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3 votes
0 answers
164 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.$...
Dan's user avatar
  • 2,997
-1 votes
1 answer
158 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 .)
Yilmaz Caddesi's user avatar
5 votes
3 answers
876 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(...
Faoler's user avatar
  • 439
2 votes
1 answer
224 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 ...
Enea Olati's user avatar
2 votes
1 answer
201 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&...
D.S. Lipham's user avatar
  • 3,105
2 votes
0 answers
206 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 ...
Max Muller's user avatar
  • 4,675
0 votes
0 answers
56 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 ...
Don Freecs's user avatar
0 votes
1 answer
173 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+\...
Bloble's user avatar
  • 3
16 votes
1 answer
529 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 ...
Joseph Iosue's user avatar
3 votes
1 answer
306 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 ...
David Corwin's user avatar
  • 15.5k
4 votes
1 answer
260 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 ...
fsp-b's user avatar
  • 461
4 votes
1 answer
169 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 ...
PHL's user avatar
  • 331
7 votes
0 answers
328 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 ...
Bruno Gavranovic's user avatar
1 vote
1 answer
615 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)....
PalmTopTigerMO's user avatar
5 votes
2 answers
350 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 $K \subset X$,...
Charles Ryder's user avatar
3 votes
1 answer
292 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)$...
Notamathematician's user avatar
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 ...
Notamathematician's user avatar
0 votes
1 answer
168 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 ...
moonlight's user avatar
2 votes
0 answers
788 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 ...
Matthieu Latapy's user avatar
6 votes
0 answers
100 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 ...
D.S. Lipham's user avatar
  • 3,105
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. ...
user avatar
1 vote
1 answer
206 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}^...
ABIM's user avatar
  • 5,079
4 votes
2 answers
329 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{\...
YanChen's user avatar
  • 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 ...
Nicholas McNeely's user avatar
1 vote
0 answers
121 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 $A$...
Enea Olati's user avatar
5 votes
1 answer
678 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 ...
Max Muller's user avatar
  • 4,675
0 votes
1 answer
278 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 ...
ABIM's user avatar
  • 5,079
1 vote
2 answers
315 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, $$\prod_{i=0}^{N}\big(i!\big)=G(N+2)$$ ...
Wiliam's user avatar
  • 155
2 votes
1 answer
755 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 ...
horropie's user avatar
  • 649
0 votes
1 answer
245 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^...
user142929's user avatar
5 votes
2 answers
409 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 $\...
user avatar
5 votes
0 answers
173 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 ...
FKranhold's user avatar
  • 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(\...
clathratus's user avatar
-2 votes
1 answer
210 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 ...
Juan Gustavo Wouchuk Schmidt's user avatar
1 vote
1 answer
383 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}...
u136536's user avatar
  • 35
1 vote
1 answer
221 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 ...
THC's user avatar
  • 4,513
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 ...
mick's user avatar
  • 763
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 $\...
Batominovski's user avatar
4 votes
1 answer
261 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 $h\not\in H_i^g\overset{\textrm{def}}{=}\left \{ g^{-1}fg\ |\ f\in H_i \right \}$ ...
Andrey's user avatar
  • 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}{...
John Finkelstein's user avatar
0 votes
1 answer
163 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 ...
Almeo Maus's user avatar
0 votes
2 answers
418 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?
daon's user avatar
  • 239
0 votes
1 answer
104 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 ...
Ingo Althoefer's user avatar
5 votes
1 answer
285 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 "...
Mike Stay's user avatar
  • 1,532
6 votes
1 answer
317 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 ...
user43326's user avatar
  • 3,051
2 votes
1 answer
449 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 $\widehat{F}=Lan_{y} F:[C^{op},Set]\rightarrow D$ ...
user84563's user avatar
  • 915
0 votes
1 answer
315 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 \...
WhatsUp's user avatar
  • 3,242
2 votes
3 answers
259 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 ...
user90369's user avatar
  • 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}...
apg's user avatar
  • 612
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}...
Sandi's user avatar
  • 55