Questions tagged [problem-solving]
Use this tag when you want to determine the thinking that is needed to solve a certain type of problem, as opposed to looking for a specific answer to a question.
4,275
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Bagging a random sample with duplicate
this is a question posed by an engineer friend who had this probability problem arise at work. It goes like this,
Say we have 5000 marbles. Of them, we know that 50 are duplicates. They are not ...
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1
answer
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matrix representation for $φ: \mathbb{R}[t]→ \mathbb{R}^2 f(t) → \begin{pmatrix} f(1) − f(0)\\ f(2) − f(1) \end{pmatrix})$ to the basis$(1,t,t^2,t^3)$
I am having problems with this exercise. Can someone help me?
a) Determine the matrix representation for $φ: \mathbb{R}[t]degree≤3 → \mathbb{R}^2 f(t) → \begin{pmatrix} f(1) − f(0)\\ f(2) − f(1) \end{...
- 464
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0
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22
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Existence of at most one matrix $A \in M_{2,3}(\mathbb{R})$ such that $BA =\operatorname{diag}(1,1,0).$
Hey I want to check my solutions to this exercise:
a) Let $B\in M_{3,2}(\mathbb{R})$ be a matrix. Show that there is at most one matrix $A \in M_{2,3}(\mathbb{R})$ such that $BA=\begin{pmatrix} 1 &...
- 464
2
votes
1
answer
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Determine the mapping matrix $A=M_B^B$ with the base $B=(1,t,t^2,t^3)$
Hey I have some problems with this problem
Consider all polynomials with $\leq \deg 3$ and the following map:
$f: \mathbb{R}[t] \rightarrow \mathbb{R}[t]$
$f(t) \rightarrow f(t+1)-f(t)$
a) Determine ...
- 464
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votes
0
answers
41
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When should you give up on a specific approach while working on a hard problem? [closed]
Over the past three days, I have been spending an unreasonable amount of time on a relatively simple problem. I listed my failed attempts in the post and it turns out that my first attempt was the one ...
- 317
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0
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How to determine which non-right angle to use when determining components of a vector
I'm learning electrostatics right now and just had a quick problem that involved calculating the electric field at a test point z units above the $x$ axis due to $2$ equal charges separated by equal ...
- 2,086
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1
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Two subsets $A$ and $ B$ of the $(x,y)$ plane are said to be $equivalent$ if there exists a function $f:A\to B$ which is both one-to-one and onto.
Two subsets $A$ and $ B$ of the $(x,y)$ plane are said to be equivalent if there exists a function $f:A\to B$ which is both one-to-one and onto.
$(i)$ Show that two line segments in the plane are ...
- 1,865
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0
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36
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Is there an algorithm to solve number manipulation problems?
There are this kind of problems where, given a certain amount of the same numbers, it's needed to manipulate them with functions or operators in a way to get a certain result. It's allowed to glue ...
- 546
4
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2
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134
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Finding the hypotenuse on a triangle with a circle inscribed. [closed]
The problem is a right-angled triangle with a inscribed circle, see figure, where you want to solve the distance CB (the hypotenuse).
The only known values are two lines which are linked to the circle ...
- 49
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1
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23
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Lagrangian of a spherical pendulum
I am working on a problem that's asking me to express the lagrangian of a mass $m$ suspended from a rigid massless rod of length $L$, but free to rotate otherwise (a spherical pendulum). The problem ...
- 2,086
2
votes
1
answer
48
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Calculating Expected Increase In Prize Line Count
Hello and thanks in advance.
I'm trying to solve a problem I'm having with a pattern matching game.
The game's board consists of a 5x5 grid of numbers where column 1 (on the far left) contains 5 ...
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1
vote
1
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Hamiltonian for a free particle in $2$D in Polar coordinates
I am working through a problem that's asking me to calculate the Hamiltonian of a free particle in polar coordinates moving in two dimensions. I believe I have the correct form of the equation, ...
- 2,086
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0
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28
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Fomin et al., Mathematical Circles Chapter 4- Pigeon Hole Principle Problem 12. Max. no. of kings that can be placed so no two put each other in check
I found this problem in Mathematical Circles in the Pigeon Hole Principle chapter:
What is the largest number of kings which can be placed on a chessboard so that no two of them put each other in ...
- 1,787
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1
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Problem solving strategy to determine who wins a game?
Consider a game on a chess board with a single counter in the top right corner. You can only move this counter to a square that is adjacent. There are two competitors and the aim of the game is to get ...
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50
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Average commute time
Roads in town looks like this:
Where $t=45$ means the path takes 45 minutes, and
$t=\frac{N}{100}$ means the path takes as many minutes as the number of drivers who chose that path divided by $100$. ...
2
votes
0
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95
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How do I determine whether our idea is novel?
Mathematics talent runs in my family, but although I enjoy learning about things in mathematics and biostatistics, I am a biochemist by training.
So I read a blog post by a mathematician, and started ...
5
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170
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Randomly pressing buttons of a calculator [closed]
There's a normal scientific calculator with only $12$ buttons: from $0$ to $9$ and $+-$
Randomly press these buttons $n$ times. What will be the math expectation of the result? (Imagine that the ...
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Solving Linear Non-Homogeneous Recurrence - where F(n) contains an exponential (2^n) and a polynomial (n+3)
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What is the reason as to why we are allowed to "solve it one by one" (highlighted in red) ? - where we go on to consider 2^n (green) and n+3 (pink) of F(n) seperately and then find ...
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Invariants principle Sikinia (Engel problem-solving strategies E4)
I'm studying Engel book and the invariant principle, but the solution of problem E4 blocks me.
E4. In the Parliament of Sikinia, each member has at most three enemies. Prove that the house can be ...
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3
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How to express the area of a right triangle in terms of altitude and bisector of its right angle?
In a right triangle $ABC$ with the right angle $C$, the altitude $CH$ is equal to $h$, and the angle bisector $CL$ is equal to $l$. Find the area of triangle $ABC$.
My attempt: Let $AC=a$, $BC=b$, ...
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3
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How to solve a trinomial with a rational exponent $x^{q}-x-c=0$
I require solutions to equations of the following form: $x^{q}-x-c = 0$ where $q$ is a rational number very close to 1 with $q$ such that: $$q=\frac{p^k+1}{p^k}$$ $$q=\frac{p^k}{p^k-1}$$ $p$ prime, ...
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Set theory exam question on infinite union of relations [closed]
Let $R$ be any relation over $\mathbb Z$ (the integers). Let $R_n $ be the set of all relations $ R $ (over the integers) such that $n$ is natural, by:
$
R_0:= R$.
$R_{n+1}:= R_n \cup (R_n ◦ R_n)$.
$...
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1
answer
69
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Elementary number theory digit sum question.
Define $f(x)$ as the digit sum function in base $10$; Then, I am tasked with finding solutions to equation:
$$x = 2017f(f(x))\;.$$
My first idea was to use the congruence $x\equiv f(x) \bmod 9$ to ...
- 3
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0
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19
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Minimum distance of a point from four other points [duplicate]
In this question in my textbook:
Let $A(0,1), B(1,1), C(1,-1), D(-1,0)$ be four points. If P be any other point, then $PA+PB+PC+PD=d$, where $[d] =$
What I have done: Taken two triangles $PAB$ and $...
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3
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How can I find the natural numbers $a$ and $b$ such that $a+b<2021$
I am trying to solve the inequality $a + b <2021$, where $1\leqslant a \leqslant 2023$ and $1\leqslant b \leqslant 2023$, $a$ and $b$ are integer numbers, $a \neq b$. I tried
with $a = 1$, then $b\...
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2
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Complex Valued Optimization Problem
I recently encountered the following optimization problem:
$$
\max_{z_i} \frac{\Big|\sum_{i=1}^n a_i z_i \Big|^2}{e + \Big|\sum_{i=1}^n b_i z_i\Big|^2}, \qquad \text{such that } |z_i| \leq 1, \quad \...
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Find the general solution for the PDE $u_{xx}(x,y)+yu_x(x,y)=0$, $u$ is $C^2$
(Edit: I redesigned to a more convenient approach)
I have to find rigurously the general solution for the following partial differential equation
\begin{equation}\tag{1}
u_{xx}(x,y) + y u_{x}(x,y) = 0 ...
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2
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Solve the following differential equation using power series :$(x-1)y''+xy'+\frac yx=0.$
I was learning to solve differetial equations, using power series. There was a problem given as:
Solve the following differential equation using power series :$$(x-1)y''+xy'+\frac yx=0.$$
I tried ...
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Knots and links of parallel arcs
Starting for this diagram, we may say the following:
$a)$ It can be coloured $\mod3$ since:
@blue: $0+1 \equiv (2*2) \mod3$
@red: $2+0 \equiv (2*1) \mod3$
@green: $1+2 \equiv (2*0)\mod3$
$b)$ Here we ...
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26
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Constrained Optimization Problem (Theoretical Question) [closed]
I have a theoretical question about a constrained optimization problem. Would it be possible for the constraint to be equal to zero?
4
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1
answer
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If $A\subset\mathbb{N}$ has positive upper density, $B\subset\mathbb{N}$ is infinite, does $\exists c$ such that $\vert\{c+a:a\in A\}\cap B\vert>k?$
Definition: A subset A of the natural numbers is said to have positive upper density if $\ \displaystyle\limsup _{n\to \infty }\frac{\lvert A\cap \{1,2,3,\dotsc ,n\}\rvert}{n}>0.$
Let $\ A\subset \...
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Looking for a more intuitive set of parameters to define a bounded random walk process based on its stationary probability density
I'm using a stochastic process that produces a continuous random variable called a Bounded Random Walk (Nicolau, 2002). The model has 4 parameters. Unfortunately, choosing appropriate parameter values ...
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50
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Solving IBVP with Laplace Transform
It is given the following BVP
\begin{align}
u_{t} &= u_{xxxx},\quad\quad x \in (0, \pi), t >0 \tag1\\
u(0,t) &= u(\pi,t) = 0 ,\quad t > 0 \tag 2\\
u_{x}(0,t) &= e^{-4t},\quad t > ...
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Geometry Problem with Isosceles Triangles and Cyclic Quadrilaterals
PQR is a right-angled triangle at P and has PQ<PR. Point T is on QR so PQ=QT. Point O is the midpoint of PT. Let X be the point on the circumference of triangle PTR so that angle PXQ=90. Prove that ...
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How to solve the equation $(1-e^{-αa})(1-e^{-βa}) = 1/2$ for a?
Here is what I tried:
$(1-e^{-αa})(1-e^{-βa}) = 1/2$ implies $e^{-ln(2)} -e^{-βa} -e^{-αa} + e^{-a(β+α)} = 0$
But from here I don't know how to get the value of "a", any help ?
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Can someone help me understand this calculus problem means by a few terms?
I have this problem below and I have a few questions.
A sphere of radius $1$ overlaps a smaller sphere of radius $r$ in such a way that their inter-section is a circle of radius $r$. (In other words, ...
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Why Mathcad didn't solve my problem?
When my teacher did this in class, it worked fine and he shared this with us but now when I'm trying on my computer it doesn't work. It is the exact same document he shared. It is supposed to look ...
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Is my solution correct for problem 1.37 in Sipser's Theory of Computation?
The problem is as follows:
Let $
\Sigma_3=\left\{\begin{bmatrix}0\\0\\0\end{bmatrix},\begin{bmatrix}0\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix},\ldots,\begin{bmatrix}1\\1\\1\end{bmatrix}\...
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1
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Let $V = \mathbb F_p^9$ and $W \subset V$ a dimension $5$ subspace. Find the number of subspaces $U \subset V$ with $\dim(U) = 6, \dim(W \cap U) = 3$
Let $K = \mathbb F_p$ be a finite field with $p$ elements where $p$ is a prime Let $V = K^9$ be a vector space, and let $W \subset V$ be a subspace of $V$ such that $\dim(W) = 5$. Find the number of ...
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Rearrange numbers [closed]
Suppose you have the three numbers $2m$, $2m+3$ and $2m+5$. Each round you have to reduce every number by one. Before each round you can add half of some even number of your $3$ numbers to any of the ...
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Step by step changing likelihood
I have a tuple of items which I am iterating over, say 10 items. I want to make decisions on each iteration based on an increasing likelihood which relates directly to the position of the item in the ...
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2
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42
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Linear Algebra: Two vectors and Unknown Ks. [closed]
Here is a problem I cannot get my head around;
The question asks to find the values of K1 and K2 in the following expression:
k1v+k2w, knowing that v=-2i+2.5j and w=[4 -0.5].
The only other ...
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1
vote
1
answer
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Let $F:C^n \to C^n$ be a linear transformation, if $\forall x \in \mathbb{C^n}: \Vert F(x) \Vert = \Vert F^*(x) \Vert$ then F is normal.
The question is in the title.
My attempt:
Since $\Vert F(x) \Vert = \Vert F^*(x) \Vert, \forall x \in \mathbb{C^n}$, then $\Vert F(x) \Vert^2 = \Vert F^*(x) \Vert^2, \forall x \in \mathbb{C^n}$. So, $\...
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2
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Is there a technique for solving the integral of the square root of a constant minus the square of the cosecant?
I have to solve this integral:
$$
\frac{1}{\pi}\int_{\theta_1}^{\theta_2}\sqrt{\Gamma^2 - \frac{J^2}{\sin^2(\theta)}}\text{d}\theta,
$$
where $\theta_1$ and $\theta_2$ are the endpoints of one of the ...
0
votes
0
answers
31
views
Rearrange formula so it solves for x instead of y
Predefined vars example: $ Y = 10^{8}, L=[1,2,3,4,5,6,7,8], P=[2,2,2,2,2,2,2,2] $
Formula:
$$ Y =\sum_{i=1}^8 {(nCr(x,9-i) * (\prod_{n=i}^8 {P[n]}) * L[i])} $$
I have tried a few things but am really ...
0
votes
0
answers
18
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Solving a differential equation problem with boundary conditions
I need to solve this differential equation with precise BC:
u′′(x)=ζ−ξ^2*u(x)
u(a)=v1
u′(a)=v1sech(χ⋅a)sinh(χ⋅a)
The solution of this problem provide the slip function u(x) in a mode - 2 Fracture ...
- 1
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0
answers
24
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Create matrix operations to compute $[y1, y2 x1, x2]$ from vectors $[x1,x2,1]$ and $[y1, y2, 1]$
I want to create a series of matrix operations that would allow me to combine the vectors $[x1,x2,1]$, $[y1, y2, 1]$ without using a Hadamard product to produce $[y1, y2 x1, x2] = [y1, y2, 1] \circ [1,...
- 1
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0
answers
43
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Using Fourier Transform to Solve Initial Value Problem
It is given the following problem
\begin{align}
u_t&=ku_{xx}-\mu u_x, \quad x \in \mathbb{R}, t>0 \tag1 \\
u(x,0)&=\varphi(x),\quad \quad \quad \ \ x \in \mathbb{R} \tag2
\end{align}
Where $...
0
votes
0
answers
30
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Heat Wave problem
In case 2 I have stuck in the second case what do I have to do? Have I to use hyperbolic function?
$$
\begin{align}
u_{t}&=k u_{xx} \quad -l \leq x \leq l,\ t \geq0, k>0 \tag1 \\
u(-l,t)&...
0
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0
answers
32
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Solving the Malthusian Growth ODE (without separation of variables)
I remember that I have seen somewhere that the simplest mathematical model of population growth
$$
\frac{{\rm d}P}{{\rm d}t}=kP
$$
[where $P$ is the population at time $t$, ${\rm d}P/{\rm d}t$ is the ...
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