# All Questions

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### how to solve this question of two functions

I tried to solve this equation in different ways but no luck find the number of points of two functions. f(x) = sin x y = a at a given a. in this case lets say a = 0.15 sin x = a sin x = (0.15)x ...
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### Is there a name for a topological space $X$ in which very closed set in $X$ is a countable union of compact sets?

Is there a name for a topological space $X$ which satisfies the following condition: Every closed set in $X$ is a countable union of compact sets Does Baire space satisfy this condition? Thank ...
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### Proving that the unities of a ring form a group under multiplication

I am presented with the following task: Show that if $U$ is the collection of all units in a ring $\langle R, +, \cdot\rangle$ with unity, then $\langle U, \cdot\rangle$ is a group. I am still ...
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### An Application of Rouche's Theorm to Two Cases

Here is my question - it is an example sheet question, completely non-examinable: [I have managed this first part, but am including it to help give a sense of where the question is going.] $(i)$ ...
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### Suppose that matric $AB+A+B=0$, how to prove that $AB=BA$?

Lets say, we have $n\times n$ matrices $A$ and $B$. Suppose that $AB+A+B=0$. How can we prove that $AB=BA$? Thank you in advance. Any help is much appreciated.
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### f is differentiable on convex domain D and Re(f')>0 implies f is injective on D

Suppose that $f$ is differentiable on a convex domain (open and connected) D and Re($f')>0$ in D. How can we prove that f is injective on D? I found some answers using the mean value theorem for ...
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### Please solve this in details inverse problem i am using complement angle formula

For any $x \in [-1,1]$, how can I prove that: $$\sin^{-1}(2x\sqrt{1-x^2})=2\cos^{-1}x$$ Also, can someone explain to me how to understand the graphs of $sin$ and $cos$ functions?
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### In a fair deck of 52 cards, how many 5 card hands contain the ace of spades?

is the answer 51 Choose 4? assuming all possible combinations of hands are equal to (52 Choose 5)
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### Finding a vector potential, i.e. given $\vec{A}$, how to find $\vec{B}$ s.t. $\vec{A} = \nabla \times \vec{B}$?

I understand that this might not be unique, but is there a (relatively) painless way to generate such a 'vector potential', so for a given field $\vec{A}$, a new field $\vec{B}$ which satifies: ...
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### Sort the array in minimum moves

Given an array A[1 .. N], which contains N integers. I need to sort this array.But the only operation that I can perform on array is : ...
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### Transfer function from Bode-plot

Could anyone lead me through the steps of getting the transfer function from this Bode-diagram? http://i.imgur.com/1kWvheO.png I've been reading up on this material but can't quite get a hang of the ...
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### Prove that if $H$ is a subgroup of index $2$ in a finite group $G$, then $gH = Hg \forall g \in G$.

Prove that if $H$ is a subgroup of index $2$ in a finite group $G$, then $gH = Hg \; \forall \; g \in G$. I know that $H$ itself is one coset of the subgroup and the other is the compliment of the ...
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### Derive the variance of values with respect to a value

As from the title, I'd like to get the following: $\frac{\partial }{\partial x_a}(\frac{\sum_{i=0}^n (a_ix_i - \mu)^2}{n})$ where $\ a_i$ is the $\ i$ -th element of another series (of known ...
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### Statistics and data spread help?

How do you tell if the inter quartile range or standard deviation is a better spread of data?
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### combinina The Normal destribution and binomial destribution

If we consider a normal distribution as an approximation of a binomial distribution with 2 conditions below: $$p=q$$ and $$n \to \infty$$ Of course a poison distribution can be considered as an ...
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### Congruences or logs

How do you know the pairs of integers $x,y$ such that $$y=\frac{ln(p(x))}{log(k)}$$ is true, where $p(x)$ is any Diophantic equation and $k$ any Natural number?
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### Upper bound for $\sum_{k=1}^n \binom{n}{k} \left(c\frac{k}{n}\right)^k$

I have been looking all over the place, but I can't seem to find a closed for expression or good upper bound for $$\sum_{k=1}^n \binom{n}{k} \left(c\frac{k}{n}\right)^k, \mathrm{where}\; c > 0$$ ...
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### Sobolev, Holder, Lp spaces continous and compact embeddings proof

I would like to know if the following proof is fine. I haven't filled in all the detail but please let me know what you think about the basic outline.(I am aware that there are posts which have dealt ...
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### Ideal in $k[x,y]$ generated by two elements

Suppose $k$ is a finite field of order $q$. Let $f = \prod_{1 \leq i \leq s} (x + b_i y)$ and let $g = \prod_{1 \leq i \leq t} (x + c_i y)$, where $b_i, c_j \in k.$ I am interested in finding out ...
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### If $[E:F]$ is finite and $\alpha \in E$ then there is an irr. polynomial in $F[x]$ with root $\alpha$

I'm studying for an exam and encountered a confusing proof of the following fact in my notes: Let $[E:F]$ be finite and $\alpha \in E$ then there is an irreducible polynomial $p(x) \in F[x]$ with ...
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### Solving the recurrence relation $a_{n+1}=a_n^2$

How would one solve the recurrence relation $a_{n+1}=a_n^2$ for, say, $a_0=2$? The solution seems to be $a(n)=2^{2^n}$, but how would one get to that conclusion? Furthermore, how would one solve a ...
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### Do you know to find the limit?

I need to find the limit of this equation : $$\lim_{x\to 0^{+}} \frac{\sqrt{1-e^{-x}}-\sqrt{1-\cos x}}{\sqrt{\sin x}}$$
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### Having trouble understanding the Tor functor

I am having trouble understanding the Tor functor as presented in Dummit and Foote. Given $\dotsb\to P_n\to P_{n-1}\to\dotsb\to P_0\to B\to 0$ as a projective resolution with homomorphisms ...
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### Terminology with zero divisors

Let $R$ be a commutative ring with identity $1$. If for some $a \in R$ there exists $b \in R$ such that $ab = 1$, then we say that $a$ is a unit and that $b$ is a multiplicative inverse or reciprocal ...
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### Integration with substitution $u=\cos(x/2)$

Please could you help me solve this integral Find $$\int \cos x \, \sqrt{1-\cos x} \, dx.$$ Hint: use the substitution $u=\cos(x/2)$. Thanks.
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I am taking a course in abstract algebra, and have a question regarding this idea. I am being asked to show, after having proven that an r-cycle is a conjugate to its own inverse (there is an x such ...
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### Classify Singularities

So, I'm trying to classify the singular points of the following function: $$f(z)=e^{\cot(\frac {1}{z})}$$ Obviously, when z is zero, the function tends to approach infinity, so that must be a ...
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### predicting interarrival and waiting times poisson process

I have been recently going through the poisson distribution and its theory of interarrvial times. I have a problem. I have a point A. Now, there is a guard stationed at that point. He is granted ...
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### Eisenbud 3.11(d) - A Uniform Bound on the Length of Certain Modules

I am trying to solve this exercise from Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. There is a hint or possibly a solution in the back, but I want to try to get some more ...
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### Nice notation for support of random variable.

does anyone have good suggestions for notation that will represent support of a random variable (r.v.)? So, for example if $X$ is r.v. operator $\text{SUPPORT}(X)$ will output the support of random ...
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### The best symbol for non-negative integers

I mean to specify the set {0, 1, 2, ...}, i.e., non-negative integers in an engineering conference paper. Which symbol is more preferable? $\mathbb{N}_0$ $\mathbb{N}\cup\{0\}$ $\mathbb{Z}_{\geq0}$ ...
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### (Can we) Determine the slope of the line from single points from parallel lines?

This is a very elementary question that I am missing out, So, given two (or more) points in parallel lines, (i.e. we only know of a single point lying in each line), (can we) determine the slope of ...
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### Proving $x_i = \dfrac{\det(A_i)}{\det(A)}$

Suppose we have an nxn matrix A, and $A\underline{x} = \underline{p}$ prove that $x_i = \dfrac{\det(A_i)}{\det(A)}$ where $A_i$ is the matrix obtained from A by replacing the $i^{th}$ column by ...
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### how can i calculate Limiting distribution $\displaystyle\frac{\sum_{i=1}^n Y_i}{\sum_{i=1}^n Z_i}$

let $Y_1,Y_2,\ldots,Y_n$ are random sample of bernoulli distribution with parameter of $\displaystyle\frac{\theta_1}{\theta_1+\theta_2}$ and $Z_1,Z_2,\ldots,Z_n$ are random sample of geometric ...
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### Can a non-extreme point be an optimal solution of a Linear Programming problem?

Consider a linear programming problem. Is it possible for an optimal solution to exist, but not at an extreme point? According to Bertsimas & Tsitsikalis ("Introduction to Linear Optimization", ...
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### Assuring the output range of neural network

I am learning some model based on examples ${((x_{i1},x_{i2},....,x_{ip}),y_i)}_{i=1...N}$ using a neural network of Feed Forward Multilayer Perceptron (newff) (using python library neurolab). I ...
I am trying to prove the following. I have seen it alluded to in other places of the internet (this site included) but without proof. Let $L,L_1\ldots L_n$ be linear functionals on a vector space ...