# All Questions

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### contour integral to show the sum of a series

Let $Γ_n$ be the cycle that traverses the square with vertices ±(n+ $\frac 12$ )(1±i). Show that there is a constant δ > 0 such that |sin(πw)| > δ for all w ∈ $U_n$ $Γ_n$. By considering the ...
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### If $\tan x=\sin x/\cos x$ then what is $\tan 3x$ equal to?

Would $\tan 3x$ be equal to $\sin 3x/\cos x$? Or perhaps $\sin 3x/\cos 3x$? Regards, Tom
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### if x is an integer, then $(x^3+1)\bmod 3 = (x+1)^3 \bmod 3$

Can anyone help me explain why if $x$ is an integer, then $(x^3+1)\bmod 3 = (x+1)^3 \bmod 3$? I know there are 3 cases. $x=0\bmod3,\ x=1\bmod3,$ and $x=2\bmod3$ totally new to this form of mathematics,...
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### Number of ways of putting n indistinguishable balls into k indistinguishable groups.

http://www.campusgate.co.in/2011/10/permutations-balls-and-boxes-related.html Can someone explain how the recurrence table for Case 4 has been obtained ?
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### Maximize the prime sums in a table

The entries of a $3×3$ table are integers from $1$ to $9$, and each number appears exactly once. Consider the row, column, and diagonal sums of numbers in the table. Find the maximum number of these ...
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### Difference between Double and triple integral?

Hi all I am going to be starting multivariable calc and I am trying to read up but I can't seem to quite grasp this exactly yet. What are the differences between double and triple integrals? I am ...
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### Roots of Unity and Primitive roots

For $\mathbb{Z}/13$, I want to find its primitive roots and 4th roots of unity. For $g$ to be a primitive root, we must have that $g^6 \neq 1 \pmod{13}$ and $g^4 \neq 1 \pmod{13}$. $2$ satisfies this....
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### Series having strict inequality implies limits having strict equality?

I was wondering if I have two convergent series, say, $\sum_{n=1}^{\infty} s_n = s$ and $\sum_{n=1}^{\infty} t_n$ = t, and for all their partial sums we have that: $s_n > t_n$. Is it then ...
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### Differential Geometry-Hodge Star

The Hodge star is given by $$*(dx^{i_1}\wedge dx^{i_2}\wedge....\wedge dx_{i_p})=\frac{1}{(n-p)!}e_{i_1 i_2....i_p i_{p+1}...i_n}dx^{i_{p+1}}\wedge dx^{i_{p+2}}\wedge....\wedge dx^{i_n}$$ The question ...
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### Show that the real projective line, P1, is orientable

I'm asked to show that the real projective line, P1, is orientable. I'm not quite sure how to define orientable to prove this. Thanks.
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### What does invariant exactly mean and how does it get the invariant?

I have read many journal about simulation of regularized long wave. In numerical test section,many researcher use invariant of mass,momentum and energy to check accuracy of their method but i found ...
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### Unwind the equation

Let $x, y, z, t$ be positive integers. Given that $$68(xyzt+xy+zt+xt+1)=157(yzt+y+t)$$ Find the value of the product $xyzt$. I couldn't even start with the problem. I just know that the expression n ...
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### Proving the recursive formula for “Virahanka Numbers”

So apparently Virahanka was an Indian mathematician that, in a way, discovered the Fibonacci sequence 500 years before Fibonacci. He was interested in finding the number of patterns of short syllables ...
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### Boundedness in a linearly ordered set

Let $X$ be a linearly ordered set and $A \subset B \subset X$ Can it happen that $A$ is bounded in $X$ but not in $B$ ? Can it happen that $A$ is bounded in $B$ but not in $X$ ? Attempt: I think ...
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### Pricing of Binary or Digital Options or Feynman-Kac Equation for $\mathbb E f(X_T)$ with diffusion $X$ and discontinuous function $f$.

I am trying to find references (books, papers, etc.) for calculating $\mathbb E f(X_T)$, where $X_T$ is a diffusion and $f$ is a real function that is not continuous by means of solving a PDE or ...
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### $(f_n)$ Borel measurable implies $\sup_n f_n$ and $\inf_n f_n$ Borel measurable

Suppose $(f_n)$ is a sequence of Borel measurable functions. Show that both $\sup_nf_n$ and $\inf_nf_n$ are Borel measurable. Attempt: Suppose $(f_n)$ is a sequence of Borel measurable functions. ...
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### Find $\lim_\limits{n\to \infty}\underbrace{{\sin\sin\cdots\sin(x)}}_{n\text{ times}}$. Why am I wrong? [duplicate]

Find $\lim_\limits{n\to \infty}\underbrace{{\sin\sin\cdots\sin(x)}}_{n\text{ times}}$. It is known that after the first sine, we get something in $[-1,1]$. If it is $0$ then it is constant and ...
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### Help Constructing an infinitely differentiable function…

Given $a<b$, I need to find a function $\psi\in C^{\infty}(\mathbb{R})$ so that $\psi(x)=0$ when $x\leq a$, $0<\psi(x)<1$ when $a<x<b$, and $\psi(x)=1$ when $x\geq b$. Previously, I ...
### Prove if $n^2$ is even, then $n^2$ is divisible by 4
I am working on this question Prove for every integer n if $n^2$ is even, then $n^2$ is divisible by 4. prove by contradiction Proof: Since there exists an integer $n$ such that $n^2$ is even,...