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

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3answers
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what does $(A\cdot\nabla)B$ mean?

I was studying a physics book and I saw this expression $$(A\cdot\nabla)B$$ where $A$ and $B$ are vectors. What's the definition of this? I've also seen this in some identities
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0answers
24 views

How to estimate the product of the $k$ largest eigenvalues of a matrix

Now I have a question which let me to prove that the product of the largest $k$ singular values of a real matrix is always larger than the one of $k$ largest eigenvalues. For $k=1$, I use the ...
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0answers
25 views

product of polynomial [on hold]

How can I calculate this? $$\prod_{i=1}^n \left(1+x_i\right) $$ Can I use vieta`s formulas?
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1answer
29 views

Value of finite product based on empty set

How does one evaluate the following product if the set S happens to be empty? \begin{aligned} f(n)= n \prod_{x \in S} \left(1-\frac{1}{x}\right) \end{aligned} Is the value simply n or is it ...
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0answers
27 views

Linearise product of two non-negative variables

Is there a trick to linearise the product of two non-negative (decision) variables in linear optimisation? Let $x_1$ and $x_2$ be these variables with $0 \leq x_1 \leq a$, $a \in \mathbb{R}_+$ and $...
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0answers
8 views

Product of B-splines

It's my first post and i'm delighted to be among you... I'm reading at the moment a paper "Some Identities for Products and Degree Raising of Splines" from Knut Morken. enter image description here He ...
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1answer
27 views

Is there anything known in general about upper and lower bounds for $\prod_{i\leq n\vee p_n>k}(p_i-k)$

I have no specific reason to ask this question other than seeing that it comes up quite often when I'm playing around with prime numbers. Let $$f(n,k)=\prod_{i \leq n\vee p_n>k}(p_i-k)$$ Where $p_i$...
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1answer
21 views

Understanding components of a vector

I learned that we can get the component of a vector in any direction using the dot product. The problem I have is the meaning of the term component itself. The component of a vector $\vec A$ in the ...
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1answer
666 views

Product of Fibonacci numbers

I'm looking for the asymptotic approximation of the product of the first $n$ Fibonacci numbers. Does there exist a tight approximation for these kind of things?
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1answer
53 views

Product as the sum of powers times a symmetric polynomial: What's the name of this property and what is it used for?

I noticed that the product of a group of positive integers $N$ with $n$ elements can be expressed as the sum of powers of the smallest member of $N$ times some (what I later found out be called) ...
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1answer
48 views

Proving ${\pi\over 2}=2\tan^{-1}\left({1\over A}\right)+\tan^{-1}\left({1\over B}\right)$

Let $A=2^{2^{-x}}$ and $B=2^{2^{-x}+1}(1+2^{2^{-1}})(1+2^{2^{-2}})\cdots(1+2^{2^{-x+1}})$ Showing $x\ge2$ $${\pi\over 2}=2\tan^{-1}\left({1\over A}\right)+\tan^{-1}\left({1\over B}\right)\tag1$$ ...
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1answer
61 views

The limit of consecutive positive integers which are the product of n primes.

The maximum length of a string of consecutive primes is 2: that is, the primes 2, 3. This is easily proven, as no even number other than 2 is prime. In contrast, consider the set of numbers which ...
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1answer
23 views

If two infinite sets have the finite closed topology, then their product does not have the finite closed topology.

Let $X_1$ and $X_2$ be infinite sets and $T_1$ and $T_2$ be the finite-closed topology on $X_1$ and $X_2$, respectively. Show that the product topology, $T$, on $X_2 \times X_2$ is not the finite ...
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1answer
34 views

Source of faulty reasoning in expectation of product of random variables?

For iid $\xi_i>0$, with $\mathbb E[\xi_i]=1$, what is $\mathbb E[\prod_i^M\xi_i]$? Approach 1: $\mathbb E[\prod_i^M\xi_i]=\prod_i^M\mathbb E[\xi_i]=1$. There is another approach for $M\gg1$ with ...
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1answer
40 views

Proof of $\sin nx=2^{n-1}\prod_{k=0}^{n-1} \sin\left( x + \frac{k\pi}{n} \right)$

I have seen this identity on Wolfram mathworld and in a comment to another similar trigonometric proof: Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ I can't seem to find a ...
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1answer
35 views

Modified Sum of Products

A given number k is to be expressed as a sum of products of integers keeping in mind that the integers used in above process do not exceed their cumulative sum as 100. For e.g., k = 19 can be ...
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1answer
79 views

Summation is to integration, as the large product operator is to…? [duplicate]

The integral is defined many ways but one that I am aware of is the Riemann Integral(?) which looks sorta like $\sum^n_{i=0} f(a +i\frac {b-a}n)*\frac {b-a} n$. An interesting thought is "is there a ...
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1answer
20 views

Product of all Square Roots, taken only Decimal Digits

How and where could I compute the decimal reminder of a product of square roots times ten: $$Dr\left( \prod_{x=1}^{k}x^\frac{1}{2} \right) \times 10$$ Where $k$ is a power of $10$. I would like to ...
2
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1answer
56 views

How to prove that $e^{-\gamma}=\prod_{n=1}^\infty\left(1+\frac1n\right)e^{-1/n}$

Suppose we defined the Gamma function $$\frac1{\Gamma(z)}=ze^{\gamma z}\prod_{n=1}^\infty\left(1+\frac zn\right)e^{-z/n}$$ where $\gamma$ is just a constant. I want to prove that $\Gamma(1)=1$, so I ...
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1answer
41 views

Prove $2^i\prod_{j=1}^{i}\sum_{k=1}^{2j-1}(-1)^{k-1}k^2=(2i)!$

Prove identities (1) $$2^i\prod_{j=1}^{i}\sum_{k=1}^{2j-1}(-1)^{k-1}k^2=(2i)!$$ (2) $$2^i\prod_{j=1}^{i}\sum_{k=1}^{2j}(-1)^{k-1}k^2=(2i+1)!$$ Is there another approach to prove (1)? $$2^i\...
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1answer
33 views

Why is $\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$?

I came across this approximation in the book Principles of Population Genetics by Hartl and Clark (page 130). $$\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$$ ...
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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 $A$ ...
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0answers
15 views

Doing a project for a CALC II class, and need help determining if we have sufficiently solved a product problem.

We finished our final in Calc II, and now we are doing math projects to pass time until the semester ends. https://docs.google.com/presentation/d/1bYMKfCqcc9zG32Zs_Wti2Fp9rXV7Nbt4QxR3Vnp33Dc As you ...
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1answer
28 views

Spectra of matrices with null product

If I consider two square matrices $A$ and $B$ such that $A B = B A =0$ and I know eigenvalues and eigenvectors of $A$, is it possible to get informations about the spectrum of $B$? In particular, I ...
2
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1answer
30 views

Weierstrass Approximation Theorem for a Product Space?

I am faced with the following problem: Let $X$ and $Y$ be compact Hausdorff spaces and $f$ belong to $C(X \times Y)$. Show that for each $\epsilon > 0$, there are functions $f_{1}, f_{2}, \cdots , ...
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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 ...
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1answer
16 views

Euclidean algorithm for dividing two products.

Say I have numbers, $a$ and $b$ represented as two products $$a = \prod_{i=0}^{N_a} a_i \hspace{1cm} b = \prod_{i=0}^{N_b}b_i$$ I do know $\{a_k\}$ and $\{b_k\}$ but can not store $a$ or $b$ in a ...
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0answers
21 views

Minimize and maximize the sum of dot products at the same time

this is the problem. I have a set of numerical positive vectors of equal length. For each pair of vectors $(\mathrm{i}, \mathrm{j})$ I define the vector $\mathrm{ij}=\mathrm{i} - \mathrm{j}$. I also ...
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2answers
37 views

Why is my answer incorrect for this differentiation question?

$$y = x* ((x^2+1)^{1/2})$$ I must find $$dy/dx$$ $$u = x, v = (x^2+1)^{1/2}$$ To do this I must use the product rule and the chain rule. To get dv/dx, $$(dv/dx) = (1/2)*(b)^{-1/2}*2x $$ $$(dv/dx) ...
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0answers
108 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(...
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2answers
83 views

Prove $\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$

I want to prove $$\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$$ if $\sum_{k=1}^n a_k\leq1$ and $a_k\in[0,+\infty)$ I have no idea where to start, any advice would be greatly appreciated!
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1answer
24 views

Find when the product would be an integer

The problem: The sequence $\{a_n\}$ is defined recursively by $a_0=1,a_1=\sqrt[19]{2}$ and $a_n=a_{n-1}a_{n-2}^2$ for $n \geq 2$. What is the smallest positive integer $k$ such that the product $a_1 ...
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1answer
45 views

Equivalence of definition of product in a category

I was reading Mitchel book on categories and the following observation without proof is given: A family of morphisms given by $\lbrace p_{i}:A \to A_{i} \rbrace$ is the product of $A_{i}$ if and only ...
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2answers
30 views

By induction prove $\prod _{i=1}^{n} \frac{n+i}{2i-3} = 2^n (1 - 2n)$

I need to prove the following by induction. $\forall n \in \Bbb N$ $\prod _{i=1}^{n} \frac{n+i}{2i-3} = 2^n (1 - 2n)$ I know the steps to take but I'm failing to come to the right side of the ...
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0answers
13 views

Parentheses and Comma Notation

I came across the following formula for normalizing Smith-Waterman scores, and I do not understand what the SW(p1, p2) part is trying to notate. Does it perhaps refer to a product? Click here to see ...
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1answer
90 views

Prove that $\prod\limits_{2 < p \leq y}\left(1-\frac{2}{p}\right)\sim\frac{D}{\log ^2 y}$ [duplicate]

I'm writing my bachelor thesis about Brun's sieve method and his theorem. In one proof I found this statement without further explanation. It is important to show that the product doesn't converge "...
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23 views

Product of directed partial orders

Is a product poset (with componentwise order) of nonempty posets a dcpo if and only if each multiplier is a dcpo? (for both binary and arbitrary products)
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25 views

Is there a constant $C$ such that $\prod_{p\leq x}\dfrac{p-1}{p}<\dfrac{e^{-\gamma}}{\log x}\cdot C$?

By Mertens' third theorem: $$\prod_{p\leq x}\dfrac{p-1}{p}\sim\dfrac{e^{-\gamma}}{\log x}$$ But does there exist a constant $C$ such that: $$\prod_{p\leq x}\dfrac{p-1}{p}<\dfrac{e^{-\gamma}}{\log x}...
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0answers
31 views

I am looking a comparison of this computation and Riemann's approach for $lcm(1,2\ldots,x)$

Looking a comparison with a reasoning due to Riemann, I ask to me about the behaviour as $x\to\infty$ of the following arithmetical function $$ \left( \prod_{n\leq x}n^{-\mu(n)}\right)\cdot \left( \...
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1answer
39 views

Why does the product of adjugates equal an adjugate of the product?

How can I show that $\mathrm{adj} (AB) = \mathrm{adj}(B)\ \mathrm{adj}(A)$? It is obvious if determinants are non-zero, but if any of the matrices are singular, I just don't get it. UPD. I've just ...
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0answers
23 views

Prove Euler characteristic satisfies $\chi(X \times Y)=\chi(X)\chi(Y)$ for polyhedra $X$ and $Y$

Prove that for any topological polyhedra, $X$, $Y$, the product $X \times Y$ has the Euler characteristic $\chi(X \times Y)=\chi(X)\chi(Y)$ I know that for polyhedron $P$ which is homemorphic to a ...
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0answers
30 views

Formula for combinations involving product notation?

So after looking at the factorial formula and learning about product notation, I recognized this relation between them: $$\prod_{n=1}^kn=k!$$ And after fooling around and doing some trial and error, I ...
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1answer
38 views

“Binary-Like” Function?; In Consecutive Products as Multi-Factorials…

Summary Is there a function $Z(a,b)$ or how would one find such a function so that for $a,b\in \mathbb N$, it would produce $0$'s on for each $a$th step for each $b$th value? For example: $a=2$, ...
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1answer
218 views

How many numbers $ N \le 10^{10}$ are the product of $3$ distinct primes?

How many numbers $ N \le10^{10}$ are the product of $3$ distinct primes? I can realistically calculate any $\pi(n), n < 10^{15} $ but I don't think it's possible to list all primes $>10^8$ in ...
2
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2answers
32 views

Gamma representation of certain sequence

I'm trying to find a gamma rep for $ 15 \cdot 13 \cdot 11 \cdot 9 \cdot 7 \cdot ... $ Steps so far: It's a simple sequence of $ n \cdot (n-2) \cdot (n-4) \cdot (n-6) \cdot (n-8)... $ and so on. ...
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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
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1answer
37 views

Complex inner product proof

I have just solved this problem in the real inner product space with $\langle \cdot , \cdot \rangle$ as the inner product. Show that in a real inner product space we have: $\langle x,y \rangle = \...
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1answer
31 views

Nice formula for a sum product

So suppose I have an ordered set of numbers: $(a_1, a_2, ..., a_n)$ and I want to express the following sum/product in an elegant manner: $ a_1 + a_1 a_2 + a_1 a_2 a_3 + ... + a_1 a_2 ... a_n $ I ...
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0answers
26 views

Resolving Zeros in Product of items in list.

Given the formula: $\sqrt [ 1/N ]{ \prod _{ n=1 }^{ N }{ { P }_{ n } } } $ where ${ P }_{ n }$ is a list of real numbers, e.g. [0.4, 0.3, 0.2, 0.1] And the ...
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1answer
45 views

Finding an upperbound for $\sum_{i=2}^{n}\bigg(\prod_{k=2}^{i}\dfrac{p_k-2}{p_k}\bigg)$

I was wondering whether there exists a known upperbound for: $$f(n)=\sum_{i=2}^{n}\bigg(\prod_{k=2}^{i}\dfrac{p_k-2}{p_k}\bigg)$$ For example: $$f(4)=\dfrac{1}{3}+\dfrac{1\cdot3}{3\cdot5}+\dfrac{1\...