0
votes
0answers
3 views

Defining second value of the interval for bisection method

To find the interest rate of a loan (I'm talking about non linear formula that calculates Pmt) by using bisection method, I set the first value of the interval a = 10 ^ (-8), but I don't know how to ...
0
votes
0answers
8 views

Aren't all digits of the $2^n$ for $n \ge 12$ even?

Define $a(n)$ is $2^n$ . All of digits of $a(11)$ (= 2048) is even. However, all of digits of $a(12)$ (= 4096) isn't even and all of digits of $a(13)$ (= 8192) isn't even. Aren't all digits of the ...
0
votes
0answers
2 views

Prove that $J_n(0)$ and $(J_n(0))^t$ are similiar

Prove that $J_n(0)$ and $(J_n(0))^t$ are similar ($J_n(0)$ is a $nxn$ Jordanian block which belongs to the eigenvalue $0$). Use your answer and Jordanian form to prove that for every matrix $A \in ...
0
votes
0answers
6 views

Is there a way to see if $\alpha \in \mathbb{C}$ is constructible at a glance?

The notion of constructibility is not too obscure but mathematically, I find the definitions tedious and not very easy to handle with. I don't know if Ian Stewart's book Galois Theory edition 4 ...
0
votes
0answers
2 views

How to prove polynomial time reduction is reflexive

How do we show that A is polynomial time reducible to itself? i.e. that A ⩽p A.
0
votes
0answers
3 views

Classification and numerical solution strategy for integral equation

Integral equation So I have an integral equation that I need to numerically solve, the integral equation is given (in it's most complicated form) by $ f(r) = -\frac{E_0r}{\epsilon(r)} + ...
0
votes
0answers
11 views

How to prove this statement on groups?

For a positive integer $n$. let $Φ(n)$ denote the number of elements $r∈Z$n such that $gcd(r,n)=1$. Show $Φ(mn)=Φ(m)Φ(n)$ for all $m, n∈N$ such that $gcd(m,n)=1$. The only thing I can come up with ...
0
votes
0answers
7 views

Expressing the Christoffel symbols of the first and the second kind in terms of the metric tensor $g$ and expressing the symmetries

I am trying to expressing the Christoffel symbols of the first and the second kind in terms of the metric tensor $g=g_{ij}$ and express the symmetries of each with respect to the permutation of the ...
0
votes
0answers
6 views

reference request: $C^k(\overline\Omega)$ as restriction of $C^{k}$ functions on $\Omega$

Let $\Omega\subset\mathbb{R}^d$ be an open set. $C^k(\Omega)$ is defined as the space of functions $f:\Omega\to\mathbb{R}$ such that $\partial^nf$ is continuous for $0\leq|n|\leq k$. There are ...
0
votes
1answer
6 views

Deduce that: $\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}[mv\frac{du}{dx}+nu\frac{dv}{dx}]$

Deduce that: $$\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}(mv\frac{du}{dx}+nu\frac{dv}{dx})$$ When I differentiate $\frac{d}{dx}(u^{m}v^{n})$ I get: $$\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}(mv+nu)$$ Is ...
0
votes
1answer
10 views

Solve the Following differential equation: $y''=2yy'$

Solve the following equation: $y''=2yy'$ My attempt: $y''=2yy'$ integrate on both sides: $y'=y^2y = y^3$ We therefore get: $y=\dfrac{1}{4}y^4$ But when I verify my answer in the original ...
1
vote
2answers
32 views

If $\sum_{n=1}^{\infty}\frac{1}{{n}^2} = \frac{{\pi}^2}{6}$ then $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$ is equal to:

If $\sum_{n=1}^{\infty}\frac{1}{{n}^2} = \frac{{\pi}^2}{6}$ then $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$ is equal to: I do not know what to try to find the solution. A hint along with the explanation ...
0
votes
0answers
8 views

Closed complex integral in an annulus

I have the function $$f(z)=\frac{(e^z-1)(1-\cos(2z))}{z^4\sin(z)},$$ and I want to find $$\oint_{|z|=1}f(z)dz.$$ What I know:  Let $A=\{z\in\mathbb{C}|r<|z|<R\}$ be the annulus with ...
0
votes
0answers
6 views

How to express two variables in two other variables

If: $A=R\cos x$ and $B=R\sin x$ Then how can I express $R$ and $x$ in terms of $A$ and $B$ in a rigorous way? Meaning that I take the domain and range in account? I tried: $$\cos x=\frac{A}{R}$$ ...
1
vote
0answers
7 views

f left continuous & strictly increasing; B Borel $\implies$ f(B) Borel?

How's it going? In an attempt to use the Radon-Nykodym theorem to bulldoze through the admission of measures by bounded variation & monotonic functions (sidestepping all that Caratheodory ...
0
votes
0answers
12 views

How to calculate $\lim_{n \rightarrow \infty} n(1- \sqrt[n]{n}) = \infty$

Problem Show that $\lim_{n \rightarrow \infty} n(1- \sqrt[n]{n}) = \infty$ by the following consequence. Consider the sequence $(s_n)$ defined by $s_n=1 + 1/2 + \cdots + 1/n$. ...
-3
votes
2answers
18 views

Fraction with power denominator

I'm very confused as to how you're supposed to solve an inequality in which there is a fraction with a power as a denominator. Example: $$2^x + 8/{2^x} > 6$$ Thank you in advance!
-1
votes
0answers
5 views

Proper map and sequences in metric spaces

Let $f:X\to Y$ be a continuous map between metric spaces. Show that $f$ is proper if the following condition holds: For every sequence $x_n\in X$ such that $f(x_n)$ is bounded, $x_n$ is also ...
0
votes
0answers
12 views

Evaluating a complex integral with a pole

I am asked to evaluate the integral $$\int_\gamma \frac{e^{2z^2}}{z^{77}}\,{\rm d}z$$ where $\gamma$ is a circle centre $0$ traversed once anti-clockwise. Clearly the integrand has a pole of ...
0
votes
0answers
6 views

Hessian and Laplacian

Let $(M^n,g)$ be a smooth Riemannian manifold with a smooth boundary boundary $\partial M$ such that $Ric^M≥0$, and the second fund. form of $\partial M$ is $II\geq c>0$. Suppose that ...
0
votes
0answers
8 views

Calculate a volume using a triple integral

I'm tring to find this volume using a triple integral in the form $dy$ $dx$ $dz$ However I think I'm evaluating the wrong integral because the result is 1 when the volume should be 1/6... Can someone ...
-4
votes
0answers
14 views

How much would this child get in the 30 days of the month of June.

This is the 10th question in my assignment. I'm not too sure if I got the common difference for this question correct. Here's the question: A child tries to negotiate a new deal for her pocket money ...
-1
votes
0answers
5 views

non-transcendental ratio of circumference to diameter

Does there exist a non-transcendental curvature of the plane such that the ratio of the diameter to the circumference of a circle in that plane is also non-transcendental?
0
votes
0answers
2 views

Convert this problem with 2-norm cost to an SOCP?

I am solving a non-convex problem via sequential convex optimization. Here is a minimal example of my problem at iteration $i$: $$ \begin{align*} \min_{\Delta t,F_i}&\left( \Delta ...
0
votes
0answers
4 views

Advantages/disadvantages of a utiliarian social welfare function

If a planner obeys anonymity and strong Pareto principle (individual preferences carry over to the group), then the social utility function will be: $W(x)=\sum_{i=1, ..., n}U_i(x)$ i.e. summing ...
0
votes
0answers
2 views

How to find spacial periodicity

I am struggling with the following question. Consider $$U(x,t)=2\cos\left(kx-\sqrt{\frac{a}{b}}k^2+(\Delta k)^2)t\right)\cos\left((\Delta k)(x-2\sqrt{\frac{a}{b}}kt\right)$$ and suppose that $\Delta ...
-1
votes
0answers
9 views

Confusion of intersection of two 2-d planes in 4-d

I've only just started a linear algebra course at my uni and I'm wondering if it is intuitive to say that two 2-d planes can't intersect in 4-d in such a way that they produce a 3-dimensional solution ...
1
vote
0answers
12 views

What is the difference between a line segment, and a directed line segment?

Is a line segment by definition directed? Does directed mean it is in movement? If its a segment of a line, doesn't it neccessarily follow the rest of the line?
0
votes
1answer
16 views

How to determine if (1,0,1,1), (1,1,0,1) , (0,1,1,1) spans $R^4$?

I set up a system where $a(1,0,1,1) + b(1,1,0,1) + c(0,1,1,1) = (1,1,1,1)$ (the standard basis of R4) then i found that $a + b = 1$ $b + c = 1$ $a + b + c = 1$ which implies that $a = c = 0,$ and ...
0
votes
0answers
7 views

Special dice generating non-decreasing sequence

Suppose that, when rolled for the first time, a special 6-sided dice shows $1,\ldots, 6$ with probability $\frac{1}{6}$ each, and then, upon rerolling, shows with equal probability a number greater or ...
0
votes
0answers
14 views

What is the difference between ${\mathbb{Z}[x_1, .., x_n]}_{( p )}$ and ${\mathbb{Z}[x_1, .., x_n]}/(p)$?

Let $p$ be any prime. What is the difference between ${\mathbb{Z}[x_1, .., x_n]}_{( p )}$ and ${\mathbb{Z}[x_1, .., x_n]}/(p)$? ${\mathbb{Z}[x_1, .., x_n]}_{( p )}$ is the localization of ...
1
vote
2answers
18 views

The sum of the 2nd and 3rd term is 12, and the sum of the 3rd term and 4th term is 60.

This is the 9th question in my assignment. It's another confusing question. Here's the question: In a geometric progression, the sum of the 2nd and 3rd terms is 12, and the sum of the 3rd and 4th term ...
0
votes
0answers
9 views

Existence of Solution to Integral Equation

How do I show that the integral equation \begin{equation*} x(t) = \ln(1+t) + 1/2\int_0^1e^{-t}\sin^2(ts)x(s)ds \end{equation*} has a solution $C[0,1]$?
1
vote
1answer
8 views

How far does does this guy walk if he empties 24 barrow fulls and returns to the load each time?

This is the 6th question in my assignment. It sounds quite confusing to me. Here's the question: A man moves a load of soil for top dressing an orchard by emptying barrow loads in a line 20 meters ...
0
votes
0answers
5 views

Convergence of Iterative First Order ODE Method

Let us suppose we have a function $f(x,t)$ which is Lipschitz wrt $x$. Then we have the following iterative method $$ x_{m+1}(t) = c+\int_{t_0}^tf(x_m(v),v)dv$$ for solving the ODE $x'(t) = ...
0
votes
0answers
6 views

Second order partial derivative of an unknown function

Given that $U=U\left(\frac{S_1}{S_2},t\right)$ and I find the derivative wrt $S_1$ of $U$ to obtain: $\frac{1}{S_2} \frac{\partial U}{\partial S_1}$, when I now want a second derivative wrt $S_1$, do ...
0
votes
2answers
19 views

If $x=\frac{2ab}{a+b}$ show that $1/a, 1/x, 1/b$ is an arithmetic progression.

This is the 5th question in my assignment. It's a really freaky looking problem. I haven't come up with an answer yet but I made an attempt with the working out. Here's the question: If ...
1
vote
0answers
10 views

Prove/Disprove: for every formula $A$: $A$ is valid or $\lnot A$ is valid for $M$.

Prove/Disprove: Given a structure $M$. For every formula $A$: $A$ is valid or (probably exclusive) $\lnot A$ is valid in $M$. Intuitively, it sounds true. I tried to prove it by contradiction; ...
0
votes
0answers
9 views

Generating family for box topology

Suppose $X_{\alpha}$, $\alpha\in\Lambda$ is a family of topological spaces and the product set $\Pi_{\alpha\in\Lambda}X_{\alpha}$ is equipped with the box topology. Can we find a generating family of ...
0
votes
0answers
12 views

For a sequence of i.i.d. (Bernoulli ?) RV we have for the partial sums $S_{n+m}-S_n=m$ i.o. almost surely

Problem: Given a sequence $(X_n)_{n \geq 1}$ of i.i.d. RV and $P(X_1=1)=P(X_1=-1)=1/2$ we have for $m \geq 2k+1 \in \mathbb{N}$ for the partial sums $S_{n+m}-S_n=m$ i.o. a.s. My approach: I want ...
0
votes
1answer
11 views

The 11th term and the 12th term have the same sum as the first 10 terms of the same arithmetic progression.

This question is part of my advanced maths assignment on arithmetic and geometric progressions. Here's the question: Q4. Given that both the sum of the first ten terms of an arithmetic progression and ...
-2
votes
0answers
17 views

why if $\frac{G} {Z(G)} \cong Z_P. Z_P$ then $G' \cong Z_P$?

why if $\frac{G} {Z(G)} \cong Z_P. Z_P$ then $G' \cong Z_P$? can you help me?
0
votes
2answers
31 views

Expected Value help

A fair coin is tossed. If a head occurs 1 die is rolled, if a tail occurs 2 dice are rolled. Let X be the total on the die or dice. What is E[X]? To be honest, I don't get this. The answer was ...
0
votes
0answers
7 views

Find the orders of zeros for a complex function

Find the orders of zeros for the following functions at z = 0: 1.$$ z^2 (e^{z^2} - 1) $$ 2. $$ 6 \sin(z^3) + z^3 (z^6 - 6) $$ The question means that I should set both functions to zero and find ...
2
votes
3answers
14 views

Does the PNT establish a connection between primes and the logarithm?

The prime number theorem states that $$\pi(x) \sim \frac x {\ln(x)}$$ Morally, this seems to suggest that there is a fundamental connection between primes and the natural logarithm. But since we're ...
0
votes
0answers
7 views

Topology of point-wise convergence

Does there exists a topology on the product space $\Pi_{\alpha\in\Lambda}X_\alpha$ other than the product topology such that the convergence and the point-wise convergence are equivalent?
0
votes
0answers
6 views

Essential singularity of the resolvent operator of an unbounded operator

Is there an unbounded operator with isolated points in the spectrum, not all of which are eigenvalues? For unbounded operators it is known that isolated spectral points are either poles or essential ...
0
votes
0answers
6 views

Weak formulation of a nonlinear problem with test functions in a dense subspace of $H_0^1$

I am reposting a question from Math Overflow, because it seems it gets no attention. Let $\Omega\subset \mathbb R^{d=3}$ is a bounded and Lipschitz domain. Let $u\in H_g^1(\Omega)$ satisfy the weak ...
2
votes
0answers
7 views

How to find the appropriate weights to maximize the third coordinate while the first two are zeros

Let's assume, that $v_1, ..., v_n \in \mathbb{R}^3 $ and $ \lambda_1, ..., \lambda_n \in [0, 1] $ The $ v_1, ..., v_n $ vectors are given. I have to find the appropriate weights ($ \lambda_1, ..., ...
0
votes
1answer
6 views

Sign of a flux surface integral

Use a parametrization to find the flux $$\iint_S F \cdot n \, d\sigma$$ across the surface in a given direction: $$F=xy\overrightarrow i-z\overrightarrow k$$ outward (normal away from the z-axis) ...

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