# All Questions

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### A complete math programme

I want to correct my biggest mistake in life, which is a complete lack of interest in math when I was a kid. Now, after learning a lot of new things, knowing science and working as a programmer I ...
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### Partial derivatives of all orders of linear map exist

If F is a linear map from R^n to R^m is it true that F is C^infinity, i.e. partial derivatives of all orders exist? My thought is that the answer should be "yes," because the derivative of F is just F ...
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### Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that the difference $X_m-X_l$ is uncorrelated with $X_k$, that is, ...
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### Value distribution into random variables

I want to distribute a large number into quantities of 100s, 50s, 25s, 5s, 1s Lets say, I'm selling apples in quantities of 100s, 50s .... 1. Based on the quantity there is different pricing. 1 apple ...
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### Induction and trigonometry

Do you know how to prove that cos(X/2) + cos(3x/2)... + cos(2n-1)/2 = sin(nx)/(2sin1/2x) with induction? I have tried with n = 1 which gives cos x/2 = sin(nx)/(2sin1/2x) I am not sure on how to ...
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### Solving Logical equivalence & propositional logic problems without truth tables

I have no particular "Logic question" in hand at the time being, but need help to understand a way that can be used to prove "Logical equivalence without using truth tables". moreover can we solve ...
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### Does $\pi \left(\dfrac{x+y}{2}\right)=\pi\left(\sqrt{xy}\right)$ hold for infinitely many values of $x$ and $y$?

The problem is (assume $\pi(x)$ to be the prime-counting function, Does there exist infinitely many solutions to the equality $\pi \left(\dfrac{x+y}{2}\right)=\pi\left(\sqrt{xy}\right)$ with ...
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### How can check that for a given norm, we can found an inner product?

Let $$\Bbb C^2=\{w=(z_1,z_2) : z_1,z_2\in\Bbb C\}$$ be the vector space of all ordered pairs of complex numbers. Can we obtain the norm defined on $\Bbb C^2$ by $$||w||=|z_1|+|z_2|$$ from an inner ...
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### How to find right child in a pyramid number?

A pyramid number: 0 1 2 3 4 5 6 7 8 9 So is there any equation like: ...
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### Surface area of a slightly deformed sphere

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates. I define a deformed sphere ...
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### How to find estimation value of $u$ s.t minimize $\sum_{i=1}^n \sum_{j=1}^m \|x_i-c_j\|^2u_{ij}^p$

I have a dataset $x=\{ x_1,x_2...x_n\}$. I will divide the data into $m$ classes with cluster center (mean of each data in a cluster) is $c_j$. The absolute square distance between each data $x_i$ ...
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### Characterizing the A-module ${M}/{S}$

I've been working through this for a little while, and I'm not 100% sure I understand what I'm supposed to be doing here, or maybe I'm not grasping correctly what they mean by "Characterize". ...
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### The “argument” of a quaternion

My question is pretty simple. I've been trying to read a pretty introductory text on Clifford algebras, and I encountered how they define the "argument" of a quaternion as an ordered quadruple ...
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### How can I calculate the formula of this fractal-like structure?

I did the following fractal-like structure manually, and I was trying to convert it to a formula to compute some parts of the drawing, but I get lost due to the complexity of the structure. Is as ...
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### G acts freely on X. G is paradoxical implies X is also paradoxical

I am proving the Banach-Tarski paradox using a series of small results. For definition of certain terms, see here. Group $G$ acts freely on $X$ i.e. $\operatorname{Stab}(x)=e, \ \forall \ x\in X$. ...
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### Is the sum of a unimodal and increasing function still unimodal?

There is no specific function. I would like to know if there is information on summation of a unimodal and increasing function? (Continuous functions)
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### Cardinality of the set of all field automorphisms of $\mathbb C$ [duplicate]

Does $\mathbb C$ have infinitely many field automorphisms? Does it have uncountably many field automorphisms?
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### Integral Test question

So this is the problem: http://postimg.org/image/5g815zgk5/ I am getting $\lim_{b\to\infty} 2\sec^{-1}(2b) - 2\sec^{-1}2$ Now what? What do I do with $\sec^{-1}(2b)$? What happens to a trig function ...
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### Matrix $B \in M_n(S)$, for $S$ an $R$ algebra, with $R$ independent entries, $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent?

Let $R$ be a field (or a domain, or a commutative ring), and $S$ an $R$-algebra. Let $B \in M_n(S)$ have $R$ independent entries. Let $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent? I am ...
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### example of multiplication of ordinals with infinite cardinality with larger value on right where we dont' take the max?

I recall reading about a rule for multiplying ordinals where at least one is infinite, and where the cardinality of the multiplier (on right) is larger than the multiplicand (on left). If I recall ...
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### Is there an alternative intuition for solving the probability of having one ace card in every bridge player's hand?

I am trying to get to know probability a little better since it's a weak point for me and I was wondering what other ways there were to intuitively think about the problem of finding the probability ...
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### Incommensurable line segments

I have an issue with a proof given in my lecture hopefully someone can help me with. It will be shown that the hypotenuse $c$ of a right-angled and isosceles triangle will be incommensurable to the ...
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### How does one maximize “mixedness” among seating charts over time?

Background: My class has $10$ students and $3$ tables; naturally, the students are distributed with $3, 3,$ and $4$ seated at the individual tables. On the second day of class, students sat in the ...
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### Can I divide $50$ cars on$5$ days? any trick? [on hold]

Can I divide $50$ cars on $5$ days, on condition that the numbers should be odd numbers? is there any trick? I'm asking for $5$ 5 odd numbers whose sum is $50$
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### Analysing Exact Sequence

I have the following exact sequence $\mathbb{Z}\xrightarrow{f}\mathbb{Z} \xrightarrow{g} K_0(\mathcal{T})\xrightarrow{h}\mathbb{Z}\xrightarrow{0}0$. From here I want to conclude that ...
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### Proving one expressions is greater than the other using limits?

In general, is it sufficient to show that one of them increases faster than the other? $$1-P_{k,1}< or > (1-P_{k,2})(M+B(1-p))/(M+B))$$ where $P_{k,1}$ and $P_{k,2}$ are decreasing with M. ...
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### Are these two compositions of two functions differentiable?

Assuming $U=\{x\in\mathbb{R}^2:x_1^2+x_2^2<1\}$ is the open unit circle in the plane and $f,g:U\rightarrow\mathbb{R}^2$ two functions with $f(0)=g(0)=0$. $f$ is Fréchet-differentiable in $0$, and ...
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### Divergence and formal adjoint operators: are they bounded/continuous?

Let $(M,g)$ be a smooth Riemannian manifold. The divergence operator is the map \begin{align*} \delta_g:\Gamma^k(S^2M)&\rightarrow\Gamma^{k-1}(T^*M)\\ ...