0
votes
2answers
348 views

Reflecting a point by a line in $\mathbb R^3$

I would like to know if it's possible, given the vector equation of a line and the coordinates of a point, whether it's possible to reflect the point by the line.
1
vote
3answers
119 views

Symmetry in the line $x=y$

In the plane, why does the symmetry in the line $y=x$ sends the point of coordinates $(x,y)$ into the point of coordinates $(y,x)$ ? P.S. : does someone have a proof without linear algebra ?
7
votes
1answer
93 views

topological group operation vs homotopy group operation

Let $X$ be a topological group. Let $\tau_1$ and $\tau_2$ representing elements of $\pi_n(X)$. Is it true that $$ [\tau_1] [\tau_2] = [\tau_1 \tau_2] $$ in $\pi_n(X)$?, where of course "$[\tau_1] ...
2
votes
2answers
101 views

gambling probability problem

We are given a fair coin. We start out with 5 dollars. We keep tossing the coin. If the outcome is different than the previous one, we are awarded another 5 dollars. However, we do not get anything if ...
15
votes
3answers
698 views

About 0.999… = 1

I've just happened to read this question on MO (that of course has been closed) and some of the answers to a similar question on MSE. I know almost nothing of nonstandard analysis and was asking ...
1
vote
1answer
262 views

Maximum Likelihood fitting of truncated, mixed, two population systems (Gaussian Examples)

TLDR: I am trying to do maximum likelihood fitting of a dataset having two mixed populations, observed over a subset of their parameter space, within it to two pdfs. I include working code with ...
3
votes
2answers
268 views

Differential equation - quick question (first order differential )

I am new to the differential equation and I need some ideas how to solve this problem. $x^2y'=x^2y^2-2$ $y'=y^2-\dfrac{2}{x^2}$
1
vote
2answers
68 views

Derivative of $\log_{x}(x^2+3)$

How to compute $f'(x)$ where $f(x)=\log_{x}(x^2+3)$ ? When we deal with $x^{x^x}$ we use $e^{x^x\ln x}$. What do we "do" with logarithms?
2
votes
2answers
149 views

probability: finding Probability density function

Let $X\sim\exp(λ)$ and $Y$ equals its decimal part. How would you find the probability density function of $Y$? I started by looking for $F_Y(y)$ but got stuch in this level: $F_Y(y)=P(Y\le ...
0
votes
1answer
203 views

I don't understand this 'semi colon' notion in regards to PDE solutions

In solving first order PDE's with solution $u(x,y)$, when constructing a graph of $u$ as a union of initial curves $C_s$ emanating from the initial curve $\Gamma$. My lecture notes say, for each $s ...
3
votes
2answers
182 views

Is this a transcendental number?

A complex number that has transcendental real part is always transcendental? How about in the case of imaginary part?
1
vote
3answers
136 views

Finding the integer solutions

Find all integer solutions of $$(a + b^2)(a^2 + b) = (a − b)^3.$$ Obviously $b = 0$ is one. But how to get other solutions?
1
vote
1answer
104 views

Probability of an exact number events over several periods

The probability of an event occuring N times in a day is $ P[N=n] = \frac{1}{2^(n)} where 0\le n $. The number of times an event ocurrs in one day is independent of the number of times the event has ...
1
vote
1answer
45 views

anyone help me with this problem, does there exist any subest of $\mathbb R^2$ such that

let $f$ be any real valued continous function on $S^1$ ( unit circle in $\mathbb R^2$). does there exist uncountably many pairs of distinct elemens $x, y \in\mathbb R^2$ such that $f(x)=f(y)$?
0
votes
1answer
39 views

Calculate the thickness of material a ray passes through crossing a metal pipe

I have a metal pipe of internal radius R and wall thickness W. I fire a ray across the pipe's cross-section, perpendicular to its longitudinal axis, so that at its closest point the ray is distance d ...
2
votes
3answers
116 views

Computing derivative $x^{x^x}$

Could you show me how to compute $f'(x)$, where $f(x)=x^{x^x}$. I know that for $g(x)=x^x=e^{x\ln x} \ \ $ $g'(x)=e^{x\ln x}(\ln x+1)$ Now, my problem is this: is $f(x)=x^{x^x}= e^{x^x \ln x}$ or ...
2
votes
1answer
220 views

About stochastic differential equations

Consider, for all $x \in \mathbb R $, the process $\left( X_t^x\right)_{t\geq 0} $ unique solution of the following SDE: $$ X_t ^x =x + \int _0 ^t \sigma\left( X_s^x\right) ~dB_s + \int _0 ^t ...
2
votes
3answers
60 views

How do i prove that this given set is open?

Let $X$ be a topological space and $E$ be an open set. If $A$ is open in $\overline{E}$ and $A\subset E$, then how do i prove that $A$ is open in $X$? It seems trivial, but i'm stuck.. Thank you in ...
2
votes
2answers
71 views

Monic polynomial divided by $x-r$

I'm trying to prove the following, and I've made some progress on (i) and am having a bit of trouble with (ii) A polynomial $p(x)$ over a field $k$ is $monic$ if the highest power of $x$ has ...
3
votes
2answers
151 views

A vanishing theorem for differential forms.

I am trying to prove that for an algebraic surface $X$ (under some extra assumptions that are probably not important) there the space $H^0(X,\Omega_X^1)$ is trivial, i.e. that there exist no globally ...
3
votes
2answers
527 views

Use Riemann Sums to Find Area bounded by Curve

How can I find the area bound by $\;x=0,\, x=1,\;$ the $\;x$-axis ($y = 0$) and $\;y=x^2+2x\;$ using Riemann sums? I want to use the right-hand sum. Haven't really found any good resources online to ...
4
votes
1answer
134 views

How to do this integral

Prove that $$\int \frac{d^{n}q}{(2\pi)^{n} }\frac{q^{2a}}{(q^{2}+D)^{b}}=D^{-(b-a-n/2)}\frac{\Gamma (b-a-n/2)\Gamma (a+n/2)}{(4\pi )^{n/2}\Gamma (n)\Gamma (n/2)}$$ The angular part is easy to do as ...
3
votes
0answers
78 views

max-n is not inherited by normal subgroups

This is exercise 3.1.9 at page 70 of Robinson, A course in the theory of groups. Prove that the property max-n is not inherited by normal subgroups, proceeding thus: let A be the additive group of ...
-1
votes
1answer
89 views

Are Periodic Orbit all Iterations?

We have, If $x$ is a periodic point of a continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the period is $k$ i.e. $f^k(x)=x$ but $f^n(x)\not=x, \forall n: 0<n<k$. (The statement up ...
1
vote
2answers
100 views

Generating function of $(1 + x + x^{10})^{20}$

I try to find the generating function of $(1 + x + x^{10})^{20}$. My main problem is that all the forumalas I know, based on sequence (meaning, $1 + x + x^2 + x^3+\dots$ and so on), and I don't ...
2
votes
1answer
72 views

Equality of two definitions of conditional expectation

Let $X,Y$ be two random variables and let $Q(x,B)$ be a transition kernel from $X$ to $Y$: $$ \mathsf P_{X,Y}(A,B) = \int\limits_{A}Q(x,B) \, \mathsf P_{X}(dx) $$ Then we can define $\mathsf E(Y ...
1
vote
1answer
28 views

Calculating the normal to a surface $S \equiv \{(x,y,u(x,y)\}$

Either I'm missing something obvious or my lecture notes are making a large jump because I can't seem to see why this is the case: For an equation $$a(x,y) u_x + b(x,y)u_y = c(x,y)$$ consider the ...
1
vote
2answers
116 views

Average error of two normally distributed measurements

There are two methods of measuring on object of length $x$. The error of the first method is normally distributed with a mean of 0 and standard deviation of $0.0056x$. The error made by the second ...
0
votes
1answer
225 views

$\lim$ in $X$ and $\lim_{n\rightarrow \infty} f_n = f$ in $C(X)$, how to follow that $\lim_{n\rightarrow \infty} f_n (x_n) = f(x)$

$X$ is a compact metric space and $( C(X), ||.||_\infty )$ the space of the continous functions on X with the maximum norm. If it holds that every sequence converges in $X$ : $$\lim _{n\rightarrow ...
2
votes
1answer
108 views

Find a basis of $(L^2((-\pi,\pi), \mathbb{R}))^2$

I need to find a basis of $(L^2((-\pi,\pi), \mathbb{R}))^2$. I believe a basis of $L^2((-\pi,\pi), \mathbb{R})$ can be produced by the eigenfunctions of $\triangle$ (see L.C. Evans: Partial ...
0
votes
1answer
142 views

Cauchy-hadamard formula

Can we get the radius of convergence of $\sum\limits_{n=1}^{\infty}b_nx^n $ is not less than the radius of convergence of $\sum\limits_{n=1}^{\infty}a_nx^n $? where, ...
2
votes
2answers
213 views

Square roots of unity modulo (N/f)^2

My question relates to square roots of unity modulo N, ie $r^2 = 1 \mod N$. I have an efficient algorithm for obtaining these for arbitrary $N$. But for a given $N$ what I really want is to obtain ...
4
votes
1answer
149 views

Effective Well ordering of reals

Is there an effective (constructive) well order on reals ? I know several questions were already asked on this topic, and the answers were very good to this well known problem. My question is more ...
36
votes
8answers
1k views

Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$

Evaluate $$\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$$
0
votes
1answer
85 views

Differential Equation $u'(t) = \sqrt{|u(t)|}$

I got the equation (exercise of an old exam) $$ u'(t) = \sqrt{|u(t)|} \quad ; \qquad u(t_0) = u_0 $$ with $u(t) \in \mathbb R$. Then I have to say on which intervals $\mathcal I$ solutions exist and ...
0
votes
1answer
89 views

There are no $f,g:\mathbb{C}\to\mathbb{C}$, entire end $e^{f(z)}=e^{g(z)}+c$.

I am trying to prove Picard's little theorem. Here are my steps: If $a,b\notin Im(\phi(\mathbb{C}))$ then $\phi_1(z)=\phi(z)-a$ and $\phi_2(z)=\phi(z)-b$ don't vanishes any where. then there are ...
3
votes
1answer
422 views

Normalisation of an algebraic curve.

I need to compute explicitly the normalisation of a singular algebraic curve $C$ which is given by an explicit equation in $\mathbb{A}^2$. This task is mostly reduced to finding the integral closure ...
2
votes
1answer
103 views

Proof of a fact about symmetric pd matrices

Several times I bumped into the following argument in my studying If $A$ is a symmetric, positive definite $n$ by $n$ matrix then there exists a nonsingular $n$ by $n$ matrix $C$ such that $A=C'C$. ...
2
votes
1answer
393 views

Computing an integral involving standard normal pdf and cdf - with peculiar limits.

I have had a look at some of the other questions on this topic but cannot quite work out the solution to this integral (or prove that there isn't a solution). Is there a way to work out: ...
2
votes
1answer
165 views

Proving an implication from a matrix equation

I am supposed to prove that, if $\eta$ is lorentz metric, $M$ is a 4 by 4 matrix and $x^tM^t \eta Mx=x^t\eta x$ for any column vector $x$, then $M^t\eta M=\eta$. What I did seems awfully clumsy. I ...
6
votes
3answers
176 views

How does one show that the set of rationals is topologically disconnected?

Let $\mathbb{Q}$ be the set of rationals with its usual topology based on distance: $$d(x,y) = |x-y|$$ Suppose we can only use axioms about $\mathbb{Q}$ (and no axiom about $\mathbb{R}$, the set of ...
1
vote
3answers
166 views

Universal property of functor category

Between two categories $\mathcal{C}$ and $\mathcal{D}$ there is a functor category $Fun(\mathcal{C},\mathcal{D})$ with functors as objects and natural transformations as morphisms. Is there a ...
1
vote
1answer
51 views

Are there other advanced topics similar with markov process?

I want to make current day depends on a random days rather than yesterday or depends on specified past days for example two days, then change bayes equations but my talent is not enough to deal with ...
3
votes
2answers
570 views

Rule of inference for proof by contradiction.

In the book "Discrete Mathematical Structures" - Kolman, author has stated that proof by contradiction is based on the tautology ((p⇒q)∧(~q))⇒(~p).And that this argument form is often applied to the ...
2
votes
1answer
155 views

Find the maximum and minimum values of $A=\frac{y}{x}-\frac{x}{y}+\frac{z}{y}-\frac{y}{z}+\frac{x}{z}-\frac{z}{x}$

Find the maximum and minimum values of $$A=\frac{y}{x}-\frac{x}{y}+\frac{z}{y}-\frac{y}{z}+\frac{x}{z}-\frac{z}{x}$$ $x,y,z$ are possitive real numbers satisfying $M\le4m$ where $M=max{(x,y,z)}$, ...
1
vote
1answer
120 views

Picard criterion: Show $\mbox{range}(T)^{\bot}=\overline{\mbox{range}(T)}^{\bot}$

The so-called Picard criterion is: Let X,Y be Hilbertspaces and $T\colon X\to Y$ is a compact operator with singular value decomposition system $\left\{(\sigma_j,u_j,v_j)\right\}$. An element ...
5
votes
1answer
47 views

Is the relative rank function with respect to an ample line bundle non-decreasing

Let me make the question in the title more precise. Let $f:X\to $ Spec $k$ be a smooth projective connected variety over a field $k$ of characteristic zero. Let $\mathcal L$ be a line bundle on $X$. ...
1
vote
1answer
150 views

Spectral radius of an operator .

I would like to know the spectral radius of the operator $T_k$ from $C[0,1] \to C[0,1]$ : $$T_k x (t)= \int_0^1 k(t,s) x(s) ds$$ where $k(x,y)\colon [0,1]^2 \to \mathbb C$ is continuous. And ...
1
vote
1answer
42 views

A Quadratic Maximum?

What does the following mean? Context: Laplace integrals Consider the integral $$\int_a^b f(t) \exp(x\phi(t))\,\,\,dt$$ where $\phi(t) $ achieves its maximum at some $t=c\in (a,b)$. Assume that ...
1
vote
3answers
109 views

If the Quotient $M/M'$ of a finitely generated module $M$ is a free module then $M'$ is finitely generated.

I am new to modules and I want to show that "If the Quotient $M/M'$ of a finitely generated module $M$ is a free module then $M'$ is finitely generated". Please help me.

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