For questions regarding identities in algebraic structures, including the construction, composition, and interpretation thereof.

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0
votes
1answer
19 views

Linear composition

can you help me with this quest? About composition $f$ and vector space $\mathbf{V}=\mathbb{Z^4_2}$ we know the following: $f \circ f = id_V$,$~~f $ $ \left(\begin{array}{ccc} 1\\ 0\\ 1\\ 0\\ ...
0
votes
0answers
23 views

Inner product identity

Let $u,v:\mathbb{R}^m\to\mathbb{R}^3$ be $C^1$ functions. I need to prove the following identity: $$ \langle \nabla u, \nabla v \rangle = \langle u,u \rangle \langle \nabla u, \nabla v \rangle - ...
1
vote
1answer
21 views

Finding the value of $y=b^2(3a^2+4ab+2b^2)$ if $a^2(2a^2+4ab+3b^2)=3$ and $a$ and $b$ are distinct zeros of $x^3-2x+c$

If $a$ and $b$ are distinct zeroes of the polynomial $x^3-2x+c$ and $$a^2(2a^2+4ab+3b^2)=3$$ $$b^2(3a^2+4ab+2b^2)=y$$ Evaluate $y$ I tried for many hours but couldn't solve this question. ...
11
votes
2answers
135 views

Solve $x^7-5x^4-x^3+4x+1=0$ for $x$

Solve for $x$ $$x^7-5x^4-x^3+4x+1=0$$ This equation has been bugging me since the past few days. I have found, using the Rational Root Theorem that $x=1$ is a root of this equation. However, ...
1
vote
0answers
19 views
+100

Commutator $[A_{p,q},A_{s,t}]$ in the pure braid group?

Let $B_n$ be the braid group; that is, a group generated by $\sigma_1,\cdots,\sigma_{n-1}$ with relations $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ for $i=1,\cdots,n-2$; ...
0
votes
1answer
14 views

Iterated sums identity

How to show that the following iterated sums are equal? $\sum_i \sum_j f(i)h(j)g(i,j) = \sum_j\bigg(\sum_i g(i,j)f(i)\bigg) h(j)$
2
votes
7answers
133 views

$f \circ g =\operatorname{ id}$ and $g \circ f \neq \operatorname{id}$?

Give two functions $f$ and $g$ s.t. $$f \circ g = \operatorname{id}$$ but $$g \circ f \neq \operatorname{id}$$ or a proof that this is impossible. This must be trivial, but I can't figure it out :) ...
17
votes
3answers
2k views

Monstrous Diophantine Equation

If $x,y\in\mathbb{Z}^+$, then find all the integral solutions to: $$x^6-y^6+3x^4y-3y^4x+y^3+3x^2+3x+1=0$$ I tried solving this question for an hour but still couldn't get it. I tried mod ...
0
votes
1answer
36 views

Proof for $∃xA⇔¬∀x¬A$

I want to prove, that $∃xA⇔¬∀x¬A$, using classic axioms. I think, I have to start with the following step: $∃xA⇔∃x¬¬A$ But I do not know, how to make this step, using axioms: $∃x¬¬A⇔¬∀x¬A$
-2
votes
1answer
44 views

Is this identity correct?

Is this identity true? Wolfram|Alpha thinks is not. $$x^{ln(x^3)} = e^{3\,[ln(x)]^2}$$ That's how I demonstrated it: $${\left(e^{ln(x)}\right)}^{3\,ln(x)} = e^{3\,[ln(x)]^2}$$ ...
0
votes
1answer
17 views

Does the following identity hold: $ [{A \times B^* + A^* \times B} ]$ = $2Re{[A \times B^*]}$

This seems to be true at first glance following that $a + a^*$ = $2Re(a)$ In any case, can someone help me verify whether this identity holds? Note: $\times$ is the cross product
0
votes
1answer
35 views

How to prove this identity (equality)?

I want to prove following equality: $$\begin{align*} &\left(\lambda_1 z_1^2+\lambda_2 z_2^2\right)\left(\frac{1}{\lambda_1} z_1^2 + \frac{1}{\lambda_2} z_2^2\right)=\\& =\frac{1}{4}\left ...
0
votes
1answer
40 views

Proof for $\forall x A \Leftrightarrow \neg \exists x \neg A$

I try to proof, that $\forall x A \Leftrightarrow \neg \exists x \neg A$ I know how to proof, that $\forall x A \Leftrightarrow \exists xA$, but I don't understand, how to get negation.
10
votes
1answer
80 views

Is there a proof of $\sum_{n=0}^x {{(-1)^n(x-n+1)^x}\over{n!(x-n)!}} = 1$ using induction?

Can someone prove (or disprove) this equality? $$\sum_{n=0}^x {{(-1)^n(x-n+1)^x}\over{n!(x-n)!}} = 1$$ where the value of $x$ can vary. This is a pattern found in derivatives and stuff but I'm not ...
1
vote
0answers
40 views

New identity for sums of Bessel functions?

I've come across the following proposed identity: $$ ...
2
votes
1answer
64 views

Are $\sin(\alpha\beta)$ and $\sin(\alpha^{\beta})$ expressible in terms of $\sin(\alpha)$ and $\sin(\beta)$?

There is a well known formula for expressing $\sin(\alpha+\beta)$ just using $\sin(\alpha)$ and $\sin(\beta)$. It is enough to replace $\cos$ in the formula ...
1
vote
1answer
29 views

Determine integral by using the following identity (which is imaginairy)?

I want to determine the following integral: $$\int_{-\infty}^\infty \frac1{x^6+1} dx$$ by using the following identity: $$\frac1{x^6+1} = \Im\left[\frac1{x^3-i}\right]$$ How in the world can I do ...
4
votes
1answer
32 views

Geometric interpretation of linear forms in the sum of four (or eight) squares identity

There is a well-known sum-of-squares identity $$(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2. \tag{1}$$ Moreover, letting $\vec{u}=[\begin{smallmatrix}a\\b\end{smallmatrix}]$, ...
1
vote
1answer
30 views

Which identity is being used to get $\sin(wa)\cos(wt)=\sin(w(a+t))+\sin(w(a-t))$?

Which identity is being used to get $\sin(wa)\cos(wt)=\frac{\sin(w(a+t))+\sin(w(a-t))}{2}$? Couldn't find it among the trigonometric identities.
0
votes
4answers
36 views

Help in proving an algebraic identity involving powers of binomials.

For some reason I found this equation: $(1 + x)^n - 1 = x \sum\limits_{k=0}^{n-1} (1+x)^k$ I think that this is an identity. If for instance one expands the powers and the sum for n = 4, the ...
0
votes
1answer
22 views

Prove this trig identity?

I'm having trouble proving the following identity: $$A\cos(\omega t) + B\sin(\omega t) = \sqrt{A^2 + B^2} \cos(\omega t - \arctan(\tfrac{B}{A}))$$ Does anyone know how this can be done? Thanks!
1
vote
2answers
40 views

Two partial fraction identities for $\frac{x^n}{x^m+k}$

Consider the following expression: $$\frac{x^n}{x^m+k},$$ for non-negative integers $n$ and $m$, $m>n$, and $k\in\mathbb{C}$. For $k=0$ the expression clearly simplifies to $x^{n-m}$. For ...
3
votes
0answers
37 views

Decide if radical expression equals a given rational number

Given a radical expression an a rational number. Is there an algorithm to decide if the expression equals the number? Example: $(\sqrt{\sqrt[5]{74} - \sqrt[14]{78}})^{356}+\sqrt[6]{63} \overset{?}= ...
0
votes
2answers
36 views

Which identity has been used here?

I have this written down in my notes, but I cannot remember how it came about: $$\sin(3t)\cos(10t) = 0.5(\sin (13t) + \sin (-7t))$$
0
votes
3answers
47 views

How is $A\sin\theta +B\cos\theta = C\sin(\theta + \phi)$ derived?

I have come across this trig identity and I want to understand how it was derived. I have never seen it before, nor have I seen it in any of the online resources including the many trig identity cheat ...
3
votes
2answers
80 views

Tough trigonometric identity

Prove that $$\cot 13^o\cot 23^o \tan 31^o\tan35^o\tan41^o = \tan 75^o$$ I managed to rearrange it to the form $$\tan 31^o\tan 35^o\cot 49^o = \cot 15^o\tan 23^o\cot 77^o$$ and in this form we have ...
0
votes
0answers
120 views

Why is Wolfram Alpha wrong?

I calculated $$\tan 75^o - [\cos 13^o\cdot \cot 23^o \cdot \tan 31^o \cdot \tan 35^o\cdot \tan41^o]$$ and I got a nonzero answer: ...
1
vote
1answer
33 views

Prove $ \nabla \cdot (f \nabla \psi ) = \nabla f \cdot \nabla \psi + f \nabla ^2 \psi $ in general curvilinear coordinates.

Prove $ \nabla \cdot (f \nabla \psi ) = \nabla f \cdot \nabla \psi + f \nabla ^2 \psi $ in general curvilinear coordinates. I have been attempting to do this using general curvilinear dot products ...
26
votes
6answers
3k views

What Gauss *could* have meant?

I was reading the Wikipedia entry on Euler's identity ($e^{i\pi}+1=0$) and I came across this statement: "The mathematician Carl Friedrich Gauss was reported to have commented that if this formula ...
3
votes
2answers
41 views

Algebraic identity I cannot solve

$$ \sum_{i<j}\sum_{j=1}^N(y_i-y_j)^2= N \sum_{j=1}^N(y_i-\bar{y})^2 $$ Where, $\bar{y}$ is the average. This is what I did: $$ ...
1
vote
1answer
33 views

Number Systems: Determining when they have closure, identities, inverses, and more.

I have the following $9$ number systems at hand and I am to determine which of them possess a particular property. I am having trouble understanding some of the subtleties between the questions and ...
5
votes
3answers
223 views

Using an identity to simplify the sum

So I ran into this problem today. It asks me to use an identity to simplify the sum. $$\sum_{j=7}^{27}\ln\left(\frac{j+1}{j}\right)$$ I have no idea where to start. I don't know any ...
2
votes
1answer
63 views

Can $2a^2+2a+2ab^2+b^2$ be written algebraically as the sum of three triangular numbers?

Let $T(n)=\tfrac{1}{2}n(n+1)$ denote the $n$th triangular number. I'm looking for an identity of the form $$ 2a^2+2a+2ab^2+b^2 = T(f(a,b)) + T(g(a,b)) + T(h(a,b))\tag{$\star$} $$ where $a,b$ are ...
0
votes
1answer
22 views

Vector calculus identities

Let $f$ be scalar potential for the vector field $\underline u $ (i.e $\underline u = -\underline \nabla f$). Prove that the vector field $$ \underline r \wedge \underline u $$ has magnetic ...
1
vote
2answers
37 views

Identity involving trigonometric sum

I have to prove that $$\overset{N}{\underset{n=-N}{\sum}} \left(N-\left|n\right|\right)e^{2\pi inx}=\left|\overset{N}{\underset{n=1}{\sum}} e^{2\pi inx}\right|^{2}=\left(\frac{\sin\left(N\pi ...
23
votes
2answers
338 views

Disproving an “almost true” trigonometric identity

The plausible looking "identity" $$\sin(\frac{\pi}{51})+\cos(\frac{\pi}{74})=\frac{3}{2\sqrt 2}$$ is not true, but it is close indeed: $$LHS=1.0606\color{blue}{598...}$$ ...
0
votes
1answer
77 views

What trig. identity would help solve $2 + \cos(2x) = 3\cos(x)$?

I need help with a homework question that has me puzzled. I need to solve the following equation: $$2 + \cos(2x) = 3\cos(x)$$ I don't see a good trig identity to apply. I tried $\cos(2x) = ...
0
votes
1answer
44 views

Prove that $\sum_{k=1}^{\frac{n-1}{2}}\cos\left(\frac{2\pi k}{n}\right)=-\frac{1}{2}$ if $n=1\mod 2$

I found out that this equality holds by accident,$$\sum_{k=1}^{\frac{n-1}{2}}\cos\left(\frac{2\pi k}{n}\right)=-\frac{1}{2}$$ if $n=1\mod 2$. However, I am not able to prove this directly with rules ...
2
votes
2answers
84 views

Product rule for simplex numbers

The $n$th triangular number is defined as $T_2(n) = n(n+1)/2$, and there is an interesting product rule for triangular numbers: $$T_2(mn) = T_2(m)\,T_2(n) + T_2(m-1)\,T_2(n-1).$$ The tetrahedral ...
0
votes
1answer
105 views

Polynomial identities

When I was about 17 our teacher showed us how polynomial identities had equal coefficients. I remember him showing that this was so by moving one polynomial "over to the other side" and showing that ...
1
vote
1answer
48 views

Proving $\sum_{k=0}^n\dfrac{x_k^{n+1}}{\prod_{j\neq k}(x_k-x_j)}=\sum_{k=0}^nx_k$

In Problems from the book by Andreescu, there's the following problem : Let $x_0,\ldots,x_n$ be distinct complex numbers. Prove $\displaystyle \sum_{k=0}^n\dfrac{x_k^{n+1}}{\prod_{j\neq ...
1
vote
0answers
57 views

Generalization of trace norm identity

Given a $2\times 2$ complex matrix $M$, the sum of its singular values (i.e. the trace norm) can be written as: $$\mathrm{Tr}\,|M|=\sqrt{\mathrm{Tr}(M^\dagger M)+2|\mathrm{Det}(M)|}$$ Is anyone aware ...
4
votes
1answer
64 views

Are there more examples of functional equations which are also valid for the identity map?

I find the co-incidence of the identity: $$\sin(A+B)\sin(A-B) = \sin^2 A - \sin^2 B$$ very pleasing. So, I was wondering if there are more of these type of identities. To make my question precise: ...
3
votes
2answers
62 views

Find the Value of Trigonometric Expression

Given $$\begin{align} \frac{\cos \alpha}{\cos \beta}+\frac{\sin \alpha}{\sin \beta}=-1 \end{align} \tag{1}$$ Find the value of $$\begin{align} \frac{\cos^3 \beta}{\cos \alpha}+\frac{\sin ...
2
votes
3answers
179 views

Deriving the sum-to-product identities

I've been asked by my textbook to derive the "sum-to-product" identities from the "product-to-sum" identities. I've attempted to to do this but i've met a dead end, and i'm quite confused. Using ...
0
votes
0answers
42 views

Proof of Lehmer's Identity

Is there any elementary proof for Lehmer's Identity ? let $n$ be a positive integer. Prove that $$\sum_{k=0}^{n}\binom{6n+5}{6k+2}B_{6k+2}=\frac{6n+5}{3}$$ where $B_n$ denotes the $n$th Bernoulli ...
2
votes
2answers
54 views

Find the value of $\frac{S_{5}S_{2}}{S_{7}}$

If $a$, $b$, $c$ $\in \mathbb R$, we define $S_{k}=\frac{a^k+b^k+c^k}{k}$ (where $k$ is a non-negative integer). Given that $S_{1}=0$, find the value of $$\frac{S_{5}S_{2}}{S_{7}}$$ I tried: ...
1
vote
0answers
49 views

Sum involving binomial coefficients

Exist a closed form for $$\left(-1\right)^{N}\underset{i=1}{\overset{N}{\sum}}\left(-1\right)^{i}\dbinom{N}{i}\dbinom{N+i}{i-1}\,\frac{1}{2i+1}?$$ I think I've to use in some way the formula of the ...
0
votes
2answers
54 views

Prove trigonometry identity for $\sin A+\cos A$

I’ve been struggling in proving this identity for hours (yes, shame on me), but I can’t see any light. $\frac { \cos(A) }{ 1-\tan(A) } +\frac { \sin(A) }{ 1-\cot(A) } =\sin(A)+\cos(A)$ I've been ...
2
votes
1answer
40 views

What is Vandermonde's formula with multisets?

I need Vandermonde's formula in multi-set form. I modified the original formula but I get a mess with too many letters everywhere, is there a nice representation? Here's the original: $$ ...