# All Questions

1answer
9 views

0answers
37 views

### The sum of $\frac{k^{2}}{k-1}$ from 6 to 12

Can you evaluate this sum by using the properties of the sigma notation ? Or I must develop this and evaluate them one by one ?
1answer
173 views

### A theorem from the theory of groups

Let $K$ be a (not necessarily normal) subgroup of the group $G$ : $K < G$ A fixed element $g\in G$ can act, from the left, on all elements of $G$, thus generating a bijection of $\,G\,$ onto ...
0answers
16 views

1answer
46 views

### From Primitive Recursive to Recursive by Iterating over more than one Argument?

Is the only way a function can be recursive and not primitive recursive by growing faster than primitive recursion allows (as with Ackerman's function)? If so, then consider the following. Primitive ...
2answers
76 views

### Prove $R$ is a finite ring [on hold]

Suppose that $R$ is a ring and $D(R)$ is the set of all elements $x\in R$ such that there exist $b,c\neq 0$ with $xb=cx=0$. Prove that if $1<|D(R)|<\infty$ then $|R|<\infty$. ($|S|<\infty$ ...
2answers
219 views

### Is transfinite induction needed to remove all the elements from an uncountable set?

Given any uncountable set S, would I need to use transfinite induction to prove if I remove single elements recursively, I will be left with the empty set? It seems like this can be thought of as an ...
1answer
9 views

### Growth of Fourier coefficients of piecewise linear function

Suppose $f$ is a periodic continuous piecewise linear function. What can be said about the growth (or decay, rather) of the Fourier coefficients $\hat f(n)$ as $n\to\infty$, other than the ...
0answers
18 views

### non-countable subset of $\mathbb 2^{\mathbb Z}$ with finite pairwise intersection. [duplicate]

Does a non countable subset of the power-set of $\mathbb Z$ exist so that the intersection of any two elements is a finite set? If we ask for the sets to be pairwise disjoint then the answer is a ...
7answers
384 views

### Intuition for a physical real line vs. a physical “hyperreal line”

As a mathematical structure, I have no problem with the hyperreals. But I came across the following from Keisler's book "Elementary Calculus: An Infinitesimal Approach". "We have no way of knowing ...
4answers
114 views

### Can we find uncountably many disjoint dense measurable uncountable subsets of $[0,1]$?

Can we find uncountably many disjoint dense measurable uncountable subsets of $[0,1]$? Obviously we may as well assume all the subsets have measure $0$. If I didn't specify the subsets were ...
1answer
355 views

### Binary constraint integer programming problem

Hi I have a question to the folowing question: Explain how to use integer variables and linear inequality constraints to ensure: A) let x and y be integer variables bounded at 1000. How can you ...
2answers
189 views
+200

### Smallest integer $k$ so that no Sudoku grid has exactly $k$ solutions

Inspired by this question, consider hints on a Sudoku board. A regular puzzle has a unique solution. It is clear that there are puzzles with 2 or 3 solutions, and therefore, I guess, puzzles with say ...
0answers
15 views

### Proof of conformal mappings onto polygons (Stein)

Recently I am reading Stein and Shakarchi's Complex Analysis and I find a great difficulty in understanding the proof of theorem 4.6 in Chapter 8 (p.242-244), which talks about conformal mappings onto ...
3answers
55 views

### recurrence relations and generating functions - I need a hint

I need to find "closed form" to the recurrence relation given by: $a_{n+1} = \sum_{k=0} ^ {n} k a_{n-k}$ and $a_0 = 1$. I have tried using generating functions but it is no good. Any help would be ...
0answers
20 views

### Lower bound for $\Pi(n)$ - viability of probabilistic theory

Can somebody check the validity of my arguments below, and tell me why its wrong or right? Consider the sequence of non-negative integers. Let $a_0=0, a_1=1, ..., a_i=i,...$ Divisiblilty of $a_i$ ...
1answer
52 views

### How to prove that G is a cyclic Group?

Suppose $G$ is a finite Abelian group and, $\forall\ n\in \mathbb{N}$, there exist at most $n$ elements in $G$ which satisfy $x^n=1$. Prove $G$ is cyclic. Thanks for your help.
1answer
21 views

### Using a linear function as a routine to determine a matrix

Let $F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ be a linear function, i.e., $$F(\alpha x + \beta y) = \alpha F(x) + \beta F(y)$$ Suppose you are given a routine that returns $F(x)$ given any ...

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