I apologize in advance if this is something that is already well-known in the literature, but I would like to ask nonetheless (for the benefit of those who likewise do not know): Are there (known) lower and upper bounds to the following arithmetic / number-theoretic expression: $$\frac{I(x^2)}{I(x)} = \frac{\frac{\sigma_1(x^2)}{x^2}}{\frac{\sigma_1(x)}{x}}$$ where $x \in \mathbb{N}$, $\sigma_1(x)$ is […]

I am struggling with combinations and permutations. One particular concept that is bugging me is selecting outcomes. I posed a few questions in a forum. \What is the probability that you are dealt a “full house”? (Three cards of one rank and two cards of another rank.)\ I received following answer “”When counting the number […]

Definition: Let $(X, \mathcal{T})$ be a topological space, where the set $X$ has more than one element. Suppose that for every pair of distinct elements $a, b \in X$, there exists a separation $(A,B)$ of $X$ such that $a \in A$ and $b \in B$. Then we say $(X, \mathcal{T})$ is very disconnected. Is this […]

I’m currently reading Sheldon Axler’s “Linear Algebra Done Right”. Can anyone recommend any good books on matrix theory at about the same level that might compliment it?

I am interested in self-studying real analysis and I was wondering which textbook I should pick up. I have knowledge of all high school mathematics, I have read How to Prove It by Daniel J. Velleman (I did most of the excercises) and I have completed a computational calculus course which covered everything up to […]

Has work been done on looking at what happens to the exponents of the prime factorization of a number $n$ as compared to $n+1$? I am looking for published material or otherwise. For example, let $n=9=2^0\cdot{}3^2$, then, $$ 9 \;\xrightarrow{+1}\; 10 $$ $$ 2^0\cdot{}3^2 \;\xrightarrow{+1}\; 2^1\cdot{}3^0\cdot{}5^1 $$ or looking just at the exponents, $$ [0,2,0,0,…] […]

Currently, I am taking a course where we defined the multidimensional Riemann Integral of a map $f:\mathbb{R}^n \to \mathbb{R}$ as the limit $$\lim_{\varepsilon \to 0}\phantom{a}\varepsilon^n \sum_{x \in \mathbb{Z}^n}f(\varepsilon x)$$ If we require that $f$ is continuous and has a compact support, it is guaranteed that the above limit exists. Unfortunately, I do not know any […]

I start reading Vector Calculus by Jerold E. Marsden, Anthony J. Tromba and I want to know if there is a book with the answers of the exercises. I like a lot this book, it seems to be made for a beginner but also for those who want to assimilate new knowledge. Thanks ðŸ™‚

in a lecture I have seen a definition of a covering space, different from what I would call the usual one (e.g. the one in Munkres): A surjective continuous map $p:E\rightarrow B$ of spaces $E$ and $B$ is a covering map (and $E$ a covering space of $B$) if for every $b\in B$ there is […]

I am interested in learning more about what the (formal) implications are when transforming a $n$-dimensional function space (e.g., the space of all $\mathbb{R}^n\to\mathbb{R}$) to a higherdimensional one (say, $\mathbb{R}^m\to\mathbb{R}$, $m>n$); in particular, in terms of consistency or range conditions (is there a difference?). A real example could be the relationship between the result of […]

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