Questions tagged [products]

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

Filter by
Sorted by
Tagged with
0 votes
1 answer
22 views

In what framework is it okay to swap the derivative in a product within an integral? [Viscoelasticity]

Dear people with an affinity for math, I am just an engineer approaching the field of viscoelasticity. Currently, I would like to understand the derivation of the generalized Kelvin-Voigt material. It ...
user avatar
  • 11
1 vote
2 answers
53 views

Expected number of rounds for a product of uniform random variables on $[1/2,3/2]$ to be for the first time below a given threshold

Starting with w=1, each time we multiply w by a number x sampled independently and uniformly from [1/2, 3/2] until it is smaller than a given value c. What's the expected number of rounds for this ...
user avatar
2 votes
2 answers
78 views

How to prove the product of first n consecutive odd numbers is a square less than another square?

I have observed for first few values of consecutive odd numbers, the result is always of the form: $m^2 - n^2$, where $m$ and $n$ are two distinct positive integers. That is: $$1\cdot 3\cdot 5 \cdot 7\...
user avatar
-1 votes
0 answers
35 views

Derivative of a product sequence

Can someone help me with this problem please? Let $f \in C^1(\mathbb{R}, \mathbb{R})$, $N \in \mathbb{N_0}$, $x_n \in \mathbb{R^+}$ and $\theta_n \in \mathbb(0,1)$. I want to obtain the derivate of ...
user avatar
  • 1
0 votes
0 answers
11 views

How many possibilities to sort products $s_is_j$ for a set {$s_i$}?

Having $n$ numbers $s_i$ with $s_1\geq s_2\geq...\geq s_n$ and combining them to all possible products with two factors (i.e. $s_1 s_1, s_1 s_2,$…) this makes $\frac{n(n+1)}{2}$ new numbers which I ...
user avatar
  • 1
0 votes
1 answer
42 views

What concise function $f$ satisfies $\prod_{i=1}^n \prod_{j=1}^n a_{i,j} = f \left( \det(\mathbf{A}) \operatorname{Adj}(\mathbf{A})^{-1} \right)$?

Let $\mathbf{A} = [a_{i,j}]$ be an $n \times n$ matrix with determinant $\det (\mathbf{A})$. Consider the equality $$\prod_{i=1}^n \prod_{j=1}^n a_{i,j} = f \left(\det(\mathbf{A}) \operatorname{Adj}(\...
user avatar
  • 1,316
1 vote
1 answer
46 views

proving an inequality based on double products and binomial

Iv been trying to prove the following inequality: let $a_1,\ldots,a_n$ be a non-increasing sequence, i.e., $a_1\geq a_2\geq \cdots \geq a_n\geq 0$ such that $\sum_i a_i=m$. Then, prove that $$\prod_{i=...
user avatar
-1 votes
0 answers
36 views

How to proof this integral with log and product result

enter image description here I found this integral in reference of Euler's work,but I can't find the proof
user avatar
0 votes
0 answers
43 views

Are the meets of the set of 2-adic numbers that terminate in $\overline{01}_2$ **not quite** homeomorphic to the free group on two elements?

I think the set $X$ of 2-adic numbers that terminate in $\overline{01}_2$ is super-close to homeomorphic to the free group on two elements, but not quite. Can you help me characterise "not quite&...
user avatar
1 vote
1 answer
46 views

Find a formula for calculate $\prod_{n=0}^N(a^n + b)$

I have an expression, in which I am looking for a formula to calculate $$ \prod_{n=0}^N (a^n +b)$$ as a summation. Is there a general name for this type of formula? Or a relevant approach to find ...
user avatar
  • 143
0 votes
0 answers
19 views

Calculating the unknown elements of a matrix based on a known matrix.

Suppose there is a 2$\times$2 matrix $M=\matrix[a\ ,b\ ;\ c\ ,d]$, which $$M \matrix[y_1\ ,\ y_2]^T=\matrix[y_1'\ ,\ y_2']^T$$ if the elements of the matrix M are known, is it possible to calculate ...
user avatar
0 votes
0 answers
31 views

Somehow the last digit of 6x7x8 or 16x17x18 is the same as 3!, can someone prove this pattern for me?

When trying to find a solution for the problem "Find the last digit of factorial". I encounter this small problem inside a solution on the internet and I can't prove it. Somehow 6x7x8 (336), ...
user avatar
1 vote
1 answer
72 views

Is a single number a product?

This may seem like a strange question, but I was wondering if a single number $a\in \mathbb{C}$ can be seen as a product under the mathematical definition? Put differently, I am wondering if $\prod_{i=...
user avatar
  • 487
1 vote
1 answer
54 views

Product of Indicator function

I was solving a question and in an intermediate step, I need to solve the following product: $$\prod_{k=1}^N\mathbb{1}_{\{y_k\geq1\}}$$ Is there a way to simplify it or write it in a closed-form ...
user avatar
  • 47
1 vote
2 answers
73 views

Solve $\frac{(2n)!}{n!}=30{,}240$

Problem:$$\frac{(2n)!}{n!}=30{,}240$$ Attempt: I know that $$\frac{(2n)!}{n!}=\prod_{k=n+1}^{2n}k=(n+1)(n+2)...(2n)$$ I've considered factoring $30240=5 \times 6 \times 7\times 8 \times 9 \times 2$ ...
user avatar
0 votes
1 answer
49 views

Why the scale factor of the product of two gaussian functions is the convolution of the same gaussian functions?

The product of two Gaussian functions $$ f(x)=\frac{1}{\sqrt{2 \pi} \sigma_{f}} \exp\left(-\frac{x^{2}}{2 \sigma_{f}^{2}} \right) \quad \text { and } \quad g(x-y)=\frac{1}{\sqrt{2 \pi} \sigma_{g}} \...
user avatar
3 votes
2 answers
128 views

What spaces are homeomorphic to $\mathbb{Q}^\omega$ = $\mathbb{Q}^\mathbb{N}$ = $\mathbb{Q}^\infty$?

What spaces are homeomorphic to $\mathbb{Q}^\omega$ = $\mathbb{Q}^\mathbb{N}$ = $\mathbb{Q}^\infty$? (The space of all rational sequences, considered with the standard product topology). I have ...
user avatar
0 votes
0 answers
31 views

Is it possible to represent these values with a single general expression?

Let $A_1$ and $A_2$ be two $2$-by-$2$ matrices over $\mathbb{Q}$. Suppose we have the following relation $$A_1A_2=\begin{pmatrix} \frac{1}{a^2} & 0 \\ 0 & 0 \end{pmatrix},~A_2 A_1=\begin{...
user avatar
  • 9,587
1 vote
1 answer
50 views

Does $\det \mathcal{H}_f = (n-1)(-1)^{2n+1} \left[ f(x_1, \dots, x_n) \right]^{n-2}$ for $f(x_1, \dots, x_n) = \prod_{j=1}^n x_j$?

Let function $f : \mathbb{R}^n \to \mathbb{R}$ be defined by $$f(x_1, \dots, x_n) := \prod_{j=1}^n x_j$$ Like any $C^2$ function, we can compute its Hessian $\mathcal{H}_f$, which will be a $\mathbb{R}...
user avatar
  • 1,316
2 votes
3 answers
59 views

Expansion of $\prod_{j=1}^{n} \left( \sum_{i = 1}^{m} x_{i, j} \right)$

I would like to know if there is a sum-of-products expansion for the following product-of-sums. A special case is given here for the difference of two entries. $$\prod_{j=1}^{n} \left( \sum_{i= 1}^{m} ...
user avatar
  • 1,316
0 votes
0 answers
8 views

Does a distribution over $\mathbb{R}^n$ s.t. $\prod_{j}^n X_j \in \mathbb{N}_0$ exist?

This question was motivated by the fact that $\mathbb{E}[U] = \sum_{k=0}^{\infty} Pr[U > k]$ if $U \in \mathbb{N}_0$. The easiest use of this fact is to simply use a single random variable whose ...
user avatar
  • 1,316
0 votes
0 answers
18 views

Term for the product of two axes?

I was searching for an unrelated factoid the other day and came across a term which was defined roughly to mean the product of quantities described by two axes. For example, in a distance vs time ...
user avatar
0 votes
1 answer
73 views

What is the use of product and co-product in category theory? What is special about categories closed under (finite) products and co-products?

What is the use of product and co-product in category theory? I am familiar with the categorial definition... A product of two objects $X_1$ and $X_2$ of a given category is a third object from which ...
user avatar
0 votes
0 answers
33 views

Canceling out two sums with inside product problem

I think that maybe I am missing that mathematically fundamental here which I cannot figure it out or the following is completely wrong. I saw somewhere this one : $$ g(x_{i})=\frac{\sum_{i=1}^{n}(x_{i,...
user avatar
0 votes
1 answer
24 views

Gelfand's corrolaries counterexample?

Gelfand's corrolaries (https://en.wikipedia.org/wiki/Spectral_radius#Gelfand_corollaries) state that, for any $2$ matrices $\mathbf{A}_1$, $\mathbf{A}_2$, the following relation is true: $ \rho(\...
user avatar
  • 5
0 votes
1 answer
65 views

Confusion on the definitions of product topology

All the books I have read so far defined the product topology $\tau$ on the finite Cartesian product of spaces $(X_i, \tau_i)_{i \le n}$ for some $n \in \mathbb{N}$ by the topology generated by the ...
user avatar
  • 2,421
1 vote
0 answers
52 views

Invariant of product [closed]

I have tried to work out this problem relating to invariants and products, but I have no idea how to continue. Can somebody write a solution to this problem? The numbers $1, 2, 2^2, \cdots , 2^{20}$ ...
user avatar
1 vote
1 answer
55 views

$X$ and $Y$ are iid. and $XY$ is Normal distributed. What distribution does $X$ and $Y$ have?

Is there a distribution, $D$, such that if $X,Y\sim D$ are iid., then the product $XY$ is Normal distributed? I natural idea would be to sample $X',Y'\sim N(0,1)$ and then take $X=\mathrm{sign}(X')\...
user avatar
  • 3,943
0 votes
0 answers
74 views

Product of Binomial coeficients

Suppose I have a set $\{n_1, n_2, \cdots, n_m\}$ such that $n=\sum\limits_{i=1}^m n_i$. Is there a way to simplify the expression $$ \prod\limits_{i=1}^m {n \choose n_i} $$ I feel like this might be ...
user avatar
  • 109
1 vote
0 answers
41 views

Proving sum/product identity without using the determinant

I want to show that \begin{align*} \sum_{\sigma \in S_{n+1}}^{} \text{sgn}\left(\sigma\right) \prod_{i = 1}^{n+1} a_{i, \sigma(i)} = \frac{1}{{a_{1, 1}^{n-1}}}\sum_{\sigma\in S_{n}}^{}{\text{...
user avatar
  • 192
1 vote
0 answers
40 views

Sum of Reciprocal of Repeated Products [closed]

For what real values of $c\in\mathbb{R}$ does the following series converge and diverge? $$ \sum_{k=1}^\infty\frac{1}{\prod\limits_{\substack{1\leq j\leq k \\ j\neq -c}}\left(1+\frac{c}{j}\right)} $$
user avatar
  • 1,155
4 votes
1 answer
66 views

Formulating an alternating sum of product combinations

Consider some list $A=(a_1,a_2,\cdots,a_n)$. I'd like to find a closed form for the following operation. $$f(A)=\sum_{k=1}^n(-1)^{k-1}s_k= s_1-s_2+\cdots(-1)^{n-1}s_n.$$ Where $s_k$ is the sum of all ...
user avatar
  • 3,726
0 votes
1 answer
47 views

convert product to summation

I would like to write the following function as a summation $F(x) = \prod_{n=1}^{N} (1-e^{-\frac{x}{\gamma_{n}}})$. I could not figure out how to expand it? Can someone please give me a hint? Thanks.
user avatar
0 votes
1 answer
31 views

Product term as a constant

I am wondering how to treat a constant in a product e.g.: $\prod_{i=1}^{3} 2$ My best guess would be: for a constant $ c \in \mathbb{R}$ $\prod_{i=l}^{u} c = c^{u-l+1}$
user avatar
1 vote
0 answers
39 views

How to estimate the value of this absolute sum?

(*) $\quad$ $\hspace{1mm} |\sum_{k=0}^{N}(\prod_{i=0}^{k-1}(\frac{n-i}{n}) - 1)\frac{z^{k}}{k!}| \hspace{1mm}$ Consider (*) for some bounded $N < n$. I want to confirm that (*) is bounded by some $\...
user avatar
3 votes
3 answers
144 views

Can we somehow compute $(z^2 - 1)(z^2 - 4)(z^2 - 9)(z^2 - 16) \cdots (z^2 - k^2)$ in half the number of operations?

Let $f(z, k) = (z^2 - 1)(z^2 - 2^2)(z^2 - 3^2)(z^2 - 4^2) \cdots (z^2 - k^2)$ in less than $1 + k + k + k + (k-1) = 4k$ arithmetic operations, specifically I need it to be roughly $2k$ operations or ...
user avatar
0 votes
2 answers
37 views

Conditional Probability of a Product of Probabilities

What is the conditional probability of a product of probabilities? More precisely, suppose that $P(C) = P(A)P(B)$, where $A$ and $B$ are any two events which are not necessarily mutually independent. ...
user avatar
  • 1
0 votes
1 answer
47 views

Suppose $A=\sum_{k=0}^{\infty}f(k)=B\prod_{n=0}^{\infty}g(n)$. If $A$ and $f(k)$ are known, how to find $B$ and $g(n)$?

My question like "some-to-product" or vice versa. See the following example (for reference, see here and here); $$\pi=\sum_{k=0}^{\infty}\frac{4(-1)^k}{2k+1}=2\prod_{n=0}^{\infty}\frac{4n^2+...
user avatar
2 votes
0 answers
46 views

Closed form for $\prod_{j=1}^n\prod_{k=1}^{m_{j}-1} (x_j-kN^{-1})$

Let $n,N\in\mathbb{N}$ and $m_1,x_1,\ldots,m_n,x_n\in\mathbb{N}$. I'm trying to rewrite following expression $$\prod_{j=1}^n\prod_{k=0}^{m_{j}-1} (x_j-kN^{-1})$$ I am only interested in the summands ...
user avatar
  • 1,732
0 votes
0 answers
39 views

Generalization of the Leibniz rule to negative power differential operator $\partial_x^{-s}\,,\, s\geq 0$

I'm trying to understand how the operator $\partial_x^{-s}\,,\, s\geq 0$ acts on a function, and on the product of two functions. In addition, do we have $$ \partial_x^{-1}(uv)=\partial_x^{-1}(u)v+u\...
user avatar
  • 121
1 vote
1 answer
87 views

Multiplication of 3D matrix by 2D matrix in Matlab without using loops

Want to multiply a $3D$ matrix $A$ of size $n$ x $m$ x $p$, ($n$: number of raws, $m$: number of columns and $p$: number of slices), by a 2D matrix $B$ of size $m$ x 1 x $p$. The results as shown in ...
user avatar
  • 177
0 votes
1 answer
27 views

Simplify product of two product operators

Given the relation $$ \prod_{j=2}^{N-1}\left( \prod_{k=1}^{j-1} \sin^2\theta_k\right), $$ I want to show that the power of $\sin^2\theta_m$ is $\sum_{j=m+1}^{N-1}1=N-m-1$, knowing that $\theta_m$ ...
user avatar
3 votes
3 answers
175 views

If $\alpha,\beta,\gamma$ are the roots of $x^3+x^2-x+1=0$, find the value of $\prod\left(\frac1{\alpha^3}+\frac1{\beta^3}-\frac1{\gamma^3}\right)$

If $\alpha,\beta,\gamma$ are the roots of the equation $x^3+x^2-x+1=0$, find the value of $\prod\left(\frac1{\alpha^3}+\frac1{\beta^3}-\frac1{\gamma^3}\right)$ My Attempt: $\alpha+\beta+\gamma=-1$ $\...
user avatar
  • 5,330
0 votes
0 answers
36 views

How to show, that a product (1-1/i^3) greater than 1/2? [duplicate]

$$ \prod_{i=2}^{2009} (1-1/i^3)\gt 1/2 $$ I was trying to use the formula for the cubes: $$ a^3-b^3=(a-b)*(a^2+a*b+b^2)$$ which gave me $$ (2-1)*(2^2+2+1)/2^3*(3-1)*(3^2+3+1)/3^3..$$ But I do not see, ...
user avatar
  • 667
0 votes
1 answer
42 views

Prove $\lim _{n \rightarrow \infty} \prod_{i=1}^{n+1} \cos \frac{\sqrt{2 i-1}}{n} a^{2}=\mathrm{e}^{-\frac{a^{4}}{2}}$ using the "fitting method"

The question is: $$\lim _{n \rightarrow \infty} \prod_{i=1}^{n+1} \cos \frac{\sqrt{2 i-1}}{n} a^{2}=\mathrm{e}^{-\frac{a^{4}}{2}}$$ A limit proof problem, from a book of mathematical analysis ...
user avatar
0 votes
0 answers
29 views

Summation over permutations

I've the following formula (taken from Hong Qian - Fractional Brownian Motion and Fractional Gaussian Noise): $$ \sum_P \prod_{j=1}^n E[X_j, X_{P_j}] $$ where the sum runs over all permutations P: j → ...
user avatar
0 votes
1 answer
74 views

Describe all path-connected $3$ fold covers of $X=S^1 \vee \mathbb{R}P^2$

Describe all path-connected $3$ fold covers of $X=S^1 \vee \Bbb{R}P^2$ (please justify why your list is exhaustive). Which are regular (i.e. normal) and why? I get that $X$ has a universal cover, ...
user avatar
0 votes
1 answer
67 views

How to simplify properly this product?

Let $\mu_n:=\{\xi \in \mathbb{C}^* \ / \ \xi^n=1\}$ and $\mu_n^*$ be the set of elements of order exactly $n$. We can also write : $\mu_n:=\{{\exp(\frac{2ik\pi}{n}) , \ k\in[\mid0,n-1 \mid]}\}$ and $\...
user avatar
  • 3,009
2 votes
2 answers
78 views

Is there a simple way to expand the following Product Sum?

$$ F(n) = \prod_{i = 1}^{n} \sum_{j = 1}^{i} v_j $$ $$ v_k \in \mathbb R $$ I would like to convert this from a Product-Sum to just a Sum of the following form: $$ F(n) = \sum_{i=1}^{C_n} g(i) $$ ...
user avatar
  • 2,204
1 vote
1 answer
175 views

I wish to solve exactly this formula involving sums and products

I was solving a physics exercise and I encountered this formula: $$\left< n_l \right>=\left[1+\sum_{k\neq l} \left(e^{bN(l-k)}\frac{\prod_{j\neq l} (1-e^{b(l-j)})}{\prod_{j\neq k} (1-e^{b(k-j)})}...
user avatar

1
2 3 4 5
35