Mathematics

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6 answers

Why is $A^TA$ invertible if $A$ has independent columns?

How can I understand that $A^TA$ is invertible if $A$ has independent columns? I found a similar question , phrased the other way around, so I tried to use the theorem $$ rank(A^TA) \le min(rank(A^T),rank(A)) $$ Given $rank(A) = rank(A^T) = n$ and…
Chewers Jingoist
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75
votes
5 answers

How to write a good mathematical paper?

I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar. A good time always goes too fast! Two years are fled. In the third year of PHD, my major is general topology and I'm facing…
Paul
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75
votes
6 answers

Fourier transform of function composition

Given two functions $f$ and $g$, is there a formula for the Fourier transform of $f \circ g$ in terms of the Fourier transforms of $f$ and $g$ individually? I know you can do this for the sum , the product and the convolution of two functions. But I…
MathematicalOrchid
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75
votes
2 answers

Is it mathematically valid to separate variables in a differential equation?

I read the following statement in a book on Calculus, as part of my mathematics course: Technically this separation of $\frac{dy}{dx}$ is not mathematically valid. However, the resulting integration leads to correct answer. The book also contains…
Devarsh Ruparelia
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75
votes
4 answers

Are there any differences between tensors and multidimensional arrays?

I see lots of references saying things like A tensor is a multidimensional or N-way array But others that say things like it should be remarked that other mathematical entities occur in physics that, like tensors, generally consist of…
rhombidodecahedron
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75
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4 answers

Does every set have a group structure?

I know that there is no vector space having precisely $6$ elements. Does every set have a group structure?
user23505
74
votes
3 answers

Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $

I've found here the following integral. $$I = \int_{0}^{1}\sin{(\pi (1-x))}x^x(1-x)^{1-x}\,dx=\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}$$ I've never seen it before and I also didn't find the evaluation on math.se. How could we…
user153012
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74
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5 answers

A new imaginary number? $x^c = -x$

Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would allow solving for logs with negative bases or with…
Warren L.
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74
votes
1 answer

Fractal behavior along the boundary of convergence?

The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, emeritus of Reed College) showed me 15 years ago,…
2'5 9'2
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74
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9 answers

Subgroup of index $2$ is Normal

Please excuse the selfishness of the following question: Let $G$ be a group and $H \le G$ such that $|G:H|=2$. Show that $H$ is normal. Proof: Because $|G:H|=2$, $G = H \cup aH$ for some $a \in G \setminus H$. Let $x\in G$. Then $x \in H$ or $x \in…
William T.
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74
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17 answers

What is a real world application of polynomial factoring?

The wife and I are sitting here on a Saturday night doing some algebra homework. We're factoring polynomials and had the same thought at the same time: when will we use this? I feel a bit silly because it always bugged me when people asked that in…
Dan
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74
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3 answers

Does non-symmetric positive definite matrix have positive eigenvalues?

I found out that there exist positive definite matrices that are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues. Does this hold for non-symmetric matrices as well?
user19653
74
votes
3 answers

How likely is it not to be anyone's best friend?

A teenage acquaintance of mine lamented: Every one of my friends is better friends with somebody else. Thanks to my knowledge of mathematics I could inform her that she's not alone and $e^{-1}\approx 37\%$ of all people could be expected to be in…
hmakholm left over Monica
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74
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2 answers

Can $18$ consecutive integers be separated into two groups such that their product is equal?

Can $18$ consecutive positive integers be separated into two groups such that their product is equal? We cannot leave out any number and neither we can take any number more than once. My work: When the smallest number is not $17$ or its multiple,…
Hawk
  • 6,818
74
votes
11 answers

Is Euclid's proof on the infinitude of primes flawed because it yields some composites?

I've expressed Euclid's proof on the infinitude of primes on Mathematica: f[x_] := Product[Prime[n], {n, 1, x}] + 1 TableForm[Table[{f[x], PrimeQ[f[x]]}, {x, 1, 20}]] Which results in: $\begin{array}{ll} 3 & \text{True} \\ 7 & \text{True} \\ 31 &…
Red Banana
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