Spook
  • Member for 9 years, 10 months
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Show that an entire function $f$ s.t. $|f(z)|>1$ for $|z|>1$ is a polynomial
Accepted answer
9 votes

If the Taylor series about 0 does not terminate, $f(1/z)$ has an essential singularity at $0$ (why?) Then from the Casorati–Weierstrass theorem (have you learnt this?) you know $f(1/z)$ cannot be ...

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Show y is odd in the equation $y^3 =x^2 +2$
8 votes

Suppose not. Then $x^2=y^3-2$, hence $x$, is even. Therefore, $2=y^3-x^2$ is divisible by $2^2$, which is a contradiction.

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If the derivative approaches zero then the limit exists
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8 votes

Picture a sinusoidal curve, but stretched in the horizontal direction more and more as $x\rightarrow\infty$. The amplitude is fixed but the velocity decreases. Something like $\sin (\sqrt{x})$.

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Factorial (Proof by Induction)
5 votes

$(n+1)n!<(n^n)(n+1)<((n+1)^n)(n+1)=(n+1)^{(n+1)}$. Try to direct your algebraic manipulations so that the expressions gradually look like the desired result.

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linear transformation and angles?
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4 votes

No. Imagine strectching the plane in the $x$-direction. Formally $(x,y)\mapsto (2x,y)$. The angle $(1,1)$ makes with $(1,0)$, which, originally is 45 degrees, is reduced.

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Summation of $i \cdot j$ from $ 1$ to$ 3$
3 votes

No, you have not understood the notation. It just means adding up all possible evaluations of ij under the restriction that $1\le i<j\le 3$.

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How can we denote the following function in terms of big-O notation?
2 votes

In the context of computational complexity, we want to describe the behaviour of $f(n)$ as $n\rightarrow \infty$. $\frac{\log 4n }{n}\rightarrow 0$ as $n\rightarrow \infty$ so $f(n)/n\rightarrow 1/3$ ...

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How to prove $4\mid n$?
2 votes

All the $x_i/x_j$'s in the sum is 1 or -1. Therefore you need an even number of them to make the sum 0, since every time you have a 1 you must have another -1 to cancel it out and vice versa. ...

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Is the ring $(\mathbb Z/n, +, \times)$ a subring of $(\mathbb Z, +, \times)$?
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2 votes

No. A subring of $\mathbb{Z}$ must contain $1$. Then it must contain all elements $1+1+\cdots+1$ and all their additive inverses. And also zero. So any subring of $\mathbb{Z}$ must be $\mathbb{Z}$ and ...

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Precedence of $\times$ and $\cup$.
1 votes

If the context is product topology, I think it is more likely to mean the latter as A and C would often be open sets of some space X and B and D would be open sets of some space Y. And it doesn't make ...

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Prove that any finite subset of a linearly independent set is linearly independent
1 votes

A set $S$ being linearly independent by definition means no non-trivial linear combination of elements in $S$ is zero. And a linear combination is a FINITE sum.Therefore if all finite subsets are LI, $...

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Prove that if $a\in [0,1]$, then $\lim\limits_{x \to a} f(x) =0$
1 votes

The key is that $A_n$ are all finite. Can you see that if an ant approaches $a$ without touching $a$, for any $n$, eventually it will leave $A_1, A_2, \cdots, A_n$? If you see that then you are one ...

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If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
1 votes

Generally the one-step subgroup test is faster but in this case you can just check the group axioms: the only non-trivial one is closure. If $a^2=b^2=e$, can you see that $ab$ is its own inverse, ...

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Inequality with exponents $x^x+y^y \ge x^y +y^x$
0 votes

Assume $x\ge y$ by symmetry. We want $$x^x-x^y\ge y^x-y^y.$$ i.e. $$x^y(x^{x-y}-1)\ge y^y(y^{x-y}-1),$$ which is then easy to show by listing all possible cases according to how $x,y$ compare to 1.

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Differentiability on vector values function
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0 votes

Hint: Express $|x|$ as a square root of scalar product of vectors. Then try to make the scalar product look like $f(x)+A(x)+o(x)$ where $A$ is a linear transformation. Then use the fact that $x\mapsto ...

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