# All Questions

70 views

### Finding the average sum of 15 numbers out of 100 numbers

I've got a question. For the numbers 1,2,...,100 we choose with the same probability 15 numbers. 1) What is the average sum of those numbers? 2) What is the average sum of those numbers when the ...
67 views

38 views

### Radical Inequality related to three variables

Prove the following. $$\sum_\text{cyclic}\sqrt[4]{\dfrac{(a^{2}+b^{2})(a^{2}-ab+b^{2})}{2}}\leq\dfrac{2}{3}\left(\sum_\text{cyclic}\dfrac{1}{a+b}\right)\left(\sum_\text{cyclic}a^{2}\right)$$
138 views

### $\infty + \infty = \infty$?

(The context is a measure-theoretic one.) I know that $\infty - \infty$ is indeterminate, but what about $\infty + \infty = \infty$? It seems this statement is true and if I input it into Wolfram ...
52 views

### To what is $f^{-1}(G_2)$ equal??

If $\displaystyle{f: G_1 \rightarrow G_2}$, then $\displaystyle{f^{-1}: G_2 \rightarrow G_1}$, right?? Could you tell me which set is $$f^{-1}(G_2)$$?? Is it the set $G_1$??
99 views

### Blowing up a Singular Point More Than Once.

I am trying to understand how $I_n$-fibres appear in an elliptic surface by performing a sequence of blow-ups. To be concrete, I am looking at the following elliptic surface given in Weierstrass ...
494 views

### Forming a simple polygon from the extrusion of a polygonal chain

Let's say I have a set of vertices connected by edges to form a polygonal chain. Each vertex may be shared by a number of edges to form various sub-chains. An example is shown below. Each edge has ...
53 views

### Question regarding morphism of ringed spaces

I have recently started studying schemes, and I have encountered this passage from the book by Kenji Ueno: My questions: i) If $(X,O_X)$ is a local ringed space, why is $(X, i_*({O_X}_{|U}))$ also ...
268 views

### What is the difference of an n-tuple and a permutation of n elements

My understanding of n-tuple and a permutation of n elements is, that both are ordered sequences of n elements. Are there differences in the objects correlating to these two terms ? I guess it ...
68 views

81 views

### Neighbourhood of a matrix

Sometimes I find definitions which say that something happens in a neighbourhood of a matrix. For example a dynamical system generated by $x'=Ax, \ A \in \mathcal{M}(n)$ is structurally stable if ...
147 views

### What is a circle's area if its radius is $\pi$?

The area of a circle equals $\pi r^2$. If a circle's radius is $\pi$, what is its area? I believe the answer is $\pi^3$, right?
128 views

### Prove that the product satisfies the universal property for coproducts in Ab.

I refer to this question. The question is Is there an intuitive way to understand why finite products and coproducts in Ab coincide, while the same is not true in Grp? The author of the ...
76 views

### $m(E∩I)≥αm(I)$ for all intervals $I⊂[0,1].$ Prove that $m(E) = 1.$

Let $E$ be a measurable subset of $[0, 1].$ Assume there is a constant $α > 0$ such that $m(E∩I)≥αm(I)$ for all intervals $I⊂[0,1].$ (Here $m(·)$ denotes Lebesgue measure.) Prove that $m(E) = 1.$ ...
811 views

195 views