# Questions tagged [products]

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### Product of a vector by an inverse of Toeplitz matrix

It is well known that using fast Fourier transform it's possible to multiply a vector by a Toeplitz matrix $A \cdot v = w$ in $n\cdot\log(n)$ operations. I read somewhere that also the product of a ...
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### $\sigma$-product of the Hilbert cube

Given a homogeneous space $X$ and $p\in X$, we define the sigma product to be the following subspace of $X^\omega$: $$\sigma X=\{\mathbf x \in X^\omega:x_n=p\text{ eventually}\}$$ ("eventually&...
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### Is there a theory of formal product series?

A few years ago, I asked a question on MSE about the existence of an infinite product representation of a functional square root of the sine function. No answers were given, though user ...
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### Conjecture on sum over permutations of products of Catalan numbers

Context In a recent paper involving entanglement in linear optics, we came across some summations involving Catalan numbers and permutations. In particular, these sums arise when doing integration ...
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### Is Spec of a ring monoidal or anti-monoidal?

Let $A$ and $B$ be rings. A very senior mathematician impressed on me the importance of writing $$\operatorname{Spec}{A \otimes B} = \operatorname{Spec}{B} \times \operatorname{Spec}{A}$$ One can ...
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### Product topology from two premetric spaces induced by sum of premetrics?

For metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, it is an exercise that the product topology on $M_1\times M_2$ is induced by the metric $d((x_1, y_1), (x_2, y_2)) =d_1(x_1, x_2) + d_2(y_1, y_2)$. Do ...
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### How far is a countably infinite reduced abelian $p$-group from being an infinite direct sum?

Question Let $G$ be a countably infinite reduced abelian $p$-group. Is it always possible to write it has an infinite direct sums of non-trivial groups? If it is not true, how far is $G$ from being an ...
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### Cartesian product is to monoidal product as pullback is to what?

I'm trying to complete the following pattern product : monoidal product : coproduct pullback : ? : pushout That is, if the monoidal product is a ...
1 vote
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### Polynomial invariant — from product formula to monomial expansion

Context This question deals with the polynomial invariant denoted by $H_{n}$ in Maksym Fedorchuk and Igor Pak's 2004 paper Rigidity and polynomial invariants of convex polytopes (sections 7.6 and 9)....
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### Product of inductive limit topologies on $C_c(X)\times C_c(X)$

I have a stupid question about a topology on $C_c(X)$. Here $X$ is locally compact Hausdorff. Can assume $\sigma$-compact if it helps. Definition (topology on $C_c(X)$): For each compact $K \subset X$,...
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### Given multiple posets, what is the probability that a randomly selected (uniform dist) subposet of their product has a max under the product order?

Given multiple totally ordered posets, how do I find the probability that a randomly selected (with uniform distribution) subposet of their product has a maximum under the product order? I have some ...
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### Fastest way to calculate the eigenvalues of a product of two Toeplitz matrices

I have the following problem: I need to find the fastest way to calculate the eigenvalues of a matrix that is the product of two Toeplitz matrices. $B = A U$. The first is a regular Toeplitz matrix $A$...
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### Is it possible to express the functional square root of the sine as an infinite product?

Cross-post from MSE. It is known that the sine can be expressed as an infinite product: $$\sin(x) = x \prod_{n=1}^{\infty} \Big{(} 1 - \frac{x^{2}}{n^{2}{\pi}^{2}} \Big{)} .$$ We can define that ...
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### Explicit examples of (probability) measures on $\prod \mathbb{R}$

Let $\prod_{n \in \mathbb{N}} \mathbb{R}$ be equipped with the Tikhonov product of the Euclidean topologies on $\mathbb{R}$ and let $B$ the corresponding Borel $\sigma$-algebra. What is are some ...
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### Closed form of $\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$

I have made a question here about closed form of the following: $$\prod_{k=0}^{N}\big(k!\big)^{{N}\choose{k}}$$ I know that there is a known closed form for, $$\prod_{i=0}^{N}\big(i!\big)=G(N+2)$$ ...
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### Definition of twisted geometries and existence of coordinate transformation for twisted $AdS_2 \times S^2$

In the paper Multiply Twisted Products by Yong Wang, general definitions for so called warped and twisted products are given: A (singly) warped product $B \times_b F$ of two pseudo-Riemannian ...
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### About infinite products and Euler Gamma functions [closed]

I am interested in knowing how to calculate infinite products like (or reading any reference about it): $$\prod_{j=1}^{\infty}\left( 1-\left( \frac{x}{a+j\pi} \right) ^2 \right)$$ Inserting it into ...
1 vote
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