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

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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 ...
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 ...
1
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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 ...
1
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0answers
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 ...
0
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0answers
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{ ...
0
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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 ...
1
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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.
0
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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)? ...
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 ...
1
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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 ...
1
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0answers
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 ...
0
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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) - ...
0
votes
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 ...
0
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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 ...
1
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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?
0
votes
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
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 ...
0
votes
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
votes
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
votes
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 ...
4
votes
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 ...
1
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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
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} ...
1
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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
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} ...
1
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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 ...
1
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1answer
27 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
votes
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
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 ...
3
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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?
0
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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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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 ...
0
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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!$
2
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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 ...
0
votes
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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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 $u$) ...
4
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1answer
47 views

example diagram of pullbacks and fiber products

I am going through Category Theory for Scientists. I am on section 2.5.1 Pullbacks. I am having trouble visualizing a pullback. Earlier in the book the author gives a nice diagram of an example of ...
0
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1answer
38 views

Summation in 104 Number Theory problems

There's a paragraph of 104 Number Theory problems, on page $9$ that says: From the formula $\prod_{i=1}^\infty\frac{p_i}{p_i-1} = \infty ,$ using the inequality $1+t \le e^t$, $t \in \mathbb{R}$ we ...
2
votes
1answer
25 views

A polynomial equality problem

$a_1,a_2,a_3,\ldots,a_n,a_{n+1}$ are fixed real numbers in $(-1,\infty)$. $x_1$ and $x_2$ are fixed real numbers in $(0,1)$. Is it possible to prove that there exists or doesn't exists a real number ...
3
votes
3answers
135 views

How find this value $\prod_{k=1}^{\infty}\left(1+\dfrac{1}{k^5}\right)$

Find the value $$\prod_{k=1}^{\infty}\left(1+\dfrac{1}{k^5}\right)$$ I know this :How find this $\prod_{n=2}^{\infty}\left(1-\frac{1}{n^6}\right)$ and maybe can find the $2k+1$? can you someone konw ...
0
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2answers
52 views

How to prove a product of k consecutive integers is always a multiple of k?

How to explain and prove that a product of k consecutive integers is always a multiple of k!.
2
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0answers
18 views

Fourier Series from product of to functions

I have to calculate the Fourier Series of $x\sin(x)$ beeing $2\pi$ periodic on $[-\pi,\pi]$and i did it the standard way. But then i wanted to solve the problem with multiplication of two fourier ...
0
votes
3answers
33 views

Dot Product and vector length

Hi! I am working on some online homework for my calc2 class that covers the dot product and I am really struggling with this one question. I understood how to solve part a, because we covered that ...
0
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1answer
17 views

Condition for the product of the ratio of the elements of two sequences to be greater than 1.

I have the following product: $$\prod_{n=1}^N \frac{A_n}{B_n}$$ , where $A_n$ and $B_n$ are the nth element of the finite sequences {$A_x$} and {$B_x$} respectively. I'd like to know the conditions ...
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1answer
43 views

Prove $F_n(z)=\frac1{2i}\left(\left(1+\frac{iz}n\right)^n-\left(1-\frac{iz}n\right)^n\right)…$

In my textbook there is a proof for the following If $n=2m+1$ with $m\in\mathbb N$, then we can write ...
2
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0answers
30 views

Is there a sort of “two-sided semidirect product”? [duplicate]

Let $G,H$ be groups. Suppose we have both an action of $G$ on $H$, and an action of $H$ on $G$, both non-trivial. Let "$\cdot$" define the former action, and $\circ$ define the latter. What can we ...
0
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0answers
28 views

How do I do the math necessary to make these five matrices multiplied together equal the result shown?

I'm currently studying the math involved with rotating vertices around an arbitrary axis in 3D space. For this, I have found the following page to be very helpful: ...
2
votes
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 ...