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

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55
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2k 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 ...
9
votes
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373 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 ...
6
votes
0answers
88 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 f(x)^...
5
votes
0answers
35 views

Prove $\dfrac{1}{2}+\sum_{n = 1}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kn) = \int_{0}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kx)\,dx$

Let $N>0$ and $a_0,a_1,...,a_N$ be any positive numbers. How to prove that $$\dfrac{1}{2}+\sum_{n = 1}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kn) = \int_{0}^{\infty}\prod_{k = 0}^{N}\text{sinc}(...
5
votes
0answers
163 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
685 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 $x_{ij}>0$....
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
111 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,\\ \text{...
3
votes
0answers
126 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: $$p_{k}(x)=-\prod_{i=1}^{k}{(x+i)}^{\left\lfloor\frac{k}{i}\right\rfloor-1}\left[p_o(x)...
3
votes
0answers
42 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
128 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
284 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
183 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
31 views

What is $\prod _{j=1}^n \left(\sqrt{j}+1\right)$?

By the Fundamental Theorem of Algebra, it is easily seen that for a monic polynomial $p(x) \in \mathbb{C}[x]$, $$\prod _{j=1}^n p(j) = \frac{\prod_{p(r)=0}\Gamma(1+n-r)}{\prod_{p(r)=0}\Gamma(1-r)},$$ ...
2
votes
0answers
42 views

Multiplication of polynomials of the same degree

Consider polynomials of the form \begin{equation} p(x)=x^{n-2r}\sum_{i=0}^ra_ix^{2i}, \end{equation} where \begin{align} r&=n/2, \quad n \quad \text{even},\\ r&=(n-1)/2, \quad n \quad \text{...
2
votes
0answers
123 views

What notation would I use to differentiate between a cartesian product and a cotangent bundle of surfaces?

If the $S^1$ is defined by $x^2 + y^2 = r^2$ , $T^2 = S^1 \times S^1$ is defined by $\left(\sqrt{x^2 + y^2} -R\right)^2 + z^2 = r^2$ , $T^3=S^1\times S^1\times S^1$ is defined by $\left(\sqrt{\left(...
2
votes
0answers
42 views

How can I find the elements generating a group in a special way?

Suppose, a finite permutation group G is given. I want to find the minimal set $x_1,...,x_n$ such that every element of $G$ can be uniquely written in the form $$x_1^{j_1}...x_n^{j_n}$$ with $0\le j_i\...
2
votes
0answers
45 views

Is $(\omega \times \omega)^{\omega}\cong \omega \times \omega \times… \cong \omega^{\omega}$? Where “$\cong$” means homeomorphic.

I'm interested in the circumstances for when we can conclude that two ordinal spaces are homeomorphic by an examination of their written form. Specifically, I'm taking an ordinal space, say $(\omega^...
2
votes
0answers
104 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 ...
2
votes
0answers
64 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 this:...
2
votes
0answers
53 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
131 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
32 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
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0answers
88 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
134 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
14 views

nth product of sequential matrices

$\forall n \in \mathbb{N}$, let: $$P_n = \left( \begin{matrix} a & 1-a \\ b_n & 1-b_n \end{matrix} \right). $$ Whereby $\{b_n\}_{n \in \mathbb{N}}$ is a monotonically increasing sequence of ...
1
vote
0answers
19 views

In which cases are the main diagonal elements of a product of positive definite matrices positive?

Let $A$ and $B$ be symmetric positive definite (pd) $n \times n$ matrices and $C = A \cdot B$. In which cases is then every $c_{ii}$, the $i$-th main diagonal elements of $C$, positive? When $A$ ...
1
vote
0answers
27 views

which values of k satisfies special property to formulate L function

Consider $x*\prod_{a=1}^{n}(1-x^a)^k$ Famously for k=24 this product satisfies the condition to be an L-Function. More information can be found here My question is for what other values of k, such ...
1
vote
0answers
15 views

Where can I find methods to evaluate products?

I found it was slightly difficult to find resources that discussed methods for evaluating products, like $\Pi_{n=0}^ka_n$ Preferably, I want to start with the basics and move through some readings on ...
1
vote
0answers
32 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 $z_{...
1
vote
0answers
20 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
28 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
33 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 \Pi_{...
1
vote
0answers
12 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 $$\|N_{n+1}(...
1
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0answers
44 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\}} \Phi_{...
1
vote
0answers
59 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 $\mathbb{N}$...
1
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0answers
45 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
64 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
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0answers
119 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
vote
0answers
75 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\cdot2\cdot3+1\cdot2\cdot4+1\cdot3\...
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
vote
0answers
125 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
vote
0answers
121 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
vote
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
31 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
vote
0answers
65 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 $\...