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

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0
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3answers
13 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 ...
3
votes
2answers
78 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 ...
-1
votes
0answers
16 views

Express $\prod \frac{a_i}{x+b_i}$ in terms of known functions [on hold]

I am interested if there is a way to express the finite product $$ \prod_{i=1}^{S} \frac{a_i}{x+b_i} $$ in terms of known functions like Gamma, Beta, etc.
2
votes
2answers
116 views

Why does $\frac{d}{d\theta} \left(\theta\prod\limits_{i=1}^nx_i\right) = \sum\limits_{i=1}^n\ln x_i$

Is this just the product rule? I have this in my notes but I didn't think anything of it and now I'm wondering how this happens? Edit: Im working with maximum likelihood estimation and in my notes I ...
18
votes
4answers
2k views

How to prove those “curious identities”?

How to prove $$ \prod_{k=1}^{n-1} \sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}}$$ and $$ \prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\pi n/2)}{2^{n-1}}$$
2
votes
1answer
50 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 ...
0
votes
1answer
27 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 ...
1
vote
1answer
47 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 ...
1
vote
1answer
41 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 ...
0
votes
0answers
27 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 $f$ ...
3
votes
1answer
39 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 ...
0
votes
0answers
54 views

Countable product of first/second countable spaces is first/second countable.

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. ...
0
votes
0answers
14 views

Solving $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $

I would like to work out the result of $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $. Here, $t, i, N_i, m_i$ are positive integers. My effort: $$ \Pi^t_i 2 m_i \left(N_i!\right)^{m_i} \implies (2 m_1 ...
0
votes
2answers
25 views

How to find no. of digits of a large product

The question is: The product of 45,454,545,454,545 and 1,234 contains how many digits? I dont know how to solve it other than typing it in my calculator, but that method is wrong too.
0
votes
2answers
48 views

Prove $n!\mid\prod_{k=i}^{i+n-1}k$

I have no idea how to prove this, I haven't yet learned much about this kind of product. $$ n!\mid\prod_{k=i}^{i+n-1}k $$
0
votes
2answers
37 views

Define Derivative of Product of Polynomials

I have a a problem with defining a certain term... The derivative of a product of polynomials is the sum of derivatives of the products of the summands of the polynomials of the original product. ...
1
vote
0answers
29 views

Product of Several Functions Becomes Very Small: Scaling?

I have the following ratio: $$\frac{\sum_{i = 1}^n Y_i \prod_{p = 1}^P \lambda_p^{z_{i,p}}}{\sum_{i = 1}^n \prod_{p = 1}^P \lambda_p^{z_{i,p}}}$$ where $\lambda_p \in (0,1]$ is a parameter, and ...
0
votes
2answers
33 views

How to multiply a vector from the left side with matrix?

I have always dealt with vector - matrix multiplication where the vector is the right multiplicand, but I am not sure how to apply the product between a matrix and a vector when the vector is the left ...
-1
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0answers
46 views

Need help computing $\prod_{n=2}^\infty(1-n^{-2})$ [duplicate]

I need some help finding product. I am new to this, so I need some help. I am trying to compute $\prod_{n=2}^\infty(1-n^{-2})$. Please help me with this.
2
votes
2answers
66 views

For $n>1$, estimate the product $(n+1)^{n+1}(n/1)^n\dots (1/n)$ from above

If $ n $ be a positive integer $>1$, prove that $$2^{n(n+1)}\gt(n+1)^{n+1}\biggl(\frac{n}{1}\biggr)^{n}\biggl(\frac{n-1}{2}\biggr)^{n-1}...\biggl(\frac{2}{n-1}\biggr)^{2}\biggl(\frac{1}{n}\biggr)$$ ...
4
votes
1answer
170 views

Properties of Weak Convergence of Probability Measures on Product Spaces

EDIT: For the Bounty, I made a substantial edit revision concerning the structure of the question, to make it more readable (hopefully). Moreover I added a question on problem 2.7 of Billingsley’s ...
28
votes
4answers
2k views

What is to geometric mean as integration is to arithmetic mean?

The arithmetic mean of $y_i \ldots y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i $$ For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and ...
0
votes
0answers
33 views

Is there a product integral that preserves zeroes?

The integral essentially takes the arithmetic mean of the range of a function multiplied by the domain, adding together each possible output weighted by the amount of the domain accounted for by that ...
2
votes
2answers
42 views

Factorial Representation of product

So I've been trying to work out if it is possible to write: $\large \Pi_{i=1}^n (3i-1)$ as an expression involving the quotient or product of two factorials, or really any expression involving ...
0
votes
0answers
22 views

A property of product order

Let $\mathfrak{A}$ be a poset, let $a\in\mathfrak{A}$. By definition $$\star a = \{ x\in\mathfrak{A} \mid \text{there exists non-least } y\in\mathfrak{A} \text{ such that } y\le a \text{ and } y\le ...
1
vote
0answers
19 views

Product of a matrix and a tensor

I need to know how to compute the following product: $M(x)\frac{\partial M(x)}{\partial x}M(x)$ $\quad$ where $x \in R^{n}$. Assuming the dimensions of the matrices are compatible,how do we take ...
1
vote
2answers
46 views

Prove $\prod\limits_{k=0}^{n-1} \left(x^2-2x\cos \left(\alpha+\frac{2k\pi}{n}\right)+1\right)=x^{2n}-2x^n\cos(n\alpha)+1$

I have read in a paper that there is a formula as follows: $$\prod_{k=0}^{n-1} \left(x^2-2x\cos\left(\alpha+\frac{2k\pi}{n}\right)+1\right)=x^{2n}-2x^n\cos(n\alpha)+1.$$ In the paper they said that we ...
0
votes
0answers
26 views

Average Sum bigger than average product?

How can I prove that $\sum_{k=1}^K p_kt_k > \prod_{k=1}^K t_k^{p_k}$ given that $t_k>0$, $0<p_k<1$ and $\sum p_k = 1$ Thank you
12
votes
1answer
477 views

New Year Combinatorics 2015

In the spirit of the festive period and in appreciation of the encouraging response to my X'mas Combinatorics problem posted recently, here's one for the New Year! Express the following as a ...
0
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1answer
29 views

Simplifying a -1 term out of a finite product

I've come up with an algorithm that relies upon the value of the following product: $$Q_{k} =\prod_{n=0}^k [f(n) - 1]$$ Where $f(n) \ge 2$ and strictly increasing integer function [see note]. ...
1
vote
1answer
34 views

An inequality involving sums and products

I am curious to know whether the following holds or not. If $n_1,n_2,n_3,m_1,m_2$ are positive integers strictly greater than 1 such that $$n_1+n_2+n_3 > m_1 +m_2$$ then $$n_1n_2n_3 \geq m_1m_2.$$ ...
0
votes
1answer
23 views

Boolean algebra how simplify products of sum Form

How Solve it to minimum number of literals i can't understand basic properties to simplify this expression $(A̅ +C)(A̅ +C̅ )(C+D)(B̅ +D)(A+B+C̅ D)(A+B̅ +C)$ explain me to understand concepts of ...
3
votes
1answer
344 views

Convergence of infinite product

This could be something which is already somewhere in the website, but I am unable to locate any. Prove $$\prod_{n=1}^{\infty} (1-z^n)$$ converges absolutely and uniformly on each compact subset of ...
1
vote
0answers
27 views

Binomial square sum and product

Given $c,n\in\Bbb N$ what is the expression for $$S(n,c)=\binom{n}c^2+\binom{n-c}c^2+\dots+\binom{x}c^2$$ and $$P(n,c)=\binom{n}c^2\cdot\binom{n-c}c^2\cdot\dots\cdot\binom{x}c^2$$ where $x-c<c\leq ...
24
votes
11answers
2k views

Is there any way to define arithmetical multiplication as other thing than repeated addition?

Is there any way to define arithmetical multiplication as other thing than repeated addition? For example, how could you define $a\cdot b$ as other thing than $\underbrace{a+a+\cdots+a}_{b ...
2
votes
1answer
39 views

Alternative factorization of $\prod\limits^{n}_{k=1}k!^{k+1}$

Question: How can I succinctly express (using the product and sum notations) the following expression? $$n^{(n+1)}(n-1)^{(n+1)+n}(n-2)^{(n+1)+n+(n-1)}\cdot\cdot\cdot ...
40
votes
0answers
1k views

When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty != \prod_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty \# = \prod_{k=1}^\infty ...
0
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1answer
23 views

General Notation for a Reductive Operation, such as Sum (Σ) or Product (Π)

In functional programming, people often use operations like "fold" or "reduce", to convert from a collection to a single object using a binary operation. This is analogous to the sum and product ...
1
vote
1answer
21 views

Fiber product with diagonal morphism [duplicate]

Stacks tag 01KR states that the diagram of schemes is "by general category theory" "a fibre product diagram". I tried to show this using the universal property, but didn't obtain anything useful. ...
3
votes
1answer
52 views

Spectrum of infinite product of rings

$\def\Z{{\mathbb{Z}}\,} \def\Spec{{\rm Spec}\,}$ Suppose $R$ a ring and consider $\Spec(\prod_{i \in \mathbb{Z}} R)$. Now for the finite case, I know that holds $\Spec(R \times R) = \Spec(R) \coprod ...
0
votes
0answers
17 views

Which is the total energy of the product of two discrete energy signals

Assume that I have two signals with finite energy. The first, $x_s$ \begin{equation} E_s= \sum_{i=0}^{N} |x_s(i)|^2 \end{equation} The second, $w$ \begin{equation} E_w= \sum_{i=0}^{N} |w(i)|^2 ...
0
votes
2answers
33 views

Proving there exists a set such that the sum of the elements equals the product

Show that for all odd positive integer $n$, there exists a set $A$ where $A= [a_1, a_2, a_3, ... , a_n]$ and $\displaystyle\sum_{i=1}^n a_i =\prod_{i=1}^n a_i$. Edit: $a_1,...,a_n$ must be distinct. ...
2
votes
2answers
29 views

Verify the Product of a Summation

Can anybody verify that the below equation equals $0$? $\prod\limits_{k=2}^{10} (\sum\limits_{i=1}^{k-1}(2(i-1)))$ Here is my work, I believe it's correct: Note: The sequence continues, I just ...
1
vote
1answer
50 views

On a step of a proof of the Borel-Cantelli lemma.

This is an excerpt taken from Probability with Martingales by Williams. The framework is probability theory. Why is the equation being discussed true if condition $\{ n \ge m \}$ is replaced ...
0
votes
3answers
58 views

Norm of matrix $M= u \otimes v^*$ if $u$ and $v$ are unit norm vectors

Let $u$ and $v$ be an $n \times 1$ and $m \times 1$ unit norm (L-2 norm) vectors, respectively. Let us define matrix $M$ (of dimension $n \times m$) as the kronecker product of $u$ and $v^*$ $$M= u ...
0
votes
0answers
15 views

Expanding two products term by term

I have the following double product $$ \prod_{a=0}^{3}\prod_{b=0}^{3-i} \Big((p-b)u + (q-a)v\Big) $$ and it does not matter what these variable really are. I want to analytically expand it so I first ...
1
vote
0answers
30 views

How to find $\sup \Pi_{i=0}^{n} (\sin(i)^2 - \frac{25}{16})$?

Let $\sup,\inf,{\rm dif}$ denote resp supremum , infimum and $\rm dif$ = supremum - infimum. Does any of the 3 below have a closed form ? $\sup \Pi_{i=0}^{n} (\sin(i)^2 - \frac{25}{16})$ $\inf ...
0
votes
1answer
41 views

total differential of product of scalar & vector functions

I've probably made mathematical mincemeat out of this but, suppose I have a product of scalar and vector functions, such as the momentum $\mathbf{p} = m \mathbf{v}$. To keep it reasonably simple but ...
0
votes
2answers
26 views

Using the chain rule to solve reciprical of a function

Here is my issue. To prove the quotient rule from the product rule, we require the use of chain rule on $(h(x))^{-1}$. Why? For example, if $h(x)=3x^{-1}$, then $h'(x)=-3x^{-2}$ However, the chain ...
4
votes
1answer
112 views

Product identity for $n^n$

I came across the rather nice identity \begin{align} &&\frac{(-n)^{n-1} \Gamma (n+1)}{(1-n)_{n-1}}&&\tag{1}&\\ \\ &=&\prod _{k=1}^{n-1} \frac{(k+1) n^2}{n^2-k ...