0
votes
2answers
54 views

Euler method(path s1s2=s2s1)

Given a differential equation $\frac{dy}{dx}=f(x,y(x)), y(x_0)=y_0$. What is the condition for function of f(x,y) such that the result of $y(x_0+S_1+S_2)$ by using Euler forward method, a step size ...
0
votes
1answer
192 views

Finding the inverse of Hadamard matrix [closed]

The $Hadamard\,\,matrices\,\,{H_0},\,\,{H_1},\,\,{H_2}, \ldots $ are defined as follows. Let ${H_0}$ be the $1 \times 1$ matrix $\left[ 1 \right]$. For $k = 1,2, \ldots ,$ let ${H_k}$ be the ${2^k} ...
2
votes
1answer
406 views

Weak derivative zero implies constant function

Let $u\in W^{1,p}(U)$ such that $Du=0$ a.e. on $U$. I have to prove that $u$ is constant a.e. on $U$. Take $(\rho_{\varepsilon})_{\varepsilon>0}$ mollifiers. I know that ...
2
votes
1answer
50 views

Non-decidable $\Pi^0_1$ (effectively closed) classes

Are there non-decidable $\Pi^0_1$ (effectively closed) classes? According to a draft of Effectively closed sets by Cenzer and Remmel, the class $$ P = \{ 0^n1^\omega \mid n \in B\} $$ is a ...
1
vote
1answer
86 views

In Whitehead & Russell's PM, if $P$ is an infinite well-ordered series, can $P$ have a last term?

If I'm not mistaken, $B‘\overset{\smile}{P}$ is the last term of $P$. If it does not exist, there is no need to put ~$(B‘\overset{\smile}{P}) \in C‘∇‘P $ in the hypothesis. Chances are I missed ...
0
votes
2answers
242 views

Mind Teasers : Difficult Brain Twister (Today Challenge)

A strange tradition is followed in an orthodox and undeveloped village. The chief of the village collects taxes from all the males of the village yearly. But it is the method of taking taxes that is ...
0
votes
2answers
68 views

What is this kind of geometry called?

I want to get Cartesian coordinates of the points of a curve (e.g. a bezier curve) based on the distance (e.g length of the arc) from the start point on the curve. To make this more clear, suppose I ...
2
votes
1answer
72 views

Spectral radius of a normal element in a Banach algebra

I know that if $A$ is a C*-algebra, then $||x||=||x||_{sp}$ for every normal element $x\in A$. I like to find an example of a normal element $x$ in a involutive Banach algebra with $||x||\neq ...
0
votes
0answers
63 views

Question about Corollary I.6.6 in Hartshorne

I am having trouble understanding something in Corollary I.6.6 of Hartshorne. Let $K$ be a function field of dimension one over $k$ (by which he means a finitely generated extension of transcendence ...
0
votes
1answer
21 views

Getting the average of values with errors.

I have five data values each with an associated error. I want to find the mean of these values but also take the errors into account. How do I do this? Lets say the data values and errors are: ...
4
votes
3answers
175 views

Calculate limit of ratio of these definite integrals

How do I evaluate the following limit? $\lim_{n\to\infty}\frac{\int_{0}^1\left(x^2-x-2\right)^n dx}{\int_{0}^1\left(4x^2-2x-2\right)^n dx}$
0
votes
2answers
26 views

Prove $P(X<2<4X)=P(\frac{1}{2}<X<2)$

Is $P(X<2<4X)=P(\frac{1}{2}<X<2)$ ?? I came across a question that requires me to find the probability that 2 lies between X and 4X. which is $P(X<2<4X)$ , so this was impossilbe ...
0
votes
0answers
22 views

My question is in which condition $M_m=P_m$?

Suppose I have a collection of operators $\{P_m\}$ on a finite dimensional Hilbert Space which satisfies $\sum_{m} P_m= I$ where $m$ is eigen value. and I have a collection of positive definite ...
4
votes
0answers
175 views

Proving the Maximum Principle and the Continuous Dependence on Initial Condition and Boundary Conditions.

This is a two part problem that uses the Maximum/Minimum Principle and the Continuous Dependence. I already got the answer for the Maximum/Minimum Principle, but now I have to apply the Continuous ...
1
vote
0answers
31 views

GMM estimation of linear regression with intercept restriction

Say I have a time series regression as follows: $$y_t = a_i + \beta_i x_t + \varepsilon_t^i \ \ ; \ \ t = 1, 2, \cdots, T \ \ \text{for each } i$$ Now say I impose the following restriction on the ...
10
votes
5answers
230 views

Evaluating $\int\limits_0^{\pi} \frac{dx}{1+2\sin^2x}$

After making u substitution two times, I am getting indefinite integral as $$\int\limits\dfrac{dx}{1+2\sin^2x} = \dfrac{\arctan(\sqrt{3}\tan(x))}{\sqrt{3}}+ C$$ I am stuck at working the bounds ...
0
votes
1answer
20 views

Is the set of polynomials an $x^n+ a_{n-1}x^{n-1}+\ldots+a_1x +a_0$ such that $2^k+1$ divides $a_k$ an ideal in $\Bbb Z[x]$?

Is the set of polynomials $a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$ such that $2^k+1$ divides $a_k$ an ideal in $\Bbb Z[x]$? I think it is true for $2^k+1$ and it will be true for all the divisors as ...
0
votes
1answer
97 views

Difference between type and similarity type

In usual terminology, is there a difference between the type and similarity type? Is there a general consensus for the definition of the two terms? Please suggest to me books where I can study these ...
0
votes
1answer
40 views

Normal modes of a drum and Kac's question: Can one hear the shape of a drum?

I consider a vibrating membrane $D\subset {\mathbb{R}}^2 $, fixed on $\partial D$. The vertical displacement $f=f(x,y,t)$ of the membrane satisfies the wave equation. I search solutions of the form ...
1
vote
2answers
98 views

Combination Transversion

Suppose we have a lottery, consisting of 5 balls. The range of balls is 1-39. In any given pick, there will be no duplicate values, and the order need not matter. The upper limit of combinatorial ...
3
votes
3answers
329 views

If $f$ is continuous on $\mathbb R$ then $\exists c\in\mathbb R: f(x)=c$ has only one solution

I have to prove that there is no continuous function $f: \mathbb{R} \to \mathbb{R}$ such that, for each $c \in \mathbb{R}$ the equation $f(x)=c$ has exactly two solutions. My attempt: We have that ...
1
vote
1answer
42 views

Kolmogorov Exponential Bounds (Upper)

This is one version of Kolmogorov exponential bound from Allan Gut's Probability: A Graduate Course (2005, p385-386). Let $Y_k$ be an independent sequence of random variables with zero mean and ...
1
vote
2answers
203 views

Minimize and maximize the function $g(a,b) = a + b$ given the constraint $h(a,b) = \frac{1}{a} + \frac{1}{b} = 1$.

I want to minimize and maximize the function $g(a,b) = a + b$ given the constraint $h(a,b) = \frac{1}{a} + \frac{1}{b} = 1$, and I want to find the values of $a, b,$ and $\lambda$. This is what I've ...
0
votes
1answer
40 views

Random process of the form $X(t)=A^t$ where $A$ is a given random variable

We’re going to look at a random process, which is a sequence of random process, which is a sequence of random variables that depend on time. Let $X(t)=A^t$ where $A$ has the density $f_A (a)=(3/8) ...
4
votes
1answer
151 views

Can the sum of two independent identical random variables be uniform? [closed]

$X$ and $Y$ are two independent and identically distributed random variables. Can $X+Y$ be uniformly distributed over interval $[0,1]$?
2
votes
4answers
316 views

Prove that a continuous function defined on an interval $[a,b]$ has a fixed point.

I have to prove that : Suppose that $f:[a,b] \to [a,b]$ is continuous. Prove that there is at least one fixed point in $[a,b]$. But I don't know how to attack it since I can't apply anything of ...
0
votes
1answer
74 views

On the definition of positive and negative crossings

In the definition of the Kauffman bracket, we have to resolve a crossing in two ways. The way to resolve these crossings involve distinguishing whether the crossings are positive or negative. But ...
0
votes
1answer
134 views

Complement of the complete bipartite graph

Hey taking the complement of the complete bipartite graph $K_{m,n}$ I think that I get a disconnected graph composed of the complete graph $K_m$ and the complete graph $K_n$ is that right?
0
votes
2answers
37 views

Finding the value of an expression with logarithms

Given that $\log_{b}a=0.74$ and $\log_{b}(a-1)=0.65$ find the value of the following expression: $$\log_{b}(a^{4}-1)-2\log_{b}(a^{2}+1)+\log_{b}(a^{3}+a)-\log_{b}(a+1)$$ I tried using log laws to ...
2
votes
1answer
144 views

Different definitions of Casimir element

I read about the Casimir element just recently and I found it a bit difficult to wrap my mind around the definition(s). In fact, I have seen two different definitions. For concreteness, let ...
2
votes
2answers
36 views

What conclusion could be drawn about the maps from their compostie?

Let $f \colon A \to B$ and $g \colon B \to C$. Then If $g \circ f$ is injective, then I know that $f$ is injective, but what can we say about the injectivity of $g$? If $g \circ f$ is surjective, ...
0
votes
1answer
50 views

Subset of samples has any effect on sufficiency of the statistic?

If we have the following iid samples $$ X_1, ..., X_n \sim N(\mu, \sigma^2) $$ where only $\mu$ is unknown. We know one sufficient statistic is the following: $$ T = \frac{1}{n} \sum_{i=1}^n X_i $$ ...
5
votes
1answer
150 views

When is the next palindrome?

Okay, this is more just for fun than anything else. I'm driving in my car today, (true story) and my odometer is about to hit $81,818$. So, being a math nerd and all, I immediately see the pattern ...
8
votes
7answers
350 views

If $\,f^{7} $ is holomorphic, then $f$ is also holomorphic. [closed]

I need some help with this problem: Let $ \Omega $ be a complex domain, i.e., a connected and open non-empty subset of $ \mathbb{C} $. If $ f: \Omega \to \mathbb{C} $ is a continuous function and ...
1
vote
2answers
62 views

Prove that $[a]$ and $[n]$ are not relatively prime if and only if there is a nonzero element $[b] \in \Bbb{Z}_n$ such that $[a][b] = 0$

Here is my attempt (1) ---> First of all I know that $[n] = [0]$ and then we assume that a and n are not relatively prime then there exists an integer $x = \gcd(a,n)$ and $x \neq 1$ and so there ...
0
votes
1answer
3k views

3 digit odd numbers that can be formed using 0,3,5,7 - no repetition

Q. How many 3 digit odd numbers can be formed using 0,3,5,7, repetition not allowed. WHAT I DID :- 3 x 3 x 1 = 9 For Hundredth place - It can be filled in 3 ways (any of 3,5,7), we cannot use 0. ...
1
vote
1answer
105 views

The definition of convex body and the Hilbert cube

I currently have a question about the definition of convex body. The formal definition is: a convex body is a convex set which has non-empty interior. By non-empty interior, we meant for a set ...
1
vote
1answer
62 views

$H$ is an orthocenter of triangle $ABC$

$H$ is an orthocenter of angle $ABC$. Angle $B$ is $60^{\circ}$. Perpendicular bisectors of $AH$ and $CH$ cross line $AC$ at points $A_{1}$ and $C_{1}$. Show that the centre of $A_{1}HC_{1}$'s ...
0
votes
2answers
26 views

Probing a particular function

I've been playing with a particular function $$Q(n) = \sum_{i=1}^n i\cdot i!$$ in C++, and I'm trying to see if it is possible to find the following in an elegant way: 1) Is it possible to rewrite ...
0
votes
1answer
47 views

Block matrix exponent proof

Let $A = \begin{bmatrix} B & C \\ 0 & I\end{bmatrix}; \tag{1}$ $B$ and $C$ are $n \times n$ matrices. Prove that $A^k =\begin{bmatrix} B^k & (B^k-I)(B-I)^{-1}C \\ 0 & ...
4
votes
2answers
33 views

Mathematical expression for map from $[0,1]$ to $S^2$

A topological space is called arcwise connected if, for any points $x,y\in X$, there exists a continuous map $f: [0,1]\rightarrow X$ such that $f(0)=x$ and $f(1)=y$. Although it is intuitively ...
2
votes
3answers
115 views

What is the meaning of $dy=dx^2$?

When I read the mathematical analysis ,I think if the differential is $dy=Adx^2$ $A$ is a function about x, what will happen? Maybe, it is not proper defined ,but I think the "function" meet ...
0
votes
1answer
48 views

The accuracy of approximating $ f(x) = x^{2/5}$ for $0.9 \le x \le 1.1$ using the cubic Taylor polynomial

For the equation $ f(x) = x^{2/5}$, $a=1$, $n=3$, $0.9 \le x \le 1.1$ I was able to approximate f by the following Taylor polynomial: $$ F_3(x) = 1 + \frac2 5 (x-1) - \frac3{25}(x-1)^2 + ...
0
votes
1answer
25 views

Last step of making integral stationary

In a problem making an integral stationary, I'm being told that this last implication is wrong, why? Is my integration wrong? $$ \frac{d}{dx} y(x) = \pm\frac{c_1x^2}{\sqrt{x^2 - c_1^2}}$$ $$\implies ...
1
vote
3answers
72 views

$(C_{[0;1]}, \Vert . \Vert_1)$ is not a Banach space

I'm going to prove that $C_{[0;1]}$ is not a Banach space w.r.s to the norm $\Vert x \Vert_1 = \int_{0}^{1} |x(t)| dt$ by consider the series $\sum_{n=1}^{\infty}x_n$ where $x_{n}(t)= t^{n} \cdot ...
1
vote
1answer
29 views

When is the product of a hermitian unitary and another unitary hermitian?

I have a Hermitian unitary Ĥ and I want to know, if Û is some other unitary, when is Ĥ Û a Hermitian unitary? Specifically, what are the conditions on Û such that ...
1
vote
1answer
39 views

Prove the uniformly continuity of a function with a certain property

I need to prove this: Suppose that $f: \mathbb{R} \to \mathbb{R}$ is continuous and has the property that for each $\epsilon >0$ there is $M>0$ such that if $|x| \ge M$, then $|f(x)|< ...
5
votes
3answers
102 views

Connectedness of $O(3)$ group manifold

A topological space is said to be connected if it cannot be written as $X=X_1\cup X_2$, where $X_1,X_2$ are both open and $X_1\cap X_2=\emptyset$. Otherwise, X is called disconnected. Is it wrong to ...
3
votes
1answer
102 views

Maximal Ideals in $R=\{a+bi:a,b\in \mathbb Z\}$

I've read similar question but please this is not duplicate of Maximal ideals in the ring of Gaussian integers because the answer to it contain PID which I've not yet done etc. $R=\{a+bi:a,b\in ...
2
votes
0answers
39 views

Does a random binary sequence almost always have a finite number of prime prefixes?

Does a random binary sequence almost always have a finite number of prime prefixes? Specifically, let $x = \sum_{1 \le i}{2^{-i} \cdot x_i}$ with $x_i \in \{0,1\}$ be a random real in $[0,1)$, $X_i = ...

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