0
votes
1answer
41 views

Help with the proof that the sum of all the roots of a complex number is zero

If a complex number $z \neq 0$ has n roots, then each root can be expressed as: $$z_j=(\sqrt[n]{r}) e^{ \left( {i \theta+2\pi j }/{n} \right) } $$ For $j=0,1,2,...,n-1$ Thus, the summation of all ...
-2
votes
0answers
20 views

Explanation of definition of normal subgroup

Recently I am studying group theory to know about Orbit-Stabilizer Theorem and I'm a novice learner.I got a definition of normal subgroup as mentioned in the link: ...
-3
votes
1answer
27 views

Time taken to give answer if probability is given.

This is a question that I am struggling with: Since the password is periodically changed, you would like to know the answer as soon as possible. So you decide to interrogate the minions in an order ...
-7
votes
0answers
33 views

How can I prove or disprove that every series converges to philosophy? [on hold]

Let $x_0$ be any wikipedia page. Let $x_n$ be the page that the first non-etymological link on $x_{n-1}$ leads to. How can I prove that $x_n$ eventually converges to philosophy? Technically it seems ...
-1
votes
2answers
19 views

Properties of bilinear forms

Let $V$ be a real vector space with norm $\|\cdot\|_V$ and $W$ a closed, linear subspace of $V$. A bilinear form $a\colon V\times V\rightarrow \mathbb{R}$ is called symmetric, if $a(u, v) = ...
3
votes
0answers
51 views

Partial sums of falling factorials

I want to know if there exists some way, approximate or exact, to do a partial sum of falling factorials of the kind: $$\sum_{k=i}^{n}(a+k)_{h}$$ where all are constants. And I'm interested too in ...
-3
votes
1answer
36 views

Verify these trig identities [on hold]

Can somebody verify these equations? $$\sin x\tan x+\cos x=\sec x$$ $$\tan x+\cot x=\sec x\csc x$$ $$\cos x\cot x+\sin x=\csc x$$
1
vote
0answers
14 views

inverse function theorem for analytic functions whose derivative might vanish

Suppose $x(t), y(t)$ are monotone increasing functions, and $f$ and $g$ are real-analytic functions that are not identically zero. If $f(x(t)) = g(y(t))$ for all $t$, does it follow that $x$ is an ...
0
votes
2answers
24 views

Financial Mathematics, Simple interest question. Help.

Laurie deposits $\$60,000$ in a bank at $5\%$ interest per annum. Andrew deposits $\$40,000$ in bank at $8\%$ per annum. How long wil it take, by simple interest, for Andrew to have more money ...
0
votes
2answers
30 views

Optimization problem: Calculus 1

A company manufactures and sells $x$ units of a product per week. The weekly average cost in dollars per unit is $C =\frac13 x^2 + 9x + 17 + \frac{1552}{x}$ and the selling price in dollars per unit ...
1
vote
0answers
13 views

linear operator vs. quasilinear operator

Let $\mathcal{T}$ be an operator defined on a linear space of complex-valued measurable functions on a measure space $(X,\mu)$ and taking values in the set of all complex-valued finite almost ...
2
votes
1answer
54 views

Why multiplying those 2 quaternions doesn't give the expected result?

Using a left-handed coordinate system, let Q = axisAngle({0,0,1}, 1/4*pi) * axisAngle({0,1,0}, 1/4*pi) be the quaternion representing the rotation "1/8 circle ...
0
votes
0answers
8 views

Best approach to determine the equivalence classes of a formal language

I created a minimum automaton for a formal language using the Myhill-Nerode theorem. The language for which I created the automaton is defined by $L=\{w \in \{a,b\}^*:w=av \text{ for a word } v ...
2
votes
1answer
21 views

Use of Laplace transform to solve initial value problem.

--Short Explanation: I have to say I am going crazy with this problem as it does not give me the same as the suggested solution in the book: Problem: $y''-7y'+10y=9\cos{t}+7\sin{t}$ $y(0)=5$, ...
1
vote
2answers
34 views

Why is the closedness of the set on which $f = g$ immediate, when proving the Identity Theorem?

It's a short proof given on Wikipedia, and I understand the argument for why the set on which $f = g$ must be open, but I'm not sure why closedness of the set is obvious. Apparently, this comes from ...
1
vote
0answers
13 views

Kolmogorov's superposition theorem for non-continuous functions

I'm trying to think about Kolmogorov's superposition theorem. This theorem states that, for each $n ≥ 2$ there exist continuous functions $ϕ_q : [0, 1] → R, q = 0, ..., 2n$ and constants $λ_p ∈ R, p = ...
1
vote
0answers
27 views

The ring of modular forms for $\Gamma_0(11)$

Let $\mathcal M(11) = \oplus \mathcal M_k(11)$ be a graded algebra of modular forms for congruence group $\Gamma_0(11)$. I want to find generators and relations between them. I proved that $\dim ...
4
votes
1answer
27 views

How do you convert different bases?

I know how to convert any number into base 10 by using the below method. Write (6712)base 8 in base 10. Ans: $6 \times 8^3 + 7 \times 8^2 + 1 \times 8^1 + 2 \times 8^0 = 3530_{10} $ However, I am ...
1
vote
1answer
19 views

Complex integrate and residues

Evaluate the integral of that $f(z)=\frac{z+1}{z^2-2z}$ around the circle $|z|=3$ oriented counterclockwise First I found that singularity points are $z=0,z=2$ ...
0
votes
1answer
17 views

Orthogonality on a circle

These theory questions always throw me for a loop. Ok, the question is: Suppose that AB is the diameter of the circle with center O and that C is a point on one of the two arcs joining A and B. Show ...
0
votes
1answer
31 views

if $\lim a_n = \infty$ and $\lim b_n = B$, then $\lim (a_n+b_n) = \infty$

I'm having trouble starting the proof not sure exactly how to go about it. So far I know that for a sequence to go to infinity it means that for all $n >0$ there exists $n_0$ for all $n$ greater ...
3
votes
1answer
44 views

A function with midpoint-linear derivative is a quadratic polynomial

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function such that $$f'\left(\frac{a+b}{2}\right) = \frac{f'(a)+f'(b)}2,\quad \forall a,b\in\mathbb{R}$$ Prove that $f$ is a polynomial of ...
1
vote
2answers
17 views

Probability of Independent Events individual vs in series

I understand that independent events (such as a fair coin flip) should not be viewed in succession. For example, if you flip heads 10 times in a row, the odds of flipping the next coin heads is still ...
4
votes
1answer
71 views

Next book in learning Differential Geometry

I have just finished the book "Manfredo P. do Carmo - Differential Geometry of Curves and Surfaces". My aim is to reach to graduate level to do research, but articles are not only too advanced to ...
0
votes
1answer
11 views

Understanding a chart about likelihood

I am learning stats from All of Statistics 1e. I understand that likelihood is the probability of a parameter, given some data. So if you flip an unknown coin 1000 times and get 500 heads a parameter ...
0
votes
1answer
32 views

Category theorem

I don't have a mathematician background (I am engineer) I understand some concepts but still very abstract for me and I have to show the following: 1.- Of what category is the set of all rational ...
3
votes
1answer
31 views

Prove that the congruence $x^2 \equiv a \mod m$ has a solution if and only if for each prime $p$ dividing $m,$ one of the following conditions holds

Let $m$ be odd and let $a \in \mathbb{Z}.$ The congruence $x^2 \equiv a \mod m$ has a solution if and only if for each prime $p$ dividing $m,$ one of the following conditions holds, where $p^{\alpha} ...
1
vote
1answer
24 views

Show that any nontrivial ideal that is also a subspace of $M_{n\times n}$ over a field $F$ is maximal.

I just need to show that if $A$ is a nonzero square matrix, then I can "build" the identity matrix $I$, Then the ideal must be maximal.
2
votes
1answer
48 views

What is a pullback in simple calculus context?

The definition of a pullback provided by my text is quite accessible Let $\phi : M \to N$, $f:N \to \mathbb{R}$, then $f\circ \phi: M \to \mathbb{R}$, where $\phi^*f = f\circ\phi$ and $\phi^*$ is ...
2
votes
0answers
36 views

Combinatorial express of n^3 [duplicate]

I know following expression $$n = {n\choose 1 } $$ $$n^2 = {n\choose 2 } + {n+1\choose 2 }$$ but how about $n^3 = $? Are there simple expression?
2
votes
2answers
39 views

Computing the $n^{\textrm{th}}$ permutation of bits.

I've seen this post about the $n^{\textrm{th}}$ permutation of a set but that is not what I need. If you have a bit string (ones and zeros only) there are algorithms to quickly permute the NEXT ...
0
votes
0answers
43 views

Polynomial: Number of solutions

Functions of polynomials often have more than one solution. For example, $x^2 = b$ with positive $b$ has two solutions for $x$. How does that work for higher polynomials? Say, I have for positive ...
0
votes
0answers
25 views

An interval covering problem [on hold]

Consider a set $A$ of intervals $[a_i,b_i)$, whose union is $[0,1)$. Prove there must exist a subset $B$ of $A$ such that the intervals in $B$ are pairwise non-overlapped and the sum of their lengths ...
0
votes
0answers
25 views

what are the principal applications of symmetric matrices in physics giving examples of it's applications? [on hold]

symmetric matrix is useful in many areas of sciences such as : physics . i'm very interested to know the suspect and some interesting applications of " Symmetric" matrix in physics or any branch ...
0
votes
0answers
14 views

minimising quadratic function subject to integer solutions

I would appreciate if one could help me to solve this problem. I have a bivariate quadratic function: $$ f(a_1,a_2)=(1-u_1^2)a_1^2 +(1-u_2^2)a_2^2 -2u_1u_2a_1a_2 $$ where $u_1^2+u_2^2=1$ and $a_1$ ...
1
vote
0answers
14 views

Branch cut jump of $\log z$

$$\int _ C \log z\,dz$$ where $C$ is a full circle in positive direction with radius $R$. I substitute $z=Re^{it}$, $dz=Rie^{it}dt$ $$\int _ 0 ^{2\pi} \log (Re^{it})Rie^{it}\,dt$$ $$\int _ 0 ^{2\pi} ...
0
votes
1answer
26 views

show that the sequence $b_n$ is monotone and find its limit [on hold]

let $b_1 >0$, $b_{n+1} = 3(1+b_n)/(3+b_n)$. show that $\{b_n\}$ is monotone, $0<b_n<3$, and deduce that $\lim b_n = \sqrt 3$. im having trouble showing the monotone part.
3
votes
3answers
28 views

Extend isomorphism of subgroups to homomorphism of groups

Given two finite groups $G_1$ ang $G_2$ with respective subgroups $H_1$ and $H_2$ satisfying $H_1\cong H_2$ via the isomorphism $\phi$, is it always possible to extend $\phi$ to an homomorphism ...
0
votes
1answer
38 views

Linear Transformations in Linear Algebra

We are given: Show how to evaluate a linear transformation for a specific vector $x$ , when the transformation is defined in the form $$T(x) = y$$ We know that a linear transformation is defined as ...
1
vote
1answer
24 views

counter example for “every ideal is contained in a maximal ideal” in non-unital case?

As known, the fact "every ideal in a unital commutative ring is contained in a maximal ideal" is proven using Zorn's lemma, but it really uses that the ring has the identity. (While using Zorn's ...
1
vote
1answer
17 views

Complex analysis, residues

Find the residue at $z=0$ of $f(z)=\dfrac{\sinh z}{z^4(1-z^2)}$. I did \begin{align} \frac{\sinh z}{z^4(1-z^2)} & =\frac{1}{z^4}\left[\left(\sum_{n=0}^\infty ...
5
votes
2answers
126 views

Prove or disprove $\int_{-\infty}^\infty \frac{dx}{\cos x+\cosh x}=\frac{1512835691 \pi}{1983703776}$

In this question, Evaluating the integral $\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x}$ , robjohn evaluates the integral to a nice summation with an approximate value. When plugged into W|A, it ...
0
votes
0answers
22 views

functions (on intervals) in vector spaces [on hold]

I'm studying mathematical methods for solving physics and engineering problems. I've looked in a few books and I'm curious about functions being manipulated like vectors. How could I find out more ...
0
votes
0answers
5 views

Coarse Graining: Correlation destroyed after coarse graining

X Y are two correlated random variable. How likely the correlation between them will be destroyed if the following coarse graining method is applied on them? We divide the range of X and that of Y ...
1
vote
0answers
6 views

Etingof problem 2.16.2: Irreps of Two-dimensional Lie algebra over a field of positive characteristic

This is problem 2.16.2 in Etingof's introduction to representation theory. Note that problem 2.16.1 is a proof of Lie's theorem. I'm having trouble with the second case, where the base field has ...
2
votes
1answer
95 views

Impossible Math Riddle [duplicate]

Mathematician A asks Mathematician B to guess the age of his three sons. Mathematician A starts off by giving Mathematician B two clues. The two clues are: The product of their ages is 72. When you ...
0
votes
1answer
27 views

Solving System of Equations modulo a prime

Consider the equation: $$ C \equiv HMH^{-1} \pmod{p}, $$ where $C,M, H$ are, say, $2\times 2$ matrices, and $p$ is an odd prime. The elements of the matrices $C, M$ are integers. The elements ...
3
votes
2answers
49 views

Existence of a non-abelian group of order $p^n$.

Question: Let $p$ be any prime and $n \geq 3$. Show that there exists a non-abelian group of order $p^n$. Attempt: Take $n = 3$. Writing $\mathbb Z_p \times \mathbb Z_p = \{e, \alpha_1, ...
0
votes
1answer
14 views

Left and Right Inverses with semigroups

Having the semigroup $(F,\circ)$ where $F=\{f: f: \mathbb{N}\to \mathbb{N}, \mathrm{Dom}(f) = \mathbb{N}\}$. The identity $e∈F$ is the function $e(n) = n$, define the function $g(n) = m$ if ...
1
vote
0answers
32 views

Limit comparison test how to choose $b_n$?

$$\sum_{n=1}^\infty \frac{2n-1}{4n^2+1}\tag{1}$$ i would like to find out if this series convergent or not so i use Limit comparison test and choose $a_n$ and $b_n=\frac{1}{n}$ why do i need to ...

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