0
votes
1answer
5 views

Proving that measure space is continuing from below

Let $(X, \mathcal{M}, \mu)$ be a measure space and $\{E_j\}_{j=1}^\infty \subset \mathcal{M}$ such that $E_1 \subset E_2 \dots $ I want to prove that $\mu(\cup _1^\infty E_j) = \lim_{j \to ...
0
votes
0answers
9 views

How are probabilities defined?

This stray thought has been bothering me for the past week. It seems that all probabilities and percentages are defined using the extremes 0% and 100%. Where: 0% is the probability that something ...
0
votes
0answers
7 views

Understanding of solution for a functional equation.

Problem For all $x,y \in \mathbb{R}$ which is $x^2 \not = y^2$, a function $f$ satisfies the following. $$(x-y)f(x+y) - (x+y)f(x-y) = 4xy(x^2 -y^2)$$ Find the function $f$. ...
0
votes
0answers
4 views

Is $F_n\to F_{\infty}$ equivalent to $\lim_{n\to\infty}\int\phi dF_n=\int\phi dF_{\infty}$ for every $\phi \in C(R)$?

If $F_1,F_2,...,F_{\infty}$ are distribution functions. Is $F_n\to F_{\infty}$ equivalent to $\lim_{n\to\infty}\int\phi dF_n=\int\phi dF_{\infty}$ for every $\phi \in C(R)$? I intuitively think this ...
0
votes
0answers
3 views

There exists a $M$ such that $\mid f^k (0)\mid \leq k^4 M^k$. Show that $f$ can be extended analytic on $\Bbb C$.

(a) Suppose that $f$ is analytic on the open unit disk $\{z: |z|<1 \}$ and there exists a $M$ such that $\mid f^k (0)\mid \leq k^4 M^k$ for all $k \geq 0$. Show that $f$ can be extended analytic on ...
0
votes
0answers
12 views

Simple Harmonic Motion under Periodic disturbing force

A particle of mass $m$ is executing a SHM in a straight line under an acceleration $n^2 \times (distance)$. If a periodic force $mk \cos{pt}$ be introduced and the time period of forced vibration ...
0
votes
0answers
6 views

Help provide some hint on a stirling approximation problem.

Use Stirling approximation to show $\mathop {\lim }\limits_{x \nearrow \infty } {e^{ - x}}\sum\limits_{x + a\sqrt x \le n \le x + b\sqrt x } {\frac{{{x^n}}}{{n!}}} = \int_a^b {\frac{{{e^{ - ...
0
votes
0answers
13 views

Irreducible polynomials in $\mathbb{Z}_3/\left\langle x^2+1\right\rangle$

We wish to find all the irreducible polynomials in $\mathbb{Z}_3/\left\langle x^2+1\right\rangle$. I came across this problem in my course on advanced algebra. I have little knowledge about the ...
0
votes
0answers
18 views

Mathematical induction.

Use mathematical induction (and proof by division into cases) to show that any postage of at least 12 cents can be obtained using 3 cent and 7 cent stamps. I thought this was the simple kind of ...
0
votes
0answers
10 views

Determining a succinct big $\Theta$ expression

Determine a succinct big-$\Theta$ expression for the growth of function $$ (\log^{50}n)n^2 + n^{2.1}(\log n^4) + 1000n^2 + 100000000n $$
0
votes
1answer
9 views

Paramaterization of paraboloid and plane.

Consider the paraboloid $z=x^2+y^2$. The plane $2x-4y+z-6=0$ cuts the paraboloid, its intersection being a curve. Find "the natural" parameterization of this curve. I have set each equation equal ...
0
votes
0answers
6 views

Evaluating and Simplifying a Double Integral

I have an integral as follows $f(t) = \int_r^\infty \frac{(sP)^{1-\rho}t^{-\alpha/2}}{1+(sP)^{1-\rho}t^{-\alpha/2}} \;dt$ I wish to get rid of the $s$ in $f(t)$ because this is an inner integral ...
0
votes
0answers
9 views

normal distribution under special condition

Given Gaussian random var U∼N(−1,1) and V∼N(1,1), are T=(U+V, U−2V) (T is a 2-element vector) and W={U with 50% chance, V with 50% chance) also Gaussian random var? If they are, what is the mean and ...
2
votes
1answer
15 views

Determining the value of A given $Z_4=Z_8\oplus Z_2/A$

Let Z define the integers and $Z_a$ define the integer group modulo a. I want to determine what A is. Given $Z_4=Z_8\oplus Z_2/A$, where $A\subset Z_8\oplus Z_2$, am I able to just say that given ...
0
votes
0answers
5 views

Determining a lower bound on the order of a group based on its presentation

I am reading Abstract Algebra book by Dummit and Foote (3-rd edition). On pages 26-27 they define a dihedral group: $D_{2n} = \langle r,s | r^n = s^2 = 1, rs = sr^{-1} \rangle$ The authors ...
0
votes
0answers
8 views

Just a little poker question i was wondering about?

There are six players remaining in an 18-player tournament. The top four players win prizemoney. All remaining players understand and utilize both equity and pot odds to make mathematically sound ...
0
votes
0answers
12 views

Combinatorial problem of choosing points inside an equilateral triangle without them being too close.

Determine the smallest integer $m_n$ which satisfies the following property: If $m_n$ points are chosen inside an equilateral triangle of sides 1, then at least two of them are at distance ...
-3
votes
0answers
13 views

if [a,b] is a subset of(c,d) what relationships exist between a,c and b,d?

I want to know what is the relationships I couldn't figure it out???? if $[a,b]$ is a subset of $(c,d)$ what relationships exist between $a$, $c$ and $b$, $d$?
0
votes
1answer
18 views

Limit $n \rightarrow \infty \frac{n}{e^x-1} \sin\frac{x}{n}$

I am just working through some practice questions and cannot seem to get this one. Plugging this into wolfram alpha I know the limit should be $\frac{x}{e^x-1}$, but I am having a bit of trouble ...
0
votes
1answer
14 views

Statistics (unsure how to do it)

A person's resting heart rate is the lowest number of heart beats per minute when fully relaxed and without distractions. Age, fitness, genetics, health status and gender affect the resting heart ...
0
votes
0answers
11 views

All Possible Pairs of 18

I will be having 18 Students in my class this year. I'd like to have them learn in pairs rotating every day with a different student in the class. What are all the possible pairings. for example on ...
1
vote
1answer
10 views

The convergence of a product of sequences converging in $L^2$.

Earlier today I found myself pondering the following question for which I do not have a reasonable answer. Suppose $f_m\to f$ and $g_m\to g$ in $L^2$. Moreover suppose that $f_m g_m\in L^2$ for ...
0
votes
0answers
6 views

Regularity of elliptic PDE in terms of weighted Sobolev space

There seems to be a whole industry of weighted ver. of the Poincaré's inequality (Weighted Poincare Inequality) I wonder if there are results like, weighted $L^2$ equivalent of (interior/boundary) ...
1
vote
1answer
7 views

Why $\Bbb{E}f$ is $\cal F_0$ measurable if $f$ is independent of $\cal F_0$?

In my professor's lecture note there is a remark saying that "$\Bbb{E}[f]$ is $\cal F_0$ measurable if $f$ is independent of $\cal F_0$". I think this should be easy, but I just don't see why. Can ...
2
votes
2answers
26 views

Show that f is surjective

So im having a little trouble proving this. Can anyone help me out? Let $A$, $B \subseteq E$. Moreover, let $$f: \mathscr{P}(E) \to \mathscr{P}(A) \times \mathscr{P}(B)$$ be defined by $$f: X ...
0
votes
1answer
12 views

Problem regarding inverstion of order of summation

Theorem 8.3 in Baby Rudin states the following: Given a double sequence $\{a_{ij}\}$, $i=1,2,3, ..., j=1,2,3, ...$, suppose that $$ \sum_{j=1}^{\infty} |a_{ij}| = b_i ~~~~~~~~~~ (i=1,2,3,...) $$ ...
1
vote
0answers
21 views

Prove that $g$ has a limit at $x_0$ if $f$ has a limit at $x_0$ and $\lim_{t\to x_0} f(t)=f(x_0)$

The problem: Suppose $f:[a,b]\to R$ and define $g: [a,b]\to R$ as follows: $g(x)=\sup \{f(t):a\le t\le x \}$ Prove that $g$ has a limit at $x_0$ if $f$ has a limit at $x_0$ and $\lim_{t\to ...
0
votes
2answers
13 views

Find the point at which the line intersects the plane. Is the intersection perpendicular?

Find the point at which the line $$x = 1 - t \\ y = 3 + t \\ z = 7 + 2t \\$$ intersects the plane $$x + 2y + z = 20$$ Is the intersection perpendicular? I have found the point of intersection to be ...
2
votes
2answers
20 views

Prove function space is linearly independent.

Let $V$ the space of all funcions $f:Ŗ\rightarrow R$. Prove that the ten functions defined by $x\rightarrow |x-1|$,$x\rightarrow |x-2|$,....,$x\rightarrow |x-10|$ are linearly independent. I need ...
4
votes
1answer
49 views

A possible dumb question about derivative

I was solving some differentiation problems when I found the function $$g(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}.$$ So I thought: If I define the function $f:\mathbb{R_{x>0}}\to \mathbb{R}$ as ...
0
votes
0answers
9 views

Normal convolved to Exp(polynomial)?

Is there an analytic solution for a Normal (normalized Gaussian) distribution of variance v convolved to e^y(x), where y(x) is an m-th order polynomial? Assume that m is even and the m-th coeff of y ...
0
votes
0answers
5 views

Determine the number of saddle points under specified conditions

Suppose a function with two variables $f(x, y)$ is smooth enough everywhere. If it has a local minimum and a local maximum, can we say that there are at least two saddle points as well? If so, how can ...
1
vote
2answers
23 views

Prove that from the equalities, $\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$ follows $x^yy^x=y^zz^y=z^xx^z$.

Problem : Prove that from the equalities, $$\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$$ follows $$x^yy^x=y^zz^y=z^xx^z$$. My approach : $$\frac{x(y+z-x)}{\log ...
1
vote
0answers
14 views

If $|H|=112$ then $A_7\cap H \lhd H$?

I posted this because Alex Clark asked in chat and I'm not sure how to proceed. Let $G$ be a group such that it has a fixed subgroup isomorphic to $A_7$, which we done simply by $A_7$. Let $H$ be a ...
1
vote
1answer
10 views

Given $k$ distinct linear operators, prove such an $\alpha$ exists

I have $k$ distinct linear operators $\{\phi_i\}$ which act on $V$, a vector space on some number field $K$ (in the sense that $\Bbb Q$ is the smallest possible one). Now I have to prove that there ...
1
vote
1answer
18 views

Random matrix theory

A random symmetric $2 \times 2$ matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{12} & a_{22}\end{pmatrix}$ is a member of the gaussian orthogonal ensemble (GOE), if it satisfies three ...
1
vote
0answers
36 views

Help in solving an integral.

I am trying to evaluate this integral, but could not find a solution. I tried it, assuming it to be product of two exponential and then tried integration by parts but it does not lead to anywhere. Can ...
1
vote
2answers
12 views

Partitioning a number as a sum of $k$ non-zero numbers, but order does not matter

I would like some confirmation regarding my logic here, which I feel is 'suspiciously straightforward'. Say I wish to express a number as the sum of $10$ non-zero numbers, where order does not ...
0
votes
0answers
7 views

What's the necessary condition for that any three vectors are parallel to the edges of a triangle in the plane?

What's the necessary condition for that any three vectors are parallel to the edges of a triangle in the plane? I answered the following: The necessary condition is that the vectors are ...
0
votes
2answers
23 views

On the Spivak's proof of the theorem 3-11 (calculus on manifolds)

In second paragraph of the case 1 within the proof: What is $U$ s.t $A\subset U$ and satisfies in the proof of the case 1 of theorem 3-11. $\psi_i$ is defined on $U_i$ and its support is not ...
0
votes
1answer
13 views

Normal Matrix with Real Eigenvalues is Hermitian

Let $A$ be a normal matrix. Then I want to show that, if $A$ has real eigenvalues, $A$ is Hermitian. (Notation: * denotes the complex conjugate, T denotes the transpose, and $\dagger$ denotes the ...
1
vote
2answers
32 views

Definition of $f \vee g$ and $f \wedge g$

In Olav Kallenberg's Foundations of Modern Probability he uses the notation $f \vee g$ and $f \wedge g$ where $f, g$ are two functions from a set $\Omega$ to $\mathbb{R}$. What does this notation ...
-1
votes
0answers
12 views

Not sure what formula to use? (what to solve for?)

The question states, "The weight of people in a certain pacific island is normally distributed with a mean of 175 lb. and a standard deviation of 33 lb. They want to design a one-person canoe that ...
0
votes
0answers
11 views

Orthogonal Position and Velocity Vectors

Is it true that if the position and velocity vectors of a moving particle are always perpendicular the path of the particle is on a sphere? If so how do I prove it? Geometrically I believe it makes ...
0
votes
0answers
9 views

Express the root of the equation by Lambert W function

Lambert $W$ function is know as the root to the transcendent equation $$ We^{W}=x $$ nevertheless, recent I found a equation of the type $$ \sqrt{F}e^{1/F}=1/x $$ where $F=F(x)$. I just wonder that, ...
0
votes
0answers
10 views

Fourier transform of $1 - \cos(xe^{-x^2})$

Is there a closed form expression or maybe an infinite series? If not is there a "good" approximation to it? Even a "good" approximation of the fourier transform close to zero frequency would do. Can ...
2
votes
3answers
74 views

Proving $\sin^2(x) + \cos^2(x) =1$ using calculus

Ok so the book in which I found this doesn't say mention the trigonometric functions by name but the question is: Let $s(x)$ and $c(x)$ be functions satisfying $s'(x)=c(x)$ and $c'(x)= -s(x)$ for ...
1
vote
0answers
17 views

Quiver algebra as a wreath product?

I'm having trouble understanding a definition of a quiver Hecke algebra. Suppose $k$ is a commutative ring, and $\Omega$ a finite set. We build a quiver $Q_{\Omega,n}$ with vertex set $\Omega^n$. ...
0
votes
1answer
12 views

Proving the asymptotic relationship between $(lg\cdot n)^{0.5}$ and $lg\cdot (n^{0.5})$?

Say $f(n) = (lg\cdot n)^{0.5}$ and $g(n) = lg\cdot (n^{0.5})$ It would appear that $f(n) = O(g(n))$ for $n \gt 55$ correct? How do I go about proving the the relationship for this?
0
votes
0answers
4 views

Using Jacobi eigenvalue decomposition for decomposition into non-eigenvalue matrix?

I am a student in computer vision struggling with the problem of camera calibration. I am having trouble decomposing a matrix, Q, according to the formula: ...

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