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

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2
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0answers
32 views

Product of Primes

Let $\mathbb{P}$ denote the set of prime integers. How would one evaluate $$\prod_{p\in \mathbb{P}}\frac{p-1}{p}$$ I do not think that the fact that ...
-1
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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
1answer
49 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
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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
53 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
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. ...
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
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 ...
-1
votes
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.
0
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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 ...
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 ...
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 ...
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 ...
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
0
votes
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
33 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 $$n_1n_2n_3 \geq m_1m_2.$$ ...
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 ...
0
votes
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 ...
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 ...
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 $$2^{n(n+1)}\gt(n+1)^{n+1}\biggl(\frac{n}{1}\biggr)^{n}\biggl(\frac{n-1}{2}\biggr)^{n-1}...\biggl(\frac{2}{n-1}\biggr)^{2}\biggl(\frac{1}{n}\biggr)$$ ...
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. ...
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 ...
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 ...
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
29 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 ...
0
votes
1answer
28 views

Is $\nabla\phi\nabla\psi$ a scalar product or a dyadic product?

After reading an introduction to vector analysis I wanted to try out some operations myself checking whether I understood everything well. I thought this $$ ...
0
votes
1answer
18 views

Is $v^* H w= h^T (w \otimes (v^*)^T)$ in this specific case?

Let $v$ and $w$ be an $n \times 1$ and $m \times 1$ unit norm vectors, respectively. Also, let $H$ an $n \times m$ matrix. We denote by vectors $a^*$ and $a^T$ the conjugate transpose and transpose of ...
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 ...
1
vote
3answers
53 views

computing the product $\prod_{n=1}^{2016} \frac{2^{2^{n-1}}+1}{2^{2^{n-1}}}$

how can i calculate the product: $\prod_{n=1}^{2016} \frac{2^{2^{n-1}}+1}{2^{2^{n-1}}}$? I can see that in the denominator it's a geometric series, but in the numerator i can't see how to simplify. ...
3
votes
1answer
51 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
1answer
24 views

Product of improper integrable riemann function and integrable function.

I have the following problem while working with linear differential equations: I'm told to proof that the following system: $t^{-\sigma}x' = A(t) x$ where $A: \mathbb{R} \to {M_N(\mathbb R)}$ is ...
0
votes
0answers
19 views

Is this product of two product factorizations correct?

I am working on an induction proof and would like to know whether this product equality is true: $$\big (\prod_{i=2}^n (\lambda_i-\lambda_1) \prod_{n\ge i > j \ge 1}(\lambda_i - \lambda_j)\big )$$ ...
0
votes
2answers
46 views

limit of $\lim_{n \to \infty}\prod\limits_{k=0}^{\frac{n}{2}-1} \left(1-\frac{1}{2k+2} \right)$

I want to calculate the limit of $$\lim_{n \to \infty}\prod\limits_{k=0}^{\frac{n}{2}-1} \left(1-\frac{1}{2k+2} \right)$$ I think, that it's $0$, but I don't know how to prove this. I can't even ...
0
votes
2answers
54 views

How to show that the determinant of this matrix is in a nice product factorization,

Show that $$det \begin{bmatrix} 1 & 1 & \cdots &1 \\ \lambda_1 & \lambda_2 & \cdots &\lambda_n \\ \lambda^2_1 & \lambda^2_2 & \cdots ...
9
votes
1answer
279 views

Why is this sum equal to $0$?

While solving a differential equation problem involving power series, I stumbled upon a sum (below) that seemed to be always equal to $0$, for any positive integer $s$. $$ \sum_{k=0}^s \left( \frac{ ...
2
votes
1answer
24 views

Does this vector product, based on indexing with a powerset, have a name?

Given two vectors $\vec{u}$, $\vec{v}$ indexed by $2^X$ for some finite set $X$, define $\vec{u} \star \vec{v}$ as the vector of similar type whose dimension indexed by $S \subseteq X$ is: ...
0
votes
2answers
26 views

Result of product with n=0

It's a simple question but I couldn't find informations about it and I'm starting to learn product sequences. I noticed (using WolframAlpha) that: $\prod_{i=x}^{0}{f(i)}|_{x > 0} = 1$ Why is ...
1
vote
0answers
11 views

How can I prove the following inequality?

Let be $N_{n+1}(x)=\prod_{i=0}^n(x-x_i)$. Now I have to prove that $$||N_{n+1}(x)||_{\infty,[-5,5]}\leq n!\frac{h^{n+1}}{4},\qquad h:=\frac{5-(-5)}{n}=\frac{10}{n}.$$ I've started with ...