0
votes
0answers
3 views

Prove that if $E(X\log X)<\infty$ then $E(\sup_n |S_n|/n)<\infty$.

This is part 2 of a two part question. In the first part, we were asked to show that if you had a non-negative sub martingale $M_n$ then $$\sup_n E(\sup_{k\leq n} M_k)\leq \sup_n 2E(M_n \log M_n)+2$$ ...
0
votes
0answers
6 views

Ordered set and ordered field

In an ordered set if $x$ and $y$ are different elements, we can still set $x=y$, right? If yes, is it true that for an ordered field, under the extra conditions, viz., $$x+z<y+z \hspace{1cm} ...
0
votes
0answers
3 views

Can anyone solve a stochastic differential equation - related to neuroscience research?

I'm a neuroscience grad student, and I'm hoping one of ya'll could help me solve this problem regarding particle diffusion. It relates to my research on molecular-level neural plasticity, but I've ...
0
votes
0answers
4 views

Need help with a probability question from Ross

Let $X,Y$ be independent $Uniform(0,1)$ random variables. Find $E(|X-Y|^a)$ where $a>0$. My working: Define $W=1$ if $X>Y$ and $W=0$ if $X<Y$. We seek ...
0
votes
0answers
3 views

Cartesian equations for the line tangent to two surface.

I am asked to find a Cartesian equation for the line tangent to both the surfaces x^2+y^2+2z^2=4 and z=e^(x-y) at the point (1.1.1) I tried to find out normal vector to both surfaces and tangent ...
0
votes
0answers
9 views

Roots of of the derivative of a polynomial are in the convex hull of the polynomial

$(def)$: The $\mathbf{Convex}$ $\mathbf{Hull}$ of a set $\{ z_1,\ldots,z_k \} \subset \mathbb{C}$ is the set $$ CH[z_1,\ldots,z_k] = \left\{ z \in \mathbb{C} : z = \sum_{j=1}^k \lambda_j z_j, \; \; 0 ...
0
votes
0answers
9 views

Prove that $d(x,y)=\sum_{i=1}^\infty \frac{|x_i - y_i|}{2^i}$ converges

Question: Let S be the set of sequences of $0$s and $1$s. For $x = (x_1, x_2, x_3, ...)$ and $y = (y_1, y_2, y_3, ...)$. Define $d(x,y)=\sum_{i=1}^\infty \dfrac{|x_i - y_i|}{2^i}$ Proof the ...
0
votes
2answers
18 views

How can I show that $f$ must be zero?

Let $f(x)$ be continuous on $[a,b]$ and suppose $\int_a^b f(x)g(x)dx = 0$ for every continuous function $g$ on $[a,b]$. Prove that $f(x)=0$ on $[a,b]$. I understand that $f(x)$ must be zero ...
0
votes
0answers
7 views

Finding neccesary and sufficient condition for uniqness of neighbours on graph

Let $(G,E)=(V_1\cup V_2,,E)$ be a bipartite graph. Find necessary and sufficient condition s.t for every vertex in $V_1$ exist two unique neighbors from $V_2$. That seems obvious (necessary) that ...
0
votes
1answer
9 views

Lie algebra of the semiorthogonal group $O(p,q)$

How do I prove this: If $\mathcal{O}(p,q)$ is a Lie algebra of the semiorthogonal group $O(p,q)$ then $\mathcal O(p,q)$ consist of all matrices of the form: $$X= \left( \begin{matrix} a ...
0
votes
2answers
20 views

Explanation of the formula $df^{-1} = df\circ f^{-1}.$

Can someone explain the formula (for sufficiently nice $f$), $$df^{-1} = df\circ f^{-1}$$ So far, I have tried working with the relation $df^{-1} = (df)^{-1}$ and the chain rule but I am not able to ...
1
vote
0answers
11 views

Proving that $\int_{\mathbb{R}} f \ d\mu = \frac{1}{N}\sum_{i=1}^N f(\lambda_i)$

I want to know if my proof is correct and if there is some easier way to prove this. Consider some fixed real numbers $\lambda_1\leq\ldots\leq\lambda_N$ and the measurable space ...
0
votes
0answers
6 views

Isomorphisms with invariant linearly independent dense subset.

If $T$ is an isomorphism acting on a separable Banach space $X$, can we find a countable, dense, linearly independent set $D\subset X$ such that $T(D)=D$? If $X$ is finite dimensional, then the answer ...
0
votes
1answer
11 views

Hopf's Umlaufsatz and winding number

The Hopf Umlaufsatz theorem states that the angle of the tangent along a simple smooth closed curve turns by ±360 degree. Now, if I do know that the winding number of a simple smooth closed curve is ...
0
votes
0answers
10 views

Help converting currencies of 3 countries (linear algebra problem)

Three countries, A, B and C trade goods and services in a closed economy. The percentage of the total production of each country which is consumed by any given country is given in the corresponding ...
2
votes
0answers
8 views

Do I have the right idea for this isomorphism of Lie algebras of matrix groups?

I previously determined that the Lie algebra of $O(3,\mathbb C)$ is the set of skew symmetric matrices and that the Lie algebra of $SL_2(\mathbb C)$ is the set of traceless matrices. I am now trying ...
0
votes
0answers
18 views

The coefficient of $x^{2n}$ in the binonmial expansion of $(x+1)^{4n}$

I though the answer would be $\binom{4n}{2n}$ but apparently that's wrong?
0
votes
1answer
6 views

Show collinear with 3 position vectors

Show that point A, B, C are collinear A (2,-1,1) B (3,2,-1) C (6,11,-7) Thanks
0
votes
0answers
12 views

self-adjoint operator over a three dimensional vector space

How do I prove that a self-adjoint operator over a three dimensional vector space, is a matrix $$X= \left( \begin{matrix} a & x\\ x^t & B \\ \end{matrix} \right),$$ ...
0
votes
0answers
3 views

How to prove that $\Gamma\vdash\forall x\psi$ if $x$ occurs free in $\Gamma$, via generalization theorem?

In Enderton's logic [page $120$], he says: Assume we wish to prove $\Gamma\vdash\phi.$ where $\phi$ is $\forall x\psi$. If $x$ does not occur free in $\Gamma$, then it will suffice to show ...
0
votes
1answer
7 views

Problems with differentiation of vector functions

This is really a homework problem for which I'd like a fresh set of eyes to look the solution over. The math isn't working out right. This means I've done something incorrect. Without further ado. ...
0
votes
0answers
9 views

What are the irreducible components of $V(xy-z^3,xz-y^3,x-z^2+y)$?

I was reading the question here, and trying to fill out msteve's answer. It's not clear to me how to break up $V(xy-z^3,xz-y^3,x-z^2+y)$ into irreducible components, of which there should be two I ...
0
votes
1answer
6 views

Area of a square inscribed in a circle of radius r, if area of the square inscribed in the semicircle is given.

If a square is inscribed in a semicircle of radius r and the square has an area of 8 square units, find the area of a square inscribed in a circle of radius r. I started by assuming that the side of ...
0
votes
3answers
29 views

$a-c = \frac{b}{2},\ a-b = \frac{c}{6},\ b+c = 32$ find $a$ =?

I am getting frustrated as I am fighting with this. Please help. I know $$ 2a = b+2c \\ 6a = 6b+c $$ but after this i get confuse what to do next ?
1
vote
2answers
11 views

Question on decreasing sequence of sets

I can't imagine this is a new question, but I was unable to find what I was looking for. I have seen it stated that if $X$ is a topological space, and $(A_{k})_{k \in \mathbb{N}}$ is a non-increasing ...
1
vote
1answer
12 views

Function continuity outside a closed subset

Let $f:M \subset \mathbb{R}^p \to \mathbb{R}^q $,continuous at $a \in M $. Show that if $f(a) \notin \overline{B} (b,r) \subset \mathbb{R}^q $, then exists $ \delta > 0 $ such as $ f(x) \notin ...
0
votes
0answers
28 views

number of linear maps from $V\to V$

Question is to find total number of linear maps from vector space $\mathbb R^3(\mathbb R)$ to vector space $\mathbb R (\mathbb R)$ which are not Onto? I think trivial map, $T (x, y,z)=0$ could be one ...
0
votes
0answers
11 views

Series, and limits proof. Show $|n b - \Sigma _{k =1}^{n} b_k| \leq \Sigma _{k = 1}^{N} |b_k - b| + M(n - N)$

Can someone please verify the proofs? Anything would help. Let {$b_k$} be a real sequence and $b \in R$. a) Suppose that there are $M,N \in N$ such that $|b - b_k| \leq M$ for all $k \geq N$ . ...
0
votes
0answers
16 views

about Heine-Borel Theorem in a function space

In Pugh's real mathematical analysis. About the Heine-Borel Theorem in a function space, it states that a subset $\epsilon$ $\in C^0$ is compact if and only if it is closed, bounded, and ...
0
votes
2answers
15 views

Find Point on the line segment (7/8) of the way connecting points P and Q

with P = (4,3,-4) and Q = (5,-4,3). My thinking is take the distance between the two, which is (1,-7,7) and taking 7/8 of it which is (-7/8,-49/8,49/8). But I feel like that is wrong and I have to ...
0
votes
0answers
9 views

Object defined as a member of some category

Given a category $\mathcal{C}$ we have a class $\mathrm{Obj}(\mathcal{C})$ of objects of $\mathcal{C}$. Depending on the category these objects can be things like groups, vector spaces (over a given ...
1
vote
1answer
9 views

What is $f(y_1\mid\sup_i Y_i=z)$ for $Y_1,\ldots,Y_n$ i.i.d. uniform $[0,\theta]$?

Suppose that $Y_1,\ldots,Y_n$ are random variables independently and identically distributed as uniform on $[0,\theta]$ for some $\theta>0$. How do I find the conditional density of $Y_1$ given ...
2
votes
1answer
27 views

Proving that $\sin^7\theta + \cos^7\theta <1$ using basic trigonometry and identities [on hold]

How do I prove $\sin^7\theta + \cos^7\theta < 1$ for an angle between $(0,\pi/2)$?
0
votes
0answers
11 views

Definition of a simplicial complex

In the book "Algebraic Topology" by E Spanier, a simplicial complex is define a follows - D1 - A simplicial complex $K$ consists of a set $\{v\}$ of vertices and a set $\{s\}$ of non empty subsets ...
0
votes
0answers
10 views

Find the lengths of the given curves

So I have a problem where I need to find the length of a given curve using integration. I've probably put about 2 hours into this question but I'm completely stumped as to solving it. Here is the ...
0
votes
0answers
10 views

Exercise 2.17: Algebraic curves - William Fulton

Let $V=V(Y^2-X^2(X+1))\subset\mathbb{A}^2$, e $\overline{X}, \overline {Y}$ the residues of $X,Y$ in $A(V)$ its coordinate ring; let $z= \dfrac{\overline{Y}}{\overline{X}}\in K(V)$. Find the pole sets ...
1
vote
1answer
16 views

Every isomorphism on a separable Banach space has a completely invariant dense subset

If $T$ is an isomorphism acting on a separable Banach space, can we always find a countable dense subset $D$ of $X$ such that $T(D)=D? $
5
votes
2answers
41 views

Elementary number theory , when is $12n^2 + 1$ a square

Prove that if $$k = 2 + 2\sqrt{12n^2 + 1}$$ is an integer then it is a square. Can anyone help me with this? All I know is that k is an integer if and only if ${12n^2 + 1}$ is a square. What do I ...
2
votes
0answers
15 views

Is the Wikipedia article on the proof for Bertrand's Postulate correct?

I was checking out the Wikipedia article on the proof of Bertrand's Postulate. For Lemma 4, the argument is made for $x\ge 3, x\#<2^{2x-3}$ Here's the proof by induction: $n = 3$: $n\# = 6 ...
1
vote
0answers
9 views

Let $B = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2| \}$. Show that B is balanced, but that its interior is not.

I have the following definitions. The interior $E^o$ of $E$ is the union of all open sets that are subsets of $E$. A set $B \subset X$ is said to be balanced if $\alpha B \subset B$ for every ...
0
votes
1answer
9 views

Covert vector parametric equation to general form

Given the equation of a plane $x$ is $$x(s,t)=(0,1,1)+s(1,0,1)+t(2,1,-1)$$ How can I convert this equation into the general form $$A(x-x_0)+B(y-y_0)+C(z-z_0)=0$$ Thank you.
2
votes
1answer
28 views

$\sum_{n=0}^\infty r^n \sin(n\theta)$

Question is to find the value of $$\sum_{n=0}^\infty r^n \sin(n\theta)\text{ for }r=0.5\text{ and }\theta=\pi/3$$ I dont know any tools which can solve this question.
0
votes
0answers
12 views

Poisson's equation with Robin boundary conditions

Explain how to define $u \in H^1(U)$ to be a weak solution of Poisson's equation with Robin boundary conditions: \begin{align} \begin{cases} \, \, \, \, -\Delta u = f & \text{in }U \\ ...
4
votes
2answers
23 views

Getting equations from a network graph

I am learning about Network Analysis in Discrete Math and I need help figuring out how to get the equations from this graph: The arrows represent how many particles are going in a given direction. ...
0
votes
1answer
17 views

Show that $d_V(x,y)$ is metric

Question: On the set of integers $\mathbb{Z}$, show that the function $d$, defined as follows, is a metric: $$d_V(x,y) = \begin{cases} 0 & \text{if}\ x=y \\ \min{\{\dfrac{1}{n!}}\}\mid\ n!\ ...
-6
votes
0answers
15 views

Xavier spent less than of an hour walking home from school. Which fraction is less 5 2 than ? 5 5 F 7 3 G 4 5 H 10 2 J 9 [on hold]

math is a awesoma and very creative subject.Please help me with this particular question. Thank yo very much
0
votes
1answer
24 views

Integration Convergence/Divergence Questtion

$$ \int\limits_0^{\pi} \frac{ dt}{\sqrt{t} + \sin t }$$ How can one tell if this integral converges or diverges? Integral of $1/(\sqrt{t}+\sin(t))$ from $0$ to $\pi$. I can't even find the ...
2
votes
1answer
15 views

Finding conditional expectation from system of equations

I have three equations: $$X_m = \beta_0 + \beta_1 \cdot X_I + \varepsilon_{BL}$$ $$W_M = X_M + \varepsilon_{MBL}$$ $$W_I = \gamma_0 + \gamma_1 \cdot X_I + \varepsilon_{RDI}$$ The $\varepsilon$'s ...
1
vote
0answers
16 views

Determine a generating function and name the coefficient you would need to count the solutions to distribution question

The Full Question Find the generating function and name the coefficient which would give us the solution to this problem: count all integer solutions to $x_1 + x_2 + x_3+x_4+x_5 = 30$ where $x_i ...
0
votes
2answers
34 views

What is the logic behind the probability of getting 'four of a kind' in poker?

This hand ($5$ cards of $52$) has the pattern $AAAAB$ where $A$ and $B$ are from distinct kinds. The number of such hands is $\binom{13}{1} \binom{4}{4} \binom{12}{1} \binom{4}{1}$. The probability ...

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