Questions tagged [products]

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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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  • 256
7 votes
0 answers
151 views

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 ...
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1 vote
1 answer
429 views

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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5 votes
2 answers
273 views

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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3 votes
1 answer
261 views

Sum with products turned into subsequences

Let $p, q \in \mathbb{Z}$. Let $\operatorname{wt}(n)$ is A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and $$n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots)$...
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1 vote
1 answer
191 views

Subsequences of odd powers

Let $p$ and $q$ be integers. Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. Then ...
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0 votes
1 answer
56 views

Upper bound for an infinite series of Pochhammer Symbol

Let $a_n = \frac{1}{n!}\prod_{i=0}^{n-1} (r+\alpha i)$, for constants $0<r, \alpha<1$. The series is convergent by the ratio test. I want to find the exact value or maybe an upper bound for the ...
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2 votes
0 answers
226 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 ...
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6 votes
0 answers
83 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 ...
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8 votes
2 answers
754 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. ...
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1 vote
1 answer
164 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}^...
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  • 860
4 votes
2 answers
315 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{\...
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  • 43
0 votes
0 answers
52 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 ...
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1 vote
0 answers
63 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 $A$...
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5 votes
1 answer
343 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 ...
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  • 3,463
0 votes
1 answer
240 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 ...
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1 vote
2 answers
301 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, $$\prod_{i=0}^{N}\big(i!\big)=G(N+2)$$ ...
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  • 155
2 votes
1 answer
433 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 ...
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  • 639
0 votes
1 answer
134 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^...
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5 votes
2 answers
226 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 $\...
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3 votes
0 answers
123 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 ...
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  • 1,563
17 votes
5 answers
1k 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(\...
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-2 votes
1 answer
182 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 ...
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1 vote
1 answer
275 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}...
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  • 35
1 vote
1 answer
145 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 ...
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  • 3,067
5 votes
1 answer
407 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) ...
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  • 441
7 votes
1 answer
771 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 $\...
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4 votes
1 answer
198 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 $h\not\in H_i^g\overset{\textrm{def}}{=}\left \{ g^{-1}fg\ |\ f\in H_i \right \}$ ...
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2 votes
0 answers
901 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}{...
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0 votes
1 answer
147 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 ...
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0 votes
2 answers
321 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?
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  • 167
0 votes
1 answer
98 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 ...
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5 votes
1 answer
242 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 "...
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  • 1,450
6 votes
1 answer
280 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 ...
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  • 2,968
2 votes
1 answer
299 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 $\widehat{F}=Lan_{y} F:[C^{op},Set]\rightarrow D$ ...
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  • 885
0 votes
1 answer
237 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 \...
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  • 3,100
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 ...
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  • 293
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}...
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  • 466
5 votes
2 answers
839 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}...
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  • 55
7 votes
1 answer
418 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+\...
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  • 1,206
0 votes
0 answers
411 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 ...
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  • 1,042
6 votes
2 answers
365 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 $p_n$ ...
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  • 3,979
3 votes
2 answers
205 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}^...
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  • 608
3 votes
0 answers
100 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,\ ...
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  • 791
4 votes
0 answers
155 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 $\kappa$ ...
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  • 1,596
3 votes
1 answer
231 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 ...
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  • 3,979
4 votes
1 answer
488 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-...
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  • 41
3 votes
0 answers
506 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 ...
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2 votes
1 answer
631 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 ...
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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)=\...
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