0
votes
2answers
73 views

Solutions of a second order nonlinear differential equation

I am in trouble solving the following differential equation: $$\ddot{x}(t)=-\alpha\dot{x}(t)\dfrac{x(t)}{\left(\beta^2+x(t)^2\right)^{\frac{5}{2}}}$$ where $\alpha$ and $\beta$ are constant. How can I ...
1
vote
0answers
79 views

Constructing a “one-way function” of two variables (a.k.a “stop my friend from hacking my game”)

This might be more of a computer science question than a mathematics one; I thought I'd start here but perhaps people might want to point me to a better forum, if this isn't the right one. ...
0
votes
1answer
41 views

Inverse matrices properties.

I know about the properties of matrix multiplication for multiplication such as $A(BC)=(AB)C$. However I'm curious if $(AC)B$ would also have the same value. I'm asked to represent $A$ in terms of $B$ ...
2
votes
1answer
83 views

Tangent space to $\mathbb{R}P^{n}$

I could not find any other question here related to this. If I have missed out, then this could be voted as a duplicate(Sorry if it is!). I was just trying to figure out the tangent space to the ...
0
votes
1answer
171 views

Problem finding the number of r-element multi-subsets of the multi-set $M=\{ a_{1},a_{2},…,a_{n},m.b \} $

Let $m,n,r \in \mathbb{N}$. Find the number of $r$-element multi-subsets of the multi-set $$M= \{ a_{1},a_{2},...,a_{n},m.b \} $$ when $r \leq m,r\leq n$. Below is the given answer. ...
8
votes
7answers
200 views

How to integrate $\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}dx$

How can I approach this integral? ($0<a \in \mathbb{R}$ and $n \in \mathbb{N}$) $$\large\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}\, dx$$ Integration by parts doesn't seem to make ...
5
votes
4answers
2k views

Finding the distance between a point and a line, can there be a negative distance?

In finding the distance from a point $(x_1, y_1)$ to a line L: Ax + By + C = 0, can there be a negative distance? Is this the formula for finding the distance? $$ d= \frac{|Ax + By + C|}{\sqrt{ A^2 ...
3
votes
5answers
679 views

Coordinates of the point on the circle inscribed in a square

I try to find a way to calculate coordinates of a point nested on a circle inscribed in a square. The available variables, are: 1) side length of the square = 100; 2) circle radius = 50; 3) angle (a) ...
1
vote
1answer
63 views

Calculating the normal vector of a surface.

Let $\alpha: I\rightarrow \mathbb{R}^3$ be a parametrized curve with non-zero curvature every where and parametrized by arc length. Let $$x(s,v)=\alpha(s)+ r(n(s)\cos v + b(s)\sin v), r\not =0, ...
2
votes
2answers
67 views

Fourier Transform of mix partial derivative

I know FT{$\frac{\partial u}{\partial x}$} = (ik)FT{u}. Give a function $U(x,y)$. Is the following true? FT{ $\frac{\partial^2 U}{\partial y \partial x}$} = FT{$\frac{\partial U}{\partial y}$} ...
1
vote
2answers
116 views

Natural numbers object via initial morphism

I assume that a natural number object (or see nLab) can be defined as an initial morphisms. (edit: as in the title, I ment initial morphism, not objects) $\hspace{1cm}$ Thoughts: Probably $X:=1$, ...
0
votes
1answer
83 views

First order differential equation with time-varying parameter

If I have: $\dot{\sigma}(t) = -\gamma \sigma(t)$ where $\gamma$ is a constant, the solution is given by: $\sigma(t) = \sigma(0) e^{-\gamma t}$ Now what if I have the differential equation: ...
0
votes
1answer
55 views

Derivative of SquareRoot with h-formula

I know the general formula for getting a derivative, and the formula for the derivative of the square root function, but I'm interested in how to do prove it using the formula for the definition of ...
1
vote
0answers
30 views

Range of sum of Normal Distribution.

May be its silly question but I was just wondering is there any way to find out the absolute range of sum of values of Random normal distribution of N numbers with mu and sigma as mean and Std. Dev. ? ...
2
votes
1answer
177 views

Why are “not bounded” operators not everywhere defined?

Let $X, Y$ be Banach spaces, $\mathcal{D}(T)$ a subspace of $X$, and $T\colon X\to Y$ a linear map. Such a $T$ is commonly called an unbounded linear operator, where unbounded just means that the ...
0
votes
2answers
91 views

proof of convergence in probability [closed]

Let $(X_n)_{n\in\mathbb{N}}$ be i.i.d. random variables taking values in the set of natural number $\mathbb{N}$. Assume that $\mathbb{P}(X_1=i)=p_i>0$ for $i\in\mathbb{N}$. Let $D_n$ denote the ...
0
votes
2answers
47 views

How to integrate $ \int\limits_0^\infty{\frac{1}{{\sqrt {({y_1} - {y_2}){y_2}} }}d{y_2}} $?

I tried to do but it does not exist. $$ \int\limits_0^\infty{\frac{1}{{\sqrt {({y_1} - {y_2}){y_2}} }}d{y_2}} $$ Thank you
3
votes
2answers
98 views

Finding $f_Y$ such that $Z=Y\cos(X)\sim\mathcal{N}_{0,\sigma}$ for $X\sim\mathcal{U}[0,2\pi]$

I need to choose the probability distribution $f_Y(y)$ of a random variable $Y$ such that the variable $Z=Y\cos(X)$ is normally distributed with zero mean, i.e. ...
0
votes
2answers
53 views

If a generated subgroup is cyclic

I would like to make a similar question to question "Exercise on generated subgroup": Let $G$ be a finite group and $H\leq G$, $H$ cyclic with $|H|=exp(G)$. If $x\in C_{G}(H)\smallsetminus H$, then ...
5
votes
1answer
70 views

Non-differentiability of a function of two variables at a point

I have problems understanding this: Function $\;g(x,y)\;$ is given, for which a) $\;g_x(0,0)=7\;$ b) $\;g(t+2t^2,\sin3t+4t^2)=5e^t\;$ c) $\;\lim_{t\to 0}\frac{g(t,2t)-g(3t,4t)}t=10\;$ They ask ...
4
votes
1answer
181 views

A derivative which is not Lebesgue integrable on any interval?

If $f=x^2\sin(x^{-2})$, then $f'$ exists everywhere (including $x=0$) but $f'$ is not Lebesgue integrable on $[0,1]$ (precisely because of the singularity at $x=0).$ I'm trying to find a function $f$ ...
1
vote
1answer
55 views

Tate's thesis - continuous map from a local field to circle group

I am currently reading Decomposition of Unitary Representations defined by a discrete subgroups of nilpotent groups, by C.C. Moore. It is metioned that if $\mathbb{K}$ is a $p$-adic field in his ...
1
vote
1answer
41 views

Power series coefficients

I've been trying for days now to find a closed form for the coefficients of the power series about $x=0$ of the function $$ f(x)=\exp\left(r^2\frac{x(n-2)-x^2(n-1)+x^n}{(x-1)^2}\right), $$ but I ...
1
vote
3answers
108 views

Solving Coin Toss Problem

If a coin is tossed 3 times,there are possible 8 outcomes. HHH HHT HTH HTT THH THT TTH TTT In the above experiment we see 1 sequnce has 3 consecutive H, 3 sequence has 2 consecutive H and 7 sequence ...
1
vote
2answers
137 views

Proving the differentiablity of a function.

Consider the differentiablity of the following function: $$f(x)=x\left(x+3\right)e^{-\frac{x}{2}}$$ My text proves the differentiability by taking 'Left Hand Derivative' and 'Right Hand Derivative' ...
0
votes
2answers
26 views

Characteristic polynomial factor over the real numbers

Ve=the set of symmetric 2x2 matrices I'm trying to show that any element of Ve has a characteristic polynomial that factors over the real numbers and has two distinct eigenvalues unless the matrix ...
0
votes
2answers
70 views

Solving using separation of variable

First of all, I'm learning how to apply separation of variable method, and the first question I came across is this. Solve $y'= 2x + y$ using separation of variables with substitution $u = 2x + y$. ...
0
votes
1answer
25 views

seperation of variables difficult integral

$x'(t)=x(t)*(a-x(t)), x(0)=x_0$ I have to solve this ODE with seperation of variables. But I have problems. $\frac{\partial x}{\partial t}=x(t)*(a-x(t)) \iff ...
1
vote
1answer
37 views

Unique linear combination and basis

Let $S \in \mathbb{R}^3$ be the following set of vectors $$ v_1=\begin{pmatrix} 1\\ 0\\ 1 \end{pmatrix} , v_2 =\begin{pmatrix} 0\\ 1\\ 1 \end{pmatrix}, v_3 = \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix} ...
28
votes
6answers
3k views

What Gauss *could* have meant?

I was reading the Wikipedia entry on Euler's identity ($e^{i\pi}+1=0$) and I came across this statement: "The mathematician Carl Friedrich Gauss was reported to have commented that if this formula ...
0
votes
2answers
42 views

Convergence of Fourier Series in $L^1(\mathbb{T})$

Suppose $f \in L^1(\mathbb{T})$ and the sequence of partial sums of its Fourier series converges (in $L^1(\mathbb{T})$) to $g$. How can I prove $f=g$?
2
votes
3answers
270 views

How prove this $\int_{a}^{b}f(x)dx=\frac{1}{2}(b-a)[f(a)+f(b)]-\frac{1}{12}(b-a)^3f''(\xi)$

Let $f(x)$ be a twice-differentiable function on $(a,b)$,show that there exsit $\xi\in(a,b)$ ,such $$\int_{a}^{b}f(x)dx=\dfrac{1}{2}(b-a)[f(a)+f(b)]-\dfrac{1}{12}(b-a)^3f''(\xi)$$ if this problem ...
2
votes
1answer
66 views

Simple proof that this sequence converges [verification]

This is a relatively simple problem. I'm just making sure I have the right idea here. I'd like to prove that the sequence $\displaystyle a_n = 1 + \frac{1}{n^{1/3}}$ converges. My proof is: We ...
0
votes
3answers
64 views

How to find the order of a permutation?

Given that $x= (1 2 3)(4 5 6 7 8)$, what is the order of $x$? (i.e the smallest integer $k$ such that
0
votes
3answers
40 views

How to determine different absolute value equation cases?

This is a question from this post. From: $$ |3x|=\left\{ \begin{align} 3x & \text{ , if }x\geq 0 \\ -3x & \text{ , if }x <0 \end{align} \right\} $$ $$ |4x+1|=\left\{ \begin{align} ...
3
votes
2answers
144 views

Do Lipschitz/Hurwitz quaternions satisfy the Ore condition?

The Lipschitz quaternions $L$ are the quaternions with integer coefficients and the Hurwitz quaternions $H$ are the quaternions with coefficients from $\Bbb Z\cup(\Bbb Z+1/2).$ A ring satisfies the ...
1
vote
3answers
76 views

Ideal generated by a subset of ring.

The definition of Ideals generated by a subset : Let $S$ be any subset of ring $R$ then an ideal $I$ of $R$ is said to be generated by $S$ if : (1) $S \subseteq I$. (2) for any ideal $J$ of $R$ ...
2
votes
3answers
365 views

Evaluating the integral $ \int{\frac{x}{\sqrt{2x^2 + 3}}}dx $

I am trying to integrate the following: $$ \int{\frac{x}{\sqrt{2x^2 + 3}}}dx $$ It seems to me to be a trig substitution; however, I couldn't seem to get it into one of the three forms, i.e., ...
0
votes
2answers
50 views

What are the tangents and asymptotes to $(x-1)(x+1)(x-3)$?

What are the tangents and asymptotes to $(x-1)(x+1)(x-3)$? The equation $$\frac{dy}{dx}=0$$ is not solvable so there are no tangents parallel to x-axis. The function is increasing and it has no ...
4
votes
1answer
203 views

Definition of cocone in category theory

I've been reading some basic category theory, and am slightly confused about the definition of a cocone. I've been looking at the notes here - ...
1
vote
1answer
50 views

Show $\frac{n}{2} \log(n!) = \Omega (n^2 \log(n))$

I am trying to show that $\frac{n}{2} \log(n!) = \Omega (n^2 \log(n))$ but I seem to get a conflicting result. What i did is: $n!=1*2*3*...*n \leq n*n*n*...*n=n^n$, so $\frac{n}{2} \log(n!) \leq ...
1
vote
1answer
121 views

Bochner: Lebesgue Obsolete?

Bochner's notion of integral: $$F\text{ Bochner integrable}:\iff \exists S_n\in\mathcal{S}:\quad \int\|S_m-S_n\|\mathrm{d}\mu\to 0\quad(S_n\to F)$$ This version totally circumvents Lebesgue's notion ...
-1
votes
2answers
24 views

arranging the variables around when using inverse

If I want to show that $a$*$x$*$a^{-1}$ = $y$, is it acceptable to show that $x$*$a$*$a^{-1}$ = $y$ which then simplifies to $x$*$1$ so $x$=$y$? If not, how could I reorder them using what property?
1
vote
1answer
181 views

Krull dimension in finite ring extensions

Let $K$ be a field and $R=K[a_1, \dots, a_n]$ a finite ring extension. Suppose that the degree of transcendence of $R$ over $K$ is $r$. Then the Krull dimension of $R$ is at most $r$. I would like to ...
0
votes
3answers
107 views

Triple Integral in Spherical Coordinates.

$\newcommand{\de}{\operatorname{d}}$A little stuck on this one. $$\iiint_V ye^{-(x^2+y^2+z^2)^2}\,{\rm d} V$$ Use Spherical Coordinates to evaluate where V is the solid that lies between y=0 and the ...
1
vote
2answers
158 views

Elias Stein : Real Analysis

I cannot understand why this particular line in the text is true: " Moreover, there are $O(k^{d-1})$ cubes in $\cal{Q}\ '$ " For the text see ...
2
votes
2answers
52 views

divisibility on prime and expression

This site is amazing and got good answer. This is my last one. If $4|(p-3)$ for some prime $p$, then $p|(x^2-2x+4)$. can you justify my statement? High regards to one and all.
2
votes
2answers
54 views

Alternative hash table analysis

Let us say that we have to hash $n$ elements to $m$ hash slots. Now what could be the average length of a chain. We can assume that prob. that 2 elements will map to a particular location will be ...
0
votes
1answer
45 views

Mean value theorem of a function in [a,b]

Is Mean Value Theorem (Rolle's Theorem) applicable for the following function: $$\log \frac{x^2 + ab}{(a+b)x}$$ in the interval $(a,b)$ My text says that it's applicable. But isn't the function ...
1
vote
3answers
60 views

How do you generally prove that the sum of a right triangle's hypotenuse and it's base have a common factor with the perpendicular side?

I recently came across this one observation in my research on right triangles: In any Pythagorean triangle, the sum of the hypotenuse and the base is a multiple of, at least one factor of the base, ...

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