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

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40
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1k 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 $$\infty != \prod_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty \# = \prod_{k=1}^\infty ...
8
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355 views

Two (strictly related) proofs by induction of inequalities.

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
5
votes
0answers
76 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
125 views

Solving a question by using special products (Students debate to Teacher)

So today,we got back our exam papers,and we found a question marked wrongly and teacher said that it is wrong.We all students do NOT believe this.So here is what happened. Before reading the next ...
4
votes
0answers
102 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
105 views

Asymptotic solutions of a sparsely perturbed recurrence relation

Recurrence relation I am trying to find approximate solutions $T(n)$ of the recurrence relation $$ p\ T(n-1) - \left[p+q+\overline{S} + \varepsilon \tilde{S}(n)\right]T(n) + q\ T(n+1) = 0,\\ ...
3
votes
0answers
123 views

Simplification of recursive polynomials

Suppose I have known polynomials $p_i, i=0\ldots k-1$. I have the following horrific looking recursive equation: ...
3
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0answers
41 views

How I can calculate this product

How I can calculate this product: $$s=∏_{j=1}^{p-3}\sinh^2[2^{j-1}\cosh^{-1}( 2)]$$ for a natural number $p>3$.
3
votes
0answers
119 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
267 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
618 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
178 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
61 views

Cleaning Up Messy Product Notation

Suppose I have the following: Let $N_1<...<N_m$. Let $T_{N_k}(x)=\sum_{i=0}^{N_k}{\frac{x^i}{i!}},$ $ t(i,j,x)=(T_{N_i}-T_{N_j})(x)$ I'm trying to define a polynomial $p_{k,m}(x)$ like ...
2
votes
0answers
49 views

How the second form of following equation is derived form first form (i.e. given first line, what are the steps involved in writing second line

How the second form of following equation is derived form first form (i.e. what are the steps involved in writing second line)
2
votes
0answers
130 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
30 views

Vectorial product analog operation in 4+ dimensions?

I am thinking 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, ...
2
votes
0answers
30 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
201 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
86 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
124 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
29 views

Product of Several Functions Becomes Very Small: Scaling?

I have the following ratio: $$\frac{\sum_{i = 1}^n Y_i \prod_{p = 1}^P \lambda_p^{z_{i,p}}}{\sum_{i = 1}^n \prod_{p = 1}^P \lambda_p^{z_{i,p}}}$$ where $\lambda_p \in (0,1]$ is a parameter, and ...
1
vote
0answers
19 views

Product of a matrix and a tensor

I need to know how to compute the following product: $M(x)\frac{\partial M(x)}{\partial x}M(x)$ $\quad$ where $x \in R^{n}$. Assuming the dimensions of the matrices are compatible,how do we take ...
1
vote
0answers
27 views

Binomial square sum and product

Given $c,n\in\Bbb N$ what is the expression for $$S(n,c)=\binom{n}c^2+\binom{n-c}c^2+\dots+\binom{x}c^2$$ and $$P(n,c)=\binom{n}c^2\cdot\binom{n-c}c^2\cdot\dots\cdot\binom{x}c^2$$ where $x-c<c\leq ...
1
vote
0answers
29 views

How to find $\sup \Pi_{i=0}^{n} (\sin(i)^2 - \frac{25}{16})$?

Let $\sup,\inf,{\rm dif}$ denote resp supremum , infimum and $\rm dif$ = supremum - infimum. Does any of the 3 below have a closed form ? $\sup \Pi_{i=0}^{n} (\sin(i)^2 - \frac{25}{16})$ $\inf ...
1
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0answers
11 views

How can I prove the following inequality?

Let be $N_{n+1}(x)=\prod_{i=0}^n(x-x_i)$. Now I have to prove that $$||N_{n+1}(x)||_{\infty,[-5,5]}\leq n!\frac{h^{n+1}}{4},\qquad h:=\frac{5-(-5)}{n}=\frac{10}{n}.$$ I've started with ...
1
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0answers
40 views

Product of Dependent Bernoulli variables

Let $B_{i,n}$ with $i=1,...,n$ be the triangular Bernoulli array defined as $$ B_{i+1,n} = B_{i,n}\,R_{i+1,n}+\left(1-R_{i+1,n}\right)\,F_{i+1,n}, $$ where $R_{i,n}$ and $F_{i,n}$ are iid Bernoulli ...
1
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0answers
23 views

The product of distribution taken over Unions

I have a probability problem as follow: $\mathbb{P}\big[\mathop{\arg\sup}_{x \in \bigcup_{i\in \{1,2\}} \Phi_{k,i} } \mathcal{f}(x )\geq y \big] = 1-\mathop{\prod}_{x \in \bigcup_{i\in \{1,2\}} ...
1
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0answers
54 views

Uncountability of $\mathbb{R}^I$ if $I$ is uncountable

Prove that if $I$ is uncountable, then $\mathbb{R}^I$ with the product topology is not countable. Based on what I have read, a set is uncountable if there is a bijection from that set to ...
1
vote
0answers
42 views

Integral over a product

How the following integral can be computed: $I = \int_0^1 (x-a_1)^{b_1}(x-a_2)^{b_2}...(x-a_n)^{b_n} dx$? Here, $a_i,b_i$ are real numbers and $n$ is a natural number. Are there any techniques for ...
1
vote
0answers
85 views

Is the product of two discrete $\sigma$-algebras necessarily discrete?

I know that the answer to this question is negative, since proving the opposite is an exercise in Terrance Tao's Measure Theory book. However, it doesn't make sense to me. In another part of the same ...
1
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0answers
60 views

How to simplify sine function

Does anyone have an idea for simplifying this formula? $$f(x)=\prod\limits_{k=2}^{14}\sin(\frac{15x\pi}{k})$$ Or even more general case: $$f(x,y)=\prod\limits_{k=2}^{y-1}\sin(\frac{xy\pi}{k})$$ ...
1
vote
0answers
97 views

How to prove taht a product of two complete residue system is not a complete residue system?

Claim. Let $n$ be a natural number and $A=\{0,1,2,3,\cdots,n-1\}$ be a complete set of residues modulo $n$. Let $\sigma$ be a permutation of $A$. Show that the set $C=\{\sigma(i)i:i\in A\}$ is not a ...
1
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0answers
73 views

Summation verification

I have a particular polynomial $$ 1-10x+35x^2-50x^3 $$ Which can be written nicely as $$1-(1+2+3+4)x+(1\cdot2+1\cdot3+1\cdot4+2\cdot3+2\cdot4+3\cdot4)x^2$$ ...
1
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0answers
39 views

Is there a name for expressions of the form $\sum_{i=1}^N \prod_{k=1}^i a_k$?

Is there a name for expressions of the form $\sum_{i=1}^N \prod_{k=1}^i a_k$? If so, where can I find the equivalent of a Wikipedia entry?
1
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0answers
120 views

Counterexamples in $R$-modules products and $R$-modules direct sums and $R$-homomorphisms (Exemplification)

Q: (Exemplification) Do examples family of $R$-modules like $\{M_i\} , \{N_i\}$ and $R$-modules like $M,N$ such that every of four question in underline is true. (in other word for every question ...
1
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0answers
109 views

Simplify the product of two sums

How can I simplify the following product of two sums: $$ \biggl(\, \sum ^{n}_{k=0}a_{k}\biggr) \biggl(\, \sum ^{n}_{k=0}\dfrac {1}{a_{k}}\biggr) $$
1
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0answers
43 views

Can I do the following when solving my integration??

I appreciate any feedback for my question. I have an integration as follows $$\int_{-\pi}^{\pi}\frac{1}{2\pi} \prod_i \frac{1}{1+ x_ig(\theta)} d\theta $$ I have that $g(\theta)$ is the defined as ...
1
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0answers
27 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] ...
1
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0answers
59 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
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0answers
34 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
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0answers
40 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
39 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
53 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
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0answers
30 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
35 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
86 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
58 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
27 views

Number of optimas of product of convex functions

I am dealing with a function, which is a product of two strongly convex functions, and trying to determine the number of its local minimum. For example, I have $$H=f(x)\cdot g(x)$$, in which both $f$ ...
0
votes
0answers
51 views

Countable product of first/second countable spaces is first/second countable.

Let $N$ be a countable indexing set, and $((X_k,\tau_k))_{k \in N}$ topological spaces. Define $X = \prod_{k \in N} X_k = X^N$ and let $\tau$ be the product topology on $X$ induced by $\tau_k$s. ...