# All Questions

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### What is the sum of all such possible numbers given the following conditions?

What is the sum of all such possible numbers given the following conditions that A 4 digit number is formed using the digits 0,2,4,6,8 without repeating any one of them.? MyApproach: If you fix ...
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### Maximal commuative subalgebra of $gl(n)$

Let $gl(n)$ be spanned by $n^2$ abstract operators $\{a_j^i\}$ satisfying the $gl(n)$ commutation relation: $$[a_j^i,a_l^k] = a^i_l \delta_j^k - a_j^k \delta_l^i$$ Why can we conclude that the ...
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### How should I incorporate $\epsilon$

If $u:\mathbb R^2\to \mathbb R$ and $u$ has continuous partial derivatives prove that (a).$u(x+h,y+k)-u(x,y)=\dfrac{\partial u}{\partial x }h +\dfrac{\partial u}{\partial y }k +\epsilon_1$ ...
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### For Functionals does Closed Graph Theorem imply Uniform Boundedness Principle??

For Functionals, Uniform Boundedness Principle can be rephrased as the following : Let ${X}$ be a Banach Space, $K$ be the field($\mathbb{R}$ or $\mathbb{C}$). Let $\mathcal{F}$ be the subset of ...
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### how $x= \cos\theta$ and $y= \sin\theta$?

I'm learning Trigonometry right now and at current about trigonometry functions. I'm little bit confused right now in a section of the chapter. Please have a look at the image. I didn't get the ...
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### Intuitive proof of the formula ${}_nC_r + {}_nC_{r-1} = {}_{n+1}C_r$

I came across this formula in combination— ${}_nC_r + {}_nC_{r-1} = {}_{n+1}C_r$. Even though I know its rigorous mathematical proof, I want a logical and elegant proof of this. For example, ...
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### how to find roots for polynomial which is of the form (1+(x^n)^0.5

how to find roots for (1+(x^n))^0.5 =0; it is actually finding poles of system , where the above expression is sitting in the denominator of that system. even matlab is finding for (x^n)+1, but not ...
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### How to denote the comma as an element of a set

I have a set whose elements include the comma character ",". But the comma is used as a separator when listing the elements of a set. Any suggestions for a sensible notation to use in this case?
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### Permutation of n things taken r at a time with repetition

I am stuck with understanding/writing all the permutation of n different things taken r at a time when repetition is allowed. Or more precisely : In how many ways 3 rings can be worn on 2 fingers ...
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### Understanding Braid Groups

Following the book "Basic Notions of Algebra"-Shafarevich, in Braid group with $n$ strings, A braid looks like a permutation on $n$ letters The composition also is similar to product of ...
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### Cyclic shift in size K of a permutatoin P.

I am stucked in this question,can anyone please help me out.P is a permutation of integers 1, 2,... N. We want to change it a little. To do this, we choose an integer K that satisfies an inequality 2 ...
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### Proving the lower bound for the tail of a sequence $\lvert (f_n(z)) \rvert$ where $(f_n(z))$ converges uniformly

I have a sequence of functions $(f_n(z))$ which converges uniformly to $f(z)$, where $\lvert f(z) \rvert \geq R \gt 0$ for some $R \in \mathbb{R}$. I should prove that there exists $N$ such that for ...
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### Evaluating $\int^{\frac\pi2}_{\frac\pi4}(2\csc(x))^{17}dx$

I saw this question in JEE Advanced. But in that we had to simplify it to $$\int^{\log(1+\sqrt2)}_{0}2(e^u+e^{-u})^{16}du$$ But I pose the following question as to evaluate this integral in closed ...
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### Definition of the smooth local convergence of a sequence of hypersurfaces

I'm reading a lecture note on mean curvature flows (by C. Mantegazza) and studying about analysis of type-I singularities. Let $M$ be a smooth closed $n$-dimensional manifold and ...
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### what is wrong with this derivation?

We have simple function : $$Y = X^2$$ Writing $X^2$ as : $X^2 = X+X+X+...........+X$ $(X times)$ We can write above equation as : $$Y = X+X+X+X+...........+X$$ Differentiating with respect to X, ...
How can I construct an entire function whose growth rate at infinity satisfies $$lim_{r \to \infty} \frac{log M(r)}{\sqrt {r}} =1$$ where M(r) = $max_{|z|=r} |f(z)|$? Based on the above limit, I ...
If I have a factored quartic polynomial in a function such as: $f(0)=6 - A(x-2)(x+1)(x-3)(x+4)$ In the form of $f(x) = A(x-a)(x-b)(x-c)(x-d)$ I would like to know how to to graph it in Wolfram ...