Mathematics Stack Exchange Community Digest

Top new questions this week:

Why does an exponential function eventually get bigger than a quadratic

I see this answer to this question and this one. My $7$th grade son has this question on his homework: How do you know an exponential expression will eventually be larger than any quadratic expression?...

algebra-precalculus exponential-function quadratics  
asked by John L 52 votes
answered by Some Guy 47 votes

Teacher claims this proof for $\frac{\csc\theta-1}{\cot\theta}=\frac{\cot\theta}{\csc\theta+1}$ is wrong. Why?

My son's high school teacher says his solution to this proof is wrong because it is not "the right way" and that you have to "start with one side of the equation and prove it is equal ...

algebra-precalculus trigonometry proof-writing solution-verification  
asked by Ramblin Wreck 27 votes
answered by Zadig 10 votes

On the parity of $\left\lfloor{\frac{3^n}{2^n}}\right\rfloor$

Let $a_n=(-1)^{\left\lfloor{\frac{3^n}{2^n}}\right\rfloor}$ and $$s_n=\sum_{k=1}^na_k.$$ Is it true that $s_n\le 0$ for all $n\geq 1$ ? (This is true for $n\le 100000$.) In other words, odd numbers ...

asked by lsr314 25 votes
answered by hgmath 10 votes

Why does ${x}^{x^{x^{x^{\,.^{\,.^{\,.}}}}}}$ bifurcate below $\sim0.065$?

When you calculate what ${x}^{x^{x^{x\cdots }}}$ converges to between 0 and 1, before approximately 0.065 the graph bifurcates. Why does this happen and is there a reason for it happens at that ...

limits tetration bifurcation  
asked by Kevin Ndayishimiye 22 votes

How to color a board such that each square has exactly two neighbors of the opposite color

Let $m≥2$, $n≥2$ integers. We want to color the squares of an $m × n$ board with black and white so that each square has exactly two neighbors of the other color. Determine all the values ​​of $m$ and ...

discrete-mathematics coloring  
asked by Qqq 11 votes
answered by Misha Lavrov 5 votes

Do there exist uniform triangular prisms with all vertices in $\mathbb Z^3$?

It's quite easy to find a regular square prism (cube) or a regular triangular antiprism (octahedron) with vertices in $\mathbb Z^3$. Take for instance, take the convex hulls $$ \begin{align*} &\...

geometry polyhedra polytopes  
asked by Peter Kagey 10 votes
answered by RavenclawPrefect 7 votes

What is the largest volume of a polyhedron whose skeleton has total length 1? Is it the regular triangular prism?

Say that the perimeter of a polyhedron is the sum of its edge lengths. What is the maximum volume of a polyhedron with unit perimeter? A reasonable first guess would be the regular tetrahedron of side ...

geometry polyhedra  
asked by RavenclawPrefect 9 votes

Greatest hits from previous weeks:

How many even numbers of four digits can be formed with the digits 0,1,2,3,4,5 and 6 no digit being used more?

My attempt to solve this problem is: First digit cannot be zero, so the number of choices only $6 (1,2,3,4,5,6)$ The last digit can be pick from $0,2,4,6$, so the number of choices only 4 Second ...

asked by akusaja 3 votes
answered by akusaja 1 vote

A loss and gain problem

This is a very simple but confusing puzzle. A customer buys goods worth $200$ rupees from a shop. The shopkeeper selling these goods makes zero profit from this purchase. The lady gives him a $1000$ ...

asked by constantlearner 8 votes
answered by fgp 8 votes

What does the small number on top of the square root symbol mean?

I just came across this annotation in my school's maths compendium: The compendium is very brief and doesn't explain what this means.

algebra-precalculus notation  
asked by Hubro 14 votes
answered by Jakube 16 votes

What is the Jacobian matrix?

What is the Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone please explain with examples?

calculus multivariable-calculus derivatives jacobian  
asked by Pratik Deoghare 113 votes
answered by Américo Tavares 73 votes

why does e raised to the power of negative infinity equal 0?

Why is it that e raised to the power of negative infinity would equal 0 instead of negative infinity? I am working on problems with regards to limits of integration, specifically improper integrals ...

calculus limits exponentiation infinity  
asked by apple.pi 8 votes
answered by DeepSea 13 votes

The difference between log and ln

$$\dfrac{1}{2}\ln(x+7)-(2 \ln x+3 \ln y)$$ Our professor let's us solve this but i do not understand how $\ln$ works. He says it has same properties with $\log$ but i still don't get it. What's the ...

asked by Zhianc 27 votes
answered by Arturo Magidin 33 votes

How to find perpendicular vector to another vector?

How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please? I have a solution to this when I have $3\mathbf{i}+4\...

linear-algebra geometry vector-spaces vectors  
asked by niko 68 votes
answered by carlop 52 votes

Can you answer these questions?

Factorization of sum of terms

I found in a solution that expands the following series, $$ =\sum_{i=1}^m{\left( \alpha _{ik+1}p_{k+1}+\alpha _{ik+2}p_{k+2}+...+\alpha _{in}p_n \right) ^T\left( \alpha _{ik+1}p_{k+1}+\alpha _{ik+2}...

linear-algebra matrices  
asked by Avra 1 vote

Permissable to define a fork function with two x values mapping to different functions but whose y-values are identical?

Say we got this simple example $$f(x) = \begin{cases} p_1(x),\text{ for }5 \leq x \leq 8,\\ p_2(x),\text{ for }8 \leq x \leq 10 \end{cases}$$ Assuming $p_1(8)=p_2(8)$, is this permissable? Arguments ...

functions definition  
asked by Sebastian Nielsen 1 vote
answered by GVT 0 votes

$I=\langle x^2+2y^2-3, x^2+xy+y^2-3\rangle$ in $\Bbb Q[x,y]$ how do I find the Grobner basis for $I \cap \Bbb Q[x]$ and $I\cap\Bbb Q[y]$?

Given the ideal $$I=\langle x^2+2y^2-3, x^2+xy+y^2-3\rangle$$ in $ ℚ[x,y]$, how do I find the Grobner basis for $I \cap \Bbb Q[x]$ and $I\cap\Bbb Q[y]$? I have learned to find a Grobner basis using ...

algebraic-geometry ring-theory ideals  
asked by user607408 1 vote
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