Questions tagged [products]
The products tag has no usage guidance.
52
questions
2
votes
0answers
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
0answers
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
2answers
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
1answer
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
2answers
308 views
The complex trigonometric function degenerates to the positive integer
For any integer $N \geq 2$, we have the identity:
$$\frac{\ \prod _{n=1}^{N-1}\ \left(2+2\sum _{m=1}^{n\ }\cos \frac{\ m\pi \ }{N}\ \right)\ }{\prod _{n=1}^{N-1}\ \left(1+2\sum _{m=1}^{n\ }\cos \frac{\...
0
votes
0answers
49 views
Given multiple posets, what is the probability that a randomly selected (uniform dist) subposet of their product has a max under the product order?
Given multiple totally ordered posets, how do I find the probability that a randomly selected (with uniform distribution) subposet of their product has a maximum under the product order?
I have some ...
1
vote
0answers
50 views
Fastest way to calculate the eigenvalues of a product of two Toeplitz matrices
I have the following problem:
I need to find the fastest way to calculate the eigenvalues of a matrix that is the product of two Toeplitz matrices. $B = A U$.
The first is a regular Toeplitz matrix $A$...
5
votes
1answer
303 views
Is it possible to express the functional square root of the sine as an infinite product?
Cross-post from MSE.
It is known that the sine can be expressed as an infinite product: $$\sin(x) = x \prod_{n=1}^{\infty} \Big{(} 1 - \frac{x^{2}}{n^{2}{\pi}^{2}} \Big{)} .$$ We can define that ...
0
votes
1answer
236 views
Explicit examples of (probability) measures on $\prod \mathbb{R}$
Let $\prod_{n \in \mathbb{N}} \mathbb{R}$ be equipped with the Tikhonov product of the Euclidean topologies on $\mathbb{R}$ and let $B$ the corresponding Borel $\sigma$-algebra. What is are some ...
1
vote
2answers
293 views
Closed form of $\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$
I have made a question here about closed form of the following:
$$\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)$$
...
2
votes
1answer
358 views
Definition of twisted geometries and existence of coordinate transformation for twisted $AdS_2 \times S^2$
In the paper Multiply Twisted Products by Yong Wang, general definitions for so called warped and twisted products are given:
A (singly) warped product $B \times_b F$ of two pseudo-Riemannian ...
1
vote
1answer
93 views
On a type of sequence of integrals inspired in the Borwein integral and an integral due to Furdui
This post is inspired in the Borwein integral, and in a problem proposed by Ovidiu Furdui in Crux Mathematicorum, that is the Problem 3707, in page 151.
I've considered integrals of the form $$\int_0^...
5
votes
2answers
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
0answers
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
5answers
970 views
Closed-form expression for certain product
$\mathrm G$ is Catalan's constant.
I recently found the product
$$
\alpha=\prod_{n=1}^{\infty}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\...
-2
votes
1answer
173 views
About infinite products and Euler Gamma functions [closed]
I am interested in knowing how to calculate infinite products like (or reading any reference about it):
$$\prod_{j=1}^{\infty}\left( 1-\left( \frac{x}{a+j\pi} \right) ^2 \right)$$
Inserting it into ...
1
vote
1answer
230 views
How to obtain a product-to-sum identity for the sinc function?
We know that
$$\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
1answer
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
1answer
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
$$\sup f_- (n) ...
7
votes
1answer
597 views
Is a categorical coproduct of epimorphisms (monomorphisms) always an epimorphism (a monomorphism)?
Let $\mathbf{C}$ be a category (that does not necessary have a coproduct for every collection of objects). Suppose that we have two families of objects $(A_i)_{i\in I}$ and $(B_i)_{i\in I}$ in $\...
4
votes
1answer
180 views
Can a commutator of a special type be conjugate to its inverse?
Let $H=H_1\ast H_2$ be a free product of non-trivial groups $H_1$ and $H_2$.
We call an element $h\in H$ hyperbolic if $h\not\in H_i^g\overset{\textrm{def}}{=}\left \{ g^{-1}fg\ |\ f\in H_i \right \}$ ...
2
votes
0answers
770 views
Is there an infinite product like this for $\cos x$?
There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example
$$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
0
votes
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
3answers
252 views
Nonzero solutions of an infinite product
Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$.
Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$
$$f(a;b):=\prod\limits_{k=1}^\infty ...
2
votes
0answers
98 views
How to estimate $\prod_{t=1}^{N}\frac{1}{2-z^t}$ for large $N$?
Based on the top answer to How to estimate of $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for large $N$?
Can anyone find an approximate closed form for
$$
\frac{\mathrm{d}^k}{\mathrm{d}z^k}\prod_{t=1}^{N}...
5
votes
2answers
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
1answer
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:
$$(1+\frac{1}{2!}+\frac{1}{2!^2}+\dots)(1+\...
0
votes
0answers
383 views
Exterior product in relative cohomology
Let us consider two pair of spaces $(X, A)$ and $(Y, B)$. We set $(X, A) \wedge (Y, B) := (X \times Y, (X \times B) \cup (A \times Y))$. Given a cohomology theory $h^{\bullet}$, we can define a ...
6
votes
2answers
359 views
Does the limit of this product over primes converge for all $\Re(s) > \frac12$?
Numerical evidence suggests that:
$$\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)$$
with $p_n$ ...
3
votes
2answers
198 views
Lower bound for Euler's function
Euler function is defined, for $|x|\le 1$, as follows:
$$\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
0answers
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
$$
AB=\{ab:\ a\in A,\ ...
4
votes
0answers
154 views
Which Topological Spaces are Powers?
Given a topological space $X$ and closed subspace $Y \subset X$, it may be the case that $X$ is a power of $Y$. That means $\displaystyle X = \prod_{i < \kappa} Y_i$ for some cardinal $\kappa$ ...
3
votes
1answer
230 views
Does $\prod_{n=2}^{\infty} \left(\frac {1}{1-\frac{\chi_k(n)}{n^s}} \right)$ converge for non-principal characters for all $\Re(s) > \frac12$?
This question loosely builds on this one.
Take the following infinite product:
$$N(s,\chi_k)=\prod_{n=2}^{\infty} \left(\frac {1}{1-\dfrac{\chi_k(n)}{n^s}} \right)$$
with $\chi_k$ a Dirichlet ...
4
votes
1answer
405 views
Is the product of two supermodular functions supermodular?
The definition of Supermodularity is that for every $x′>x$ and $y′>y$, we have
\begin{equation*}
f(x′,y′)+f(x,y)>f(x′,y)+f(x,y′).
\end{equation*}
Suppose $f$ and $g$ are supermodular, non-...
3
votes
0answers
505 views
Two (strictly related) proofs by induction of inequalities
This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
2
votes
1answer
617 views
building a product of two categories [closed]
MacLane, as well as probably any other category book, does not hesitate to define a product of two categories as a category consisting of pairs of objects, etc.
Now my question is: what law of nature ...
4
votes
2answers
2k views
Distribution of a product of two discrete i.i.d. variables
The problem is to estimate the distribution of product of two $\textit{discretized Gaussian}$ random variables with zero means. The discretized Gaussian means that the p.m.f. looks like
$D_s(x)=\...
5
votes
2answers
421 views
Function with zeros plus/minus the primes
While playing with Cohen's pari script prodeulerrat found a function.
For $s \in \mathbb{C}$ define
$$ f(s) = \prod_{p \text{ prime}} (1-\frac{s^2}{p^2})$$
The product converges everywhere, no poles ...
2
votes
0answers
148 views
Two products over primes
For $k \in \mathbb{N}$ define
$$ 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)$$
By the product for zeta $f(1)...
2
votes
2answers
1k views
product 1+1/p in terms of Chebyshev's theta or psi function
I would like to know if there is any formula for
$
\prod_{x<p\leq y}\left(1+\frac1p\right)
$
in terms of $\theta$ or $\psi$ functions
$
\theta(x)=\sum_{p\leq x}\log p
$
and
$
\psi(x)=\sum_{p^\nu\...
2
votes
0answers
404 views
Morphisms of Spectral Sequences and alternating products
Let $E_{a,b}^{r}, F_{a,b}^{r}$ be two (co)homologica first quadrant spectral sequences of vector spaces over a field $K$, and $f : E \to F$ be a morphism of spectral sequences.
Assume that morphisms $...
0
votes
2answers
271 views
Integrating a product
By trying to find a marginal distribution I came accross integration of the product series. For the sake of generality, lets assume the integral is of following form:
$$\int \prod_{k=1}^{n}\left ( x+...
3
votes
2answers
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
$$f \otimes g : (X_0 \times X_1) \rightarrow (Y_0 \times Y_1), \mbox{ such ...
4
votes
2answers
1k views
Condition to ensure that the product of closed maps be closed
If $f_i : X_i \to Y_i$ with $i=1,2,\ldots,n$ are closed maps between topological space it is known that their product map
$$f : X_1 \times \cdots \times X_n \to Y_1 \times \cdots \times Y_n : (x_1, \...
