0
votes
1answer
19 views

Rewriting a complex matrix as a real matrix and changing basis to $(z,\bar z)$

If we have a complex $n\times n$ matrix $A$ and we rewrite it as a real $2n\times 2n$ matrix $B$ by using the identity $z=x+iy$. I don't get how if we change basis from $(x,y)$ to $(z,\bar z)$ that is ...
0
votes
0answers
4 views

find a holomorphic function satisfying specific equality

Let $h$ be holomorphic function on a simply connected domain $\Omega$ with no zero in $\Omega$.Show in detail that there exists a holomorphic function $g$ on $\Omega$ where $h\left(z\right)=e^{g\left(...
0
votes
0answers
3 views

Discrete subgroups of $SU(m) \times SU(n) \times U(1)$

Is anything known about which permutation groups are subgroups of $SU(m) \times SU(n) \times U(1)$? Alternatively, how can one possibly go about finding them. I am trying to find discrete groups of ...
0
votes
0answers
5 views

Proofing that this relation is an order relation on $\Bbb N$?

$S := \{(x, y) \in \Bbb N / \{1\} \times \Bbb N\ / \{1\}$ $x$ and $y$ have the same number of prime factors and $|x - {100\over{3}}| \leq |y - {100\over{3}}|$$ \}$ Is $S$ an order relation on $\Bbb ...
0
votes
2answers
34 views

The expression $(x+a)(x+1991) +1$ can be factored as a product $(x+b)(x+c)$ where $b,c$ are integers

I need to solve following problem, but don't know to start with, please provide hints and solutions. Find all integer values of $a$ such that the quadratic expression $(x+a)(x+1991) +1$ can be ...
0
votes
0answers
1 view

What is the meaning of having a position vector $\vec r = z_1 \hat i +z_2 \hat j +z_3 \hat k \ where \ z_1,z_2 \ and \ z_3$ are complex numbers?

What is the geometrical interpretation of this? I came across this as a solution to a second order differential equation when I was solving a problem including $m\vec a = f(\vec r)$. Thank you for ...
2
votes
1answer
14 views

Proof of the convergence of $\int_0^\infty\sin{(x^4)} dx$ with Riemann-Lebesgue lemma

In this question, a comment from Lucian asserts that the convergence of the integral $$ I=\int_0^\infty\sin{(x^4)} dx $$ is due to the Riemann-Lebesgue lemma. However, I don't immediately see how to ...
0
votes
0answers
5 views

Showing $f(a) \in V$ but $f(x_n) \notin V$ for every $n$

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces and let $f: X \rightarrow Y$ be a function. Let $a \in X$ and suppose $f$ is not continuous in $a$. Prove that there exists an open subset $V$ in $Y$ ...
0
votes
1answer
17 views

Meaning of $Ax \leq b$

I continue to come across $Ax \leq b$ or $Ax= b$ in optimization problem, but I am having trouble interpreting the meaning of this. Does this have a similar meaning to the following (Cramer's Rule) ...
1
vote
1answer
31 views

Improper integral - check convergence

Does the following improper integral converge ? $$\int _0^{\frac{1}{2}}\:\cfrac{1}{\sin\left(x\right)\ln\left(x\right)}dx$$ I have tried to compare it to some known improper integrals but with no ...
0
votes
0answers
32 views

The Jeep Problem and Nash's Friends

The classical jeep problem is the following. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is ...
0
votes
1answer
12 views

possible cyclic group from fundamental theorem of finite abelian

Give a representative of each Isomorphism class of Abelian group of order 225. By the Fundamental theorem of finite abelian group: $\left | G \right |=225=3^{2}5^2$ Now, the possible isomorphism ...
0
votes
0answers
17 views

Surface area of a guitar

How would I go about finding the surface area of the guitar? Any hints would help. In particular, how could calculus be used to solve this problem?
0
votes
0answers
4 views

The Jeep Problem with Equally Spaced Stations

Consider the following problem. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is required to cross a ...
0
votes
1answer
17 views

solution of the ODE $u du =ydx+xdy$

In this case $u=u(x,y)$. When I saw this I just went on to taking improper integral both sides yielding $ u^2=4xy+K $. Yet, the book I am using now got $udu=d(xy)$, which yields $ u^2=2xy+K$. I'm I ...
1
vote
1answer
9 views

Confusion regarding uniform continuity

I was trying to check the validity of the following: If $f:\mathbb R\rightarrow\mathbb R$ and its derivative $f'$ are unbounded, then $f$ is not uniformly continuous on $\mathbb R$. To me,the ...
2
votes
2answers
599 views

What is the name of this pattern? Is there a name?

I have no background in math. I am a writer and have developed a storytelling technique I'm starting to find out may have strong roots in math. What is the pattern 1,2,3,4,5,6..2,1,4,3,6,5..3,4,5,6,1,...
0
votes
0answers
34 views

Solving $2^x+2^y+1=3^z$ in integers

I have reasons to believe that there should be an elementary, relatively simple way to find all solutions of the equation in the title in positive integers $x>y$ and $z$. Any ideas?
0
votes
0answers
9 views

A property related to free groups and topological groups

Let $G$ be a non abelian and non discrete topological group. Suppose it has a dense free subgroup countably infinite with rank at least $2$. Say $\{g_i\}$ with the $g_i$ distinct is its system of free ...
4
votes
6answers
265 views

How to Find a rational number between two irratonal number? [on hold]

Find the rational number between $\sqrt 2$ and $\sqrt3$. I try to solve by using some methods in my book but can not understand steeps.
0
votes
2answers
33 views

quadratic function positive

Put constraints on a quadratic function. I know that for $x > 0$ then $ax^2 + bx + c > 0$ I read around but I just found positive for all $x$. Thanks a lot
0
votes
4answers
42 views

How many possible “words” can be made from the first seven letters of the alphabet, allowing for repetition and enforcing alphabetical order?

Using letters from the alphabet $A = \{a, b, c, d, e, f, g\}$, how many words of length $5$ are possible when repetition is allowed but the letters must occur in alphabetical order? Not sure how to ...
0
votes
0answers
38 views

General strategies solving non linear congruence

I am trying to complete a nice summary for solving non-linear modular equations as i couldn't find a good one. I mean the specific (but still very wide) case of the form $f(x)\equiv0 \mod m$ where $f(...
0
votes
2answers
28 views

Minimal polynomial of projection on a plane?

If $g: \mathbb{R}^3 \to \mathbb{R}^3$ is the projection on a plane, what is the minimum polynomial of g? Related to this what is the minimum polynomial of a reflection with respect to a plane?
-1
votes
1answer
16 views

How to compute the inverse of a rank-$1$ matrix

I have a rank-1 matrix $R \in \mathcal{C}^{m \times m}$, how to compute another matrix X, such that $RX=I$, where $I$ is an identity matrix.
0
votes
1answer
14 views

Normalised Basis for vector space V.

Let $V = \{ (x_{1}, x_{2}, x_{3})' \in \mathbb{R}^{3} | \, 3 x_{1} + x_{2} = 0 \text{ and } 2 x_{1} - x_{3} = 0\}$ What is the normalised basis for V? I tried it two different ways: $x_{2} = -3 x_{...
1
vote
1answer
11 views

Calculating the convergence radius of a Taylor expansion

Find the radius of convergence for the Taylor series $$\left(\cot\dfrac{\pi}{100}z\right)=\sum^{\infty}_{n=0}a_{n}\left(z-20\pi\right)^{n}$$ The singularities of this function are the $100n$ where $n$...
0
votes
1answer
382 views

Find a transformation from tetrahedron to cube in $R^3$ to calculate a triple integral?

I would like to calculate the triple integral of a function $f(x,y,z)$ over a region given by a tetrahedron with vertices $(0,0,0)$, $(a,0,0)$, $(0,b,0)$ and $(0,0,c)$. I am trying to do this by ...
2
votes
2answers
32 views

Deduce the relation from the given trigonometric relation

If $$\frac{\tan3A}{\tan A}=k$$ Then prove that $$\frac{\sin3A}{\sin A} = \frac{2k}{k-1}$$ I tried this, $$ \tan3A = \frac{3\tan A-\tan^3 A}{1-3\tan^2 A}$$ then divided by tan A on both sides and ...
1
vote
0answers
35 views

When to use “:=”?

I know that "$:=$" is used in definitions. For example, one can write: $A\times B:=\{(a, b)\mid a\in A, b\in B\}$ But would you use "$:=$" in the following example: Let a function $f\colon A\...
-1
votes
0answers
24 views

Can we show $\int_0^tf(s){\rm d}B_s=-\int_0^tf'(s)B_s{\rm d}s$ for $f\in C^1(\mathbb R)$? [on hold]

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ and $f\in C^1(\mathbb R)$. Can we show that $$\int_0^tf(...
1
vote
3answers
23 views

Find the $\sum_{sym}ab$ maximum of the value

Let $a,b,c,d,e\in (0,1)$ and such $$a+b+c+d+e=1$$ find the maximun of the value $$S=ab+ac+ad+ae+bc+bd+be+cd+ce+de$$ I Conjecture the maximun is $\dfrac{2}{5}?$,such $a=b=c=d=e=\dfrac{1}{5}$,so $$S\...
-2
votes
1answer
10 views

Prove that $x \pmod n = 0$ if $x \pmod n = (k*x) \pmod n $ and $ k \ne n$

I need a quick formal proof for : Prove that $x \pmod n = 0$ if $x \pmod n = (k*x) \pmod n $ and $ k \ne n$ Thanks
4
votes
2answers
30 views

What algebra is generated by $\mathrm{O}(2)$?

The unit complex numbers can be identified with the $2 \times 2$ special orthogonal matrices $\mathrm{SO}(2)$. The problem with $\mathrm{SO}(2)$, however, is that its not closed under $\mathbb{R}$-...
0
votes
2answers
20 views

If a relation is built with $=$, is the relation always an equivalence relation?

$R \subset \Bbb R \times \Bbb R$ I have now encountered a couple of relations that have the following form: $$R=\{(a,b)\in \Bbb R\times \Bbb R\,:\,a^2 = b^2\}$$ They seem to be always equivalence ...
1
vote
0answers
5 views

Geometric equivalent of the degree zero divisor class group of an algebraic function field (in the singular case)

In an algebraic function field $F/k$ we have the degree zero divisor class group $\text{Cl}^0(F/k)$. Now since any such function field corresponds to the function field of a normal projective curve ...
0
votes
0answers
51 views

Any underlying reason why these equations look similar?

Questions Is there any way to go from either of these equations to the other? Or any more fundamental reason for their similarities? $$ \frac{1}{\zeta(s)} = \sum_{r=1}^\infty \frac{\mu(r)}{r^s} = \...
1
vote
2answers
79 views

Nonlinear 2nd order ODE

I have been looking at numerical solutions to the following nonlinear Bessel-type ODE: $$ xy'' + 2 y' = y^2 - k^2, $$ where k is a constant. In general, $y = \pm k$ is an asymptotic solution, and as $...
4
votes
0answers
15 views

The pigeonhole principle - how to solve questions like that?

We have two sequences , $(a_i)_{i=1}^{2n}$ and $(b_i)_{i=1}^{2n}$ such that $1\leq a_i, b_i\leq n$ for every $i$. Show that there are two sets of indexes $I, J \subseteq \left \{ 1,2, ... 2n \right \...
31
votes
3answers
675 views

Triple Euler sum result $\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$

In the following thread I arrived at the following result $$\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$$ Defining $$H_k^{(p)}=\sum_{n=1}^k \frac{1}{n^p},\,\,\, H_k^{(1)}\...
-2
votes
0answers
24 views

What is the partitive proportion of the given below?

Here's the given: $a$ is $42.08$%, $b$ is $26.25$% , $c$ is $15.42$% , $d$ is $11.33$% and $e$ is $3.12$%. If that's the case, what's the partitive proportion of $a:b:c:d:e$? I don't know how to get ...
0
votes
0answers
17 views

On a certain inequality for the power sum

Let $a_k \ge 0$ and $b_k \ge 0$ for $ k \in N$ such that the following two conditions (i) $0 \le \sum_{k=1}^{\infty}\frac{a_k-b_k}{k^{\alpha}}$ for $0<\alpha \le \alpha_0 <1$, (ii) $ \...
2
votes
5answers
85 views
+50

Why work with squares of error in regression analysis?

In regression analysis one finds a line that fits best by minimizing the sum of squared errors. But why squared errors? Why not use the absolute value of the error? It seems to me that with squared ...
-4
votes
0answers
8 views

so if I have a normal distribution with z=783 cm and sigma x = 150 cm and sigma y = 50 cm can I scale these sigmas for z=950? if so how?

so I have a problem that says if I have a plane at z=783 cm, measure the sigma (standard deviation) of the distribution in the x and y directions. from the graphs projection in the y-axis, projection ...
2
votes
1answer
28 views

What happens with negative plurigenus?

It is a well known result that for a smooth, projective k-variety, the dimension of the global section $H^0(X,K_X^j)$ of $j$-powers of the canonical bundle. Also called plurigenus, are birational ...
14
votes
3answers
604 views

Prove $\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$

$x,y,z >0$ and $x+y+z=3$, prove $$\tag{1}\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$$ My first attempt is to use Jensen's inequality. Hence I consider the function $...
0
votes
0answers
11 views

A question on extension of rings which related to their direct summands

I read "Foundations of Module and Ring Theory" of Robert Wisbauer and I got stuck in this problem: *Show for a ring $R$. The following assertions are equivalent: (a) $R$ has a unit. (b) If $R$ is ...
1
vote
2answers
21 views

Vector field and its components

If I have a cartesian vector field just denoted $\mathbf{E}=\mathbf{E}(x,y,z)$ (e.g. a electric field), does it mean: $$ \mathbf{E}(x,y,z)=x\mathbf{\hat{x}}+y\mathbf{\hat{y}}+z\mathbf{\hat{z}} \tag{1} ...
0
votes
0answers
13 views

Detailed explanation needed for basic query regarding expectation

I need to find the expectation of following random variable $$g=[\log_2(\frac{1+x}{1+y})]^+$$ where $[x]^+=max(x,0)$ and both $x,y$ variables depend on variable $z$. I know the conditional pdf's and ...
0
votes
0answers
9 views

Is it possible to use Lusin's Theorem to derive Frechet's Theorem?

Frechet's theorem states that every measurable function $f$ on $\mathbb{R}$ is the limit of a sequence of continuous functions converging almost everywhere. Frechet's theorem is then used to prove ...

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