Questions tagged [products]
The products tag has no usage guidance.
70 questions
3
votes
0
answers
175
views
Does $\prod_{k=1}^\infty\left[1-\big((k+1)^{1/3}-k^{1/3}\big)^3\right]$ have a closed form?
In my MSE question, "Conjectured connection between $e$ and $\pi$ in a semidisk", the answer included
$$\prod_{k=1}^\infty\left[1-\big((k+1)^{1/3}-k^{1/3}\big)^3\right]\approx 0.96454\ldots.$...
- 3,507
-1
votes
1
answer
178
views
Categories that admit all finite products but not all finite coproducts
What are examples for categories that admit all finite products but not all finite coproducts?
(See also this question: Categories that admit all products but not all coproducts .)
5
votes
3
answers
942
views
How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$?
I tried to find the indefinite integral
$$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx) \, dx$$
by using Euler's formula and put $x=\frac{\ln y}{2i}$ I got
$$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int y^{-\frac{n(...
- 513
2
votes
1
answer
299
views
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 ...
- 21
3
votes
1
answer
226
views
$\sigma$-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&...
- 3,317
2
votes
0
answers
250
views
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 ...
- 4,829
0
votes
0
answers
57
views
Probability of polynomials products to be bounded by a given bound
I am given a quotient ring $R=\frac{\mathbb{Z}[x]}{\left< x^n +t\right>}$ for $t\in\mathbb{Z}$,
and two polynomials from $R$, $A$,$B$ and let $C$ to be there product.
Defining the norms $$\Vert ...
- 133
0
votes
1
answer
205
views
Inhomogeneous Markov chains and the product-integral as a solution to the Kolmogorov forward equation
We have a inhomogeneous continous $K$-State Markov chain $X(t)$ with transition intensity matrix $Q(t)$. Therefore its entries are:
$$q_{ij}(t)= \lim_{\delta \to 0} \frac{1}{\delta} \mathbb{P}(X(t+\...
- 3
16
votes
1
answer
545
views
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 ...
- 163
3
votes
1
answer
312
views
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 ...
- 15.4k
4
votes
1
answer
279
views
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 ...
- 463
4
votes
1
answer
182
views
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 ...
- 343
7
votes
0
answers
358
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 ...
- 302
1
vote
1
answer
616
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)....
- 149
5
votes
2
answers
359
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 $K \subset X$,...
- 675
3
votes
1
answer
295
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
$$n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots)$...
- 4,928
2
votes
1
answer
234
views
Subsequences of odd powers
Let $p$ and $q$ be integers.
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then ...
- 4,928
0
votes
1
answer
178
views
Upper bound for an infinite series of Pochhammer Symbol
Let $a_n = \frac{1}{n!}\prod_{i=0}^{n-1} (r+\alpha i)$, for constants $0<r, \alpha<1$. The series is convergent by the ratio test. I want to find the exact value or maybe an upper bound for the ...
- 31
2
votes
0
answers
902
views
Confusing notation for sets of unordered vs ordered pairs
Given two finite sets $X$ and $Y$, one may consider the ordered pairs $(x,y)$ with $x\in X$ and $y \in Y$. Then, $(x,y) \not= (y,x)$, and $(x,x)$ exists if $x\in X$ and $x\in Y$.
One may also consider ...
- 1,444
6
votes
0
answers
107
views
Existence of stable spaces
An element $X$ of a class of topological spaces is called the stable space for that class if for every space $Y$ in the class we have that $X\times Y$ is homeomorphic to $X$. Note that a stable space ...
- 3,317
8
votes
2
answers
1k
views
An interesting infinite product involving the factorial function with connection to the K and gamma function
I have posted this question in StackExchange, but it didn't get any answers there. This question is important for my research. I got stuck on an infinite product which even WolframAlpha can't answer. ...
user167505
1
vote
1
answer
208
views
Is every homeomorphism approximately a product of homeomorphisms?
Let $\phi$ be a homeomorphism on $\mathbb{R}^{n+m}$, $\epsilon>0$, and $K\subseteq \mathbb{R}^n$ be a non-empty compact. Does there necessarily exist homeomorphisms $\phi_1,\phi_2$ on $\mathbb{R}^...
- 5,407
4
votes
2
answers
332
views
The complex trigonometric function degenerates to the positive integer
For any integer $N \geq 2$, we have the identity:
$$\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{\...
- 43
0
votes
0
answers
56
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
126
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 $A$...
- 11
5
votes
1
answer
761
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 ...
- 4,829
0
votes
1
answer
282
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 ...
- 5,407
1
vote
2
answers
318
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,
$$\prod_{i=0}^{N}\big(i!\big)=G(N+2)$$
...
- 155
2
votes
1
answer
824
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 ...
- 679
0
votes
1
answer
266
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 $$\int_0^...
5
votes
2
answers
441
views
Intuitive explanation of regularized products
I've come across some regularized product during study of zeta regularization .
We can prove various results like :
$ \infty != \prod_{k=1}^\infty k = \sqrt{2\pi} $
I also know the proof using $\...
user145059
5
votes
0
answers
185
views
Dual Steenrod squares
Fix the ground ring $\mathbb{F}_2$ and let $X$ be a space with finite homology. Then we have an isomorphism $\Phi^i_X:H_i(X)\to H^i(X)^*,a\mapsto \langle-,a\rangle$ which allows us to define the dual ...
- 1,623
17
votes
5
answers
1k
views
Closed-form expression for certain product
$\mathrm G$ is Catalan's constant.
I recently found the product
$$
\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(\...
- 283
-2
votes
1
answer
214
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
397
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:
$$\prod _{n=1}^k \cos \left(\frac{x}{2^n}...
- 35
1
vote
1
answer
229
views
Sabidussi theorem for morphisms between graphs
Sabidussi proved that if a finite graph $X$ is isomorphic to a Cartesian product of connected graphs $X_1,\ldots,X_m$ which are pairwise relatively prime with respect to Cartesian multiplication, then ...
- 4,547
10
votes
2
answers
1k
views
Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?
This question is an old question from mathstackexchange.
Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $
And let
$ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $
It appears that we have
$$\sup ...
- 769
7
votes
1
answer
1k
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 $\...
- 507
4
votes
1
answer
264
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 $h\not\in H_i^g\overset{\textrm{def}}{=}\left \{ g^{-1}fg\ |\ f\in H_i \right \}$ ...
- 41
2
votes
0
answers
1k
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
$$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
0
votes
1
answer
168
views
Is "square" functor monomorphic on objects?
I am trying to find whether the polynomial (monomial) functor $P : X \rightarrow X\times X $, i.e. $P(X) = X^2$, is monomorphic on objects, in other words, that if there exists an isomorphism $A\times ...
- 517
0
votes
2
answers
434
views
Ordered measurable spaces
Let $(X, \leq)$ be a partial order and $\Sigma_X$ a $\sigma$-algebra on $X$. Is the set $\{(x, y) \in X\times X \mid x \leq y\}$ measurable with respect to the product $\sigma$-algebra?
- 239
0
votes
1
answer
106
views
Maximise specific infinite product
We consider the infinite product:
$$\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$$
For which real value of $c > 1$ has ...
5
votes
1
answer
299
views
Does the category of Lawvere theories have products?
I know Law has a tensor product, is closed with respect to that tensor product, and it has coproducts. Does it have products?
My best guess at the cartesian product of Lawvere theories is the "...
- 1,532
6
votes
1
answer
327
views
Homology of the product of spaces with integer coefficients and the Massey products
Consider
$H_*(X\wedge Y;Z)$, where $X=Y=BZ/2$ for concreteness' sake. If we write $e_i$ the generator of $H_i(BZ/2;Z/2)$., we see that the $E_2=E_{\infty}$ term of the Bockstein spectral sequence
...
- 3,051
2
votes
1
answer
460
views
Yoneda extension preserving finite products?
Let $C$ be a category and let $F:C\rightarrow D$ be a functor with $D$ locally presentable and cartesian closed. When does the Yoneda extension $\widehat{F}=Lan_{y} F:[C^{op},Set]\rightarrow D$ ...
- 913
0
votes
1
answer
324
views
$Ext$ functor over a product of groups
Let $G_1$ and $G_2$ be two groups (of some kind, e.g. finite groups).
Let $M_1, N_1$ be $G_1$-modules, and $M_2, N_2$ be $G_2$ modules, always with coefficients in $\mathbb{C}$.
Write $G = G_1 \...
- 3,432
2
votes
3
answers
260
views
Nonzero solutions of an infinite product
Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$.
Definitions: $$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 ...
- 293
2
votes
0
answers
112
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}...
- 640
5
votes
2
answers
1k
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}...
- 55