1
vote
2answers
48 views

Which set of points are defined by the relation $x/|x|=y/|y|$?

Which set of points are defined by the relation $x/|x|=y/|y|$? I think the answer is a straight line bisecting the first and third quadrants through the origin ( ie the line x=y). However wolfram ...
0
votes
1answer
29 views

regarding pseudo-prime numbers.

If $W$ is an odd composite number and $-1+2^{W-1}$ is divisible by $W$ yet not by $W^2$, then $W^2$ does not divide $-1+ 2^{W(W-1)}$. Is this true? (forgive my use of symbols,I have no good math ...
1
vote
1answer
15 views

Column space of stochastic matrix.

Consider an arbitrary matrix $M \in \mathbb{R}^{n \times m}$. Denote the column space of $M$ as $\mathcal{C}(M)$. Is it always possible to construct a right stochastic matrix $S$ such that ...
1
vote
1answer
40 views

Compute limit of a function

Compute: $$\lim_{x \rightarrow 0^+} \frac{\arctan(e^x+\arctan x)-\arctan(e^{\sin x}+\arctan(\sin x))}{x^3}$$ WolframAlpha tells me it's 1/6. Any nice idea how to rewrite that expression? Thanks!
9
votes
0answers
63 views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\left\{\frac{1}{x_{1}\cdots x_{n}}\right\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_{0}^{1} \! \cdots \! \int_{0}^{1} \left\{\frac{1}{x_{1}x_{2} \cdots ...
0
votes
0answers
4 views

Notation for concatenation of indexed vectors/matrices

Is there any standard notation for concatenation of matrices/vectors where their indices are taken from a set. I have matrices $A_{ij}$ where $(i,j)\in S$. I want to denote a matrix $A$ which is the ...
1
vote
1answer
25 views

How to show that $\Phi(1-x)^{-1} =O(\sqrt{\log{x^{-1}}})$

In the middle of some proof, I have faced an expression $\Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}})$, where $\Phi(\cdot)^{-1}$ is a quantile function of the standard normal distribution and $x \in (0,1)$. ...
0
votes
1answer
25 views

Functional equation with strange property about irrational numbers

Let $f:\mathbb R \to \mathbb R$ be a function such that for any irrational number r, and any real number x we have $f(x)=f(x+r)$. Show that f is a constant function. It's easy to see any constant ...
1
vote
1answer
59 views

A proof that if the product of spaces is Hausdorff, each of them is Hausdorff

Is my approach to this question right? Question: Prove that if $$\prod_{\alpha \in J} X_\alpha (\neq \emptyset) $$ is Hausdorff, each $X_\alpha$ is Hausdorff. Attempt to answer: It is enough to ...
2
votes
0answers
13 views

Derivative of terminal state w.r.t. the inital conditions.

Let $x\in R^n$ and consider the system $$ \dot{x}=f(t,x) \;\;\mbox{with}\;\; x(0)=x_0 $$ and suppose that we know it's exact or very accurate solution $x(t)$ for the time interval $[0,T]$. I'm ...
0
votes
4answers
62 views

Questions about solving inequality: $2 < \frac{3x+1}{2x+4}$

Solve the inequality: $2 < \frac{3x+1}{2x+4}$ Step 1: I simplified $\frac{3x+1}{2x+4}$ into: $3x+1-2x-4= x-3$. Step 2: $2>x-3$ Here I subtracted $2$ from both sides into: $x>-5$ or ...
0
votes
1answer
42 views

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$ I want to use dominated convergence theorem obviously. However, not sure how to dominate it. ...
4
votes
0answers
49 views

$\int_0^\infty f(x) \; dx < \infty$ implies $\lim_{x \rightarrow \infty} x f(x) = 0$. [duplicate]

Let $f$ be non-negative, monotone decreasing such that $$\int_0^\infty f(x) \; dx < \infty$$ Show that $$\lim_{x \rightarrow \infty} x f(x) = 0.$$ I have the following solution, but wonder if ...
3
votes
1answer
28 views

Basis change for gradient approximation

This is specifically a linear algebra question, but I kind of need to explain context. I suspect that this is simply a basis change problem, but I'm not entirely sure. Also, please feel free to ...
1
vote
1answer
48 views

Asterisk Notation

I am trying to read through the Cech Cohomology section in these notes: http://pub.math.leidenuniv.nl/~edixhovensj/teaching/2011-2012/AAG/lecture_14.pdf I would be really grateful if someone could ...
1
vote
1answer
33 views

Need help with application of Hardy-Littlewood inequality (Marcinkiewicz space and distribution functions)

I am going over this work here. I couldn't understand the equality where the Hardy-Littlewood inequality is used. I think $\delta$ here is a weight so we can take it to be $1$ for simplicity. Would ...
2
votes
1answer
17 views

Does “Expected Absolute Deviation” or “Expected Absolute Deviation Range” exist in stats and have another name?

So everyone is familiar with Variance and Standard Deviation from high school, but it seems no one has any familiarity with a philosophical justification for such weird, seemingly arbitrary measures. ...
2
votes
2answers
94 views

Is the Axiom of Choice necessary to prove $\mathbb R \approx \mathcal P(\omega)$?

I am self-studying Horst Herrlich, Axiom of Choice (Lecture Notes in Mathematics, Vol. 1876), and I'm getting quite confused about cardinal arithmetic without AC. Here (Which sets are well-orderable ...
2
votes
1answer
43 views

Functional equation with cyclic function.

Find all functions $f:\mathbb R \to \mathbb R$ that satisfy: $$ f(x) + 3 f\left( \frac {x-1}{x} \right) = 7x.$$ Some progress: I plugged-in $\dfrac{x-1}{x}$ and $\dfrac{1}{1-x}$, got a system of ...
2
votes
3answers
28 views

Integration Trig Substitution

After making the correct trig substitution what does the integral of $\dfrac{1}{\sqrt{9-x^2}} dx$ reduce to without solving the equation? I reduced it down to the integral of ...
2
votes
2answers
36 views

is there a generating function for $H_{2n}$?

I have been wondering if anyone knows if there is a generating function for harmonic series of the form $H_{2n}$?. That is, we are familiar with ...
3
votes
3answers
61 views

how to find matrix from its exponential form

I know about the relation $$\frac{d}{dt}e^{At}=Ae^{At}$$ Is the only way to use it is to find the inverse of $e^{At}$ and then post-multiply on both sides? Is there a better approach?
4
votes
0answers
23 views

Compute the induced map on $\mathbb{CP}^n$

Let $d>0$ and $f:\mathbb{C}^{n+1}\rightarrow \mathbb{C}^{n+1}$ be given by $f(z_0,...,z_n)=(z_0^d,...,z_n^d)$. Let $F:\mathbb{CP}^n \rightarrow \mathbb{CP}^n$ be the induced map by $f$. Compute ...
3
votes
2answers
24 views

Basic limit question to understand the methods

I have a very basic question about proving limits with the epsilon-delta method. So i want to prove $\lim _{x\to 0}\left(\frac{1}{1-2x}\right)\:=\:1$ . first, i write it like that: ...
1
vote
1answer
79 views

Evaluation of $ \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$

Evaluation of $\displaystyle \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$ $\bf{My\; Try::}$ Given $\displaystyle \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx = \int ...
0
votes
0answers
19 views

Signing the attendance

Imagine N students sitting in a straight row, students are numbered 1 to N. Attendance sheet is first given to student 1, who uses his own pen to sign the sheet. Then he passes the sheet to student 2, ...
6
votes
0answers
80 views

Will Mathematical discoveries slow down? [on hold]

With more and more Mathematics being discovered through time is it not the case that to make new discoveries, old theorems would have to be learned? Therefore would this not mean that there is a point ...
3
votes
3answers
41 views

Average distance from a point in a ball to a point on its boundary

What variety of methods are readily available to find the average distance from a point in $\{ (x,y,z) : x^2+y^2+z^2 \le r^2 \}$ to the point $(0,0,r)$? I just worked this out and got $6r/5$. Later ...
1
vote
2answers
22 views

Equation of a Tangent Plane

Find the equation of the tangent plane to the given surface at the given point. $x=u^2, y=v^2, z=uv$ at $u=1, v=1$ How would you find the tangent plane when the surface is in this format? Thanks.
1
vote
4answers
58 views

Classic Counting Problems

Does anyone have some good, classic, counting problems? I want things that are interesting, as well as instructive- more than just compute the number of way to get a flush, etc. (Not that those aren't ...
9
votes
0answers
111 views

How much math is there?

Among other things I teach high school-level math, and one question that often comes up is: "How long would I have to study math in order to know all of it?" I usually tell them that it's like ...
2
votes
2answers
19 views

Question on Green's Theorem

Consider the vector field $\textbf{f}(x,y)=(ye^{xy}+y^2\sqrt{x})\textbf{i}+(xe^{xy}+\frac{4}{3}yx^{\frac{3}{2}})\textbf{j}$. Use Green's Theorem to evaluate $\int_C\textbf{f} \dot d\textbf{r}$, where ...
0
votes
0answers
17 views

Can I evaluate polynomials with prime numbers to find possible irreductible factors?

Let $p(x,y)$, $c(x,y)$ and $d(x,y)$ be two variable polynomials with integer coefficients which satisfy $p(x,y)=c(x,y)\cdot d(x,y)$. Given $m, n$ positive prime numbers and given $e(x,y)$ another ...
2
votes
1answer
34 views

Differentiation under the integral if and only if we have an $L^1$ dominator

Let $f(x)\in L^2(\mathbb{R})$ and define $$g(t) = \int_\mathbb{R} f^2(x)\exp(-tx^2)dx$$ for $t\geq0$. We want to show that $g(t)$ is continuously differentiable if and only if $xf(x)\in ...
1
vote
2answers
109 views

roots of a polynomial inside a circle

I am asked to show that for $n$ larger or equal to $2,$ the roots of $1 + z + z^{n}$ lie inside the circle $\|z\| = 1 + \frac{1}{n-1}$ Attempt1: Induction for the case $n = 2,$ the roots of $1 + z + ...
0
votes
0answers
13 views

Proving or disproving continuity of a function

Consider a function $f:\mathbb{R}^{n}\times \mathbb{R}^{+} \rightarrow \mathbb{R}$, with the property that for a fixed vector $a:=(a_1,a_2,\cdots,a_n) \in \mathbb{R}^{n}$, there exist a finite ...
2
votes
4answers
167 views

Line Integral Around a Triangle

Let $R$ be the interior of the triangle with vertices $(0,0), (4,2),$ and $(0,2)$. Let $C$ be the boundary of $R$, oriented counterclockwise. Now evaluate the integral below. $$\int_C(y+e^\sqrt{x}) ...
0
votes
1answer
4 views

Find the acute angle made by vector $OC$ and the x-axis.

Given that vector $OA$ = $3i+5j$, $OB$ = $-2i+6j$ and that $OC$ = $OA + OB$, calculate i) |OC|, ii) the acute angle made by vector $OC$ and the x-axis. I found i) $\sqrt122$ Please help me in ...
0
votes
5answers
52 views

How to get the following limit into indeterminate form?

I am struggling to get the following limit into its indeterminate form so that i can apply the l'Hopitals rule: $$\lim_{x\to 0^+}(\sin x)^x$$ A solution would be greatly appreciated, been struggling ...
0
votes
1answer
23 views

Crossing edges at space

Let's say I have Graph $G(v, e)$ I want to draw the graph without crossing edges on space. By giving $(x, y, z)$ for any Vertex. How can I check if one edge crosses another?
1
vote
0answers
12 views

Bifurcation in PDE

How do we characterize bifurcation in nonlinear PDE instead of ODE i.e. ht=f(x,h,hx,hxx,hxxx,...)? For example, study the temporal evolution of a regular pattern into a chaotic one. Can someone please ...
0
votes
1answer
21 views

Characterization of Sobolev Space

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
1
vote
1answer
40 views

Irreducible action of a group on a set

Let $p$ be a prime. I am solving a problem and I am told that I should use that the action of $\text{SL}_2(p)$ on $\mathbb{F}_p^2$ is irreducible. But I don't know what this means?
1
vote
1answer
11 views

Find the probability $P[ x(t) \le 1]$ where $x(t)$ is a filtered Poisson process (rect pulses)

I can't understand the following question: "The random process x(t) is defined as $$x(t) = \sum_{n=- \infty}^{+\infty} rect(\frac{t-\tau_{n}}{T}) \quad ,\quad t \ \epsilon \ (R)$$ where {$\tau_{n}$} ...
1
vote
0answers
32 views

Proving a set is of measure zero.

Let $C\subset A\times B$ be a set of content zero. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B: (x,y)\in C\}$ is not of content zero. Show that $A'$ is a set of measure zero. ...
0
votes
0answers
6 views

Is $\text{Id} = \chi_{\{ |u| \leq k\}} + \chi_{\{|u| > k\}}$ well defined for $u \in L^p(0,T;L^q)$?

Is the decomposition $$\text{Id}(z) = \chi_{\{ |u| \leq k\}}(z) + \chi_{\{|u| > k\}}(z)\tag{1}$$ well defined for $u \in L^p(0,T;L^q(\Omega))$? I guess (1) holds a.e. So the problem is, is the set ...
1
vote
2answers
31 views

Parametric formula for figure 8 mobius strip

I'm making 3D prints with Mathematica, and am interested in a parametric formula for a mobius strip that is in the form of a figure 8, rather than simply a circle with a twist in it. Can someone help ...
0
votes
4answers
47 views

How to solve Absolute Value Inequality: |x-1| ≥ 3-x

I am learning the topic of solving absolute value inequality question. I had mostly understood the steps in order to solve for an inequality. However, I'm still clueless of a step to solve the ...
1
vote
0answers
18 views

Integral closure of 1-dimensional noetherian local domains

Let $(R,m)$ be a $1$-dimensional noetherian local domain and $S$ its integral closure. Clearly $S$ is $1$-dimensional noetherian semi-local domain. Is $mS=J(S)$, where $J(S)$ is the Jacobson radical ...
0
votes
1answer
33 views

Are distance-related paradoxes limited by the size of an atom?

See these 2 paradoxes: Coastline paradox The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. ...

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