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

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Products of Functions that Don't Depend on Index

Is there a general property of products that allows you to simplify $$ \prod_{i = 1}^n f(x) \, g(i) $$ where $f(x)$ does not depend on $i$? Would it just be $$ f^{n}(x) \prod_{i = 1}^n g(i) $$ ...
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1answer
26 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
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1answer
105 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
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1answer
72 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 ...
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1answer
40 views

Multiplying Sigmas(sums)

I would be grateful if someone please rewrite or expand this please. I have problem multiplying two sigmas ($\sum $) $$ (d(n)-\sum_{k=-\infty}^{\infty} h_k x(n-k)) \times ...
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1answer
22 views

Let $a_i$ , $1\le i\le n$ be non-negative real numbers. Let S denote their sum.Pick out the true statements:

Let $a_i$ , $1\le i\le n$ be non-negative real numbers. Let S denote their sum.Pick out the true statements: (a)$\prod_{k=1}^{n}{(1+a_k)\ge1+S}$ ...
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1answer
17 views

Exponential equivalent for geometric space

I'm just starting a foray into geometric algebra and calculus so that I can develop a geometric version of the standard arithmetic neural net. Specifically when calculating the error function for a ...
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1answer
24 views

Boundary of product cartesian

What's the boundary of $\Omega\times (a,b)$, where $\Omega$ is an open bounded subset of $\mathbb R^n$ ?
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1answer
41 views

Showing an indentity with a cyclic sum

let $z_1,z_2,..,z_n$ be non equal complex numbers for any $n\geqslant2$, for any $k\in \mathbb{N}$ $$ \sum_{i=1}^{n}\frac {{z}_{i}^{n-1+k}} { \prod \limits_{\substack{j = 1\\j \ne i}}^{ n }{ ...
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0answers
421 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 ...
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415 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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0answers
94 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
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0answers
68 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
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0answers
208 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
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0answers
303 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
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0answers
157 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
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0answers
76 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
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0answers
56 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
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0answers
81 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 ...
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0answers
16 views

Remainder term for Weierstrass products

Is there a remainder term for Weierstrass products like there are for Taylor series? I have been learning about the integral form and Lagrange form of the remainder for Taylor series, which I liked ...
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0answers
25 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 ...
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0answers
29 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) ...
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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: ...
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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 ...
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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 ...
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0answers
73 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 }{ ...
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0answers
78 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 ...
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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 ...
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0answers
39 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 ...
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0answers
42 views

How can nested for-loops be expressed in mathematical notation?

Apologies if this is an obvious question; I'm not very familiar with mathematical notation for algorithms. I was coding a solution for Project Euler #4, and I came up with an interesting way of ...
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0answers
22 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{ } &+ \\ ...
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0answers
24 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 ...
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0answers
14 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 ...
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0answers
38 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.
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0answers
64 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: ...
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0answers
28 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 ...
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0answers
31 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 ...
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0answers
46 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 ...
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0answers
18 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 ...
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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)} = ...
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0answers
47 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 ...
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0answers
70 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 ...
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0answers
38 views

Calculated by multiplying the arithmetic progression terms

$$\prod_{k=1}^n(a+(k-1)d)=a\cdot(a+d)\cdot(a+2d)\cdot(a+3d)\cdots(a+(n-1)d)=\text{?}$$ Please help me! What is the formula?
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0answers
22 views

Simplify $(y + z')' (x' + y)'$ to Sum of Products

This is my working $r: (y + z')' (x' + y)' = (y)' (z')' (x')' (y)'=xy'y'z=xyz$ The answer sheet says $= xy'z$ Please explain where I've gone wrong and show working