# All Questions

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### Hyperbolic Coxeter Systems

This question is regarding the proof of a proposition in the text book reflection groups and coxeter groups by Hymphreys section 6.8. Prop: Let $(W,S)$ be an irreducible coxeter system with graph ...
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### sum of integral parts of real number and fraction

For any real x and positive integer n ,show that [x] + [x +1/n] + [x + 2/n] + .... + [x + n-1/n] = [nx] I have used the fact that x-1 < [x] <= x,for all terms and added,but not able to get ...
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### $\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a}\geq 4$

$a, b, c, d$ are positive reals. How would I prove the inequality $$\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a} \geq 4$$ I have tried using the rearrangement inequality with ...
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### How to Calculate Flat rate using Effective Rate?

Following calculations are made thriugh an Excel calculator in my company. I had to input EMI to calculate Flat Rate , but I want asolution to calculate Flat rate straight from Full/Effective rate ...
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### Is there any large diffeomorphisms of $S^{n}\times S^1$like Torus?

We know that a Torus is mapped onto itself in a special discontinuous transformation given by $PSL(2,\mathbb{Z})$. Thinking of torus as $S^{1}\times S^{1}$ and thus as a lattice, we can easily show ...
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### find the values of a, b and c from sin graph

The diagram shows part of the graph of a function whose equation is $y = a\sin(bx^{\circ}) + c$ (a) Write down the values of a, b and c. (b) Determine the exact value of P ...
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### Check if a given coordinate lies in path of a ray (coordinate geometry)

As shown in the image I have two known coordinate pair A and B and few other known coordinate pairs (RED blob) on the graph. I need to know if any of the other given coordinates fall in line of the ...
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### Cost prices, selling prices

I have a question in maths regarding GST, cost prices and selling prices. (GST is government services tax, the amount added on to an amount for the government, so it is basically tax) There are two ...
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### Inequality with a square root

If the inequality $(x+2)^{\frac{1}{2}} > x$ is satisfied. what is the range of x ? My approach - I squared both the sides and proceeded on to solve the quadratic obtained in order to solve the ...
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### Convex polyhedron with minimum faces

A convex polyhedron has at least three faces which are pentagons. What is the minimum number of faces the polyhedron might have? I have a polyhedron with seven faces but I don't know whether it is ...
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### Let $w = \log(u^{2} + v^{2})$ where $u=e^{(x^{2}+y)}$ and $v= e^{(x+y^{2})}$

Then $\frac{\partial w}{\partial x}$ for $(x=0,y=0)$ is ? I got answer as 0 since on partial differentiation I got, ...
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### What is the name of the group $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$?

I know, that $\mathbb Z_2\times \mathbb Z_2$ is the Klein four-group. Is there a nice name for $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ too?
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### Equivalent conditions for $\mathfrak{F}$ to be a differential ideal

Heres the question: Let $\mathfrak{F}$ be an ideal of forms on a manifold $M$ locally generated by $r$ independent $1$-forms. Say $\mathfrak{F}$ is generated by $\omega_1,\ldots, \omega_r$ on $U$. ...
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### Why are is partitions counting technique wrong?

I recently heard about partitions. I tried to count them using the following technique: 1) Ways to write $5$ as a sum of five positive integers: $$1+1+1+1+1$$ 2) Number of ways to write $5$ a sum of ...
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### Initial value problem $y'=e^{-y^{2}}-1.$

Let y be a solution of $y'=e^{-y^{2}}-1$ on $[0,1]$ which satisfy $y(0)=0$. Then how to prove that $y=0$ on $[0,1]$? According to me as the function $e^{-y^{2}}-1$ is Lipsctiz and continuos on $[0,1]$ ...
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### Mass of lamina defined in y ≥ 0, with edges given by y = 0, y = (4-x^2)/3 and x = −y + 2y^2, and density is y.

I've been trying work this out, but I'm stuck on the the integral calculation. I've drawn a diagram, got all the points of intersection and relevant points, but I still can't get it. I had a go at ...
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### Example of a “abrupt function”

I need example of a simple function to show that cubic spline gives better result than Lagrange's interpolation in case of some special functions. Thank you
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### A problem on random q-colourings of a graph for randomly chosen vertex

Here is an exercise from Olle Haggstrom's "Finite Markov Chains and Algorithmic Applications" from the chapter "Fast Convergence of MCMC Algorithms". The exercise is based on random $q$-colorings of ...
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### Is there a combinatorial identity for the following sum?

Let $a,b,c$ be integers. Is there by any chance a neat combinatorial identity for the following sum? $$\sum_{j=0}^c{a + jb \choose j}.$$ Thanks!
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### Elementary number theory with some arithmetic progression.

Let A={n∈N┤n is the sum of seven consecutive integers}. B={n∈N┤|n is the sum of eight consecutive integers}. C={n∈N┤|n is the sum of nine consecutive integers}. Find A∩B∩C. I tried ...
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### Is there a name for differential operators with certain homogeneity

Is there a name for the ODEs of the following form ? $$\sum_{m_k+n_k= N, }a_{m_k,n_k}u^{m_k}u^{(n_k)}=0,$$ where $u^{m_k}$ denotes the $m_k-$th power of $u$ and $u^{(n_k)}$ denotes the derivative of ...
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### Looking for a easy proof of the fact that $H^i_m(R)=0$ if $i<n$?

Let $R=k[x_1,x_2,...,x_n]$ and $m=(x_1,...,x_n)$ be an ideal of $R$. Is there any easy (without any higher machinery ) proof of the fact that $H^i_m(R)=0$ if $i<n$ ? I know that the above ...
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### What can you say about the $k$-th cohomology group of a closed orientable $n$-manifold?

What can you say about the $k$-th cohomology group of a closed orientable $n$-manifold for: (1) k=n, and (2) k=n-1 Poincare Duality tells us that for $M$ a closed $R$-orientable $n$-manifold, ...
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### Isomorphic Galois groups imply isomorphic field extensions?

Suppose we have two field extensions $K/k$ and $L/k$. I am able to show that if these field extensions are isomorphic, then their corresponding Galois groups Aut$(K/k)$ and Aut$(L/k)$ are also ...
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### Prove that for the matrix $A$, $A^TA$=$A$ and for the projection matrix $P^2=P$

I'm not sure how under what conditions the proofs are possible. I'm begging for help. This is not a homework question.
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### Reducing a Boolean function

I have the following boolean function: f(x,y,z) = xyz + xyz' + xy'z + x'yz + xy'z' I could reduce it to the following: f(x,y,z) = xy + xy'z + x'yz + xy'z Im not sure what to do next, i know it can ...