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

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In how many ways can you paint 90 distinct buckets?

In how many ways can you paint 90 distinct buckets, if 25 of them must be painted red, 40 of them must be painted blue, and 25 of them must be painted green? I am right to assume that these object ...
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2answers
18 views

How many ways can the school choose a President Vice President?

There are n >= 4 students. The school has a Board of Directors, consisting of one president and three vice-presidents. The entire board consists of four distinct students. How can I prove that ...
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1answer
39 views

Product Of Series With Increment Powers

I found this interesting aptitude question and I don't know how to solve this genre of question. Any help is welcome :) $$\prod_{n=1}^{49}n^n=1¹\cdot 2²\cdot\ldots\cdot49^{49}=?$$ Thanks.
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2answers
42 views

Closed form formula for the given product

I'm working on a recurrence which give me the following solution: $$ f(n)=(2+1)(2+\tfrac12)(2+\tfrac13)\cdots\left(2+\tfrac1{\lg(n)}\right) $$ so for $n=16$, $f(n)$ is just like: $$ ...
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0answers
26 views

Sum of products of numbers in a list

For N numbers in a list, is there a formula to get the sum of products of the numbers in the list? Example: {3, 3, 2} = (3 * 3) + (3 * 2) + (3 * 2) Right now I'm ...
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65 views
+100

Compositions of filters on finite unions of Cartesian products

Let $\Gamma$ be the lattice of all finite unions of Cartesian products $A\times B$ of two arbitrary sets $A,B\subseteq U$ for some set $U$. See this note for other equivalent ways to describe the set ...
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1answer
33 views

being $\mathbf{w}$ a vector, how do I calculate the derivative of $\mathbf{w}^T\mathbf{w}$?

Let's say that I have a vector $\mathbf{w}$. How can I calculate the derivative in the following expression? $\frac{\mathrm{d}}{\mathrm{d}\mathbf{w}}\mathbf{w}^T\mathbf{w}$
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25 views

being $\mathbf{a}$ and $\mathbf{b}$ two vectors with same length, how do I expand $(\mathbf{a}^T\mathbf{b})^2$?

Let's say that I have two vectors $\mathbf{a}$ and $\mathbf{b}$. Assuming that they have same length, their product $\mathbf{a}^T\mathbf{b}$ and its square $(\mathbf{a}^T\mathbf{b})^2$ are scalars. ...
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2answers
62 views

How does the max of $\prod_i a_i$ work?

Here are two succinct statements of the 'same' question: Statement 1: Take $a>0$ and $S \subseteq \mathbb{R}^N; S=\{(x_1,\dots,x_N)| \frac{1}{N}\sum_i x_i = a; x_i>0\}$. Define a 'product ...
1
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1answer
11 views

Could the multiplication of matrix X (with dimensions [d+1 x N]) and its transpose simplify to a matrix with [d+1 x d+1] dimensions?

In a machine learning course I'm taking, one of the lectures deals with matrix multiplication. Could anyone explain why the dot product of these two matrices would "shrink" to [d+1 x d+1] ...
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36 views

Values of $x$ for convergence

I was posed this problem, it took me a while to solve it – but, I did nevertheless. I shall pose it for all of you, too. In my opinion it is a great exercise. For what values of $x$ is the series ...
4
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1answer
46 views

Nested Sum Encountered in Maclaurin Expansion of $e^{-x^2}$

In the course of working out the Maclaurin expansions of $e^{-x^2}$ and $cos(x^2)$, I ran into the following nested sum: $$ \underbrace{ \sum_{a=0}^1 \left( a \sum_{b=0}^{a+1} b \left( ...
1
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2answers
46 views

Proper way to express 0 in this case?

If 0=(x-a)(x-b)(x-c)...(x-x)..=0. So it's a product sum that we write with pi instead of sigma but how? There should be indexes but I'm not convinced that I understand what notation to use. $$\prod_{ ...
4
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4answers
196 views

Finding $\frac{\mathrm d}{\mathrm dx} x!$

I'm trying to differentiate $x!$ but I just can't seem to do it right. I define the function as follows: $$x! = \prod_{r = 0}^{x}(x-r) \quad,\quad x \in \mathbb N$$ I've tried attempted to try it by ...
1
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1answer
36 views

Simplifying a product written in Capital Pi Notation

I'm having some trouble figuring out how to simplify Capital Pi Notation. What I tried was to expand the multiplication with various n and tried to find a pattern. Could someone point me in the ...
2
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2answers
57 views

Show that $H_i=H_{n-i}$ and $\sum H_i=1$

We define $$H_i=\frac{1}{n}\frac{(-1)^{n-1}}{i!(n-1)!}\int_{0}^{n}\prod_{j=0,j\neq i}^{n}(x-j)dx$$ This is called the Newton-Cotes coefficient. Here is the exercise: First, convince yourself that ...
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1answer
22 views

Inequality involving products

One is given two intervals $I_{a-\epsilon,b+\epsilon}$, $I_{a,b}$ of $\mathbb{R}^n$, and is asked to show that $\lambda(I_{a-\epsilon,b+\epsilon}) - \lambda(I_{a,b}) \leq c\epsilon$ for some constant ...
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0answers
47 views

How to compute product integrals?

From the wikipedia article about product integrals I can see that if our function is scalar, then to compute type I product integral we can just take exponential of a usual integral: $$\prod_a^b ...
1
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1answer
22 views

Existence of 3 Matrices with given restrictions

Would it be possible to have 3 square matrices (preferably 2x2 or 3x3) $A$, $B$ and $C$ such that: $A\neq B \neq C$; The product $A\cdot B\cdot C$ equals the Identity Matrix; All 3 matrices are ...
2
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1answer
57 views

Do we need $\mu, \nu$ to be $\sigma$-finite to show $\int fg \ d(\mu \otimes \nu) = \int f \ d\mu \int g \ d\nu$?

The problem statement: Let $(X, \mathcal F, \mu), (Y, \mathcal G, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu), g \in \mathcal L^1 (\nu)$. Show that $fg \in \mathcal L^1 (\mu \otimes ...
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0answers
45 views

ZigZag product - A simpler definition?

I have been fiddling with the ZigZag product and constructing expanders for a while now. I was wondering if the following definition of a ZigZag product is the same as the original article: Lets ...
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0answers
28 views

Product of tuples vs cartesian product of set

If $\left ( X_{i} \right )_{1\leq i\leq n}$ is an ordered n-tuple of sets their Cartesian product is defined as: $$\prod_{i=1}^{n}X_{i}:=\left \{ (x_{i})_{1\leq i\leq n} :x_{i}\in (X_{i}) \; \text{ ...
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1answer
23 views

dot product of vectors with not orthogonal basis

The dot produt (inner product in the context of Euclidean space) of two vectors $\mathbf{a}=\left [ a_{1},a_{2},...,a_{n} \right]$ and $\mathbf{b}=\left [ b_{1},b_{2},...,b_{n} \right ]$ is defined ...
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1answer
60 views

Proof of an inequality involving $(N-1)!$

How is it possible to prove the following inequality? ...
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2answers
43 views

Finding (or rather expanding) the product $(5-xy)(3+xy)$

Given the product $(5-xy)(3+xy)$ I tried the following, As we know, $(x+a)(x+b)=x^2+(a+b)x+ab$ Here $x$ is $xy$. But $xy$ has two signs$-$ and $+$. How do I solve this.
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1answer
44 views

tuple of tuples notation

Is the following notation right for indicating a $\mathit{m}-$tuple of $\mathit{n_{j}}-$tuples (I mean that each tuple of the $\mathit{m}-$tuple has a different number of elements)? ...
0
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1answer
28 views

Product of Chebyshev polynomials of the second kind?

So Wikipedia has this formula for a product of two Chebyshev polynomials of the second kind evaluated at a fixed $x$ with different indices: $$ U_n(x)U_m(x)=\sum_{k=o}^{n}U_{m-n+2k}(x) $$ Which would ...
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1answer
83 views

If $a + b + c = 0$ prove that

If $a + b + c = 0$, prove that 1)$$ \sum_{\text{cyc}}{\frac{4bc - a^2}{bc + 2a^2}} = 3 $$ 2)$$ \prod_{\text{cyc}}{\frac{4bc - a^2}{bc + 2a^2}} = 1 $$ There is a solution that uses two cubic ...
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0answers
116 views

Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k}$

For $k = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate $n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k}$ So the product goes up to $k$ and I ...
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0answers
17 views

computing equation with vectors

I've got this equation to compute. In fact i'd like to be able to compute every weight $w(k+1)$ knowing its past value $w(k)$. the equation is : $$\bf w\rm_{ij} (k+1) = \bf w\rm_{ij} (k) - ...
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1answer
38 views

Find all possible multiplicand who results in given number

I have some random figure let's say 400, I need equation to find all possible combinations (of integer) whose multiplication will results in 400. Condition is number of factors (multiplicand) will be ...
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0answers
30 views

Discrete external product of chains

everyone! Studying algebraic topology I've stumbled on a doubt about (multi)vectors and chains. There exists an external product of vectors, for instance $v_1^1 \wedge v_2^1 = v^2$ where the upper ...
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1answer
27 views

$\prod_{i\in I}(1+x_i)=\sum_{J\subseteq I}\prod_{j\in J}x_j$

I have found this equality: $$\prod_{i\in I}(1+x_i)=\sum_{J\subseteq I}\prod_{j\in J}x_j$$ Do you think is it true?
0
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1answer
26 views

How to find subbase and base for $X\times Y$?

Let $\tau :=\{X,\emptyset,\{a\},\{b,c\}\} $ on $X=\{a,b,c\}$ and $\tau^*:=\{Y,\emptyset,\{u\}\}$ on $Y:=\{u,v\}$ i) Find a subbase for the product topology on $X\times Y$ ii) Find a ...
1
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1answer
51 views

Number of product pairs equal to or less than a number

I would like to figure out how many ways there are to create product pairs equal to or less than a certain number. In other words, find a pair of integers $(n,m)$ such that $nm \le N$ for a given ...
0
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1answer
86 views

A frightening sum [duplicate]

Let $x_1,\ldots,x_r,y_1,\ldots,y_p,z_0,\ldots,z_r,t_0,\ldots,t_p$ be complex numbers. Let $A$ be the ring generated by these numbers. Prove the following holds in $\mathbb C(A)$. ...
3
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1answer
49 views

Euler's Basel Problem Rigorous Proof

In Euler's proof he uses the formula: $$\sin z = z \prod_{n \mathop = 1}^\infty \left({1 - \frac {z^2} {n^2 \pi^2}}\right)$$ and compares coefficients of the $z^3$ term in the Maclaurin series of ...
0
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1answer
34 views

Product-σAlgebra of Lebesgue sets on $R$ is subset of the Lebesgue sets on $R^2$

I want to show that $\Gamma(R)\times\Gamma(R)$ is a subset of $\Gamma(R^2)$, where $\Gamma(\cdot)$ are the lebesgue sets of $R$ or $R^2$ respectively. What can i do for that and why is it a subset and ...
4
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2answers
100 views

inequality $\prod\limits_{k=1}^n\frac{2k-1}{2k}\lt\frac1{\sqrt{3n}}$

$n$ is a positive integer, then $$\prod_{k=1}^n\frac{2k-1}{2k}\lt\frac1{\sqrt{3n}}$$ with mathematical induction, we can prove this. But I would love to find a wonderful method without ...
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2answers
64 views

Product identities

I need to use the following identities for poisson integral but i can't guz i don't know how to prove them. $$\alpha^{2n}-1=\prod_{k=0}^{k=2n-1}(\alpha-e^{i\frac{2k\pi}{2n}})$$ ...
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2answers
139 views

Product of Gamma functions II

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{10} \Gamma\left(\frac{k}{10}\right) \end{align} and can it be shown that \begin{align} \prod_{k=1}^{20} ...
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0answers
18 views

Vectorial product analog operation in 4+ dimensions?

I am thinging about a such operation. Which it need to have: It needs to be $\mathbb{R}^n\times{\mathbb{R}^n}\rightarrow\mathbb{R}^n$ The result needs to be perpendicular to the arguments (thus, ...
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2answers
53 views

Product of Gamma functions I

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{8} \Gamma\left( \frac{k}{8} \right) \end{align} and can it be shown that the product \begin{align} \prod_{k=1}^{16} ...
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1answer
44 views

question application product

can any one help me in this questions The perimeter of a square is equal to four times the length of a side of the square. Find the perimeter of a square whose side $s$ measures $2.7$ meters? thank ...
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0answers
19 views

Is there a algorithm to extract the minimum number of Cartesian products from a set of formulas?

For example, we have a set of formulas as below: B*2*j B*3*i B*3*j C*2*j C*3*i C*3*j D*2*i D*2*j D*3*i D*3*j And we could have three Cartesian products to ...
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1answer
38 views

Limit of the “productory”

With the term "productory" I just mean $\Pi_{i=m}^nx_i$ but I do not know the english term. My question is: is there a limit for such an expression in the same sense as the limit of a sum is an ...
4
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2answers
116 views

Prove that $\prod\limits_{k=1}^n(4-\tfrac{2}{k}) \in \mathbb{N}$.

How to prove that $$\prod\limits_{k=1}^n\left(4-\dfrac{2}{k}\right) \in \mathbb{N}.\tag{1}$$ Moreover, that it is even number. Update: sos440 give me great hint on $(1)$. And how about this one: ...
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2answers
76 views

I don't know how to interpret this strange $\prod$

I have got a $\prod$ that is exactly as follows: $$\prod\limits_{k=0, k \ne k}^n \frac{x-c_k}{c_k-c_k}$$ I am not sure how to interpret this. My guesses are that it equals either $0, or ,1, or ...
3
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2answers
211 views

Category with no product?

Is there a family of objects in some category which has no product? If so is there a simple reason for it?
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0answers
33 views

How to derive finite formula for PRODUCT of terms from 1 to n?

should i use limits? I totally forgot how to work with them, i can imagine doing summation(sigma) but not product