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.

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1answer
58 views

how these elements come? - finite field with polynomial $x^2+1$

I am given this finite field. $$K=\mathbb{Z_3}[x]/p$$ with $$p(x)=x^2+1$$ In my textbook, i see that the elements of this field are: $$\{0, 1, 2, x, x +1, x+2, 2x, 2x + 1, 2x + 2\}$$ I dont ...
0
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1answer
38 views

how do I find all Elements of a Group?

I am given a Group $\mathbb{Z_{11}^*}$. a multiplicative group. How do i find all elements of this Group?
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1answer
30 views

how comes this step ? inductive proof with binomial

I am sitting on this problem for couple of hours now but still cannot get the reason why the step which i marked is possible... how has $(n+1)$ disappeared here?
1
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1answer
31 views

how comes $s=4$ and $t=3$ for $4=7s-8t$

i am given these two problems: $x\equiv 1 (\bmod 7 )$ and $x \equiv 5( \bmod 18)$ I tried this way: $x\equiv 1 (\bmod 7 )$ is basically $x = 1 + 7s$ and $x\equiv 5 (\bmod 18 )$ is $x=5+18t$ then ...
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0answers
40 views

On the solution of a stacking/spending optimisation problem

Description of the problem Each week our hero, say John, can stack a minimum of $2.5$ hours and a maximum of $6.5$ hours if he works more than $8$ hours per day. Each day John cannot stack less than ...
2
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1answer
28 views

Question of how to write certain response

What is the domain of the function $$f(x,y)=\frac{1}{x^2+y^2-1}$$ The answer is clear, right?! $$x^2+y^2-1\neq 0\Longrightarrow x^2+y^2\neq1$$What is the point contained in the circle of ...
1
vote
2answers
135 views

Problems with the existence of limits of several variables

I started some time studying calculus of two variables, and I'm having difficulty in knowing (and prove) that a limit does not exist, how could I resolve, for example, these, whose statement asks to ...
0
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0answers
64 views

Solving an equation with several summations and a falling factorial

Can you help me to solve quite a complicated inequation that contains summations? I'd like to solve it for b (leaving balone on ...
22
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10answers
1k views

Puzzles or short exercises illustrating mathematical problem solving to freshman students

At high school, the solution method to almost all mathematical exercises is to apply some technique or algorithm you have learned before. At the university, the situation is fundamentally different. ...
2
votes
2answers
183 views

Allocation optimization problem

Imagine that I have $1$ million dollars which I want to invest. I have a set of $N$ elements in which I can put the money and obtain a revenue. Each element has a function that determines how much ...
1
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0answers
71 views

Recommendation for a good book on equation solving theory from the basics

I'm relearning calculus but I often find myself applying algebraic operations to equations mechanically without having a solid understanding of the side-effects of those operations. Such as extra or ...
0
votes
2answers
58 views

how to prove this: $f(A)=B$

I am given two sets: $A$ and $B$ and a function $f: A \rightarrow B$. I am asked to show and prove whether $f(A)=B$ is true or false. I am stuck not knowing how to do this. How can I do this?
1
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1answer
114 views

A proof about affine subsets

Let $V$ be a vector space and $S$ a nonempty subset of $V$. I want to show that $S$ is an affine subset (a translated subspace of the form $\{v\} +$ $U$, for $U$ a subspace of a vector space $V$ and ...
1
vote
1answer
68 views

equation with series

Hello to everybody I have a problem because I can't solve this equation: $$960 - \frac{84.60}{(1+x)^{\frac1{12}}} - \frac{84.60}{(1+x)^{\frac2{12}}} - \cdots - \frac{84.60}{(1+x)^{\frac{11}{12}}} - ...
4
votes
3answers
274 views

How to solve $e^x = 2$

I know that $\ln(x)$ is the inverse of the exponential function $a^x$. So I thought that $$ e^x=2 \Leftrightarrow x = \ln(2) $$ but my calculator says $x = \ln(2) + 2 i \pi n$, where $N \in ...
0
votes
0answers
28 views

Writing an integer as a sum of four integers, subject to restrictions [duplicate]

Let $a_n$ be the possible number of ways to write $n=i_1+i_2+i_3+i_4$ where $i_1,...,i_4$ are all integers. Here's their restrictions: $$0<i_1<3\\i_2>0\\i_3=2\text{ or }4\\i_4>1$$ Write ...
2
votes
3answers
202 views

Probability of winning the game 1-2-3-4-5-6-7-8-9-10-J-Q-K

A similar question to mine was answered here on stackexchange: Probability of winning the game "1-2-3" However, I am unable to follow the formulas so perhaps someone could show the ...
0
votes
1answer
37 views

simple intersection understanding problem

I am reading theoretical computer science subject and came across this small step. in the textbook, they defined two contextfree languages: $L_1 = \{ a^i b^j c^j | i,j>0\}$ and $L_2 = \{a^i b^i ...
1
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3answers
61 views

Distance Rate Time problem

One morning, Ryan remembered lending a friend a bicycle. After breakfast, Ryan walked over to the friend’s house at 3 miles per hour, and rode the bike back home at 7 miles per hour, using the same ...
0
votes
2answers
104 views

Nonexistence of Limit of Sum of Prime Factors

In trying to prove the following problem, I find great difficulty in proceeding to generalizing some results: Let $s(n)$ be the sum of prime factors of an integer $n$. Prove that $\lim_{n \to \infty} ...
14
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3answers
300 views

On a Putnam's 2009 problem [duplicate]

Find all even natural numbers $n$ such that the following is true: There is a non-constant function $f : \Bbb{R}^2 \longrightarrow \Bbb{Z}_2$ such that for any regular $n$-gon $A_1...A_n$, $f(A_1) + ...
0
votes
1answer
75 views

Finding speed from little information?

The problem I've been given to solve is read as follows. Two-thirds of the way across a narrow railroad bridge, Willy Gope hears a train coming towards him. He knows that the train's speed is 45 ...
12
votes
1answer
209 views

Teaser or fun calc equation to surprise husband (physicist/EE) at work

I am a geneticist and unfortunately have not worked much with advanced calc since undergrad. In genetics, as you likely know, a male is denoted as XY and a female as XX. I plan to leave a riddle for ...
0
votes
1answer
103 views

I do not know where to start this challenge, help please

If $x$, $y$ and $z$ are positive numbers such that $1\leq xy+yz+zx\leq3$ which is the set of values ​​of $xyz$? And $x + y + z$?$$$$Knowing that $x, y$ and $z$ $\in\mathbb{R^*}$
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2answers
99 views

Preimage of a function

The only way to get better at this sort of thing is to practice, and now I'm also trying to ask myself (and try to answer) more conceptual questions. If a circle with radius $r$ is given in ...
1
vote
2answers
68 views

Comparison between Bessel's coefficients

The spatial solution is written as $$\Phi_k(r) = r^{1-\frac{d}{2}} \left(c_1 J_{1-\frac{d}{2}}(k r) + c_2 Y_{-1+\frac{d}{2}}(kr)\right).$$ In the case $d=3$, the solutions can be written as ...
4
votes
1answer
271 views

The Principle of Mathematical Induction

The question is Let $( F_0, F_1, F_2,... )$ be the Fibonacci sequence defined by $F_0=0,\, F_1=1, and F_{n+1}=F_n+F_{n-1}$, n greater than or equal to 1. Prove the following identities. ...
0
votes
1answer
911 views

The cardinality of the power set with $N$ elements is equal to $2^N$ [duplicate]

Let $\mathcal{P}(X_N)$ be the power set of a set $X$ with $N$ elements. I am trying to prove by induction that its cardinality $\mid \mathcal{P}(X_N) \mid = 2^N$. Firstly, I think it helps to ...
0
votes
1answer
43 views

Finding the length

How to determine the length of PR? We can determine the angle QSP and the length $SR$ then what to do?
2
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5answers
54 views

Help with minimization problem

help me, if $x$ and $y$ are real such that $3x-4y = 12$, determine the minimum value of $z = x ^ 2 + y ^ 2$?$$$$I thought of $$3x-4y = 12\Longrightarrow x=4\frac{y+3}{3}\\z = x ^ 2 + y ^ ...
0
votes
1answer
106 views

How to solve the problem that determines the age of Diophantus?

How to solve the problem that determines the age of Diophantus?$$$$"God gave him to be a boy for the sixth part of his life, and adding one to it twelfth part covered her cheeks fluff, He gave him the ...
6
votes
3answers
485 views

Computing Infinite Continued Fractions

I am looking for "tricks" used to compute infinite continued fractions. For example, $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}$$ is the golden ratio since if we denote it by $x$, then we have ...
2
votes
1answer
84 views

Generate a formula for x in my case

Kindly see the following, ...
0
votes
0answers
2k views

Prove that if the sum of two numbers is irrational then at least one of the numbers is irrational.

Question: Prove that if the sum of two numbers is irrational then at least one of the numbers is irrational. Is your proof direct, by contradiction, or by contrapositive? State the converse. Prove or ...
1
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0answers
50 views

Solving equations involves digamma function

I am trying to derive the EM-algorithm of mixtures of negative binomial distribution $Neg\;Bin(r,p)$. I have the updating equations for updating the E-step as well as $p$ and the mixing coefficients ...
0
votes
1answer
194 views

Checking for graph isomorphism by hand

I'm working through "A First Look at Graph Theory" by Clark & Holton, and in the first exercise, there are problems asking to check whether different graphs are isomorphic to each other. I find ...
1
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0answers
83 views

Solution by of nonlinear equation

$$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + \sin u = 0$$ From the sine-Gordon equation we can easily solve, \begin{equation} \phi(x) = \pm 4 \tan^{-1}\left[e^{\frac{x-t ...
2
votes
2answers
58 views

Find lattice points on a planar curve

I have the following curve in the plane: $$y = \frac{c-x}{6x+1}$$ Given a constant value $c \in \Bbb N$; is there a technique(s) I can apply to find lattice points on this curve?
2
votes
2answers
103 views

$1^\infty=1$ or is indeterminate?

$$\underbrace{1\cdot1\cdot1\cdot1\cdot1\cdot1\cdot1\cdots1\cdot1}_{\infty}=1$$ because of $1^\infty$ is indeterminate?$$$$ And $$\lim_{x\to\infty}1^x=\lim_{x\to\infty}1=1?$$
0
votes
1answer
21 views

delivery performance

I need to calculate average speed of a driver as a performance indicator. I have the total miles driven and the total hours they took. Simple division of mi/hr is the speed. But I want to factor in ...
0
votes
1answer
55 views

Solving $Ae^{Bx}+Cx+D=0$

Is it possible in general to solve equations of the form $Ae^{Bx}+Cx+D=0$? The most I have been able to do is rearrange the equation as ...
4
votes
1answer
185 views

What is a logical solution to this problem?

Monkey beans. An urn contains 23 white beans and 34 black beans. A monkey takes out two beans; if they are the same, he puts a black bean into the urn, and if they are different, he puts in a white ...
1
vote
1answer
301 views

Proving the transitivity of a relation

I want to prove that the relation $\sim$ on fractions given by $\frac{a}{b} \sim \frac{c}{d}$ if $ad = cb$, where $a, c \in \mathbb Z$ and $b, d \in \mathbb Z_{> 0}$, is transitive. (My last ...
2
votes
3answers
98 views

Given $x \wedge y=\mathbf{F}$, how to simplify $x \wedge \lnot y$?

Given that the boolean expression $x \wedge y=\mathbf{F}$, how to simplify $x \wedge \lnot y$? Is the above question equivalent to the following question? Find z so that $\lnot(x \wedge ...
0
votes
1answer
27 views

Equation matching after inserting a value.

A little confusion I have got, In this question (in the middle), $$\chi''(r) + \frac{d - 1 + 2\beta}{r}\chi'(r) + \left(k^2 + \frac{\beta(\beta + d - 2)}{r^2}\right)\chi(r) = 0.$$ Form this line he ...
14
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3answers
451 views

Decipher the greeting card $( X^2 +Y^2 -1 ) ^3 - X^2 Y^3 = 0$

A friend of mine just got a rather weird congratulations card through the door They have zero idea what it means, I have tried graphing it and nothing spectacular comes out. Is there a standard ...
2
votes
1answer
39 views

Probability problem without replacement and series

I have a probability problem but I am unsure about the result. I would like to confirm my guesses. During an exam, 10 students are drawing one by one a question (among 12 questions) without putting ...
0
votes
1answer
31 views

Same attribution depending on salary

there's a house hold that consists of a husband and a wife, the husband earns $1500$ and the wife earns $2000$ and their rent is $2000$, what's the calculation to determine what percentage of their ...
4
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0answers
82 views

Is there a better way to solve this problem in Linear Algebra?

Well I have the following problem: Let $\alpha = \{v_1,v_2,v_3\}$ and $\beta=\{u_1,u_2,u_3\}$ be two bases of $\mathbb{R}^3$ such that $v_1=(1,0,1)$, $v_2=(1,1,0)$ and $v_3=(0,1,1)$. It's known that ...
1
vote
1answer
69 views

Need help with boundary conditions of a differential equation.

QUESTION: A particle $A$ is moving along the $X$ axis at a constant horizontal velocity $u\hat{i}$. Another particle $B$ is moving such that its velocity vector always points towards the particle ...