# All Questions

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### Does a compact set with non-empty interior have a limit point?

My Question: Let $U\subseteq \mathbb{C}$ open and $K\subset U$ be a compact set with nonempty interior $K^{o}$, then $K$ must have a limit point in $U$. Remark: I think that the statement is true. I ...
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### Find the homothetic transformation

In $\mathbb{R^3}$: Find the homothety $\Phi$, such that the following transformations are possible: $$\Phi(P)=\Phi(1,0,-1)= (2,5,0)$$ and $$\Phi(Q)=\Phi(0,1,2)= (0,5,2)$$
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### Relating $11111111…1$, $11$ ,and $1010101..1$.

How can we prove the pattern we see in the following observations: $$\frac{11}{11}=1$$ $$\frac{1111}{11}=101$$ $$\frac{111111}{11}=10101$$ .... It seems that if you have $2n+2$ numbers of ...
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### Question about non trivial zeros of Riemann zeta function

Riemann zeta function is $$\zeta(s)=\sum\frac{1}{n^s}$$ I read at wiki that the first nontrivial zero is located at $14.134725...$ As long that I could understand it means ...
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### Matrix rank and number of linearly independent rows

I wanted to check if I understand this correctly, or maybe it can be explained in a simpler way: why is matrix rank equal to the number of linearly independent rows? The simplest proof I can come up ...
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### Is is possible that three countries have three points in common?

On the world map, there are several instances of three countries that have two points in common. For example, China, Russia and Mongolia. Is there any arrangement of three (fictional) countries such ...
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### Trapezoid rule for finding coefficient

If we know that $\int_{a}^b t(x)=h \sum_{k=1}^2 dk * t(a+kh)+O(h^m)$ where $h=\frac{b-a}{3}$, how do we find the coefficient d1, d2 and m in the equation? Answer says that d1=3/2, d2=3/2, m=3 I ...
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### Showing the vectors $v_1$, $v_2$, $v_3$ are orthonormal.

I have a differential equation $$\frac{dv_i}{ds}=\sum^3_{j=1} a_{ij} v_j$$ for $i=1,2,3$ and where $a_{ij}$ is skew symmetric matrix. For some point $s_0$, $v_1 (s_0)$, $v_2(s_0)$ and $v_3(s_0)$ are ...
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### Determine logarithm versus known values of $a,b$

We know $\log_{30}(3)=a, \log_{30}(5)=b.$ How to determine $\log_{30}(16)$?
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### I guess I have found a possible explanation of “Collatz conjecture”. So, how can I publish it?

Please anyone help me in this. This is going to be my first one. So, can anyone good me in this.
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### Finding equation of tangent at given point

I am trying to find the equation of the tangent line for the given point, given $f(x) = log x$ and point $P(2, log2)$. I know the derivative of $log x$ is $\frac{1}{x ln10}$, and as a result the ...
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### Set theory and well-ordering

Let $X \ne \emptyset, X \subseteq \omega$. Show that there is $n \in X, n \cap X = \emptyset$ I am trying to solve this, but I'm a bit confused. It's stated in relation to the wellordering ...
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### Find $\det(A)$ of Matrix and condition on a and b

Let $$A=\begin{matrix} a & b & 1 \\ b & 1 & b \\ 1 & a & a \\ \end{matrix}$$ Find $\det(A)$ in terms of $a$ and $b$, and write down ...
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### Finding the maximal product of numbers of permutations

Let $n\geq 1$ be a total number of objects that must be taken from $m\geq 1$ sets of objects. For all $i \in \{1,\cdots,m\}, \ M_i \in \mathbb{N}^*$ denotes the number of objects present in the set ...
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### Find the inverse Laplace transform of $L(s)= \frac{s}{s^2 + 25} e^{-\pi s}$

$$L(s)= \frac{s}{s^2 + 25} e^{-\pi s}$$ I never seen such function. Can exponential function appear in Laplace transform? Help required
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### Find the volume of the following room [1]

I was working on a project which required me to calculate the volume of the room. The picture of the room is given below: I tried splitting the shape across the diagonals but each time end up ...
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### Second order total derivative

Suppose we have a function $g: \mathbb R^2 \to \mathbb R$ and $$\nabla g(u,v)=(5v^4-2u\exp(v-u^2), \exp(v-u^2)+20uv^3), (u,v)\in\mathbb R^2$$ Can the function $g$ be twice differentiable, i.e. does ...
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### Chinese New Year Equation 2016

In the spirit of Chinese New Year, here's a problem to commemorate the year. $\color{black}{\text{Solve the following equation for positive integers$a$and$b$:}}$ ...
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### A problem of decimals..

The exact problem: For any natural number n>1, write the infinite decimal expansion of $\frac{1}{n}$ (for example, we write 1/2 = $0.4\overline9$ as it's infinite decimal expansion, not 0.5). ...
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### Logical equivalence implication between Kleene and Classical logic

For any propositional assertions, $\phi$ and $\psi$, expressed using only the standard propositional logical connectives $\{\lnot,\land,\lor,\rightarrow,\iff\}$, if $\phi$ and $\psi$ are logically ...
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### A question on polunomial

Let $m\in (0,1)$ and ${a_n}{x^n} + .... + {a_1}{x^1} - f(m) = 0$ and $x\in \mathbb{C}$ $f(m)$ is continuous decreasing function of $m$. $a_i\ge0$ for all $i$. $k(m)$ is positive zero of first ...
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### Finding temperature after time.

A thermometer is taken from a room where the temperature is $20^\circ\,\mbox{C}$ to the outdoors, where the temperature is $-7^\circ\,\mbox{C}$. After one minute the thermometer reads ...
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### Is the following statement equivalent?

Is the following statement equivalent: $((A \vee \neg C) \wedge ((\neg B) \leftarrow C)) \vee \neg (A \wedge B) \equiv (\neg A \vee \neg B \vee \neg C)$ Our prof gave us an exercise with this ...
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### CG-homorphism proof. Stuck at the end!

I am trying to work on some questions back from my uni days, and one has gotten the better of me at the moment! Let $G$ be a finite group and $V, W$ finite-dimensional $\mathbb{C}G$-modules. Let ...
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### Check the proof of $\Bbb R$ as set of subsequential limits

I want to prove that there is a sequence in $\Bbb R$ that has all of $\Bbb R$ as its set of subsequential limits. Could someone help me check my proof? If it's not correct, could someone give a proof? ...
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### Plane integral for continuous curves

I'm trying to understand complex path integral $\int_C f(z)dz$ for continuous closed curve $C$. Is it necessary that $C$ is rectifiable and not just generally continuous? Do we get all the ...
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### another follow up question: modeling with exponential distributions

This a follow up question to the previous two: modeling with exponential distributions a follow up question about modeling with exponential distributions I'm trying to do (c). Denote the ...
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### L2 Norm of Inverse of Non-square Matrix Multiplication

Consider a matrix $A\in\mathbb R^{n\times m}$ with $n<m$. Given that $\|A\|_2 = \gamma_0$ and $AA^T$ is invertible, can we find any equality/inequality for $\|(AA^T)^{-1}\|_2$ in terms of ...
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### Prime factorization and hcf

For any given integer $n$, we prime factorize it as follows $$n = p_1^{k_1} \cdot p_2^{k_2} \cdots p_r^{k_r}.$$ Let $g = \gcd(k_1, k_2, \ldots, k_r)$ and $m_i = k_i / g$. The function $F$ is ...
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### What is wrong with this infinite sum

We know that: https://www.youtube.com/watch?v=w-I6XTVZXww $$S=1+2+3+4+\cdots = -\frac{1}{12}$$ So multiplying each terms in the left hand side by $2$ gives: $$2S =2+4+6+8+\cdots = -\frac{1}{6}$$ This ...
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### Is there K and an infinite amount of different primes $a_i,b_i$ so that min|$a_i^y-b_i^x$| <K on natural x,y for all i?

First of all I know that it was proved recently that prime gaps are less than like 7 million for an infinite amount of primes, but I'm not smart enough to follow the proof. I am looking for a ...
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### Differential? equation in car dashboard problem

I stumbled upon this question while I was driving my car. On my dashboard I have fuel gauge and engine temperature gauge next to each other, look at the pic: http://i.stack.imgur.com/aDgKj.png Fuel ...
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### Anti-derivative of continuous function $\frac{1}{2+\sin x}$

I use tangent half-angle substitution to calculate this indefinite integral: $$\int \frac{1}{2+\sin x}\,dx = \frac{2}{\sqrt{3}}\tan^{-1}\frac{2\tan \frac{x}{2}+1}{\sqrt{3}}+\text{constant}.$$ ...
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### $A$ has more columns than rows and has full row rank, show there exist infinitely many $B$ s.t. $AB=I$

If A $\in M_{m\times n}(R)$ such that $n>m$. Prove that if $\text{rank} (A) = m$ then there are infinitely many matrices $B \in \ M_{n\times m} (R)$ such that $AB = I_m$ So the question is ...
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### Mathematical theory for equally distributed dipole structures with inner equilibration

I'm looking for a mathematical theory for equally distributed dipole structures with inner equilibration. I know, that there exist two magnetic clusters, where the north and the south poles equally ...
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### Let $f$ holomorphic funcion in $U$ such that $\left|f\right|$ constant on the border of $K$. Show that $f$ is constant or $f$ have a zero in $K^{0}$.

Let $U\subseteq\mathbb{C}$ be an open and connected set and $K\subset U$ a compact subset with nonempty interior $K^{o}$. Let $f:U\rightarrow \mathbb{C}$ holomorphic funcion such that $\left|f\right|$ ...
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### Why hough Transformation detect Same line twice

After using canny edge detector my image looks like Image After Canny Edge detector Then I use Hough transform to extract line. Sometimes I able to find four line. But sometimes Same line detect as ...
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### Is a metric's form determined by its signature?

Suppose that we define a 4-dimensional vector space over the real field with a metric with signature (3, 1). Is the scalar product map determined only with this information? For example: a Minowsky ...
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### Difference between joint density and density function of sum of two independent uniform random variables

I am not able to understand the difference between the joint density function and density function for a random variable Z = x1 + x2 where x1, x2 are uniform rvs in [0,1]. I think joint density in ...
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### Even-odd multiplicative algebraic structure with two distinct identity elements?

What is the algebraic structure for the multiplication of even elements and odd elements? even times even is even. even times odd is odd. odd times odd is odd. i.e. the matrix is ...
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### What is the significance of using prime numbers in proving: $x$ is a multiply of $y$?

I came to a problem where it asks me to prove, for example, $n^4-n^2$ is a multiple of $12$. Now, factorize the multiple: $n\times n\times (n-1)\times (n+1)$. Here we have $3$ consecutive integers. ...
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### How many ways we can choose items from different boxes

I searched through the internet but couldn't find my answer, which can either be a very simple or a hard one. Assume there are $3$ boxes, which carry, respectively, $1$, $4$, $2$ items. My question ...
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### Pulley and chain

Find the length of an endless chain which will hang over a. Circular pulley of radius a so that it is in contact with two thirdsof the. Circumference of the pulley? I saw this question in a test. I ...
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### Warshall algorithm for transitive closure

has someone seen a way of applying Walshall's algorithm for transitive closure by finding the '1' elements for e.g. Alpha 1,2, Alpha 2,3 and then: 1. adding row 2 to row 1 2. adding row 3 to row 2 and ...
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### Integral of a measurable function

I do not know what should i keep as title for this question... Question goes like this.. Let $f:\mathbb{R}\rightarrow [0,\infty)$ be a measurable function. If $\int_{-\infty}^{\infty}f(x)dx=1$ prove ...
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### Probability to reach final state

Let $~~m,n>0~~$ be some positive integers. We have some system of states. Each state is pair $~~(i,k)~~$ where $~~0\leq i \leq m~~$ and $~~0\leq k \leq n~~$. Starting state is $~~(m,n)~~$. For ...
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### What subfields use computers the most and least? (soft question)

What areas within (research) mathematics use computers the most and least? What programming languages are commonly used?