1
vote
1answer
22 views

How many n-tuple to genereate zero from some random several variable equation that use constant power and a variable as the base?

Define algebraic number tuple as the aphabetical order sequence of variables that use in the equation. How many algebraic number n-tuple (x,y,z,...) are able to genereate zero to input into a several ...
0
votes
1answer
25 views

how to solve the gaussian brackets?

could someone please explain how the following expression has been reshaped,particularly the gaussion brackets. $(-1)^n * n +(-1)^{n-1} * \lceil \frac{n-1}{2} \rceil$ has been reshaped to $(-1)^n ...
1
vote
0answers
78 views

Solving an 8th degree polynomial

I know that through the Abel Ruffini Theorem the general solution to a polynomial of degree five or more cannot be found explicitly. But are there are any other ways to find the roots of such a ...
0
votes
1answer
37 views

Algorithms for factoring multivariate polynomials

I am wondering if there are any algorithms to factor polynomials in multiple variables, when you know that the factors are other polynomials with rational or integer coefficients. I know you have the ...
1
vote
0answers
53 views

Problem in solving quartic equations

There is only one step in Descartes' method for solving the quartic that confuses me. Consider the depressed quartic $x^4+px^2+qx+r=0$, note I am assuming that the coefficients $p,q,r\in\mathbb{C}$. ...
0
votes
1answer
26 views

What will be the time complexity of insertion if a queue is implemented using two stacks?

A Queue could be implemented using two Stacks. So what will be the time complexity for insertion and deletion in this queue? Thanks in advance.
2
votes
2answers
111 views

Does Fermat's Little Theorem work on polynomials?

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$? I'm trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} ...
1
vote
2answers
29 views

Polynomials through successive differences

Let $h_0:\Bbb{N}\rightarrow\Bbb{N}$ be any function. Define recursively, for $m>0$, $$h_{r+1}(m)=h_r(m)-h_r(m-1).$$ Suppose that for some $k>0$ we have $h_k(m)\equiv d$ constant. Is this ...
1
vote
3answers
111 views

How can efficiently derive $x$ and $y$ from $z$ where $z=2^x+3^y$.

How can efficiently derive $x$ and $y$ from $z$ where $z=2^x+3^y$. Note. $x$,$y$ and $z$ are integer values and $z$ is $4096$ bits integer or even more. For all $z>1$. And if equation be ...
0
votes
1answer
40 views

Finding Percentage Contribution of a Variable in an Equation

I have an equation, for example: $$ y=a-b+c $$ I am actually confused how exactly to find the contribution of the variables individually to the entire equation. Due to the negative sign, following ...
2
votes
2answers
75 views

About linear transformations

Let $V$ be the real vector space of 2x2 matrices and $End (V)$ the space of all linear transformations of V in V. $$T: V \rightarrow End (V)$$ $$T(A)(B)=AB-BA$$ I have to prove that this is a linear ...
2
votes
4answers
110 views

Why do we have to do the same things to both sides of an equation?

Forgive me in advanced if this is a trivial question. This convention makes perfect sense to me intuitively, but is there any rigorous underpinning to it? I'm beginning to read through an abstract ...
0
votes
4answers
67 views

Proof: Two polynomials $P(x)$ and $Q(x)$ attain same value for every $x \in \mathbb R$ if and only if coefficients $p_i = q_i$ are equal for every $i$

Proof: Two polynomials $P(x)$ and $Q(x)$ attain same value for every $x \in \mathbb R$ if and only if coefficients $p_i = q_i$ are equal for every $i$ I've been thinking how to prove this. I know we ...
-1
votes
3answers
108 views

Group theoretic confusion on different group structures induced on the real line by ordinary multiplication and division

This is a followup to one of my responses to a question by my old teacher Ravi Kulkarni found at A question about groups: may I substitute a binary operation with a function?. I attempted to make a ...
2
votes
1answer
93 views

Associativity of a composition $x ∘ y = xy+\sqrt{(x^2-1)(y^2-1)}$

For many hours I had been stucked at this problem. For the following composition $x ∘ y = xy+\sqrt{(x^2-1)(y^2-1)}$ I have to demonstrate that this composition is associative($(x ∘ y)∘z=x∘(y∘z)$ and ...
12
votes
1answer
288 views

When $\sin x, \cos x$ are $\mathbb{Q}$-linear combinations of square roots

Suppose $x\in\Bbb R$ is such that $$\sin x=\sum_{i=1}^m x_i\sqrt{r_i},\quad \cos x=\sum_{j=1}^n y_j\sqrt{s_j}$$ for some $x_i, r_i, y_j, s_j \in\Bbb Q \ , \ |x_i|=|y_j|=1$. Show that ...
1
vote
1answer
89 views

How expand the domain of the $\log$ function from $\Bbb Z$ to $\Bbb Q$?

Let function $f(N)=\log N$ while $N \in \Bbb N$ One can expand the domain from $\Bbb N$ to $\Bbb Z$ by a clear mathematical approach $$f(N) = \left\{ \begin{array}{l l} \log N & \quad ...
7
votes
4answers
238 views

How to justify the existence of a function, in general?

Maybe I'm too naive in asking this question, but I think it's important and I'd like to know your answer. So, for example I always see that people just write something like "let $f:R\times ...
32
votes
4answers
2k views

Does there exist rational $a,b,c$, such that $\sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$

Let $w = \sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}$. How to prove that there are no triples $(a,b,c)$, such that $a,b,c \in \mathbb{Q}$; $a \leqslant b \leqslant c$; $(a,b,c)\ne (1,2,4)$; $w = ...
3
votes
0answers
167 views

Recognizing subadditivity

Let $f: (0,\infty) \to \mathbb{R}$ be some continuous function. We say that $f$ is subadditive if the bound \begin{align} f(x+y) \leq f(x) + f(y) \tag{1} \end{align} holds. I was attempting to ...
1
vote
1answer
27 views

Algebra / Equation with 2 or more vaiables

A train moves past a telegraph post and a bridge 264 m long in 8 seconds and 20 seconds respectively. What is the speed of the train? I'm really confused if there are 2 or more variables. Can ...
2
votes
2answers
68 views

Why does $\sum_{k=n_0}^n{P(k)}$ is a polynomial in $n$ of degree $d$?

Let $P(X)$ be a polynomial over $\mathbb{Z}$ of degree $d-1$ and $n_0$ be some constant positive integer. Then why does $\sum_{k=n_0}^n{P(k)}$ is a polynomial in $n$ of degree $d$?
1
vote
1answer
63 views

Localization of a ring which is not a domain

Let $A$ be a ring (commutative with $1$), let $S$ be a multiplicatively closed subset of $A$, i.e $S$ is contained in $A$ , $1\in S$ and $a,b\in S$ implies $ab\in S$, for every $a,b\in A$. Consider ...
4
votes
3answers
275 views

There isn't a product operation that is commmutative on $ \mathbb{R}^{n} $ that satisfies all the field axioms for $ n \geq 3 $.

This proof is broken down into simple easy algebra and vector questions. I would like to discuss different answers and approaches. Please see pg 162-163 on books.google.ca/books?isbn=0387290524 ...
0
votes
1answer
56 views

Simplifying equation from an aggregate one

I have got the 4th order coefficient of $\epsilon$ from the equation (9) from the paper. I got : $$\phi_4+ \ddot \phi_4+\omega_2 \ddot \phi_2 - \Delta \phi_2+g_2 \phi_2^2+2 \phi_1 \phi_3+ 3g_3 ...
0
votes
1answer
92 views

I want to find a closed form of the sum $\sum_{i=1}^n x^{i^2}$

Is there any closed form of the following sum? $\sum_{i=1}^n x^{i^2}$, where $x$ is a variable. Calculating the sum for a first few $n$ does not give any pattern.
3
votes
2answers
822 views

Division algorithm for multivariate polynomials?

We know that if $F$ is a field and $f(X)$ a non-zero polynomial in $F[X]$, then for every polynomial $g(X)$ we can find $q(X),r(X)$ such that $$g(X)=f(X)\cdot q(X)+r(X)$$ with $r(X)$ the zero ...
1
vote
1answer
51 views

Extending a Bijective Map

I have a problem I’m working on that I know the answer to, but I still need some help understanding some things. The problem is as follows: Show that $S = \{ x \in \mathbb{R} \space | \space 0 ...
0
votes
1answer
98 views

Weak direct product

I am just reading the book "Algebra" by Hungerford and on one page it says that if $G_i$ is a family of groups $\forall i\in I$ then $\prod_{i\in I}^{w}G_i\unlhd\prod_{i\in I}G_i$ where ...
3
votes
2answers
752 views

Direct sums and direct products

This question has been in my head for a while. And today it appears again when I am reading Arveson's book on $C^*$-algebras. He says Countable direct products of Polish spaces are Polish. ...
1
vote
1answer
86 views

Simplify the expression

The below expression has three summations (sigmas) and $L$ is a real-matrix and symmetric, $X$ is a real matrix with $n$ rows and $X_{p\mathbb{.}},X_{q\mathbb{.}}$ denote the $p$ and $q$ rows of ...
0
votes
0answers
35 views

Characterization of n-grade, m-variables polynomials $P$ over $\mathbb{R}$, with $P(x_1,\ldots,x_m)\in [0,1]$ if $\forall x_i \in [0,1]$ .

I'll write an introduction. This problem came to me while I was doing an experiment with the prisoners dilemma. I'm codifing the agents behavior in genes (I want to measure the correlation between the ...
0
votes
1answer
144 views

Ratio between two sets of numbers?

I have two sets of numbers. The first set is 1 to 4. The second set is 0 to 190. What is the calculation to get the proportionate number in the second set if a number in the first set is, for example, ...
3
votes
4answers
421 views

Showing that $a+b\sqrt{2}$ is associative over multiplication

I need to show that $a+b\sqrt{2}$ is associative over multiplication. This is what I have so far. I may be taking a wrong route so just please let me know. $$(a+b\sqrt{2})*((c+d \sqrt{2})*(e+f ...
2
votes
2answers
160 views

Integers on a blackboard [duplicate]

Possible Duplicate: Why everytime the final number comes the same? Suppose we write the integers 1 thru $n$, choose 2 random ones, erase them, and replace them with the single integer that ...
2
votes
0answers
122 views

symmetries of families of polynomial functions

The family of quadratic functions $F_2(a,b,c)$, consisting of all functions of the form $f(x)=ax^2+bx+c$, has the nice property (call it P) that given any $f,g\in F_2$, there is a sequence of function ...
4
votes
1answer
171 views

How to find out if two solutions are equivalent or different?

Given 5 different numbers ($\in \mathbb N$) in a specific brackets pattern like: $$\left(\left(\left(x_1 + x_2 \right) - x_3\right) \times x_4 \right) / x_5 = \text{result}$$ Only the brackets are ...
0
votes
1answer
89 views

Different solutions under distributive and commutative equivalence

Given 5 numbers: $x_1, x_2, x_3, x_4, x_5 \in \mathbb N$ all the 4 operations: $+ - \times /$ a specific brackets pattern: $\left(\left(\left(x_1 + x_2 \right) - x_3\right) \times x_4 \right) / ...
1
vote
1answer
669 views

What is the intuition behind the proof of Abel-Ruffini theorem in abstract algebra?

Is there a way to explain this proof in Wikipedia without knowing the abstract algebra too much or deep function experience? In addition, I don't how the abstract algebra work even after I look at an ...
7
votes
4answers
2k views

If $a + 1/b = b + 1/c = c + 1/a$, how to find the value of $abc$?

If $a, b, c$ be distinct reals such that $$a + \frac1b = b + \frac1c = c + \frac1a ,$$ how to find the value of $abc$? The answer says $1$ but I am not sure how to derive that much of simple and ...
5
votes
1answer
194 views

How many triplets of real numbers $(x, y, z)$ which satisfy these $3$ restriction:

How many triplets of real numbers $(x, y, z)$ which satisfy : $$(x + y)^3 = z$$ $$(y + z)^3 = x$$ $$(z + x)^3 = y$$ I need some approaches for solving this problem.
9
votes
2answers
510 views

Motivating algebra from quadratic equations

This question gave me pause for thought. We have a quadratic equation $ax^2+bx+c=0$. How much algebra can be motivated from the standard solution. Comments point out that the formula does not apply in ...
8
votes
2answers
297 views

Convergent rational series: which ones remain rational?

Due to closure under addition, it is obvious that a finite sum of rationals is rational. The infinite ones, however (assuming they don't diverge), may remain rational, such as $\sum_{n \in \mathbb{N}} ...
5
votes
3answers
3k views

How to find roots of $X^5 - 1$?

How to find roots of $X^5 - 1$? (Or any polynomial of that form where $X$ has an odd power.)
2
votes
1answer
479 views

When addition distributes over multiplication

Everyone is familiar with distributivity of multiplication over addition of real numbers. The distributivity of two binary operations sometimes goes both ways (e.g. max and min, or for lattices in ...
18
votes
5answers
719 views

How does one actually show from associativity that one can drop parentheses?

I've always heard this reasoning, and it makes obvious sense, but how do you actually show it for some arbitrary product? Would it be something like this? ...