1
vote
2answers
9 views

Find the time that must elapse for the object to reach 98% of its limiting velocity?

I am given the initial value problem $$ \frac{dv}{dt} = 9.8 - (\frac v5) $$ and you are given $v(0) = 0$ I was looking at the solution to this problem. They first solved the differential ...
3
votes
3answers
40 views

How to solve $y' = -2x -y$

My thought: $\displaystyle\frac{dy}{dx}+x^0y=-2x$ Considering it as the form of linear equation, $\displaystyle\frac{dy}{dx}+P(x)y=Q(x)$ Multiplying $e^{\int1dx} = e^x$ on both sides, ...
9
votes
4answers
169 views

Solving a system of non-linear equations with 10 equations and 10 unknowns

I'm working on a problem where I seem to have run into a system of non-linear equations. I have ten equations and ten unknowns. In the equations below, all of the $\phi_{ij}$'s are known, but all of ...
2
votes
3answers
50 views

How do i solve this equation ${\mathbb{R}}$: $3 \sin^3x+2 \cos^3x=2 \sin x+\cos x$?

How do I solve this equation ${\mathbb{R}}$: $3 \sin^3x+2 \cos^3x=2 \sin x+\cos x $? Note : I have tried using trigonometric transformation but it seems very complicated to get the result .. may ...
0
votes
0answers
10 views

Find angles between sides of triangle and coordinate planes (xy,yz,zx planes) using three 3d vectors .

Given the following, three vectors: a⃗ =3i−2j+5kb⃗ =i−6j+6kc⃗ =2i+3j−k, find the angles between sides of triangle and coordinate planes. I calculated the sides to be 4.58 , 11.45 and 7.87. I also ...
3
votes
0answers
18 views

Interpretation of a certain transform

I'm having troubles with understanding the physical meaning of a certain transform. If $u$ is a solution to the wave equation $$\partial_t^2u-\Delta u=0\ \mathrm{in}\ ...
1
vote
0answers
10 views

does an exponential bound on a Lyapunov candidate implies asymptotic stability?

if I have a Lyapunov candidate $V:[0,\infty)\rightarrow \mathbb{R}$ and I'm able to show that $$ V(t)\le k e^{-\eta t} V(0),\qquad \forall t\in[0,\infty) $$ can I conclude something about ...
2
votes
0answers
39 views

Graham's Number versus another large number

I recently read this article about the most damage you can do in a single turn in Magic the Gathering. According to the current version of the deck, that damage is about a) $4 \rightarrow 17 ...
0
votes
3answers
29 views

Show that $\ker \hat{T} = \text{ann}(\text{range } T)$

This is an old exam problem: Let $V$ and $W$ be finite dimensional vector spaces over a field $F$ and let $T: V \to W$ be a linear transformation. Define $\hat{T}: W^* \to V^*$ by ...
1
vote
1answer
19 views

*$G$-invariant* symmetric bilinear form & $G'=\Bbb Z_2\times\Bbb Z_2$.

I got a problem with the last point I solved all the points, from (a) to (h), but I have no idea how to solve (i): how can I associate a bilinear form to a represtation? What is a $G$-invariant ...
2
votes
1answer
28 views

What is a linear functional on continuous functions on the real line not given by a measure?

What is a positive linear functional on continuous functions on the real line not given by integration against a measure? I know that the dual of $C_c(\mathbb R)$ is the set of Radon measures, ...
2
votes
2answers
20 views

Basis of a field extension

Let $K$ be a field, and let $A$ be a $K$-algebra such that $\alpha \in A$. Then the natural homomorphism $$ \phi: K[x] \to K[\alpha], \hspace{3mm} (x \mapsto \alpha )$$ has a kernel which is a ...
2
votes
6answers
61 views

Prove that $\{(x,y)\mid xy>0\}$ is open

I need to prove this using open balls. So the general idea is to construct a open ball around a point of the set. A point $(x,y)$ such that $xy>0$. Then we must prove that this ball is inside the ...
0
votes
5answers
85 views

What happens to an integral if the 'dt' disapears? (Integral with no dt)

I have been struggling with convolution yet have come up with my own method that involves cancelling the dt from the integral. I want to ask - If there is a term that is - ''Integral'' with limits ...
1
vote
0answers
10 views

$\limsup$ of Brownian Motion Time Integral

The following are well-known: $\limsup_{t\rightarrow \infty} \frac{B(t)}{t} = 0$ $\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt t} = \infty$ $\limsup_{t\rightarrow \infty} \frac{B(t)}{\sqrt ...
0
votes
2answers
15 views

Operations with probability distributions

I had an idea that passes by declaring a new type of computer variable (like Integer, Double, etc.) that represents a statistical probability distribution (PDF), for that I would need to define the ...
2
votes
2answers
76 views

Need help with $e^x=1/x$

I've tried everything. I expressed $x$ and I got $x=\ln{1\over x}$, and don't know what to do. Original question is to find $e^x-{1\over x}=0$. There is a solution I've typed it in Wolfram
0
votes
0answers
4 views

Is every bijective pseudograph homomorphism a pseudograph isomorphism?

The term pseudograph describes a graph that may have parallel edges and loops. Formally this is a triple $G = (V,E,\delta)$ with $V,E$ sets and a map $\delta \colon E \to (V \times V)/\sim$, where ...
3
votes
1answer
36 views

Is a = 0 a valid counterexample to this statement?

This is an exercise in a text I am reading for a ring theory course. Suppose the ring R contains element a such that 1) a is idempotent and 2) a is not a zero divisor of R. Deduce that a serves as a ...
0
votes
0answers
27 views

How to manipulate the bound on the summation

$$ B_n^{f^2}(x) = \sum_{k=1}^n\sum_{j=0}^{n-k} 2^{k-j} {j+k \choose j} \frac{d^j}{df^j}[f^k] B_{n,j+k}^f(x) $$ I am looking to have the bounds switched, can someone show me exactly how this is done? ...
-4
votes
1answer
19 views

Two full-length [on hold]

A high school basketball player is $75$ inches tall. The basketball hoop is $10$ feet above the court. Find the distance in feet between the top of the players head and the basketball hoop.
1
vote
0answers
24 views

Calculating the volume bounded by $z = 5$ and $z^2=x^2+y^2$ in 2 ways

I don't understand where is my mistake on calculating the volume by the second way. The volume that I want to calculate is bounded by $z = 5$ and $z^2=x^2+y^2$, so it is the upper part of the cone, ...
2
votes
3answers
56 views

Hints on solving $y'=\frac{y}{3x-y^2}$

$$y'=\frac{y}{3x-y^2}$$ My attempt: $$\frac{dy}{dx}=\frac{y}{3x-y^2}$$ $$dy\cdot(3x-y^2)=dx\cdot y$$ $$dy\cdot3x-dy\cdot y^2=dx\cdot y$$ Any direction? I need hints please ...
0
votes
3answers
36 views

complete the proof for this statement

$$\forall x \in \mathbb{R}, x \neq 0 \implies \frac{1}{x^2\:+3}\:<\:\frac{4}{5}\: $$ I thought of doing the contrapositive but not sure what to do next. $$ \frac{1}{x^{2\:}+3}\:\ge ...
1
vote
2answers
76 views

On a scale of 1 to 10 how far is this manifold from being a normal bundle?

(DIS)-CLAIMER: All the manifolds considered in this post are completely humble and have no additional structure beyond the smooth structure. Let $Y$ be a submanifold of $M$ and let $(-)^0$ denote the ...
0
votes
2answers
17 views

second derivative of the composition of two multivariable functions

Let $U \subset \mathbb{R}^n$ be open, and let $\gamma: \mathbb{R} \to U$ and $f: U \to \mathbb{R}$ be to functions that are differentiable at least twice. I want to show that $\frac{d^2}{dt^2}(f ...
0
votes
2answers
28 views

Multivariable function as a set of functions

Consider a function $f:\mathbb{R}^n \to \mathbb{R}^m$. I've understood that it can be seen as: $f_i = (f_1,f_2,\ldots ,f_m)$, where $f_i: \mathbb{R}^n\to \mathbb{R}$. What are $f_i$ exactly? ...
5
votes
1answer
42 views

General notions of basis

Free groups, free abelian groups, and vector spaces all have a notion of 'basis': a subset $B$ of the structure such that everything in the structure can be written uniquely as a finite combination of ...
1
vote
3answers
32 views

probability of exactly one out of N events occuring

I have N events. Each "i" event has probability $P_i$. What is the probability of $n$ events occuring? I have seen this answered for two and three events, but not for an arbitrary N. In principle, ...
0
votes
0answers
11 views

Existence of subgradient of a quasiconvex function

Does a continuous quasiconvex function always have a subgradient? More strongly, is it true that if $f$ is a continuous quasiconvex function, then for each $x$ there is a vector $c$, such that for ...
2
votes
4answers
107 views

What is wrong in my $f'(x)$?

We have $f:\mathbb{R}\rightarrow\mathbb{R}, f(x)=\frac{x^2-x+1}{x^2+x+1}$ and we need to find $f'(x)$. Here is all my steps: ...
0
votes
0answers
8 views

Please recommend some reading materials for me on pseudovarieties.

I would like some reading materials for finite semigroups, specifically in pseudovarieties. I have read a few papers on Simon's Theorem and, of course, have covered the basics leading up to ...
0
votes
2answers
39 views

Am I solving these initial value problem correctly?

I was just hoping someone could check my work and tell me if I'm solving these types of problems correctly? (Large image version)
-1
votes
3answers
48 views

Representing a vector in $\mathbb{R}^{3}$ as sum of only two vectors in $\mathbb{R}^{3}$

Is it possible? Or more generally can any vector in $\mathbb{R}^{n}$ can be represented as sum of (n-1) or less vectors in $\mathbb{R}^{n}$? -----EDIT----- What I basically want to ask is that can ...
4
votes
1answer
44 views

Complex integral with Fourier

So, I've been scratching my head over this for the whole day. I'm trying to solve the following integral $$\int^\infty_{-\infty} \frac{e^{i \alpha (X-\xi)}}{\sqrt{\alpha^2+ \beta^2 }} \, d\alpha$$ ...
-2
votes
1answer
23 views

Select certain elements from a vector array MATLAB [on hold]

I am new to MATLAB. I have an array consisting of real numbers. I want to select the elements that are closest to integer values. I want one element per integer. I am using the "if" statement and ...
1
vote
1answer
39 views

The real projective line and $1/\infty$

so I came up with this idea: the real projective line defines that $\infty = - \infty$. What if I divide any value $x$ (not equal to $\infty$) by infinity? Would that be 0? or "something" between ...
0
votes
0answers
28 views

Best way to quantify the difference between two vectors

There are plenty of ways of showing an error, or rather a deviation, between two vector quantities. What is the best choice? Specifically, at every timestep, I am comparing two vectors of curvature ...
4
votes
1answer
27 views

Proposition $1.3$ in Bondy & Murty's Graph Theory.

Let $G[X,Y]$ be a bipartite graph, with no isolated vertices, and $d(x) \ge d(y)$, $\forall$ $xy \in E$ (where $E$ denotes the set of edges in $G$). Then: $|X| \le |Y|$, with equality iff $d(x) = ...
1
vote
0answers
13 views

Hesse matrix is negatively semidefinite if a function has a local maximum

Let $U \subset \mathbb{R}^n$ be open, and let $f: U \to \mathbb{R}$ be at least twice continuously differentiable. Also, we assume $f$ has a local maximum $a \in U$. I now want to show that the Hesse ...
1
vote
1answer
39 views

Find recursive formula - Question from exam, check my answer

We want to demolish and move a bridge from one location to another. The bridge is made out of $m$ road segments all connected $[0,1]$, $[1,2]$, $[2,3]$...$[m-1,m]$ We have a given function $f$ which ...
0
votes
0answers
23 views

Does this modulo formula change odds?

Good day, Consider 5 people each choosing a different number between [0 and A] randomly At the end, we add all the numbers: $n_1+n_2+n_3+n_4+n_5 =N$ Then modulo A+1: $W = N \pmod{A+1}$ My ...
1
vote
3answers
88 views

Improving Mathmatical Skill [on hold]

I am a student of computer science and engineering. My understanding of mathematics is not very good. I am getting very hard time studying subject that require a background on mathematics. So, I ...
1
vote
1answer
17 views

Cholesky decomposition and variance

Well, I've been reading about simulating correlated data and I've come across Cholesky decomposition. Everything seemed clear until I found a couple of posts on this site and Cross-Validated that ...
0
votes
3answers
58 views

Solve $3^{2x} -2 \cdot 3^{x+5} + 3^{10} = 0$ for $x$

Here's the question: Solve for $x$ in $$3^{2x} - 2 \cdot 3^{x+5} + 3^{10} = 0$$ I know that you have to factor something out, I'm just not sure what that something is. Thanks in advance
5
votes
2answers
176 views

Evaluating the indefinite integral $\int\log\!\left(x+\sqrt{x^2-1}\right)\!dx$

I came across the following integral, and I don't know how to solve it. $$ \int\log\left(x+\sqrt{x^2-1}\right)dx $$ I tried the "obvious" substitution of $x=\tan\theta$, which gives you: $$ ...
1
vote
0answers
87 views

Sum of natural numbers

$$1+x+x^2+x^3+x^4+...=\sum_{k=0}^{\infty} \left(x^k\right)=\frac{1}{1-x}, |x|<1$$ $$\frac{d}{dx} \left(x^n\right)=nx^{n-1}\Longrightarrow$$ $$1+2x+3x^2+4x^3+5x^4+...=\sum_{k=0}^{\infty} ...
3
votes
1answer
36 views

How to show that $\int_{S}\frac{1}{z}\frac{1}{\cos(2\pi i a)-\cos(2\pi z)}dz\rightarrow 0$ as the sides of the square $S$ go to $\infty$.

I have a question where I am asked to show that the following sum is $$\sum_{k=-\infty}^\infty\frac{1}{a^2+k^2}=\frac{\pi}{a}\frac{e^{2\pi a}-e^{-2\pi a}}{e^{2\pi a}+e^{-2\pi a}-2}$$ by integrating ...
4
votes
1answer
59 views

Under what conditions does $M \oplus A \cong M \oplus B$ imply $A \cong B$?

This question is fairly general (I'm actually interested in a more specific setting, which I'll mention later), and I've found similar questions/answers on here but they don't seem to answer the ...
0
votes
1answer
41 views

Implicit equation. Can it be solved?

Is it possible to find a function $x:[0,T]\to [0,x_0]$ such that, for a fixed $0<\lambda<1$ we have: $$\dfrac{1}{1+\lambda}\left (1-\dfrac{x(t)}{x_0}\right )^{1+\lambda} +\dfrac{1}{1-\lambda} ...

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