2
votes
3answers
21 views

What is the dimension of a certain vector subspace of invariant matrices?

$\newcommand{\Sig}{\Sigma}$ Let $\Sig$ be a diagonal matrix with strictly positive entries on the diagonal. Define $V=\{B \in M_n\mid B\Sig +\Sig B^T=\Sig B +B^T \Sig \}$ (where $M_n$ is the vector ...
7
votes
2answers
846 views

Metric assuming the value infinity

If we instead define a metric as $d:X\times X \rightarrow [0,\infty]$, do we lose any nice properties of metric spaces? The reason I ask is that I saw this theorem: Given a finite measure space ...
0
votes
0answers
4 views

Proof of identity relating composition of permutations and sign in alternating tensors.

I was having some confusion trying to prove the following identity: Let $A: \mathcal{L}^{k} \to \mathcal{A}^{k}$, where $\mathcal{L}^{k}$ denotes the set of all $k$-tensors and $\mathcal{A}^{k}$ ...
0
votes
2answers
18 views

joint infima involved in relation to distance between sets

Consider a metric space $(X,d)$, and let $A,K \subseteq X$ such that A is closed and K is compact. I have to show that there exists an element $k_0\in K$ which achieves the minimum between the sets, ...
0
votes
2answers
17 views

Prove by induction: A tree on n≥2 vertices has ≥2 leaves

this is what I have. I'm pretty sure this is quite incorrect, but am I at least headed in the right direction? Base Case: P(2): Tree on 2 vertices can only have one edge, the edge connecting the 2 ...
3
votes
1answer
55 views

The sequence $(-1)^n\binom{\alpha-1}{n}$ converges.

I need to show that for $n \in \mathbb N_0$ and $\alpha \ge 0$ the sequence $(-1)^n\binom{\alpha-1}{n}$ converges. It can be shown that the sequence convereges to zero using a theorem claiming that ...
0
votes
3answers
11 views

Proof that dim(L(V,W))=dim V *dim W, when V, W are finite dimensional.

I've been looking a few proofs of this, and I have a problem with the standard one, which uses the concept of isomorphism. Suppose dim V=n and dim W=M.The proof uses the function M from L(V,W)(the ...
0
votes
1answer
14 views

Bounds on a quadratic form

I am currently in the middle of a proof where it would be nice to have some estimates on the size of a quadratic form. In particular, I am looking at $$x^TAx$$ where $A$ is "small" (in the analyst's ...
1
vote
0answers
9 views

Sum of sum of binomial coefficients

I know there is no simple way to solve the sum: $$\sum_{k=0}^{j}{{n}\choose{k}}$$ But what about the equation: $$\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$$ Are there any simplifications or ...
4
votes
0answers
20 views

Why is the Fundamental Theorem of Arithmetic so important?

I've recently read about the Fundamental Theorem of Arithmetic and I think that I have just about understood the proof. What I found quite interesting at first was the "Fundamental" part in the name. ...
0
votes
0answers
30 views

Division of Normal and Poisson Distribution

I am trying to understand the following: If $X$ is normally distributed and $Y$ is distributed according to the Poisson distribution, how to find out which distribution $Z=X/Y^2$ has? Is there any ...
0
votes
0answers
5 views

Laplace Method to solve Differential equation

Use the Laplace Method to solve the following boundary value problem in the form of a definite integral: $x \frac{d^5 y}{dx^5} + 4y = 0 \\ \ \\ y(0) = 1 , \ \ y(\infty) = 0, \ \ 0 \leq x \lt \infty ...
1
vote
0answers
7 views

Proving the Secant Angles in the Circle

Ok, I know this is a very easy circle geometry problem, but I want to know that how to prove the theorem of angles in the circle. Like this image here: How can I prove that the angle X is the half ...
0
votes
1answer
13 views

Construct a bump function for upper hemisphere

When reading section 13.1 of Loring Tu book, I came across this problem on constructing a bump function. Write down an explicit function $f : S^2 \to \mathbb{R}$ such that $f(p) = 1$ for all ...
1
vote
1answer
26 views

Mathematical Notation of Sequence of Functions

Let's say I have a finite set of functions $F=\{f_1,f_2,f_3,...,f_n\}$ and I want to show a recursive function that is constructed by an arbitrary sequence of applications of functions in $F$ to input ...
0
votes
3answers
30 views

What is the difference between a linearly independent set and a set that spans $\Bbb{R}^m$?

This is more of a conceptual question. Here's what I know about a linearly independent set of vectors: A set of vectors $\{v_1, ..., v_p\}$ is linearly independent if the equation $$x_1v_1 + x_2v_2 + ...
1
vote
0answers
20 views

Friedman's urn is a supermartingale or a submartingale?

Here is the urn model: At time zero there are $r$ red and $g$ green balls in an urn. At each time-step, we draw out a ball at random and replace it along with $c$ of the same color and $d$ of the ...
2
votes
1answer
20 views

If $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 0$ and $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} a_{n}$ converges.

Let $a_{n} \geq 0$ and $b_{n}>0$ for each $n$ in $\mathbb{N}$ and suppose that $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 0$ and $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} ...
0
votes
0answers
8 views

How to shift the weighted mean of a monotically increasing series of values

I have a monotonically increasing series of values $X, x\in[0,1] \ \forall i\in1,2,...,60$ the weighted mean of which is defined by $$\bar{x}=\frac{x_i+\sum_{i=2}^{60} i(x_i-x_{i-1})}{x_{60}}.$$ Given ...
0
votes
0answers
15 views

Quotient involving $\pi$-subgroups

Let $G$ is a finite group and $\pi$ be a set of primes. Suppose that $P$ is a normal $\pi$-subgroup. Is it true that the quotient group $G/P$ is $\pi$-group? I know that since $P$ is a ...
2
votes
1answer
30 views

Idempotent and nilpotent matrices are defined differently. Why?

We call $A$ idempotent if $A^2$ is $A$. But we call A nilpotent if $A^k$ is $0$ for some integer $k$. Why are not they defined uniformly like both with power 2 or both with power some integer $k$.
0
votes
1answer
16 views

Proving a conclusion (Logic)

I had a question on how to prove a conclusion with a series of premises using deduction. From a statement such as the one below: If you eat carefully then you will have a healthy digestive system. If ...
2
votes
0answers
29 views

f left continuous & strictly increasing; B Borel $\implies$ f(B) Borel (or at least Lebesgue Measurable)?

How's it going? In an attempt to use the Radon-Nykodym theorem to bulldoze through the admission of measures by bounded variation & monotonic functions (sidestepping all that Caratheodory ...
1
vote
0answers
17 views

Part of proof of Levi's Convergence Theorem for Series

I want to prove the theorem of term-by-term integration for lebesgue integrable functions (denoted as $L^1$ functions): Suppose $(g_n)$ is a sequence of $L^1$ functions over a measure space $(X,\sigma ...
0
votes
0answers
12 views

Convergence of a function of two variables

The following question has been posed to me by a student in an analysis class. For which real numbers $\alpha \gt 0$ is the function $f : \Bbb R^2 \to \Bbb R$ given by $f(x, y) = (x^2 + y^2)^\alpha$ ...
0
votes
0answers
6 views

If G is a k-connected graph, then how do I prove that if H is obtained from G by contracting an edge, then H is k-1 connected?

I'm confused as to where to start. I think it has something to do with Mengar's theorem, but I'm not sure. Do I show how contracting an edge takes away at most one internally disjoint path between two ...
1
vote
1answer
12 views

Prove $O(f(n)+g(n)) = O(f(n))$ when $g(n)=O(f(n))$

Given $g(n) = O(f (n))$, how can I prove that the following expression is true: $O(f (n) + g(n)) = O(f (n)) \tag1$ So I just write down what it says: $g(n) = O(f (n)) <=> f(n) \le c_1 ...
7
votes
2answers
89 views

How can I get f(x) from its Taylor series ???

I know how to get a Taylor series when $f(x)$ is given. I have to find $f^{(k)} / k!$. But how can I get $f(x)$ from its Taylor series? The problem is $$f(x) = \sum_{n=0}^{\infty} C_n x^n,$$ where ...
0
votes
0answers
5 views

Change of basis, transition matrix and coordinate vector

Check that B={u1 , u2, u3} and B'={u' 1 , u' 2, u' 3} are bases in R3, where u1=[-3,0,-3]^t u2=[-3,2,-1]^t u3=[1,6,-1]^t and u' 1=[-6,-6,0]^t u' 2=[-2,-6,4]^t u' 3=[-2,-3,7]^t. a)find the transition ...
3
votes
3answers
39 views

How many $5$-digit numbers (including leading $0$'s) are there with no digit appearing exactly $2$ times?

How many $5$-digit numbers (including leading $0$'s) are there with no digit appearing exactly $2$ times? The solution is supposed to be derived using Inclusion-Exclusion. Here is my attempt at a ...
1
vote
0answers
10 views

Operator with kernel $|s-t|^{-\alpha}$ compact.

I need to show the following: Let $\alpha\in(0,1)$. Then $Tf(s) = \int_0^1\frac{f(t)}{|s-t|^\alpha}\mathrm dt$ defines a compact operator $L^2[0,1]\to L^2[0,1]$. I already managed to show that $T$ is ...
0
votes
0answers
18 views

Smallest distance of a point to a surface

Let $P$ be a hyperplane of dimension $n-1$ in the space $\mathbf{R}^n$, given some integer $n\ge 3$ (let's call the first axes $x,y,z,\ldots$). Then, fix a point $A \in P$ and define the surface ...
-4
votes
1answer
33 views

Is this subset of the unit cube compact?

Given $x_1,x_2 \in \{y \in \mathbb{R}^n: y\ge0, 1^Tx=1\}$. I have the set $S=\{0 \le q\le 1, x_1^Tq \le c\}$ where $c \in [0,1]$ and the inequalities are to be understood component-wise. The set ...
2
votes
1answer
495 views

Find the complete integral of $(p+q)(px+qy)=1$.

I am stuck on the following problem that says: Find the complete integral of $(p+q)(px+qy)=1$,where $p={ \partial z \over \partial x},q={ \partial z \over \partial y}$. My Attempt: The ...
0
votes
1answer
27 views

Decompose sum into reversible pairs

Is there any efficient way to find if a sum can be decomposed into reversible pairs?And if it does can we find these numbers? For example 66 can be decomposed into 24+42 or 66666=12345+54321. One ...
2
votes
2answers
39 views

How to read this problem from Dummit-Foote's “Abstract Algebra”?

Problem 14.6.1 on page 617 says Show that a cubic with a multiple root has a linear factor. Is the same true for quartics? Let $f \in F[x]$ be a cubic. If $f$ has a root in $F$, let alone a ...
0
votes
1answer
17 views

What is the proof of this equation? (Stochastic process)

$$\mathbb{E}[g(X)h(Y)]=\mathbb{E}[h(Y)\,\mathbb{E}[g(X)|Y]]$$ I am reading the book "An Introduction to Stochastic Modeling". This equation appears a lot but I can not see why. Can anyone please ...
0
votes
1answer
26 views

Analytic function on upper half plane

Here is the problem: Let $f$ be an analytic function defined on $\mathbb{H}$:={$z \in \mathbb{C}:Imz > 0 $}. Suppose that $|f(z)|<1$ for all $z \in \mathbb{H}$. Prove that for every $z \in ...
1
vote
0answers
6 views

Reducing sequential correlations in Metropolis Algorithm

In our last lab, we use MCMC method to simulate a walker walking in the phase space. Using the Metropolis method, a walker at its currect position will sample another point inside a cube (centered at ...
4
votes
0answers
18 views

Measure on $\omega$ defined in the generic extension by an atomless measure algebra is atomless

Work in Cantor space with standard probability measure $m$. Suppose we are given a sequence of measurable sets $\bar{A}=\langle A_n : n\in \omega\rangle$ and a non-principal ultrafilter $U$ and the ...
1
vote
1answer
17 views

Modues over rings which are NOT a PID, or NOT a UFD

I am interested in studying the properties of modules over rings which are not Principal Ideal Domains or are not Unique Factorization Domains, but I am finding it very difficult to find any material ...
0
votes
1answer
28 views

Bayesian probability

recently I bumped into following puzzle and I would like to validate(or correct) my results as I asked several people and got several different answers. You are planning a picnic with your friends ...
3
votes
0answers
23 views

Can we find prime numbers with any sum of digits (except those divisible by three)

I guess that this question is not something new and that there must be people who wanted to know if this question has an affirmative answer, but I would like to share it with you, because I really do ...
0
votes
1answer
18 views

How do I show that if T*T = Id$_{V}$ then TT* = Id$_{V}$, where T is a linear transformation and T* is its adjoint operator?

We can use that (T*)*=T because I have shown that while trying to work on this one. I am just plane stuck on this one. V is also finite dimensional
0
votes
1answer
21 views

Describe the orbits of the action.

So $L$ denotes the set of oriented straight lines through the origin in $\mathbb{R}^{2}$ (that is, straight lines with a preferred direction, indicated by an arrow). The group $(\mathbb{R},+)\cong ...
1
vote
1answer
14 views

Determine if the following composition function is onto

Define $f: \Bbb{Z}\times\Bbb{Z} \to \Bbb{Z}\times\Bbb{Z}$ and $g: \Bbb{Z}\times\Bbb{Z}\to \Bbb{Z}\times\Bbb{Z}$ by: $$f((a,b))=(a+b,2a)$$ $$g((c,d))=(c+2d,c)$$ Determine if $g \circ f$ is onto. I ...
-1
votes
0answers
29 views

Torelli Theorem [on hold]

Good Morning! I would like to know what are the minimum mathematical prerequisites for study the Torelli Theorem (algebraic geometry). I also accept bibliographic suggestions for this walking. Thank ...
1
vote
2answers
51 views

Scheme over a not algebraically closed field

I'm trying to make a scheme out of an affine variety $V\subset k^n$, when the base field $k$ is not algebraically closed. I wonder if $V$ is the spectrum of its ring of regular functions ...
-4
votes
2answers
40 views

How many $4$-sequences on $\{1, \ldots, 9\}$ are there?

How many $4$-sequences on $\{1, \ldots, 9\}$ are there? Please explain how to solve this problem as I am lost, as the text book does not explain that well.
0
votes
0answers
14 views

Bit increase when averaging?

I have a given number $N$ of binary numbers, that are stored using a given number $B$ of bits. $B$ is the same for all the numbers. For example, thease values where $N = 4, B = 4$. ...

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