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Proof of Wronskian relation using induction

We have the following linear homogenous DE system $X' = AX, \tag 0$ I wanna prove with induction that $dW/dx = Tr(A)*W$ So for n=2 based on the above, we get, $A = \begin{bmatrix} a_{11} & a_{12} \...
0 votes
0 answers
2 views

How to determine if the ideal I=\langle x-1, y\rangle is a maximal ideal of $\mathbb{Q}[x, y]$

How do I determine whether $\mathbb{Q}[x, y] / I$ is a field? Where I is generated by the Gröbner basis $I=\langle x-1, y\rangle=\{a(x,y)(x-1)+b(x,y)y \quad| \;a,b\in\mathbb{Q}[x,y]\}$ I have a ...
0 votes
0 answers
24 views
+100

Boundedness of singular integral operator on normalized bump functions implies boundedness on Schwartz functions controlled by suitable seminorms

In the book "Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory integrals" by E. Stein, the author considers on page 294 a singular integral operator $T:\mathcal{S}(\...
0 votes
3 answers
25 views

Six-letter ‘words’ are formed using the letters A, B, C and D. In how many of them does each letter appear at least once?

Six-letter ‘words’ are formed using the letters A, B, C and D. In how many of them does each letter appear at least once? In the answer, it says $$ 4 \cdot 6 \cdot 5 \cdot 4+\left(\begin{array}{l} 4 ...
0 votes
2 answers
26 views

Rank of matrix - Linear Algebra

I have this question: Assume that $A,B$ are $n \times n$ matrices. $BAB = 0$ Assume that $A$ is an invertible matrix. Prove that $\rho(B) \leq \frac{n}{2}$ Here's my take: If $A$ is an ...
0 votes
1 answer
25 views

Find parameter p such that the points $ A(1,−1, 0), \ B(2 ,0 ,1), \ C(1, p, 3), \ D(2 , 2 p, 5)$ lie in the same plane.

I'm not sure how to get p with the limited amount of information i have about the system. Any suggestions on how to approach something like this?
0 votes
0 answers
2 views

Conditioning in Event Language VS Proposition Language

According to this video, one can freely decide to conceptualize probabilities in terms of either event language or proposition language. It states, "the mathematical rules are applied the same ...
0 votes
0 answers
2 views

Cardinal of a set (proof verification)

Denote by $[n]$ the set $\{1,\dotsc,n\}$, for any real $n$. Consider the set $$S=\{(l_1,l_2,l_3,l_4)\in[n]^4:\forall a\in[4]:\exists b\in[4]\setminus\{a\}:\lvert l_a-l_b\rvert\leq cM_n\}.$$ Does the ...
2 votes
4 answers
159 views

Prove that the integers $n$ and $n, 2^{2^n} + 1$ are relatively prime.

I need help to prove that $$\gcd(n, 2^{2^n} + 1)=1,\ n = 1,2,\dots$$ I have no idea how start the proof.
0 votes
3 answers
52 views

Find the general solution of the recurrence relation $3x_{n+2} − x_{n+1} − 2x_{n} = 5$.

Find the general solution of the recurrence relation $3x_{n+2} − x_{n+1} − 2x_{n} = 5$. Attempt First I found the auxiliary equation: $3 \lambda ^ 2 - \lambda - 2 = 0$. To get the solutions: $\lambda ...
1 vote
0 answers
19 views

Proof: $C^\infty[0,1]$ is dense in $L^2[0,1]$

Intro: I would like to know if my demonstration of $C^\infty[0;1]$ is dense in $L^2[0,1]$ is correct because I didn't find any complete demonstration of that statement. -(i) As we know from here all ...
2 votes
1 answer
93 views

What are some nice topics in undergraduate mathematics to write about?

As a high school student, I will soon be going on to do the IB(college equivalent). As part of the IB curriculum we are expected to write an Extended Essay(EE) that counts towards our grade. For the ...
0 votes
0 answers
7 views

What set of matrices $X,Y$ satisfy $X \circ Y = X P Y$ for at least one permutation matrix $P$?

Suppose we have three $n \times n$ matrices $X,Y,P$ where $X,Y \in \mathbb{R^{n \times n}}$ and $P$ is a permutation matrix. Let us take $\circ$ to be the element-wise product between two matrices. ...
1 vote
1 answer
16 views

Simplify $x/(x-2)+(x-1)/(x+1)=-1$ step by step

Simplify $$x/(x-2)+(x-1)/(x+1)=-1$$ step by step. So I clear the fractions by multiplying by the common denominator of $(x-2)(x+1)$ and you have $$x(x+1) + (x-1)(x-2) = -(x-2)(x+1)$$ $$x^2 + x + x^2 - ...
0 votes
0 answers
6 views

Doubt regarding the radius of convergence of Cauchy Product

Let $\sum \limits _{n=0}^\infty a_nz^n$ and $\sum \limits _{n=0}^\infty b_nz^n$ be two power series with complex coefficients. If the radius of convergence of the first series is $R_1$ and the radius ...
0 votes
0 answers
9 views

An interesting recurrent equality, possibly easier to solve in its differential form?

I encountered an interesting inequality that I'm not sure how to approach. Here $c$ is a positive constant. $$f(n+1) - f(n) = c f(n)\sum_{m=0}^n f(m)$$ I am not familiar with techniques to solve ...
8 votes
0 answers
63 views

Collatz conjecture but with $n^2-1$ instead of $3n+1.$ Does the sequence starting with $13$ go to infinity?

Let's consider the following variant of Collatz $(3n+1) : $ If $n$ is odd then $n \to n^2-1.$ $1\to 0.$ $3\to 8\to 1\to 0.$ $5\to 24\to 3\to 0.$ $7\to 48\to 3\to 0.$ $9\to 80\to 5\to 0.$ $11\to 120\to ...
1 vote
1 answer
57 views

Does the lognormal distribution satisfy the Fokker-Planck equation for geometric brownian motion?

I've been trying to understand the Fokker-Planck equation and have hit a road block on what is basically the second simplest possible example, geometric Brownian motion. Consider this Ito SDE for GBM, ...
0 votes
1 answer
948 views

Fuel and Oil Ratio Problem

Joel mixes petrol and oil in the ratio 40:1 to make fuel for his leaf blower. (i) He pours 5 litres of petrol into an empty container to make fuel for his leaf blower. How much oil should be added to ...
1 vote
2 answers
71 views

Are $\frac{p^2+1}{2}$ and $\frac{p^{5n}(p^5-1)}{2}$ are coprime to each other, $n \in \mathbb{N}$?

Let $p$ be a prime integer greater than $2$. Then I want to prove the followings: $(1)$ $\frac{p^2+1}{2}$ and $\frac{p^5-1}{2}$ are coprime to each other. $(2)$ $\frac{p^2+1}{2}$ and $\frac{p^{5n}(p^5-...
0 votes
0 answers
15 views

How many integer values of $m\in[-20;22]$ are there such that the equation $\log_2(x^2+m+x\sqrt{x^2+4})=(2m-9)x-1+(1-2m)\sqrt{x^2+4}$ has a solution?

How many integer values of parameter $m$ on $[-20; 22]$ are there such that the equation $$\log_2(x^2 + m + x\sqrt{x^2 + 4}) = (2m - 9)x - 1 + (1 - 2m)\sqrt{x^2 + 4}$$ has a solution? [For context, ...
1 vote
0 answers
9 views

Why does Galois theory most naturally take place in the context of fields?

At least as far as I can tell, historically Galois theory was a more computational tool than it appears now, and https://hsm.stackexchange.com/questions/8099/how-did-the-modern-understanding-of-galois-...
0 votes
2 answers
40 views

Maximize (z-x) such that x^2 + y^2+ z^2 =1.

Here I tried the coordinate geometry, as in the equation represents a sphere. From there (z-x) would imply the distance between the z coordinate and x coordinate so that the difference is maximum. ...
0 votes
1 answer
40 views

Given, $a,b,c\in\mathbb{R}$. Let $p(x)=ax^2+bx+c$. Find equivalent condition that always has an integer value ($p(x)$'s value) for $x\in\mathbb{Z}$.

Problem : Given, $a,b,c\in\mathbb{R}$. Let $p(x)=ax^2+bx+c$. Then, find an equivalent condition that always has an integer value ($p(x)$'s value) for an arbitrary integer $x$. [Problem is represented ...
0 votes
2 answers
3k views

Integral of Complex Numbers

I was wondering about doing integration the unorthodox way. For example I took up $$\int\sqrt{1-x^2}\,\mathrm dx$$ and instead of substituting $x$ for $\sin t$ I tried doing it for $\sec t$ which ...
3 votes
1 answer
19 views

Are invertibly cobordant manifolds diffeomorphic

Let $M$ and $N$ be oriented, closed, $n-1$ manifolds and $F$ a cobordism from $M$ to $N$ and $G$ a cobordism from $N$ to $M$ such that the composite cobordism $G\circ F\cong M\times I$ and $F\circ G\...
-1 votes
2 answers
42 views

Show $\int_0^1 \frac{1}{2x^{\frac{1}{4}}-x^3}dx <\infty$ [closed]

Show $$\int_0^1 \frac{1}{2x^{\frac{1}{4}}-x^3}dx <\infty$$ $$\int_0^1 \frac{1}{2x^{\frac{1}{4}}-x^3}dx =\lim_{c\rightarrow0}\int_c^1 \frac{1}{2x^{\frac{1}{4}}-x^3}dx$$ exists by integrating and ...
4 votes
1 answer
32 views

Still not getting difference bettwen Implies and Entails and the role of "interpretation"

Background I am trying to understand the answers to the question Implies ($\Rightarrow$) vs. Entails ($\models$) vs. Provable ($\vdash$). In their answer, ryang wrote: material conditional $\left(\to\...
0 votes
0 answers
5 views

Exemple where tower property of conditional expectation is NOT verify

Question: Let Ω={a,b,c}. Give an example for $X, F_1, F_2$ in which $E(E(X|F_1)|F_2) \neq E(E(X|F_2)|F_1)$ My ansmer: I am not at all sure of my answer. If you have any shorter and niver answer i will ...
0 votes
0 answers
3 views

Find $\mathbb{E}[\hat{\alpha}]$. Suggest a new estimate $\check{\alpha}$ of $\alpha$ which is unbiased, such that $\mathbb{E}[\check{\alpha}]=a$

I'm currently working on the following question: $$Define \: m=\sum_{i=1}^n log(\frac{\theta+X_i}{\theta}) \sim Pareto(n,\theta)\: and \: i.i.d$$ $$Let \: \hat{\alpha}=\frac{n}{\sum_{i=1}^n log(\frac{\...
1 vote
0 answers
10 views

General form of rank $2$ tensor invaiant under certain rotations

I have the problem of determining the most general form of a rank 2 tensor $t_{ij}$ (in 3 dimensions) satisfying: $t_{ij}$ is invariant under any rotation about the $z$-axis $t_{ij}$ is invariant ...
0 votes
0 answers
14 views

Proof that markov chain equilibrium using Farkas' lemma

Given a transition matrix for markov chain $ P \in \mathbb R^{dxd} $ such that $$ P_{i,j} \geq 0,\quad 1 \leq (i,j) \leq d, \quad \sum_{j=1 \in d }P_{i,j} $$ and $i=1,....,d$. Let $ x_{0}$ be ...
2 votes
0 answers
11 views

What do the elements of the chains of a simplicial complex represent?

I've just started to learn homology and I don't quite understand why we define chains the way we do. For a simplicial complex $S$ we define $C_k$ to be the $k$-chains on $S$ given by an abelian group ...
0 votes
1 answer
11 views

Theorem of the Maximum for discrete sequences of constraint sets?

Suppose that $\{X_{n}\}_{n=1}^{\infty}$ is a sequence of sets that converges to $X$ in some sense. Let $f$ be a real-valued function. I am interested in conditions under which $$ \lim_{n \rightarrow \...
1 vote
1 answer
430 views

Prove that the Minkowski functional associated with a set $K$ satisfies the triangle inequality if and only if $K$ is convex.

Taking analysis, have a problem set that's tilting toward topology. The problem asks for proof that the Minkowski functional associated with a set $K$ satisfies the triangle inequality if and only if $...
0 votes
2 answers
688 views

In $\Delta ABC$, prove $A'D=\frac{(c^2- b^2)}{2a}$ with $A'$ be the midpoint of $BC$, $AB>AC$, and other Information given.

Problem In $\Delta ABC$, we have $AB>AC$. Let $A'$ be the midpoint of $BC$ $D$ be the foot of altitude from $A$ to $BC$ Internal and external Angle Bisectors of $\angle A$ meet $BC$ at $X$ and $X'...
0 votes
0 answers
9 views

Goldbach Conjecture and Triangular Number

Goldbach conjecture states that every even number greater than 2 is sum of two prime numbers. We know that every positive integer can be represented as a sum of three triangular numbers. Is it ...
2 votes
2 answers
111 views

Two fair dice tossed simultaneously till combination $(4,3)$ is obtained. Obtain the probability that number of tosses required is atmost $2$.

Original Question: Two fair dice are tossed simultaneously till a combination $(4,3)$ is obtained. Obtain the probability that the number of tosses required is atmost $2$. My Reasoning : The dice are ...
4 votes
0 answers
86 views
+200

Does Lipschitz-kind "non-scalar ODEs" and PDEs could stand having finite-duration solutions?

Does Lipschitz-kind "non-scalar ODEs" and PDEs could stand having finite-duration solutions? Intro Recently I have found on these papers by Vardia T. Haimo (1985) Finite Time Controllers ...
1 vote
0 answers
23 views

Hyperplane contains an inversible matrix

Let $H$ be an hyperplane of $M_n(\Bbb K)$ $(n\ge 2)$ Show that there exists $A \in M_n(\Bbb K)$ such that $H=\{M\mid \text{tr}(AM)=0\}$ Deduce that $H$ contains an invertible matrix
2 votes
0 answers
7 views

Calculation on Riemannian manifolds

I am learning the variational calculation of Yang Mills functional, but I can't understand 2 steps in the following calculation: Given a variation of the connection $A$ in local coordinates: $A\to A+\...
-1 votes
0 answers
20 views

If H is a normal subgroup of $G$ and $|H| = 2$, prove that H is normal in G [closed]

Having some trouble getting started on this one
0 votes
0 answers
13 views

Prove that there exists $x$ such that $\lvert\det J\phi_x\rvert=1$

I would like some help on the following question. Let $\phi:B^k \rightarrow \mathbb{R}^k $ be a diffeomorphism and $\phi(B^k) = B^k$. Prove that there exists $x\in B^k$ such that $\lvert\det J\phi_x\...
-1 votes
0 answers
28 views

Discussion, a sieve of co-primes that contains all primes.

Discussion Claim: "All prime numbers are contained in the set [P]n ∪ [G]": [P]n is composed of the set ...
0 votes
2 answers
48 views

Solution of $\theta$ when $\tan(\theta)-\sin(\theta)=\frac{\sqrt3}{2}$

I came across this trigonometry problem. If, $$\tan(\theta)-\sin(\theta)=\frac{\sqrt3}{2}$$ What is the value of $\theta$ I got the solution that $\theta$ will be $\frac{\pi}{3}$ by expanding the ...
0 votes
0 answers
5 views

Help on this article of Centralizers of Groups

I'm studying this article and and I'm having doubts about understanding two statements at the end of the proof of Lema 3.3 Why does $G = UP$ implies $C_G(U) = U \mathrm{core}_G(P)$? And why does $\...
-1 votes
1 answer
18 views

Tangent plane to the surface and the Taylor polynomial

Let $f(x,y) = a \sin(x+y^2) + e^{bx+y^2}$, what are the values of $a$ and $b$ such that the tangent plane to the surface $z = f(x,y)$ at $(0,0,f(0,0))$ be horizontal and the second order Taylor ...
2 votes
0 answers
10 views

Can we calculate the opponent's hidden values in this statistical battle?

First-order statistical battle Imagine there is a game in which the user should guess what values the opponent is hiding from the user. In the first battle, the opponent has two hidden values ...
5 votes
3 answers
140 views

Inequality involving sums with binomial coefficient

I am trying to show upper- and lower-bounds on $$\frac{1}{2^n}\sum_{i=0}^n\binom{n}{i}\min(i, n-i)$$ (where $n\geq 1$) in order to show that it basically grows as $O(n)$. The upper-bound is easy to ...
0 votes
0 answers
20 views

What conditions are required to guarantee that my matrix is skew-Hermitian?

Consider the equation $$ \frac{\partial \boldsymbol{T}}{\partial t}=\kappa \space \frac{\partial^2 \boldsymbol{T}}{\partial x^2} $$ with $$ \boldsymbol{T}=(T_1,T_2,...T_N)^T $$ Let $$ T_i=\sum_{j=1}^...

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