(x,y,z), subject to a constraint g=c, is there something more to say about this solution -- or we just ...
0
votes
0answers
13 views

Probability of Level Crossing

I am kind of stuck on how to proceed on this. $X_n$ is an IID process with $$f_{X_n}(y)= \frac\lambda2 e^{-\lambda |y|}$$ There is a stationary autoregressive process $Y_n$ defined as $$Y_n=\rho ...
0
votes
0answers
6 views

Get parameters for given point on quadratic bezier triangle

I have a 2 dimensional quadratic bezier triangle described by the position of its corners $v_0$, $v_1$ and $v_2$ and a handle for each side $h_0$, $h_1$ and $h_2$. The parametric equation with the ...
1
vote
0answers
22 views

Prove that $F_1$ and $F_2$ are continuous and that $\int_{\gamma_1}F_1(z) dz = \int_{\gamma_2}F_2(z) dw$

Let $\Omega_1, \Omega_2 \subseteq \mathbb{C}$ and let $\gamma_1: [a,b] \to \Omega_1$, $\gamma_2: [c,d] \to \Omega_2$ be paths. Let $f$ be a continuous function defined on $\gamma_1 \times \gamma_2$ ...
2
votes
0answers
48 views

What is $\int_0^{\pi} \frac{e^{\sin x}\cos(x)}{1+e^{\tan x}} \, dx$?

I read this question. The integral has a special property, so it might possibly be evaluable? No one tried evaluating it, so I created this. Not very often I ask question like this, but here it is. ...
2
votes
0answers
21 views

Symplectic group and Quaternion Inner product

I have a problem understanding a passage from "Naive Lie theory"(by Stillwell), here is the passage from section $3.9$ ,page $71$: The idea of treating orthogonal, unitary, and symplectic groups ...
0
votes
0answers
29 views

what is the difference between statiscal averagre and average?

I'm reading a book on synthetic aperture radar and it is said that: The term $\sigma^{\circ}$ is the averaged radar cross section per unit area, also called the scattering coefficient or ...
0
votes
0answers
5 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula results in small errors in relation to the standard matrix inverse operation after each application, as shown here. From what I understand, the Sherman-Morrison identity ...
2
votes
0answers
19 views

Proofs by analysing games

I recently read the following article giving a novel proof of the uncountability of $\mathbb{R}$ by analysing a particular game, amongst other results. ...
2
votes
1answer
42 views

$m(E)=0$ or $m(E^c)=0$

The question comes from former qualifying exam of the graduate school I'm going to attend. Q: Suppose $E$ is measurable and $E=E+\frac{1}{n}$ for every natural number $n\geq 1$. Show that either ...
2
votes
0answers
16 views

Field norm of $F(\sqrt[n]{a})$

Let $F$ be a field of characteristic zero that contains a primitive $n^{th}$ root of unity. Pick $a$ such that $K=F(\sqrt[n]{a})$ is a cyclic extension of $F$ of degree $n$. Let $\sigma$ be a ...
4
votes
0answers
30 views

Spaces whose all their metrizations are complete [duplicate]

Which metrizable topological spaces $(X,\tau)$ posses the following property: Every compatible metric (i.e one which induces the same topology $\tau$) is complete. Compact metrizable spaces satisfy ...
0
votes
3answers
51 views

Formula that's only satisfiable in infinite structures

What formula in first order logic can I write that's only satisfiable over infinite structures, over a dictionary without the = sign?
0
votes
0answers
36 views

Does my derivation work?

I've been totally engaged with exponential integrals for a while. I came across to this limit in my work. I started to calculate the limit as below: currently, I am not sure about my handouts. would ...
1
vote
1answer
59 views

Finding the definite integral $\int_{0}^{2\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx$

$$\int_0^{2\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx$$ My try: $$I=\int_0^\pi \frac{e^{\sin x}\cos(x)}{1+e^{\tan x}} dx+\int_\pi^{2\pi} \frac{e^{-\sin x}\cos(x)}{1+e^{\tan x}} dx$$ also ...

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