3
votes
2answers
270 views

Calulus of variations Euler Lagrange equation

How can I find an extremal for the following functional: $$J[y] = \int_0^1 e^y (y')^2 dx $$ with the boundary conditions $y(0)=1, y(1)=\log(4)$? I have tried computing the Euler-Lagrange equation but ...
0
votes
1answer
180 views

Definition of quotient space

Let $W \subset V$ be vector spaces. I don't understand the quotient space $V/W$. I read the Wikipedia and searched this site. I would have thought: say the vector space operation is $+$. let $Q = ...
3
votes
1answer
133 views

$X_n\overset{\mathcal{D}}{\rightarrow}X$, $Y_n\overset{\mathbb{P}}{\rightarrow}Y \implies X_n\cdot Y_n\overset{\mathcal{D}}{\rightarrow}X\cdot Y\ ?$

The title says it. I know that if limiting variable $Y$ is constant a.s. (so that $\mathbb{P}(Y=c)=1)$ then the convergence in probability is equivalent to the convergence in law, i.e. ...
1
vote
3answers
96 views

suppose$x\in V$ , $V$ is a vector space. if $3x=0$, then $x=0$

suppose$x\in V$ , $V$ is a vector space. if $3x=0$, then $x=0$ It seems very trivial for me but i am not so sure how it works in vector space
0
votes
1answer
128 views

Prove that if $f(n) \in \mathcal{O}(h(n))$ and $g(n) \in \mathcal{O}(h(n))$ then $f(n) + g(n) \in \mathcal{O}(h(n))$

Prove that if $f(n) \in \mathcal{O}(h(n))$ and $g(n) \in \mathcal{O}(h(n))$ then $f(n) + g(n) \in \mathcal{O}(h(n))$. I know that $\mathcal{O}(g(n))=\{f\space | \space\exists ...
3
votes
1answer
166 views

Skew-triangular (?) matrices and their properties

I'm asking this just out of curiosity because a brief googling failed to give me the answer. By skew-triangular matrices I mean matrices with this $$ \begin{bmatrix} \times & \times & \times ...
3
votes
2answers
669 views

Every Hilbert space has an orthonomal basis - using Zorn's Lemma

The problem is to prove that every Hilbert space has a orthonormal basis. We are given Zorn's Lemma, which is taken as an axiom of set theory: Lemma If X is a nonempty partially ordered set with the ...
2
votes
3answers
178 views

Expected Number of Successes in a Sample

$200$ calculators are ordered and of those $200$, $20$ are broken. $10$ calculators are selected at random. Calculate the expected value of broken calculators in the selection. Solution: Chance ...
0
votes
1answer
360 views

Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook ...
1
vote
2answers
101 views

How to reduce the limit one gets when deriving the derivative of the general exponential function?

When applying the definition of a derivative to $\frac{d}{dx}b^x$ and a little algebra one arrives to $$b^x\times\lim\limits_{h \to 0}\frac{b^h - 1}{h}$$ where of course that limit equals $\ln(b)$. ...
1
vote
3answers
81 views

topology question in Hartshorne's Algebraic Geometry

On page 15, in Lemma 3.1 it is claimed: "A subset Z of a topological space Y is closed if and only if Y can be covered by open subsets U such that Z $\cap$ U is closed in U for each U." How do I ...
3
votes
3answers
126 views

Boolean Algebra

There seems to be some discrepancy between my answer and the solution's. Can somebody please tell me what I have done wrong? Thanks! $$\begin{align} \left(A \lor B\right) \land \left(B \lor C\right) ...
0
votes
0answers
237 views

How can a large matrix be compressed?

I have a large matrix, around 10x10. Each individual element in the matrix itself is also a very large number, roughly of the order 10^30. I know that matrices can be used to solve linear equations. ...
2
votes
1answer
56 views

About the boundedness of the derivative of a function which is in special function space.

If $f \in C^1 ([0,T] , L^2) \cap C^0 ([0,T] , W^{1,2} )$, then how can I conclude that $$ \left \| \frac{\partial f}{\partial t} \right \|_{L^\infty([0,T] \times \Bbb R^n )} < \infty ?$$ Here $f$ ...
8
votes
1answer
954 views

Prove: If $A \subseteq C$ and $B \subseteq D$, then $A \cap B \subseteq C \cap D$

Is the form and correctness of my elementwise proof of this correct? I don't have any other way of getting feedback for my proofs and I want to improve. Proof. Suppose $A, B, C, D$ are sets such ...
1
vote
2answers
2k views

How to prove volume and surface area of sphere [duplicate]

Possible Duplicate: Why is the volume of a sphere $\frac{4}{3}\pi r^3$? We know that the surface area of a sphere is $4\pi r^2$ where $r$ is the radius of the sphere, and the volume is ...
0
votes
3answers
46 views

Example of no inclusion in set of closures

Let $(X,d)$ be a metric space and $A_j \subseteq X$ for $j = 1,2,...$ . Let $B = \bigcup_{j=1}^\infty A_j$. Find an example for which $\bigcup_{j=1}^\infty \bar{A_j} \neq \bar{B}$. Also, in general ...
-1
votes
4answers
142 views

Is $O(n^2) = O(n^3)$? Prove your answer.

I am not sure how to go about doing this, I know that: $$O(g(n))=\{f : \exists \ c \ \in \Bbb R_+, \ \exists \ n_0 \in \Bbb N, \ \forall \ n\geq n_0 :f(n) \le c·g(n)\},$$ but how do I go about using ...
1
vote
1answer
638 views

Stratified random sampling

Which of the following is NOT a characteristic of stratified random sampling? (A) Random sampling is part of the sampling procedure. (B) The population is divided into groups of units that are ...
1
vote
1answer
92 views

Lebesgue measure: show that $\mu^*(k\,A)=k \mu^*(A)$

For $k\gt 0$ and a subset $A$ of $\mathbb{R}$, let $k\,A=\{kx\mid x∈A\}$. Show that $$\mu^*(k\,A)=k \mu^*(A)$$ and that $A$ is measurable if and only if $k\,A$ is measurable.
0
votes
3answers
253 views

Use $ \epsilon - \delta $ definition to prove $\lim\limits_{(x,y)\rightarrow (0,0)} \frac{(xy)^4}{ (x^2 + y^4)^3}$ exists.

As the topic, Use $\epsilon - \delta$ definition to prove $\lim\limits_{(x,y)\rightarrow (0,0)} \frac{(xy)^4}{ (x^2 + y^4)^3}$ exists. I tried to use the inequalities $|x+y|>|xy|$ and ...
21
votes
3answers
502 views

Math contest: Find number of roots of $F(x)=\frac{n}{2}$ involving a strange integral.

Edit summary: A good answer appeared. CW full answer added, based on given answers. Removing my ugly-looking attempts, as they still remain in the rev. history. Here's a final-round calculus ...
5
votes
1answer
3k views

Relations between p norms

The p-norm is given by $||x||_{p} = (\sum_{n=1}^{\infty }|x_{n}|^{p})^{1/p}$. For $0 < p < q$, it can be shown that $||x||_{p} \geq ||x||_{q}$ (1, 2). It sppears that in $R^{n}$ a number of ...
0
votes
2answers
91 views

functional analysis-normed linear space

Can somebody please help me to find the answer for this problem... Let $V$ be a norm linear space and let $x\in V\setminus\{0\}$. Also let $W$ be a linear subspace of $V$. Show that if there is ...
1
vote
2answers
119 views

Coordinates of point inside of circle

I'm sorry if what I'm asking has already been answered or is really easy but I struggled a little and haven't been able to come up with an idea. TThe context is as follows: a robot is being placed ...
1
vote
3answers
277 views

How do I prove the field $F[x]/(p(x))$ is an extension of a field $F$

Let $F$ a field and $p(x)$ an irreducible polynomial in $F[x]$. I'm trying to prove that $F[x]/(p(x))$ is an extension of $F$. I know there are two approaches. Either we can prove that $F$ is a ...
2
votes
1answer
133 views

Number of powers of $2$ having leading digit $1$

How many of the numbers $2^m$ (where $0\le m\le M)$ have leading digit $1$? My trial - Since leading digit $=1$, whenever $2^m$ reaches or just crosses a $10^x$ and is less than $2 \cdot 10 ^ x$, ...
1
vote
3answers
318 views

Cauchy Riemann equations for real functions.

Let $f(x+iy) = x^2 + i.0$. Then $u(x,y) = x^2$ and $v(x,y) = 0.$ Hence $u_x = 2x$, $v_y = 0$, $u_y = 0$, and $v_x = 0.$ Clearly this doesn't satisfy Cauchy Riemann equations, and hence is not ...
7
votes
1answer
135 views

The number of curves of given genus over a field

Let $k$ be a field. Let $g\geq 0$ be an integer. I have an elementary question. Let $N$ be the "number" of $k$-isomorphism classes of smooth projective geometrically connected curves over $k$ of ...
7
votes
2answers
273 views

Find the possible values of $a$, $b$ and $c$?

Given $(a,\space b,\space c)\in \mathbb Z^3$ and that $$\sqrt[3]{\sqrt{a}+\sqrt{b}} + \sqrt[3]{\sqrt{a}-\sqrt{b}} = c$$ Find the possible values of $a $, $b$, and $c$.
0
votes
1answer
43 views

problem understanding $a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$

$a\times(b\times(a\times b))=a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$ can anyone expand on how the final answer is derived? I try to expand but ended up scratching my head. the best I can ...
2
votes
1answer
112 views

Tangent bundle $TS^2$

Let $$TS^2 = \{ (a,b) \in \mathbb{R}^3 \times \mathbb{R}^3: ||a||^2=1 , a\cdot b=0 \}$$ be the tangent bundle of $S^2$. How do I see that $TS^2$ with the subspace topology of $\mathbb{R}^3 \times ...
1
vote
2answers
246 views

Finite free rings over complete local rings are direct products of local rings

I came across the following statement: Let $R$ be a complete local Noetherian commutative ring. If $A$ is a commutative $R$-algebra that is finitely generated and free as a module over $R$, then $A$ ...
0
votes
1answer
84 views

Finding the vertices of an Ellipse

Trying to find the vertices of a ellipse. This is what I got And so used WolframAlpha just to test it out, this is my third time using it. This is the solution that I got So as you can see in ...
1
vote
2answers
264 views

Reverse Engineer from digits sum

This question is about sums of digits of a number. Given that sum of digits of $19 = 1+9 = 10$. However, is there a way that, given $10$, we can come out with $19$? In this case, we assume that ...
1
vote
4answers
75 views

Acummulation point

Which one of these points is accumulation point, which not and why? I read the definition x-times but I'm quite confused :-/ I also found this post which is relevant to my question but it seems to me ...
4
votes
4answers
307 views

Understanding matrices as linear transformations & Relationship with Gaussian Elimination and Bézout's Identity

I am currently taking a intro course to abstract algebra and am revisiting ideas from linear algebra so that I can better understand examples. When i was in undergraduate learning L.A., I thought of ...
1
vote
1answer
73 views

Finding the fixed point

Trying to solve this question, got this answer but have a gut feeling that this might not be the way to do it, by the way this topic is related to fixedpoints The solution that I came up with
4
votes
1answer
395 views

Generator for homology of surface of genus $g$ - Hatcher 2.2.29

Consider the surface $M_g$ of genus $g$, embedded in $\Bbb{R}^3$ in the standard way. It bounds some compact region $R$. Two copies of $R$ are glued together by the identity map between their boundary ...
3
votes
2answers
273 views

Some homework questions about a Lipschitz function (cauchy sequence)

Do you want to help me with my homework? The exercise is as follows: Consider a Lipschitz function, $h:\mathbb{R}\rightarrow\mathbb{R}$, satisfying for every $x, y$: $$\left| h(x)-h(y) \right| ...
4
votes
1answer
151 views

Constructing Riemann surfaces using the covering spaces

In the paper "On the dynamics of polynomial-like mappings" of Adrien Douady and John Hamal Hubbard, there is a way of constructing Riemann surfaces. I recite it as follow: A polynomail-like map ...
4
votes
2answers
171 views

What is smoothness needed for?

We can either define a knot to be (1) a smooth embedding $S^1 \hookrightarrow \mathbb R^3$ or (2) a piecewise linear, simple closed curve in $\mathbb R^3$ Then these two definitions are ...
1
vote
1answer
74 views

The Image of T = Column Space of A

$A$ is an m x n matrix: $T(\vec{x})=A\vec{x}$ $ im(T) = {A\vec{x} | \vec{x} \epsilon \mathbb{R}^n } $ = column space of $A$ Can someone illustrate this fact with an example? (The fact that the ...
1
vote
2answers
167 views

$N$ dice are rolled, find the probability of the following events

The probability that $n$ dice are rolled is $1/2^n$. Let $S_N$ denote the sum of the numbers shown on $N$ dice. Find the probability of the following events: a) $S_N = 4$, knowing that $N$ is even; ...
1
vote
3answers
139 views

What step is wrong? [duplicate]

Possible Duplicate: Finding the error in a proof $a=b$ $ab=a^2$ $ab-b^2=a^2-b^2$ $b(a-b)=(a+b)(a-b)$ $b= a+b$ Reminder the first step where $b = 2b$ So, $1=2$ In this case in my ...
1
vote
1answer
190 views

irreducible polynomial of degree 2 or 3 without roots in an integral domain.

It is well-known that a degree 2 or 3 polynomial over a field is reducible if and only if it has a root. But what about integral domains? Can we have a reducible polynomial over an integral domain ...
8
votes
1answer
461 views

What is reductive group intuitively?

I am studying Geometric invariant theory and wonder how I should understand linearly reductive algebraic group. We say that an affine algebraic group $G$ is linearly reductive if all finite ...
2
votes
2answers
74 views

Bounded function on $\mathbb R$

Is it true that if $0< f(x)$ is a continuously differentiable function on $\mathbb R$ with $\int_{-\infty}^{\infty}|f(x)|^{2}dx<\infty$ then $|f(x)|$ must be bounded above on $\mathbb R$?
5
votes
2answers
510 views

Question about Lee's Introduction to Topological Manifolds

From page 2 in Lee's Introduction to topological manifolds: Question 1: What does "describe parametrically" exactly mean? Is it a synonym for "global coordinate chart"? (that is, an atlas ...
2
votes
1answer
101 views

measurability of a function w.r.t. the $\sigma$-algebra generated by a stochastic process?

Here is the question: Let $(\Omega, \mathcal{F}, \mathrm{P})$ be a probability space. Let $T$ be an arbitrary (possibly uncountable) index set, and $\forall t\in T$, let ...

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