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Questions tagged [products]

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

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2answers
32 views

Product Pi Notation [on hold]

I wonder what is the properties of Product Pi Notation? I can't found anywhere about the properties. First of all, i have: $X=\beta\alpha\\ X^2=\beta^2\alpha(\alpha + 1) \\ X^3=\beta^3\alpha(\alpha +...
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1answer
20 views

Is a function on a product measurable space measurable iff it is componentwise measurable?

Let $(\Omega_1, \mathcal{G}_1), (\Omega_2, \mathcal{G}_2), (\Omega, \mathcal{G})$ be measurable spaces and let $$(\Omega_1 \times \Omega_2, \mathcal{G}_1 \otimes \mathcal{G_2}) $$ the canonical ...
0
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0answers
22 views

$X$ is an zero object if and only if $(X,1,1)$ is the product of $X$ and $X$

Let $\mathcal{C}$ be a category with zero object. Show that the following conditions are equivalent: $X$ is zero object $(X,1,1)$ is the product of $X$ and $X$ This exercise had been obtein of the ...
1
vote
1answer
20 views

Given 3x nxn matrices A,B,C. Can i easily find entry [i,j] of product?

I am reading my Linear Algebra book, and one of the exercises is asking me to find the entry (2,2) of a product of 3 3x3 matrices ABC without calculating the full product. My current approach would ...
4
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6answers
111 views

Writing $x^2+xy+y^2$ as a product [closed]

How can this polynomial be written as a product of two complex factors? I know it has something to to with the n th root of 1 but i got stuck.
0
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1answer
50 views

Inequalities involved in the proof of transcendence of $[0,10,10^{2!}, \dots]$

In proof of transcendence of the simple continued fraction $[0,a_1,a_2, \dots]$ in which $a_k=10^{k!}$ (Hardy, et al's Theory of Numbers) it uses the following two inequalities: i. $(1+\frac{1}{10})(...
1
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0answers
21 views

product convert to polynomial to obtain derivative

Supposing we have a function like $$\prod_{n=0}^{5} \frac{(x+n)}{(1+n)}$$ and want to obtain the derivative at x = 0 Is it valid to convert to a polynomial form $$\ \frac{x^6}{720}+\frac{x^5}{...
1
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1answer
31 views

Comparing product definitions in well-pointed categories vs general categories

I'm new to category theory and trying to convince myself that the definition of a product makes sense. Ultimately my question is: in what sense does the category theory definition of a product ...
1
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0answers
44 views

Prime numbers and divisors

I am trying to prove that $\prod_{p \text{ is prime}}$($a_i$+1) where $p \equiv 1 \mod 4 \\ $ is = [The number of divisors d of n such that $d \equiv 1 \mod 4 \\ $] - [The number of divisors d of n ...
2
votes
1answer
98 views

How can I express $3 \cdot 7 \cdot 11 \cdots (4n+3)$ in terms of factorial?

This is the work I have done so far: $\prod_{k=0}^n(4k+3) = \frac{(4n)!}{2^n(2n)!}\cdot\prod_{k=0}^n\frac{1}{4k+1}$. I would really appreciate a clever trick how to reduce the latter product that ...
0
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1answer
26 views

What function symmetric and has unique solution?

I have multiple operands, say a, b, and c. I want an operator acting on a, b, and c, but the result should be invariant of order of operands (symmetric), and there should be no other pair with same ...
1
vote
2answers
20 views

Product Symbol for sequences in decrementing order

What is the symbol for a product of elements in some range $\displaystyle \prod_{i=4}^{1} i = 4 \times 3 \times 2 \times 1$ in reversed order? I thought of $\displaystyle\coprod_{i=1}^{4} i$ but it ...
1
vote
1answer
32 views

Proving $G(\mathbf{u},\mathbf{v})\geq0\,,\,\forall \mathbf{u},\mathbf{v}>\mathbf{0}$

Given $f(\mathbf{x})=\sum_{i=1}^{n}x_i\ln(x_i)$, I want to prove that the function: $$G(\mathbf{u},\mathbf{v})=f(\mathbf{u})-f(\mathbf{v})-(\mathbf{u}-\mathbf{v})^{T}\nabla f(\mathbf{v})$$ is non-...
3
votes
1answer
22 views

Sum operator precedence

I'm trying to read some simple equations and in order to interpret them in the right way I need to know $\sum$ and $\prod $ operator range/precedence. $$ \sum p(s, a) +\gamma $$ is equal to $\sum(p(...
0
votes
1answer
22 views

Product of independent random variables following different distributions

I need to find the CDF of the product of two independent random variables $Z=XY$. $X$ is defined in $\left ( -\infty,0 \right )$ and $\left ( 0, \infty \right)$. Y is defined in [$0,A_o^{2}$], whith $...
0
votes
1answer
30 views

Alternative form of this equation without the product symbol?

For a programming case, I need to redefine this equation, but for any value of 'n' and without the 'product' symbol $$\prod_{i=1}^n \Bigg(1-exp\bigg(-0.5\Big(\frac{c}{s_{i}}\Big)^2\bigg)\Bigg)^{m_{i}...
2
votes
2answers
42 views

Defining a kind of “projection of a measure” in a precise way

Suppose we have a probability measure $$\mu: \mathcal{S} \times \mathbb{R}\rightarrow [0,1],$$where $\mathcal{S}$ is some countable set. In some personal, handwritten lecture notes I'm reading it is ...
0
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0answers
36 views

There is a General form for Inner Product?

I am studying Linear Algebra and my professor proposed an exercise: Let $\mathbb{R^n}$ a vector space, $v=(v_1,...,v_n)$ and $u=(u_1,...,u_n)$, then there is a general form for Inner Product between $...
0
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0answers
37 views

Removing specific terms from a product of sums

Consider the following product: $$ \prod_{i=1}^N (a_i + b_i + c_i) $$ Question is: Is there a subtle way to get rid of all terms containing $\ldots a_{i}b_{i+1}\ldots$ $\mathbf{while}$ generating the ...
0
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2answers
36 views

Differentiation of a sum and product with respect to a constant

I would like to differentiate the following expression with respect to b: $$\sum_{i=1}^M (\prod_{j=1}^{i-1} (b+1+a_j))$$ $$a_j\in\mathbb{R}$$ $$b\in\mathbb{R}$$ aj is a small number between between -...
0
votes
1answer
15 views

Represent a sequence with a Product sign

I have 2 sequences I have to represent with a Product symbol (Separately of course). a) $2*\sqrt{3}*\sqrt{4}^3*\sqrt{5}^4$ b) $\dfrac{1}{n}*\dfrac{3}{n+2}*\dfrac{7}{n+3}$ I am clueless on what I ...
3
votes
1answer
64 views

Simplifying $\prod\limits_{k\neq j=0}^{n-1}\frac1{\lambda_{n,k}-\lambda_{n,j}}$ for $\lambda_{n,k}=\exp\frac{i\pi(2k+1)}{n}$

I have been able to show that for $n\in\Bbb N_{\geq2}$ $$\phi(n)=\int_0^1\frac{dx}{x^n+1}=\sum_{k=0}^{n-1}\Gamma_{n,k}\log\frac{\lambda_{n,k}-1}{\lambda_{n,k}}$$ Where $$\lambda_{n,k}=\exp\frac{i\pi(...
0
votes
0answers
33 views

What is the result of a Weibull distribution to the power 2?

It is frequent to model wind speed distribution as a Weibull distribution. I would like to know what is the distribution of the kinetic energy associated to the wind distribution which is ...
0
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0answers
47 views

What is the difference between “Product” and “Twisted product”?

I am reading Allen Hatcher's book of Vector bundle and K-theory. Here I have gotten that : "Mobius band is the twisted product of a circle and a line. On the other hand annulus the product of a ...
1
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0answers
31 views

Dimension of product of varieties

I've got this exercise asking me to prove first that the product of quasi projective varieties $X$ and $Y$ (henceforth just "varieties") is irreducible iff both $X$ and $Y$ are. I managed to solve ...
6
votes
0answers
89 views

Is there a way to classify all finite groups $G$, such that $\pi(G) := \Pi_{H \triangleleft G} |H| = |G|^2$?

Is there a way to classify all finite groups $G$, such that $\pi(G) := \Pi_{H \triangleleft G} |H| = |G|^2$? For abelian groups this problem is quite simple. All such abelian groups are exactly $C_{p^...
4
votes
1answer
58 views

Notation: Using $\prod A_i$ for noncommuting $A_i$

I am writing up some math and realized that I do not know whether $\prod_{i=1}^m A_i=A_mA_{m-1}\cdots A_1$, OR whether $\prod_{i=1}^m A_i=A_1A_2\cdots A_m$. Is there accepted consensus on this ...
0
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0answers
30 views

Prove that $\sin(\pi z)$ can be written as infinite product [duplicate]

Prove that \begin{align} \sin(\pi z) = \pi z \prod_{n=1}^{\infty} \left( 1-\frac{z^2}{n^2}\right) \, \, \, \, \forall \, z \in \mathbb{C} \end{align} The hint I had it's to use the Fourier series, ...
0
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3answers
42 views

How do you compute ab+i in terms of a and b

Given that $a+bi=p$ and $a^2-b^2=q$, how can I compute $ab+i$? Note $i=\sqrt{-1}$ is the complex number $a$ and $b$ can be any integer
1
vote
2answers
39 views

Is it always true that $\prod_{j=1}^{w}{(2s_j + 1)} \equiv 1 \pmod 4$?

My question is pretty basic. Here it goes: Is it always true that $$\prod_{j=1}^{w}{(2s_j + 1)} \equiv 1 \pmod 4$$ where the $s_j$'s are positive integers, and may be odd or even? We can ...
2
votes
2answers
420 views

Does every functor from Set to Set preserve products?

In general, not all functors preserve products. But my question is, is it at least true that all functors from Set to Set preserve products? If not, does anyone know of a counterexample?
0
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0answers
43 views

Solving Product for X

I have a question about Products written with Product signs. Can I simplify that and if yes how can I do that? I tried various ways but I didn‘t figured out. $$ \prod _{n=0}^{x-1} x-n $$ $$ \prod _{n=...
2
votes
3answers
147 views

Limiting value of a sequence when n tends to infinity [duplicate]

Q) Let, $a_{n} \;=\; \left ( 1-\frac{1}{\sqrt{2}} \right ) ... \left ( 1- \frac{1}{\sqrt{n+1}} \right )$ , $n \geq 1$. Then $\lim_{n\rightarrow \infty } a_{n}$ (A) equals $1$ (B) does not exist (C) ...
1
vote
2answers
29 views

Log of products and densities

By taking the log of the function: $$\prod_{t=1}^T n_t! \prod_{i=1}^{n_t} \frac{f_v[b_{i:n_t}]}{[1-Fv(r)]},$$ it is possible to end up with the following expression: $$ \log(n_t!)+\sum_{i=1}^{n_t} \...
9
votes
1answer
56 views

If integration is a continuous analog of summation (Addition), what is the continuous analog of multiplication (Product)?

One definition of integration over a continuous interval [a,b] into n subintervals with equal width $\Delta x$, and from each interval choose a point $x_i^*$. Then the definite integral of $f(x)$ ...
1
vote
1answer
35 views

Efficient method for computing the product of the first 8 terms of a recursive sequence

The problem I am trying to solve is the following: Let $x_1=97,$ and for $n>1,$ define $x_n=\frac{n}{x_{n-1}}.$ Calculate $x_1x_2 \cdots x_8.$ I tried the painstaking fail safe method for the ...
0
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0answers
26 views

What is the limit of the series (summation) of the q-Pochhammer symbol or the ~q-Pochhammer symbol?

I am interested in knowing if the following series converges or not: \begin{equation} \sum_{n=1}^{\infty} \prod_{i=1}^n \left(1-e^{-\sqrt{i}} \right) \qquad Expression \; 1 \end{equation} If that is ...
0
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0answers
15 views

How to write a function in R that will obtain this product?

Kaplan-Meier estimator: I've just started using $R$, so I don't understand how to build this function $($it is modified Kaplan-Meier estimator$)$. It should be something like: ...
2
votes
2answers
32 views

Prove that $ (\frac{\sum_{i=1}^n x_i}{n})^{\sum_{i=1}^n x_i} \le \prod_{i=1}^n {x_i}^{x_i}$ $, \forall x_i>0, n\ge1 $

Prove that $ (\frac{\sum_{i=1}^n x_i}{n})^{\sum_{i=1}^n x_i} \le \prod_{i=1}^n {x_i}^{x_i}$ $, \forall x_i>0, n\ge1 $ (The second sum in the left-hand side of the inequality is an exponent) I've ...
1
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1answer
29 views

Is the following equation valid for a binomial coefficient?

From my notes on the binomial series I deduced a formula for the value of a binomial equation using product notation: $ {n \choose r} = {\frac {\prod_{\lambda=0}^r(n-\lambda)} {r!}}$ I believe this ...
-3
votes
1answer
68 views

Is there a formula for a summation divided by a product of its terms?

$$\frac{\sum_{i=1}^{n}x_{i}}{\prod_{i=1}^{n}x_{i}}= \frac{1}{x_{2}x_{3}x_{4}...}+\frac{1}{x_{1}x_{3}x_{4}}+\frac{1}{x_{1}x_{2}x_{4}}...$$ There is a very clear pattern that each consecutive result ...
1
vote
1answer
31 views

Inequality for finite product

I'm recently came across the following theorem: Let $f$ be a function in $C^{n+1}[a, b]$ and let $p$ be a polynomial of degree $\leq n$ that interpolates the function at $n+1$ distinct points $...
0
votes
1answer
40 views

$X$ is connected and separable. $X=Y\times Y$. Does $Y$ has to be also connected and separable?

$I$ is a finite set. It is not hard to see that, if $X=\prod_{i\in I}Y_i$ is separable, then $Y_i$ does not have to be separable. But for this special case such that $Y_i=Y_j \ \forall i,j\in I$, I ...
2
votes
1answer
44 views

Simplifying a product of a series

I tried simplifying the product $$\prod_{k=1}^{\infty}\left[1-x^k\right]$$ by factoring it into $$\prod_{k=1}^{\infty}\left[\left(1-x\right)\sum_{i=0}^{k-1}x^i\right].$$ I am not very experienced in ...
1
vote
2answers
47 views

Converting a Product to a Sum

How can I convert $$\prod_{i=1}^k\left[\sum_{j=0}^{i-1}x^j\right]$$ to a sum? I have been trying to solve this product by inductive reasoning but I figured it was too complex... Is there an agebraic/...
2
votes
0answers
41 views

Cross product in 4 dimensions [duplicate]

Cross product is defined in three dimensions the resulting vector have the same magnitude as the area of the parallelogram formed by the 2 multiplied vectors and its direction direction is ...
1
vote
1answer
38 views

The derivative of an integral of a product of functions.

I'm trying to comprehend the following result, which is required for fractional calculus: Let $w(x,y)$ and $f(z)$ be two real functions, such that they both vanish at a point $a$. Then the ...
1
vote
1answer
51 views

In the derivative of the product of two functions , why (dx)² is ignored?

I was digging deeply in the fundamentals of calculus, I found that in the famous '3Blue1Brown' channel, when demonstrating the process of finding the derivative of the product of two functions , (dx) ²...
0
votes
1answer
44 views

Fastest structured way to get max(abc) if a+b+c=30

What is the fastest and structured way to get maximum of abc if a+b+c=n, say n=30? a,b,c are non-negative and can be non-integer.
0
votes
1answer
51 views

Inverse function of a product space

I want to prove the continuity of a function $f: (X_1,\tau_1) \times (X_2,\tau_2) \rightarrow (X'_1,\tau'_1) \times (X'_2,\tau'_2)$ where $f(x,y) = (f_1(x),f_2(y))$ and my question is: What is $f^{-...