# Questions tagged [products]

The products tag has no usage guidance.

52
questions

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votes

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132 views

### Confusing notation for sets of unordered vs ordered pairs

Given two finite sets $X$ and $Y$, one may consider the ordered pairs $(x,y)$ with $x\in X$ and $y \in Y$. Then, $(x,y) \not= (y,x)$, and $(x,x)$ exists if $x\in X$ and $x\in Y$.
One may also consider ...

**6**

votes

**0**answers

79 views

### Existence of stable spaces

An element $X$ of a class of topological spaces is called the stable space for that class if for every space $Y$ in the class we have that $X\times Y$ is homeomorphic to $X$. Note that a stable space ...

**8**

votes

**2**answers

628 views

### An interesting infinite product involving the factorial function with connection to the K and gamma function

I have posted this question in StackExchange, but it didn't get any answers there. This question is important for my research. I got stuck on an infinite product which even WolframAlpha can't answer. ...

**1**

vote

**1**answer

151 views

### Is every homeomorphism approximately a product of homeomorphisms?

Let $\phi$ be a homeomorphism on $\mathbb{R}^{n+m}$, $\epsilon>0$, and $K\subseteq \mathbb{R}^n$ be a non-empty compact. Does there necessarily exist homeomorphisms $\phi_1,\phi_2$ on $\mathbb{R}^...

**4**

votes

**2**answers

308 views

### The complex trigonometric function degenerates to the positive integer

For any integer $N \geq 2$, we have the identity:
$$\frac{\ \prod _{n=1}^{N-1}\ \left(2+2\sum _{m=1}^{n\ }\cos \frac{\ m\pi \ }{N}\ \right)\ }{\prod _{n=1}^{N-1}\ \left(1+2\sum _{m=1}^{n\ }\cos \frac{\...

**0**

votes

**0**answers

49 views

### 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 ...

**1**

vote

**0**answers

50 views

### 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 $A$...

**5**

votes

**1**answer

303 views

### 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 ...

**0**

votes

**1**answer

236 views

### 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 ...

**1**

vote

**2**answers

293 views

### 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,
$$\prod_{i=0}^{N}\big(i!\big)=G(N+2)$$
...

**2**

votes

**1**answer

358 views

### 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 ...

**1**

vote

**1**answer

93 views

### On a type of sequence of integrals inspired in the Borwein integral and an integral due to Furdui

This post is inspired in the Borwein integral, and in a problem proposed by Ovidiu Furdui in Crux Mathematicorum, that is the Problem 3707, in page 151.
I've considered integrals of the form $$\int_0^...

**5**

votes

**2**answers

198 views

### Intuitive explanation of regularized products

I've come across some regularized product during study of zeta regularization .
We can prove various results like :
$ \infty != \prod_{k=1}^\infty k = \sqrt{2\pi} $
I also know the proof using $\...

**3**

votes

**0**answers

117 views

### Dual Steenrod squares

Fix the ground ring $\mathbb{F}_2$ and let $X$ be a space with finite homology. Then we have an isomorphism $\Phi^i_X:H_i(X)\to H^i(X)^*,a\mapsto \langle-,a\rangle$ which allows us to define the dual ...

**17**

votes

**5**answers

970 views

### Closed-form expression for certain product

$\mathrm G$ is Catalan's constant.
I recently found the product
$$
\alpha=\prod_{n=1}^{\infty}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\...

**-2**

votes

**1**answer

173 views

### 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

**1**answer

230 views

### How to obtain a product-to-sum identity for the sinc function?

We know that
$$\text{sinc}(x)=\prod_{n=1}^\infty\cos\left(\frac{x}{2^n}\right)$$
and for some truncated $k$ we can write the following product-to-sum identity:
$$\prod _{n=1}^k \cos \left(\frac{x}{2^n}...

**1**

vote

**1**answer

99 views

### Sabidussi theorem for morphisms between graphs

Sabidussi proved that if a finite graph $X$ is isomorphic to a Cartesian product of connected graphs $X_1,\ldots,X_m$ which are pairwise relatively prime with respect to Cartesian multiplication, then ...

**5**

votes

**1**answer

389 views

### Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?

This question is an old question from mathstackexchange.
Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $
And let
$ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $
It appears that
$$\sup f_- (n) ...

**7**

votes

**1**answer

597 views

### Is a categorical coproduct of epimorphisms (monomorphisms) always an epimorphism (a monomorphism)?

Let $\mathbf{C}$ be a category (that does not necessary have a coproduct for every collection of objects). Suppose that we have two families of objects $(A_i)_{i\in I}$ and $(B_i)_{i\in I}$ in $\...

**4**

votes

**1**answer

180 views

### Can a commutator of a special type be conjugate to its inverse?

Let $H=H_1\ast H_2$ be a free product of non-trivial groups $H_1$ and $H_2$.
We call an element $h\in H$ hyperbolic if $h\not\in H_i^g\overset{\textrm{def}}{=}\left \{ g^{-1}fg\ |\ f\in H_i \right \}$ ...

**2**

votes

**0**answers

770 views

### Is there an infinite product like this for $\cos x$?

There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example
$$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...

**0**

votes

**1**answer

139 views

### Is “square” functor monomorphic on objects?

I am trying to find whether the polynomial (monomial) functor $P : X \rightarrow X\times X $, i.e. $P(X) = X^2$, is monomorphic on objects, in other words, that if there exists an isomorphism $A\times ...

**0**

votes

**2**answers

268 views

### Ordered measurable spaces

Let $(X, \leq)$ be a partial order and $\Sigma_X$ a $\sigma$-algebra on $X$. Is the set $\{(x, y) \in X\times X \mid x \leq y\}$ measurable with respect to the product $\sigma$-algebra?

**0**

votes

**1**answer

93 views

### Maximise specific infinite product

We consider the infinite product:
$$\frac{1}{c}\prod_{n \geqslant 1} \frac{c^n}{c^n+1} = \frac{1}{c}\frac{c}{c+1}\frac{c^2}{c^2+1}\frac{c^3}{c^3+1} \cdots$$
For which real value of $c > 1$ has ...

**5**

votes

**1**answer

221 views

### Does the category of Lawvere theories have products?

I know Law has a tensor product, is closed with respect to that tensor product, and it has coproducts. Does it have products?
My best guess at the cartesian product of Lawvere theories is the "...

**6**

votes

**1**answer

259 views

### Homology of the product of spaces with integer coefficients and the Massey products

Consider
$H_*(X\wedge Y;Z)$, where $X=Y=BZ/2$ for concreteness' sake. If we write $e_i$ the generator of $H_i(BZ/2;Z/2)$., we see that the $E_2=E_{\infty}$ term of the Bockstein spectral sequence
...

**2**

votes

**1**answer

249 views

### Yoneda extension preserving finite products?

Let $C$ be a category and let $F:C\rightarrow D$ be a functor with $D$ locally presentable and cartesian closed. When does the Yoneda extension $\widehat{F}=Lan_{y} F:[C^{op},Set]\rightarrow D$ ...

**0**

votes

**1**answer

223 views

### $Ext$ functor over a product of groups

Let $G_1$ and $G_2$ be two groups (of some kind, e.g. finite groups).
Let $M_1, N_1$ be $G_1$-modules, and $M_2, N_2$ be $G_2$ modules, always with coefficients in $\mathbb{C}$.
Write $G = G_1 \...

**2**

votes

**3**answers

252 views

### Nonzero solutions of an infinite product

Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$.
Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$
$$f(a;b):=\prod\limits_{k=1}^\infty ...

**2**

votes

**0**answers

98 views

### How to estimate $\prod_{t=1}^{N}\frac{1}{2-z^t}$ for large $N$?

Based on the top answer to How to estimate of $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for large $N$?
Can anyone find an approximate closed form for
$$
\frac{\mathrm{d}^k}{\mathrm{d}z^k}\prod_{t=1}^{N}...

**5**

votes

**2**answers

712 views

### When does the radius of convergence of the product of two $p$-adic power series increase?

Let $p$ be a prime number and denote by $R(f)$ the radius of convergence of a power series $f(x) \in \mathbb{C}_p[[x]]$, where $\mathbb{C}_p$ is the completion of the algebraic closure of $\mathbb{Q}...

**7**

votes

**1**answer

413 views

### Factorial-based constant

Am looking for a name for:
$$\prod\dfrac{1}{1-\dfrac{1}{n!}}$$
$$=2.529477472079152648180116154253954242$$
Wolfram|Alpha
Expanding the formula gives:
$$(1+\frac{1}{2!}+\frac{1}{2!^2}+\dots)(1+\...

**0**

votes

**0**answers

383 views

### Exterior product in relative cohomology

Let us consider two pair of spaces $(X, A)$ and $(Y, B)$. We set $(X, A) \wedge (Y, B) := (X \times Y, (X \times B) \cup (A \times Y))$. Given a cohomology theory $h^{\bullet}$, we can define a ...

**6**

votes

**2**answers

359 views

### Does the limit of this product over primes converge for all $\Re(s) > \frac12$?

Numerical evidence suggests that:
$$\displaystyle F(s):= \lim_{N \to \infty}\, \ln^s\left(p_N\right)\, \prod_{n=1}^N \left(\dfrac{\left(p_n-1\right)^s}{p_n^s-1} -\frac{1}{p_n^s}\right)$$
with $p_n$ ...

**3**

votes

**2**answers

198 views

### Lower bound for Euler's function

Euler function is defined, for $|x|\le 1$, as follows:
$$\phi(x)=\prod_{i=1}^\infty(1-x^i)$$
Upper bounds for $\phi$ can be simply derived from ending the product early, e.g.
$$\phi(x)<\prod_{i=1}^...

**3**

votes

**0**answers

99 views

### Cardinal of a set cinsist of product of two sets?

Let
$$
A=\{1,2,\ldots,p-1\},\qquad B=\{1,2,\ldots,q-1\}
$$
where $p,q$ are primes not necessarily distinct.
Is there any elementary way to find the cardinal of the following set
$$
AB=\{ab:\ a\in A,\ ...

**4**

votes

**0**answers

154 views

### Which Topological Spaces are Powers?

Given a topological space $X$ and closed subspace $Y \subset X$, it may be the case that $X$ is a power of $Y$. That means $\displaystyle X = \prod_{i < \kappa} Y_i$ for some cardinal $\kappa$ ...

**3**

votes

**1**answer

230 views

### Does $\prod_{n=2}^{\infty} \left(\frac {1}{1-\frac{\chi_k(n)}{n^s}} \right)$ converge for non-principal characters for all $\Re(s) > \frac12$?

This question loosely builds on this one.
Take the following infinite product:
$$N(s,\chi_k)=\prod_{n=2}^{\infty} \left(\frac {1}{1-\dfrac{\chi_k(n)}{n^s}} \right)$$
with $\chi_k$ a Dirichlet ...

**4**

votes

**1**answer

405 views

### Is the product of two supermodular functions supermodular?

The definition of Supermodularity is that for every $x′>x$ and $y′>y$, we have
\begin{equation*}
f(x′,y′)+f(x,y)>f(x′,y)+f(x,y′).
\end{equation*}
Suppose $f$ and $g$ are supermodular, non-...

**3**

votes

**0**answers

505 views

### Two (strictly related) proofs by induction of inequalities

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...

**2**

votes

**1**answer

617 views

### building a product of two categories [closed]

MacLane, as well as probably any other category book, does not hesitate to define a product of two categories as a category consisting of pairs of objects, etc.
Now my question is: what law of nature ...

**4**

votes

**2**answers

2k views

### Distribution of a product of two discrete i.i.d. variables

The problem is to estimate the distribution of product of two $\textit{discretized Gaussian}$ random variables with zero means. The discretized Gaussian means that the p.m.f. looks like
$D_s(x)=\...

**5**

votes

**2**answers

421 views

### Function with zeros plus/minus the primes

While playing with Cohen's pari script prodeulerrat found a function.
For $s \in \mathbb{C}$ define
$$ f(s) = \prod_{p \text{ prime}} (1-\frac{s^2}{p^2})$$
The product converges everywhere, no poles ...

**2**

votes

**0**answers

148 views

### Two products over primes

For $k \in \mathbb{N}$ define
$$ f(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k+1)}\right)$$
$$ g(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k-1)}\right)$$
By the product for zeta $f(1)...

**2**

votes

**2**answers

1k views

### product 1+1/p in terms of Chebyshev's theta or psi function

I would like to know if there is any formula for
$
\prod_{x<p\leq y}\left(1+\frac1p\right)
$
in terms of $\theta$ or $\psi$ functions
$
\theta(x)=\sum_{p\leq x}\log p
$
and
$
\psi(x)=\sum_{p^\nu\...

**2**

votes

**0**answers

404 views

### Morphisms of Spectral Sequences and alternating products

Let $E_{a,b}^{r}, F_{a,b}^{r}$ be two (co)homologica first quadrant spectral sequences of vector spaces over a field $K$, and $f : E \to F$ be a morphism of spectral sequences.
Assume that morphisms $...

**0**

votes

**2**answers

271 views

### Integrating a product

By trying to find a marginal distribution I came accross integration of the product series. For the sake of generality, lets assume the integral is of following form:
$$\int \prod_{k=1}^{n}\left ( x+...

**3**

votes

**2**answers

638 views

### Functions that can be written as direct products of other functions; question about terminology and notation

Let
$$f : X_0 \rightarrow Y_0, \;\;\; g:X_1 \rightarrow Y_1$$
and define that the "direct product" of $f$ and $g$ is a map
$$f \otimes g : (X_0 \times X_1) \rightarrow (Y_0 \times Y_1), \mbox{ such ...

**4**

votes

**2**answers

1k views

### Condition to ensure that the product of closed maps be closed

If $f_i : X_i \to Y_i$ with $i=1,2,\ldots,n$ are closed maps between topological space it is known that their product map
$$f : X_1 \times \cdots \times X_n \to Y_1 \times \cdots \times Y_n : (x_1, \...