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

learn more… | top users | synonyms

3
votes
1answer
48 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 ...
2
votes
1answer
28 views

Union of Cartesian squares

I am trying to find sufficient and necessary conditions for a relation to be representable as a union of Cartesian squares: $\bigcup_{i\in I}(X_i\times X_i)$ for some family $X_i$ of sets. One ...
2
votes
1answer
109 views

Why is this a differentiable structure on the product manifold?

Suppose $M$ en $N$ are differentiable manifolds with differentiable structures $\{(U_a,x_a)\}$ and $\{(V_b,x_b)\}$ resp. Consider $M\times N$ and the mappings $z_{ab}(p,q):=(x_a(p),y_b(q))$ with $p\in ...
2
votes
1answer
75 views

How to calculate a bound for this product?

Consider the following product: $$ \prod_{i=1..n} {\left(1 - {1 \over 2^i}\right)} $$ A numeric calculation, up to $n=20$, gives $0.288788370496567$. But how can I calculate its limit when $n$ goes ...
1
vote
1answer
30 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 ...
0
votes
1answer
33 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
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 ...
16
votes
0answers
572 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 $$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty ...
5
votes
0answers
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 ...
4
votes
0answers
95 views

Product-Decomposition of distributive lattices

Every nontrivial (bounded) distributive lattice arises as a direct power of a certain number of nontrivial product-irreducible (bounded) distributive lattices. My question is how this number can be ...
3
votes
0answers
78 views

Product of Summations for All Subsets

We have a set $X$ of $n$ integers $\{$$x_1$, $x_2$, .. , $x_n$$\}$, for which there are $2^n$ total subsets. The summation $s$ of a subset $X'$ is simply the sum of all integers present in $X'$, ...
3
votes
0answers
217 views

Product of sines

I am looking to evaluate $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n}$$ without using complex numbers. I can show the result if $n$ is a power of $2$, but if $n$ is anything else I reach a point where I ...
3
votes
0answers
357 views

How to avoid numerical overflow while computing a sum of products?

Suppose we have $N$ vectors $\vec{x}_1, \vec{x}_2,\dots,\vec{x}_N$. $\vec{x}_i$ is a $M$-dimensional vector: $\vec{x}_i = \left[ x_{i1}\;\; x_{i2}\;\; \dots \;\;x_{iM}\right]^T$ with all ...
3
votes
0answers
166 views

Infinity Product Equality.

Let $\{I_n\}_{n\in\mathbb{N}}$ be a sequence of intervals in the form $$ I_n = \Big [ \frac{q_n}{b_n}, \frac{q_n + 1}{b_n} \Big),$$ where $q_{n}$ is some integer, for all $n\in\mathbb{N}$. Define ...
2
votes
0answers
113 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 ...
2
votes
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 ...
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 ...
2
votes
0answers
80 views

Multiplicative group into ring operation

My question is simple, though it proves to be much more difficult than it sounds. Suppose I want to find a binary operation to add extra structure to a multiplicative group (so it becomes a ring). ...
2
votes
0answers
57 views

Why does $\frac{d}{d\theta} \theta\prod_{i=1}^nx_i = \sum_{i=1}^nx_i$

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 ...
2
votes
0answers
92 views

“Product” bundle notation.

Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively. Then there is an induced ...
1
vote
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 ...
1
vote
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, ...
1
vote
0answers
31 views

any approximation for $\sum_{i_3=3}^{n-k+3}\sum_{i_4=i_3+1}^{n-k+4}\sum_{i_5=i_4+1}^{n-k+5}\cdots\sum_{i_k=i_{k-1}+1}^{n} 1, (n \gg k)$?

Is there any approximation for $$\sum\limits_{i_3=3}^{n-k+3}\sum\limits_{i_4=i_3+1}^{n-k+4}\sum\limits_{i_5=i_4+1}^{n-k+5}\cdots\sum\limits_{i_k=i_{k-1}+1}^{n} 1, \quad (n \gg k)$$ ? We know that ...
1
vote
0answers
31 views

Efficient Cartesian product which ignores classes of elements

Given $n$ sets $X_1,X_2,..,X_n$, and what I am calling an ignore set $I = \{I_1, I_2,..,I_m : \forall i \in I_i, i \in \bigcup X_i\}$. I would like to find the cartesian product $X_1 \times X_2 ...
1
vote
0answers
30 views

Simplifying products

Sorry for the very general title, but I don't even know how to name my question. I got a formula which is: $f(n)=\prod_{i = 0}^{\infty} ((n \; \mathrm{rem} \; p^{i + 1}) \; \mathrm{div} \; p^i + 1) ...
1
vote
0answers
50 views

Proving the convergence of a product

I have become interested in taking the $n^{th}$ term of a series and evaluating a product whose $n^{th}$ term is $(1+a_n)$. After looking around I came across the following inequality: ...
1
vote
0answers
29 views

Product of numbers and gaussian function

Trying to approximate a gaussian function $g(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left(-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}\right)}$ with another function I found the product ...
1
vote
0answers
31 views

Efficient way to compute $n$ products of $n$ numbers

Say I have a set of $n$ numbers ${a_1, ..., a_n}$. I want to compute $n$ products, where the $i$th product is defined as the product of all elements in the set, except $a_i$. For example, for $n=5$, I ...
1
vote
0answers
76 views

Pi identity with sum and product

Please prove this identity $$\sum_{ n=1 }^{\infty }\left({\left(-1\right) }^{ n }\frac{\prod_{ j=1 }^{ n }{\left(\frac{ 3 }{ 2 }-j\right) }}{\left( 2n+1\right)\left( n!\right) }\right) =\frac{\pi }{ ...
1
vote
0answers
81 views

Bounding the product of a sequence

I am trying to find an upper bound for the following sequence: $$(1-p_1)(1-(p_1+p_2))\cdots(1-(p_1+\cdots+p_n))$$ with $n$ groups to multiply. I have written it like this: $$\prod_{i=1}^n \left({1 ...
1
vote
0answers
54 views

Analytic Integration of product of exponential families

I'm happy to join your community and I hope you can help me solve this seemingly straightforward dilemma I am facing. For my thesis, I am trying to solve analytically a product of two distributions ...
1
vote
0answers
40 views

What is the meaning of a surface approximation equation?

Given a set of $n$ points $P$, a point $p_i\in{P}$, $1\leq i\leq n$ and a number $k<n$, I define the group $N_k(p_i)$ as the group containing $p_i$'s $k$ nearest neighbors. In addition, each point ...
0
votes
0answers
23 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
votes
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
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 ...
0
votes
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 ...
0
votes
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
0
votes
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
votes
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: ...
0
votes
0answers
52 views

Nested sums and products

I am trying to determine how to express this series in a general form as a summation of products: \begin{equation} \begin{aligned} (p_{i1}p_{j1})(1-p_{i2}p_{j2})(1-p_{i3}p_{j3}) \mbox{ } &+ \\ ...
0
votes
0answers
28 views

How to derive this inequality containing power series? (equations are contained in the body) (Changed)

As I read a paper, I don't know how do I derive inequality, $$\frac{\prod^N_{i=1}4^{b_i/N}(1-4^{-b_i})^{1/N}}{12}\ge \frac{4^{R/N}}{16} \\ \frac{\prod^N_{i=1}(4^{b_i}-1)^{1/N}}{12}\ge ...
0
votes
0answers
16 views

Distribution of Matrix and Vector products

Given the following expression: $$ \vec w = (\mathbf M\cdot\vec u) + (\vec v\cdot\vec u) $$ Where $\mathbf M$ is a matrix of dimension $n\times m$, $\vec v$ and $\vec u$ are vectors of dimension ...
0
votes
0answers
45 views

Way to split up product of summation

If I have $\sum_{n=1}^{\infty}f(x)g(x)$, is there any way to split this up? Thanks.
0
votes
0answers
90 views

Product of standard normal and uniform random variable

I'm trying to find the PDF of the product of two random variables by first finding the CDF. I don't know where I'm going wrong. Let $X\sim N(0,1)$ and $Y\sim Uniform\{-1,1\}$ and let $Z = XY$, then: ...
0
votes
0answers
33 views

Can we convert a sum-of-products into strictly sums?

So I start with a sum-of-products: $$f = \sum_{k=a}^b{\prod_{j=c}^d{g(j,k)}}$$ I'm wondering if we can somehow convert this into a sum of sums, ie: $$f = \sum_{k=a}^b{\sum_{j=c}^d{h(j,k)}}$$ It's ...
0
votes
0answers
76 views

Notation for Kronecker product of a matrix and itself?

What is the notation for the Kronecker product of a matrix and itself? In other words, is there a short-hand way I can express the following: $X⊗X$ $X⊗X⊗X$ $X⊗X⊗X⊗X$ Where $X$ is a matrix? What ...
0
votes
0answers
24 views

Product of Consecutive Terms of a Geometric Sequence

Suppose $a_n=aq^n$, where $a>0$ and $q>0$. So $(a_n)_{n=1}^\infty$ is a geometric sequence with positive terms. The product of its consecutive terms, say, $$a_0,a_2,\ldots,a_n$$ equals to ...
0
votes
0answers
37 views

Product of consecutive integers as alternating sum

In the derivation of the Indian Buffet Process (Indian Buffet Process paper -- see page 1218), we have the following step: $\prod_{k=1}^{K_+}{(K-k+1)} = ...
0
votes
0answers
51 views

Bound on product of degrees

Is there any more or less sharp bound on the product of the out-degrees of vertices in a directed graph (except for the ones with no leaving edges)? The graph may have multiple edges between two ...
0
votes
0answers
74 views

Direct products in a partially ordered category

Consider a category, whose set of objects is a poset. Let $f=(f_0,f_1)$ is an indexed family of objects of this category. (Thus $f$ is a 2-indexed family, we can extend it to any index set in an ...