# Tagged Questions

For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.

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### Prove that $\prod\limits_{2 < p \leq y}\left(1-\frac{2}{p}\right)\sim\frac{D}{\log ^2 y}$ [duplicate]

I'm writing my bachelor thesis about Brun's sieve method and his theorem. In one proof I found this statement without further explanation. It is important to show that the product doesn't converge "...
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### Product of directed partial orders

Is a product poset (with componentwise order) of nonempty posets a dcpo if and only if each multiplier is a dcpo? (for both binary and arbitrary products)
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### Why does the product of adjugates equal an adjugate of the product?

How can I show that $\mathrm{adj} (AB) = \mathrm{adj}(B)\ \mathrm{adj}(A)$? It is obvious if determinants are non-zero, but if any of the matrices are singular, I just don't get it. UPD. I've just ...
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### Prove Euler characteristic satisfies $\chi(X \times Y)=\chi(X)\chi(Y)$ for polyhedra $X$ and $Y$

Prove that for any topological polyhedra, $X$, $Y$, the product $X \times Y$ has the Euler characteristic $\chi(X \times Y)=\chi(X)\chi(Y)$ I know that for polyhedron $P$ which is homemorphic to a ...
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### Formula for combinations involving product notation?

So after looking at the factorial formula and learning about product notation, I recognized this relation between them: $$\prod_{n=1}^kn=k!$$ And after fooling around and doing some trial and error, I ...
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### “Binary-Like” Function?; In Consecutive Products as Multi-Factorials…

Summary Is there a function $Z(a,b)$ or how would one find such a function so that for $a,b\in \mathbb N$, it would produce $0$'s on for each $a$th step for each $b$th value? For example: $a=2$, ...
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### How many numbers $N \le 10^{10}$ are the product of $3$ distinct primes?

How many numbers $N \le10^{10}$ are the product of $3$ distinct primes? I can realistically calculate any $\pi(n), n < 10^{15}$ but I don't think it's possible to list all primes $>10^8$ in ...
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### Gamma representation of certain sequence

I'm trying to find a gamma rep for $15 \cdot 13 \cdot 11 \cdot 9 \cdot 7 \cdot ...$ Steps so far: It's a simple sequence of $n \cdot (n-2) \cdot (n-4) \cdot (n-6) \cdot (n-8)...$ and so on. ...
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### Where can I find methods to evaluate products?

I found it was slightly difficult to find resources that discussed methods for evaluating products, like $\Pi_{n=0}^ka_n$ Preferably, I want to start with the basics and move through some readings on ...
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### Which kind of product do we have here?

The following GAP-output ...
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### Simplify Product of sines

Is there a way simplify this product? $$\sin\left({n} \frac{\pi}{2}\right) \sin\left({n} \frac{\pi}{3}\right) \sin\left({n} \frac{\pi}{4}\right) ...\sin\left({n} \frac{\pi}{n-1}\right)$$ And, is ...
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### Elementary proof: division by integer makes real number smaller.

It's been so long since I took discrete math, maybe someone here can help me with what ought to be a straight forward and simple proof. Effectively I want to show this: Let a and b be positive ...
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### Linear approximation to the product: $\prod_{k=0}^r\left(1+\frac12\left(\frac{\frac12+k+1}{\frac12+k}-\frac{\frac12+k}{\frac12+k+1}\right)\right)$

I have come upon with the next expression: P_r=\prod_{k=0}^r \left(1+\frac{1}{2}\left(\frac{\frac{1}{2}+k+1}{\frac{1}{2}+k} -\frac{\frac{1}{2}+k}{\frac{1}{2}+k+1}\right)\right) \end{...
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### Why the usage of $H$ in the dot product $x^H y$ instead of $T$ or $'$ for transpose?

Given two vectors $v, u \in \mathbb{R}^n$ (i.e. column vectors), then the dot product of them is defined like this $$\sum_{i=1}^n v_i * u_i$$ Or usually it can be expressed in matrix-product ...
Suppose, a finite permutation group G is given. I want to find the minimal set $x_1,...,x_n$ such that every element of $G$ can be uniquely written in the form $$x_1^{j_1}...x_n^{j_n}$$ with $0\le j_i\... 3answers 39 views ### Prove that$\prod_{n \in \mathbb{N}}{(1-a_n)} \geq 1 - \sum_{n \in \mathbb{N}}{a_n}$The following proof is obtained from this paper. My question is how to obtain the inequality. My guess is because of the following inequality: $$\prod_{n \in \mathbb{N}}{(1-a_n)} \geq 1 - \sum_{... 0answers 11 views ### To clear for variable 'a' in a sum of dependent products I can't seem to find a way to clear this equation for variable a: E[k] = \displaystyle\sum_{k=1}^nk\frac{a}{n+a-k}\displaystyle\prod_{i=0}^{k-1}1-\frac{a}{n+a-i} Do you think it's possible? Any ... 0answers 45 views ### Is (\omega \times \omega)^{\omega}\cong \omega \times \omega \times… \cong \omega^{\omega}? Where “\cong” means homeomorphic. I'm interested in the circumstances for when we can conclude that two ordinal spaces are homeomorphic by an examination of their written form. Specifically, I'm taking an ordinal space, say (\omega^... 0answers 35 views ### speed of divergence of \prod_{m=1}^n (\frac{z}{m} + 1)^m I would like to find the speed of divergence of \prod_{m=1}^n (\frac{z}{m} + 1)^m for any z. For example, if |z| < 1, doing taylor expansion we know that it is roughly e^{zn}. But I need it ... 3answers 47 views ### universal property of product: must any map satisfying it be a morphism I am thinking about the universal property of products: Let X and Y be objects of a category D. The product of X and Y is an object X \times Y together with two morphisms \pi_1 : X \... 2answers 90 views ### Product of Primes Let \mathbb{P} denote the set of prime numbers. How would one evaluate$$\prod_{p\in \mathbb{P}}\frac{p-1}{p}$$I do not think that the fact that$$\prod_{n=2}^{\infty}\frac{n-1}{n}=\lim_{n\to\infty}... 1answer 67 views ### Evaluation of$\prod^{n}_{r=1}\sin \left(\frac{(2r-1)\pi}{2n}\right)$Find value of $$\prod^{n}_{r=1}\sin \left(\frac{\left(2r-1\right)\pi}{2n}\right)$$ Where$n\in \mathbb{N}$and$n>1\bf{My\; Try::}$Let $$P = \sin \left(\frac{\pi}{2n}\right)\cdot \sin \left(\... 1answer 30 views ### What is the name for this product? I have a vectors like: \vec{a} = [a_1, a_2] \vec{b} = [b_1, b_2] And I need a vector of products of unique combinations like: \vec{p} = [a_1 b_1, a_1 b_2, a_2 b_1, a_2 b_2] does exist a ... 1answer 59 views ### Product of two sums, one finite and one infinite I'm working on a problem and I'm not sure how to find the product of these two sums: \left(\sum_{k=0}^{\infty}\text{something}\right)\left(\sum_{k=n}^{n}\text{something else}\right) The "something"... 1answer 98 views ### What symbol is used for product topology? Let ((X_k,\tau_k))_{k \in N} be topological spaces. The product topology \tau on X = \prod_{k \in N} X_k is the coarsest topology that makes all projections \pi_k:X \to X_k continuous. Is ... 0answers 31 views ### Number of optimas of product of convex functions I am dealing with a function, which is a product of two strongly convex functions, and trying to determine the number of its local minimum. For example, I have$$H=f(x)\cdot g(x)$$, in which both$f$... 1answer 41 views ### Simple formula for the$n$-ary version of$(x,y) \mapsto \frac{x+y}{1-xy}$Let$x * y = \frac{x + y}{1 - xy}$. I want a single formula for$x_1 * x_2 * \ldots * x_n$, for all natural$n$. In order to generate plausible candidates, let's see what happens at small values of$...
Let $N$ be a countable indexing set, and $((X_k,\tau_k))_{k \in N}$ topological spaces. Define $X = \prod_{k \in N} X_k = X^N$ and let $\tau$ be the product topology on $X$ induced by $\tau_k$s. ...