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

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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 ...
5
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44 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 ...
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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 ...
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0answers
29 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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1answer
45 views

A closed form for $\sum_{i=1}^{n} \prod_{k=1}^{i+2} (3k+2)$

I need to calculate the following expression. Is there any explanation to convert this expression into normal expression without those letters for sum and the product? Just normal expression. $$ ...
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44 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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21 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
22 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
40 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)? ...
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1answer
59 views

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

How is it possible to prove the following inequality? ...
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2answers
39 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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4answers
3k views

Product of two cyclic groups is cyclic iff their orders are co-prime

Say you have two groups $G = \langle g \rangle$ with order $n$ and $H = \langle h \rangle$ with order $m$. Then the product $G \times H$ is a cyclic group if and only if $gcd(n,m)=1$. I can't seem to ...
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2answers
200 views

Showing an indentity with a cyclic sum

Let $n\geqslant2$, and $k\in \mathbb{N}$ Let $z_1,z_2,..,z_n$ be distinct complex numbers Prove that $$ \sum_{i=1}^{n}\frac {{z}_{i}^{n-1+k}} { \prod \limits_{\substack{j = 1\\j ...
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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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2answers
77 views

An identity for the product of even numbers (double factorial)

I'm unable to prove this identity: Prove that: $2\cdot 4 \cdot 6 \cdot 8 \cdots 2n = 2^n \cdot n!$ Wouldnt it be like this? $ 2(1 \cdot 2\cdot 3\cdot 4 \cdots n)= 2 \cdot n!$
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112 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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1answer
82 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 ...
3
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1answer
59 views

Direct product commutes with direct sum?

Do direct products commute with the direct sums of vector spaces? Basically is $\underset{i \in I}{\prod} \underset{j \in J}{\bigoplus}M_{i,j} \cong \underset{j \in J}{\bigoplus}\underset{i \in ...
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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
29 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
26 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?
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569 views

Trying to prove that $\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$

How could one prove that: $$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$ This is about as far as I got: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ ...
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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 ...
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1answer
524 views

Prove that the product of some numbers between perfect squares is $2k^2$

Here's a question I've recently come up with: Prove that for every natural $x$, we can find arbitrary number of integers in the interval $[x^2,(x+1)^2]$ so that their product is in the form of ...
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1answer
49 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 ...
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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 $\mathbb C(A)$. ...
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1answer
46 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
32 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 ...
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2answers
97 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
63 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
52 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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133 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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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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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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0answers
190 views

The logarithm of a product

Let $p$ be a prime number, $C\in \mathbb{N}$ and C is not a square. Then define $$F=\prod_{|z| \leq \sqrt{\frac{x}{2}} \atop |y|\leq \sqrt{\frac{x}{2D}}}{|z^2-Cy^2|}.$$ Note that we omit the term with ...
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4answers
142 views

The product $\prod_{m=1}^{11} (x^m - m)$

What would be the co-efficient of $x^{60}$ in the expansion of $\space$ $\prod_{m=1}^{11} (x^m - m)$ ?
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1answer
28 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 ...
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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: ...
6
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3answers
229 views

How to find the value of $\sqrt{1\sqrt{2\sqrt{3 \cdots}}}$?

I thought up this question recently, and I think I've figured out the partial sum: $$ S_n := \left(n\prod_{k=2}^{n-1} k^{2^{n-k}}\right)^{2^{-k}}. $$ But I don't even quite know if I'm on the right ...
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4answers
1k views

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

The arithmetic mean of $y_i ... y_n$ 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 using ...
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2answers
75 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 ...
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2answers
210 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
32 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
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3answers
49 views

What is the dot product of two or three vectors graphically or visually?

I don't understand what the dot product actually is. I understand when and where to use it, but when it comes to proving things with it, I don't really grasp what it actually is making it difficult ...
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0answers
17 views

How we can calculate the power of an interval?

We know if two intervals are uncorrelated like $X=[a,b], \; Y=[c,d]$ the product of $X$ and $Y$ is: $X \times Y = [\min(S),\max(S)], \; S = (ac,ad,bc,bd)$ But for powers, if the intervals are ...
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1answer
57 views

Kronecker product and outer product confusion

I have two column vectors: \begin{equation} u = \left[\matrix{ 1 \cr 2\cr }\right] \end{equation} \begin{equation} v = \left[\matrix{ 4 \cr 4\cr }\right] \end{equation} I'm trying to ...
18
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2answers
3k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using n_th root of unity $$\Large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
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2answers
206 views

What is the mistake in this proof of product rule of differentiation?

I was trying to derive the product rule of differentiation which states: If $y=u\cdot v$, then, $y'=u'\cdot v+v'\cdot u$. So I assumed it like this: $y=u+u+u+\cdots$ ($v$ number of terms of $u$) ...