0
votes
2answers
21 views

Vector space judgment

Define $(a_1,a_2)+(b_1,b_2)=(a_1+b_1,0)$ and $c(a_1,a_2)=(ca_1,0)$ With these operations, the following conditions (1)There exists an element in $V$ denoted by $0$ such that $x+0=x$ for each $x$ in ...
2
votes
2answers
387 views

For homework questions what is the difference between being asked to verify something and being asked to prove something?

I've always been curious if there is a definite difference between the terms or if they just depend on the context of a problem.
2
votes
1answer
413 views

Solutions to $(z+1)^n = z^n$ using conformal maps.

I'm doing a homework problem where I have to find all roots of $(z+1)^7 - (z)^7 = 0$ using the roots of unity for $z^7$ I noticed that if $a$ is a root of unity for $z^7$, then $1/(a-1)$ maps the ...
0
votes
1answer
228 views

If $T\subset\mathbb{R}$ is bounded and $S \subset T$, then $\sup S \leq \sup T$ and $\inf T \leq\inf S$

Let $S$ and $T$ be nonempty sets of $\mathbb{R}$, with $T$ a bounded set and $S \subset T$. Prove that $\sup S \leq \sup T$ and $\inf T \leq\inf S$.
1
vote
0answers
30 views

Orthogonal intersection in a Riemannian manifold

Following is a question which I also asked in stats.stackexchange where I haven't got a response yet. Here is a link. Let $S$ be the set of all probability distributions on $\mathbb{R}$ and ...
11
votes
3answers
878 views

The equivalence between Cauchy integral and Riemann integral for bounded functions

Definitions Suppose $P\colon a=x_0<x_1<\dotsb<x_n=b$ is a partition of $[a,b]$. Let $\Delta x_k=x_k-x_{k-1}$ and $\lVert P\rVert$ denotes $\max_{0<k\le n}\Delta x_k$. The Cauchy integral ...
3
votes
1answer
256 views

Chi-Square goodness-of-fit test on sample space or quantiles?

I think there are two ways to perform the chi-Square goodness-of-fit test: Divide the sample space into bins of equal size and see how many observed values fall in each bin. where the expected per ...
1
vote
1answer
182 views

Sequence of Car Value at the End of Each Year

I bought a new car for (approximately, before taxes and what not, but we'll just say...) $18,000. At the end of n years, the value of my car is given by the sequence vn = 18000(3/4)^n, n = 1, 2, 3, ...
0
votes
0answers
24 views

If $a \in \mathbb{R}$ and $S = \{q \in\mathbb{Q}:q<a\}$, then $\sup S=a$ [duplicate]

Let $a \in \mathbb{R}$ and $S = \{q \in\mathbb{Q}:q<a\}$. Show that $\sup S=a$.
1
vote
3answers
58 views

Graph of a matrix and a positive power for the the matrix

A graph has a path from node $j$ to node $i$ if and only if its adjacency matrix has a positive element $(i,j)$ of $A^k$ for some integer $k.$ A proof for this statement will be highly appreciated.
5
votes
1answer
255 views

$f\left ( x+m \right )\left ( f\left ( x \right )+\sqrt{m+1} \right )=-\left ( m+2 \right ),\forall x\in \mathbb{R},m\in \mathbb{Z^+}$

Find all the function $f:\mathbb{R}\rightarrow \mathbb{R}$ sastisfied that $f$ continuous on $\mathbb{R}$ and $$f\left ( x+m \right )\left ( f\left ( x \right )+\sqrt{m+1} \right )=-\left ( m+2 ...
5
votes
1answer
397 views

Billingsley's solution to 3.18 (b)

Problem 3.18(b) in Billingsley's Probability and Measure (3e) is Show that if $\lambda^*(E)>0$, then $E$ contains a nonmeasurable subset. [Here $\lambda^*$ is the Lebesgue outer measure.] ...
2
votes
2answers
141 views

Multiply the matrices A1, A2, A3 of sizes 20 × 5, 5 × 50, 50 × 5? Ans is 1750 multiplications? how?

What is the most efficient way to multiply the matrices A1, A2, A3 of sizes 20 × 5, 5 × 50, 50 × 5? Answer is : A1(A2 A3), 1750 multiplications. how did they get the answer ? could someone show me ...
3
votes
1answer
88 views

Decomposition of semisimple Lie algebra via its roots

Exercise 14.33 in Fulton and Harris's Representation Theory claims that If the roots of a semisimple Lie algebra lie in a collection of mutually orthogonal subsets, one sees that the Lie algebra ...
2
votes
1answer
39 views

Solve non homogeneous rational equation

Is there a close form solution for an equation of this type $\frac{dx}{dt}=\frac{a}{b+c\cdot t}-d\cdot x$ with $a, b, c, d$ positive constants ?
6
votes
1answer
5k views

The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. [duplicate]

If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $. The $ ...
1
vote
1answer
103 views

function that cannot be expressed using finite characters

There are functions that cannot be expressed using finite characters. For example while function $x^3$ can be written using finite characters there exists a sequence of cartesian pair, describing ...
3
votes
3answers
952 views

Is any finite set necessarily closed and compact?

If $X$ is a finite set, then $X$ is compact since any finite set is compact. $X$ is closed since its complement $\emptyset$ is open (in any topology on $X$). Something wrong?
4
votes
2answers
426 views

example of a totally bounded but not bounded

I have got an question:- is there any counter-example of totally bounded but not bounded space can anyone help me please.thanks for your help
2
votes
2answers
64 views

What are the relations between these two minimizations

What are the relations between the minimization problems $\arg\min_{\mathbf{y}=A\mathbf{x}}\left\Vert \mathbf{x}\right\Vert _{2}$ and $\arg\min_{\mathbf{x}}\left\Vert A\mathbf{x-y}\right\Vert _{2}$ ?
17
votes
4answers
870 views

Is every open subset of $ \mathbb{R} $ uncountable?

Is every open subset of $\mathbb{R}$ uncountable? I was crafting a proof for the theorem that states every open subset of $\mathbb{R}$ can be written as the union of a countable number of disjoint ...
4
votes
1answer
448 views

How to prove $C[a,b]$ with the sup-metric is complete while with the $L^1$-metric is not?

Let $C[a,b]$ be the space of all continuous function defined on interval $[a,b]$. Consider these two norms and metrics: $$\|f\|_\infty= \sup_{x\in[a,b]}|f(x)|\text{ and metric ...
3
votes
0answers
168 views

Runge-Kutta Error Analysis

Could anyone explain to me how to reduce the error propagated by using Runge-Kutta of order 4? Or can anyone give me a nice reference to it.
19
votes
4answers
741 views

How would you explain why $e^{i\pi}+1=0$ to a middle school student?

Hi I was asked by a friends child who is in middle school why $e^{i\pi}+1=0$. Now I couldn't think of a way to explain it so he would understand. Albert Einstein once said “If you can't explain it ...
2
votes
1answer
37 views

Algebraic equation translation

One scoop of ice cream has 5mg of cholesterol less than the amount of cholesterol in one hamburger. Together, they have 37mg of cholesterol. How much cholesterol is in each food?
7
votes
2answers
251 views

Bounds on a sum of gcd's

Does there exist a positive real number $C$ and a positive integer $M$ such that for all $n > M$ we have: $$\sum_{i=1}^n\sum_{j=1}^n\gcd (i, j)\ge Cn^2 \log n$$ This originally appeared as an ...
4
votes
0answers
143 views

Understanding Blackwell's Approachability Theorem

I'm not super solid on my linear algebra, so I am getting lost in the discussions of halfspaces. Can someone give me an intuitive explanation (possibly with a concrete toy problem) of Blackwell's ...
3
votes
1answer
565 views

A locally compact subset of a locally compact Hausdorff space is locally closed

Let $X$ be a locally compact and Hausdorff space. Show that if $Y \subset X$ is locally compact, then $Y$ is locally closed, in essence $Y$ is an open subset of $\overline{Y}$, where $\overline{Y}$ ...
1
vote
0answers
86 views

Volume form on complex space, notation

If $\Omega = dz^{1}\wedge..\wedge dz^{n}$, $z^{1}$,..,$z^{n}$ are complex numbers. What does the notation $\Omega (x^{1}\wedge.. \wedge x^{n})$ means? $x^{1},..,x^{n}$ are real parts of ...
3
votes
3answers
77 views

Convergence of a sequence function

Show that the sequence of function $F_n(z)=\frac{z^n}{z^n-3^n},\ n=1,2,...,\ $ converges to zero for $|z|<3$ amd to $1$ for $|z|>3$. How can I show this? I can see why it will converge to ...
1
vote
1answer
136 views

Determine the vectors $u, v, w$

Let there be $w, u, $ and $v$, such that: $$w \times u = \langle1, 3, 5\rangle$$ $$w \times v = \langle 2, 4, 6\rangle$$ Find: $$v \cdot (((u \times w) \times v) + \text{VP}uv(w)) + ((u ...
6
votes
3answers
210 views

Show that the subset $D= \{(x,y)~|~x \ne 0, y \ne 0\}$ of the plane is open

$D= \{(x,y)~|~x \ne 0, y \ne 0\}$ I'm thinking I can show that the set is open for $x \gt 0$ and $y \gt 0$ using disks. Maybe I could do the same for $x \lt 0$ and $y \lt 0$. But this process seems ...
2
votes
3answers
130 views

How can I define a function to show that $\{3^n\mid n\in\mathbb{Z}\}$ is countably infinite?

I have to answer the following question for an assignment: Is the set $A=\{3^{n}\mid n\in\mathbb{Z}\}$ finite, countably infinite, or uncountable? I've defined what I originally thought to be a ...
3
votes
0answers
118 views

Question about solvability of the minimal polynomial of the element in extended field.

Let $F$ have characteristic $0$, and assume we have fields $L\supset K\supset F$. Now suppose $\alpha\in L$ be solvable by radicals over $K$, and the coefficients of the minimal polynomial of ...
2
votes
2answers
157 views

What is a Ramsey Graph?

What is a ramsey graph and What is its relation to RamseyTheorem? In Ramsey Theorem: for a pairs of parameters (r,b) there exists an n such that for every (edge-)coloring of the complete graph on n ...
1
vote
1answer
55 views

About vector addition

I'm teaching myself vector space and there are some phrase that I can't understand. About vector addition, the book says, Either $v$ or $w$ may be applied at $P$ and a vector having the same ...
2
votes
1answer
270 views

Covariance of Student's t-distribution

Another integral (this time it looks like a lot of work but maybe it can be simplified). I have the Student's t-distribution $$\int_{-\infty}^\infty ...
0
votes
1answer
151 views

On a characterization of primitive polynomials over a finite field

Let $K$ be a finite field. Let us define a primitive polynomial as an $f \in K[X]$ s.t. the multiplicative order of $X$ in $K[X]/(f)$ is equal to $|K|^{\deg f} - 1$. I want to show that $f \in K[X]$ ...
0
votes
2answers
28 views

Polar coordinates that uses $\frac { 1 }{ Z_1 }$

I am doing polar coordinates, and I am stuck when my book asks to do $\frac { 1 }{ Z_1 }$. I have no problems with $\frac { Z_1 }{ Z_2 }$ and $Z_1Z_2$. Here is the values for $Z_1$ I'm not so much ...
7
votes
1answer
232 views

$\sum_{i=0}^m \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$ Combinatorial proof

Is there a simple combinatorial proof for the following identity? $$\sum_{0\leq i \leq m} \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$$ where $m,j \geq 0$, $k \geq n \geq 0$.
1
vote
1answer
318 views

Stochastic processes for beginers (good links and books)

I've a syllabus like that.. Markov chains with finite and countable state space. Classification of states. Limiting behavior of n state transition probabilities. Stationary distribution. Branching ...
4
votes
2answers
102 views

$\sum_{i=1}^n i\cdot i! = (n+1)!-1$ By Induction

I am trying to prove the following by Mathematical Induction: $$\sum_{i=1}^n i\cdot i! = (n+1)!-1\quad\text{for all integers $n\ge 1$}$$ My proof by Induction follows: First prove $P(1)$ is true, ...
0
votes
0answers
85 views

disjoint and inverse domination

The disjoint domination number $\gamma\gamma(G)$ is defined as $$\gamma\gamma(G)= \min\{|S_1| + |S_2| : S_1\text{ and }S_2\text{ are disjoint dominating sets of }G\}.$$ Just want to verify if this ...
0
votes
3answers
147 views

Calculating matrix norm

I'm trying to follow the lecture notes for a dynamical systems class, and they say the norm of $$\begin{bmatrix}0 & -3 \mu x_2^2\\-3 \mu x_1^2 & 0\end{bmatrix}$$ is $3 \mu x_1^2 x_2^2$. How ...
0
votes
1answer
75 views

Prove that P is irreducible with $P(t)=t^{n}-A$

$Hello, everyone$ :D , I have this problem Let $p$ the polynomial $p \in Q[t]$ given by $p(t)=t^{n}-A$ where $A$ is prime. Prove that $p$ is irreducible. I did this: Using Eisenstein's criterion: ...
2
votes
1answer
908 views

Integration by Parts and Leibniz Rule for Differentiation under the Integral Sign

Basically a friend of mine and I have had this hot debate for a little too long, I contend that these two tools are not only logically unconnected but they require different assumptions (I believe one ...
0
votes
1answer
66 views

proof that this is an isometric map (on a $C^*$-module)

Are my steps right? I'm not sure about the statement in bold below. Let $A$ be a $C^*$-algebra. Let $X$ be an $A$-module. Let $x\in X$, let $a= \langle x,x \rangle $ Define $\lambda _a (z) = az$, ...
3
votes
1answer
60 views

How to calculate $\int_{-a}^{a} [f'(x)\log(1+e^x)\,]dx$

The complete problem says: Let $f(x) \in C^1 [-a,a]$ and even, calculate $\displaystyle \int_{-a}^{a} \left[\frac{f(x)}{1+e^x} + f'(x)\log(1+e^x) \right]dx$ The first part is easy to see. I don't ...
0
votes
2answers
133 views

Why we can assume this radical extension is a splitting field of some polynomial?

Let $F\subset L\subset M$ where $F\subset L$ is the splitting field of $f\in F[x]$ and $F\subset M$ is radical. It is said that we can assume $F\subset M$ is a splitting field of some $g\in F[x]$. ...
1
vote
0answers
88 views

Permutation and combination with males/females

There are $6$ males and $6$ females in the finals of a talent competition. A contest is held to pick the top $3$ winners in both the male and female categories in order of merit. How many different ...

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