Articles of reference request

Are there (known) bounds to the following arithmetic / number-theoretic expression?

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 […]

Books with more problems on card/urn and ball problems

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 […]

Topological Conditions Equivalent to “Very Disconnected”

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 […]

Matrix Theory book Recommendations

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?

Books for starting with analysis

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 […]

Manipulating exponents of prime factorizations

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,…] […]

Defining the multidimensional Riemann Integral as a limit of certain sums

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 […]

about a good book – Vector Calculus

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 🙂

Alternative definition of covering spaces.

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 […]

Consistency/range conditions for (integral) transform mapping into higher-dimensional space

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 […]