# All Questions

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### Complex number multiplication

Z1= cos(4π/3) + isin(4π/3) Z2= cos(π/3) + isin (π/3) I want to find out Z1Z2. I know that (x +iy)(u + iv) = (xu - yv) + i(xu + yv) So I want to simplify [cos(4π/3)cos(π/3) - sin(4π/3)sin(π/3)] + ...
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### A question about the category $C_{A,B}$

Take the category $C_{A,B}$. I'm afraid I don't know how to draw a commutative diagram, but imagine maps between $Z_1$ and $A$ and $B$, and $Z_2$ and $A$ and $B$. Let the morphism between $Z_1$ and ...
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### What does “relatively closed” mean in this context?

Please have a look at this page. What does "relatively closed" mean? I have the same problem but I don't quite understand how the set $U=\{x\in U \mid u(x)=M \}$ is both open and ...
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### Are there combinatorial games of finite order different from $1$ or $2$?

Are there any combinatorial games whose order (in the usual addition of combinatorial games) is finite but neither $1$ nor $2$? Finding examples of games of order $2$ is easy (for example any ...
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### What is the difference between Monte Carlo Simulation and Law of Lig Numbers

As far as I understood, Monte Carlo simulation uses several randomly chosen candidates to see where the result converges. On the other hand, Law of Large Numbers theorem describes the result of ...
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### Is there a formula for a simple sum of sums?

Given f(n) = 1 + (1 + 2) + (1 + 2 + 3) ... + (1 + 2 + 3 ... + n), I am wondering if there is a straightforward formula to compute ...
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### Square root - primitive question

The answer to my question might be obvious to you, but I have difficulty with it. Which equations are correct: $\sqrt{9} = 3$ $\sqrt{9} = \pm3$ $\sqrt{x^2} = |x|$ $\sqrt{x^2} = \pm x$ I'm ...
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### Is $\sqrt{-1}$ equal to $i$ or $\pm i$?

In complex numbers is $\sqrt{-1}$ equal to $i$ or $\pm i$ ? In both cases how do we explain it? The question arose when I saw it in Lathi's book (Linear Systems and Signals).
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### Proving isomorphisms from posets.

An isomorphism from a poset $(S_1,R_1)$ to a poset $(S_2,R_2)$ is a bijection $f: S_1 \rightarrow S_2$ such that, for all $x,y \in S_1$ $(x,y) \in R_1 \leftrightarrow (f(x), f(y)) \in R_2$ When ...
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### Modelling function

The amount you pay for your water usage is calculated based on the following rule: a base rate of 50 dollars a month, plus 2 dollars per litre for the first 1000 litres of water used, and 3 dollars ...
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### Limit of binomial distributions whose expectation tend to 0

According to the Poisson limit theorem, if $np\to\lambda$, then $\text{Bin}(n,p)\to\text{Poisson}(\lambda)$ (all when $n\to\infty$). Does that mean, in particular, that if $np\to 0$, the limit ...
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### I just cannot really beleive math [on hold]

Math, is of course useful and powerful, yet I can't believe in the numbers, just as it feels unreal. Numbers are abstract, I just can't deal with them, the deep and horrible distrust in my heart.
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### Completion of a measure space

Studying Measure Theory in University, I came across the following definition for the completion of a measure space: let $(X,\mathcal{E},\mu)$ be a measure space; then the set ...
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### On the use of “generic” and “general” in algebraic geometry

I have learned that in algebraic geometry, when an 'object' can be put in a family which is in a bijective correspondence with some projective variety, the generic object in this family is one which ...
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I am trying to find the inverse laplace transform of $(s^2+4) \over (s-2)(s+2)$. The solution is ${2\over(s-2)} - {2\over(s+2)} + 1$. But I can't figure out how to break it up so I can find the ...
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### Profit maximisation Problem for Firm

I was wondering if someone could help me with the following problem. To be honest, I don't really know where to start ... A profit maximising firm producing output using a single input according to ...
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### How prove this inequality $3b+8c+abc\le 12$ if $a^2+4b^2+9c^2=14$

let $a,b,c>0$ and such $$a^2+4b^2+9c^2=14$$show that $$3b+8c+abc\le 12$$ My idea: since \begin{align*}3b+8c+abc&=3b+c(8+ab)=3b+\dfrac{1}{9}\cdot 9c(8+ab)\le ...
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### If $AB=BA$ then they are diagonal

I saw this statement If $AB=BA$ then both matrices must be diagonal. Why is that?
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### proof of this theorem

$$\boxed{\mathbf{Theorem:}\textit{ If }\alpha \textit{ is a root of }Q(x)\textit{ then }(x-\alpha)\textit{ divides }Q(x).}$$ What is the proof of this theorem? I've tried to do euclidean division of ...
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### “Differentiability” implies continuity

Suppose $f$ has a derivative $\forall x\in(a,b)$ (including infinite derivatives). Does it imply $f$ being continuous at $[a,b]$ or only at $(a,b)$ or neither?
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### A question on odd perfect numbers

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, there are $48$ known examples of even perfect numbers -- on ...
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### $f$ continuous, monotone, what do we know about differentiability?

I am interested in knowing what we can say in general about when a continuous function $f:\mathbb{R} \to \mathbb{R}$ is differentiable. To my mind, there are various ways a continuous function can ...
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### How many triangles may a connected simple graph with m edges have at most?

Suppose a connected simple graph has m edges and there are at most n vertices and the degree of each vertex is at least ...
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### Topology on the set of infinite sequences

Help, who can give me an example to show the following definition of $\lim$ and closed set let $\Sigma$ be a set of states, let $\Sigma^\omega$ denote the set of all infinite sequences of elements in ...
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### Matlab Creating function for ode45

i want to create odefun from reading data from the table ...
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### Induction proof of $n^{(n+1) }> n(n+1)^{(n-1)}$

The question statement from my homework booklet goes: Prove by mathematical induction that $n^{n+1} > n(n+1)^{n-1}$ is true for all integers $n \geq 2$. I've managed to come up with this ...
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### Periodic absolute value function

Define $$h(x)=|x|$$ on the interval $[-1,1]$ and extend the defintion of $h$ to all of $\mathbb{R}$ by requiring that $h(x+2)=h(x)$. Now define the function: $$h_n (x)=\frac{1}{2^n} h(2^n x)$$ ...
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### Fourier transform of random binary vector

Consider a uniformly chosen random binary vector $V$. That is we say $V_i \in \{0,1\}$. What is the distribution of the Fourier transform of $V$? I have searched online but have not managed to find ...
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### Relation between reflexivity and FPP

It is known that if $X$ be a Banach space and $X$ be reflexive then $X$ isomorphic to a Banach space with FPP. I have two questions 1: Is there a reflexive Banach space but does not FPP? 2:It is ...
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### I need an even function that grows faster than cosh(x)

Does anyone know of any even special functions that grow very fast, faster than $\cosh(x)$? (Not the exponential) (Further info): ...
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### 2nd order inhomogeneous DE, particular solution

$$2y''-5y'-3y=2e^{3x} .$$ When I was doing the particular solution, I got $$y_p=Ae^{3x} , \, y'_P=3Ae^{3x} , \, y''_P = 9Ae^{3x} .$$ Substituting them into the DE, I get ...
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### Normal Bundle of a Manifold

I was reading "Morse Theory" by J.Milnor and at page number 32 there is remark "It is not difficult that N is an n-dimensional manifold differentiably embedded in $\mathbb{R}^{2n}$ ( N is the total ...
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### Abel's test: how to prove it

Consider the following statement: Let $b_n$ satisfy $b_1 \ge b_2 \ge \dots \ge 0$ and let $\sum_{n=1}^\infty a_n$ be a series for which the partial sums are bounded i.e. there exists $A > 0$ such ...
When two sets are positively separated we know that $\mu(A \cup B)=\mu(A)+\mu(B)$. My question is what happens when their intersection is null. Will the above equation be invalid? My Try:It has to be ...