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Solution to the Helmholtz equation with a specific form

I want to find a function $f$ that solves the Helmholtz equation $$ \Delta_w f(x,w)+\lambda f(x,w)=0, $$ and satisfy the natural Neumann boundary condition. However, I am looking for a specific form ...
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Let $f_1:A \rightarrow B$, $f_2:C \rightarrow B$ functions. Prove that, if $f_1(x)=f_2(x)$ for all $x\in A\cap C$, then $f_1 \cup f_2$ is function.

I want to test the above, so my idea is as follows: take $f=f_1 \cup f_2$ and prove that $f:A\cap C \rightarrow B$ is function. To do this, we start with $x \in A\cap C$, then $x \in A$ and $x \in C$, ...
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  • 633
-2 votes
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prime factors of $n=x^2+y^2$

Let $n$ be a positive integer. Show that there exist integers $x$ and $y$ such that $n=x^2+y^2$ if and only if each prime factor of $n$ of the form $4k+3$ appears an even number of times
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Smooth functions on a compact set

Let $K:=\overline{U}$ be the closure of $U$ open bounded connected set in $\mathbb{R}^n$. A definition of $C^k(U)$ is (Evans, Partial differential equations, p. 618): $$S_1=\{u \in C^k(U), \partial_iu ...
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  • 105
-5 votes
0 answers
11 views

how to find the following sets

In the following cases find the sets $$\bigcup_{k∈N} B_k$$ and$$\bigcap_{k∈N} B_k$$ when $$B_k = \{k-1,k,k+1\}$$ I think the correct answer is $$\bigcup_{k∈N} B_k = \{k-1\} \cup\{k\} \cup\{k+1\}$$
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8 views

Sum of $2$-torsion points on a particular elliptic curve

I'm running into some trouble trying to get some practice working with elliptic curves. Say $F$ is some field and $\lambda\in F$ is an element with $\lambda^3 \neq 27$. Let $E$ be the elliptic curve ...
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  • 1
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P-negligible sets and completeness

Obviously, every process $X$ is adapted to $\{ \mathscr{F}_t ^X \}$. Moreover, if $X$ is adapted to $\{ \mathscr{F}_t \}$ and $Y$ is a modification of $X$, then $Y$ is also adapted to $\{ \mathscr{F}...
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3 views

The Hilbert norms of KRR solutions

Assume that we solve a kernel ridge regression problem for data set $X$ of size $n$ $$ \hat{g} = \mathrm{arg} \min_g \sum_i^{n}(y_i - f(x_i)) + ||g||^2_{\mathcal{H}} $$ and for extended dataset $X_*$ ...
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14 views

Algorithm verification: Get all the combinations of possible words

I wanted to know if the algorithm that i wrotte just below in python is correct. My goal is to find an algorithm that print/find all the possible combinaison of words that can be done using the ...
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show there are at least two students a and b so that a scores at least as many as b for each problem

49 students solve a set of 3 problems. Each problem is marked from 0 to 7. Show that there are two students A and B such that A scores at least as many as B for each problem. I saw this post on AoPs ...
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  • 339
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6 views

Positivity of adjoint of unbounded operator

Suppose $A$ is a densely defined, symmetric, positive operator on a Hilbert space. Is there any additional condition in terms of $A$ that forces the adjoint $A^*$ to be also positive?
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  • 107
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How to rigorously show that the maximum variance of an hermitian matrix is $\left( \frac{h_{max}-h_{min}}{2} \right) ^2$?

Suppose we have an hermitian matrix $H$ with eigenvalues $h_i$, and I want to get the minimum value of its variance, i.e. $\vec{v}^{\dagger}H^2\vec{v}-\left( \vec{v}^{\dagger}H\vec{v} \right) ^2$. How ...
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  • 145
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1 answer
11 views

Prove. If $r$ and $s$ are bisquare, then $rs$ is bisquare

I am currently learning direct proofs. I couldn't solve the following exercise. Define an integer $m$ to be bisquare iff, $\exists a \in Z, \exists b \in Z, m = a^2 + b^2$. Let $r$ and $s$ be fixed ...
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  • 161
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26 views

Find integer solution of $\frac{1}{x} + \frac{1}{y} = \frac{1}{2020}$ [duplicate]

Find all pairs of integers $(x,y)$ for $$\frac{1}{x} + \frac{1}{y} = \frac{1}{2020}$$ Attempt: Notice $x,y \ne 0$. $$ \frac{1}{x} + \frac{1}{y} = \frac{1}{2020} $$ $$ \frac{x+y}{xy} = \frac{1}{2020} $...
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  • 4,330
1 vote
0 answers
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Particular patterns for zeros of high-degree named polynomials

A polynomial may have its roots at any location in the complex plain. However various famous polynomials have their roots on certain curves. My question is why, and I am looking for references in this ...
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  • 3,543
1 vote
0 answers
9 views

Orientation of an $n$-pseudomanifold

I was reading the following paper which had the following useful definition of an oriented $n$-pseudomanifold (Recall, an $n$-pseudomanifold is a simplicial complex where each $n-1$-simplex is ...
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  • 11
1 vote
0 answers
9 views

Two Stage Least Squares and Instrument Variable

I have a question I am little unsure about We have linear regression model $y_{i}=B_{0}+B_{1}x_{i1}+B_{2}x_{i2}+u_{i}$ where $x_{i1}$ is an endogenous regressor and $x_{i2}$ is an exogenous regressor (...
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1 vote
1 answer
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Is the cosine angle between two R.V. an (approximation) not equality to the correlation coefficient?

I have seen in websites that given two R.V. $X,Y$, if $$ \cos(\theta)=\frac{X\cdot Y}{\|X\|_2\|Y\|_2} $$ and $$ \rho=\frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)\text{Var}(Y)}} $$then $$ \cos(\theta)=\...
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  • 1,100
1 vote
0 answers
10 views

Two questions about convolution

Good day to everyone, I have the following two questions about convolutions of smooth functions with compact suppport ($C_c^k(\mathbb{R}^d)$ or $C_c^\infty(\mathbb{R^d})$). If $f$ from $C_c^m(\mathbb{...
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  • 11
2 votes
0 answers
24 views

$\int_{-x}^x f^\alpha\leq f(-x)+f(x)$

Let $0\leq f\in C(\mathbb{R})$, and for some $\alpha>1$, we have \begin{equation*}\begin{aligned} \int_{-x}^x f^\alpha\leq f(-x)+f(x), \forall\ x\in [0,+\infty). \end{aligned}\end{equation*} Prove ...
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  • 2,052
1 vote
0 answers
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From an explicit function to an implicit one

I've seen a lot of people asking questions about how to go from an implicit expression and get an explicit function. I wonder if we can do the opposite: starting with an explicit function $d(x,y)$ and ...
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1 vote
0 answers
22 views

I have a question and a solution for a Pigeonhole Principle question. Can you explain to me why?

I was quite sure that I have understood the pigeonhole principle, but this question does not make sense. Or does it? The question: A drawer contains $5$ blue socks, $7$ red socks and $6$ black socks. ...
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  • 11
0 votes
0 answers
7 views

Understanding a geometric condition

I am having a hard time understanding the following mathematical formulation regarding Voronoi tessellations. This is inspired by Chapter 2.6.1 in this book. Recall that a Voronoi tessellation is a ...
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  • 3,410
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8 views

Showing a set is open in a topology defined by invariant metric

Let $U\subset\mathbb{C}$ be an open domain and $H(U)$ the space of holomorphic functions on $U$. Define the invariant metric $d(f,g)=\sum_{m=1}^{\infty}\frac{1}{2^{m}}\frac{\rho_{K_{m}}(f-g)}{1+\rho_{...
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  • 2,509
2 votes
1 answer
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I cannot figure out why I get differing results for $\int _\gamma z\,dz$, $\gamma (t)=te^{it}$ when using integration by parts and primitive of $z$

Let $\gamma(t) =te^{it}$, $t\in [0,\pi ]$ and consider $\displaystyle \int \limits _\gamma z\,dz$. Since the primitive of $z$ is $\frac{z^2}{2}$, we can evaluate $\displaystyle \int \limits _\gamma z\,...
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Sequence of orthogonal spaces and projection

The problem is: Let {Mi} be an orthogonal sequence of complete subspaces of a pre-Hibert space V, and let Pi be the projection on Mi. Prove that {Piξ} is Cauchy for any ξ in V Many thanks!
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How does the interpretation of the 1st betti number in 3 dimensions work?

The $0$th Betti number $b_0$ represents the number of connected components. The $1$st Betti number $b_1$ represents the number of holes and the $2$nd Betti number $b_2$ the number of cavities. While ...
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-8 votes
0 answers
26 views

Find the following sets

In the following cases find the sets $$\bigcup_{k∈N} B_k$$ and$$\bigcap_{k∈N} B_k$$ when $$B_k = \{0,1,2,3...2k\}$$
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0 answers
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Why is the given set no topological manifold? Am i right?

Why is the following set no topological manifold? equppied with the subspace topology from $ \mathbb{R}^2 $ $ X = \{ (x,y) \in \mathbb{R}^2| x * y= 0 \} $ X contains all all points like $ (x,0) and ...
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  • 1
0 votes
1 answer
21 views

Flexibility with integration limits

I have an EE background and I am trying to integrate the equation for the voltage across an inductor. I am just playing around with the limits and want to check if all 3 of these equations below are ...
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-1 votes
0 answers
17 views

Strong convergence does not implies weak star convergence

Let $E$ be a normed vector space and $E^*$ its dual space. Denote by $\mathcal{T}_{E^*}$ the norm topology in $E^{*}$, by $\sigma(E^*,E^{**})$ the weak topology in $E^{*}$ and by $\sigma(E^*,E)$ the ...
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  • 299
0 votes
0 answers
28 views

Does the sequence $(Z_n)$ converge or diverge?

Let $(c_n)$ be a real and strictly monotone sequence that converges to a limit $c > 1$. Let's assume that $c_n > 1$ for each $n \in \mathbb{N}$. Let's $\beta$ be a real constant such that $\beta ...
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0 votes
0 answers
20 views

Property of the greatest common divisor

Suppose $1\le a\le n$ and $1\le b\le m$ for which $\gcd(a,n)=\gcd(b,m)=1$. Can we produce an upper bound for $\gcd(a,b)$? I was thinking that there must be some way of solving this given the ...
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2 votes
0 answers
34 views

Can any positive even number be expressed as an XOR of two prime numbers?

I just came up with this question when I was thinking about the Goldbach conjecture. For example, $$2=5 \oplus 7$$ $$4=3 \oplus 7$$ $$6=3 \oplus 5$$ $$8=3 \oplus 11$$ $$10=7 \oplus 13$$ $$12=7 \oplus ...
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4 votes
0 answers
18 views

A question on conditional probability in sde

Let $s \in[a, b]$ and $x \in \mathbb{R}$ be fixed and consider the following SIE: $$ X_{t}=x+\int_{s}^{t} \sigma\left(u, X_{u}\right) d B(u)+\int_{s}^{t} f\left(u, X_{u}\right) d u, \quad s \leq t \...
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-1 votes
0 answers
13 views

Question on proof involving functions and factorisation.

Assume that f(xy) = f(x) + f(y) for all positive integers x and y. Show that if the positive integer n has the factorization n = p^a× q^b× r^c , then f(n) = a f(p)+ b f(q)+ c f(r)
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  • 1
-2 votes
1 answer
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If $H , K \trianglelefteq F_2$ with $F_2/H\cong F_2/K$ then $H=K$

This is probably a basic fact of group theory but I am not able to prove it: Let $F_2$ be the free group generated by 2 elements and $H,K$ be two normal subgroups of $F_2$. If $F_2/H$ is isomorphic ...
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  • 529
2 votes
1 answer
24 views

Computing the log of a sum of exponentials

in a Coursera course by UW I've come across this piece of code computing the log of a sum of exponentials. ...
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  • 677
0 votes
1 answer
46 views

Why is my solution incorrect, where am I missing a concept?

We have a problem from Khan Academy: To solve it I had used online integral calculator(https://www.integral-calculator.com/) in order to solve the integral, and than evaluated the integral at 5. ...
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0 votes
0 answers
21 views

Prove $\forall a,b \in \mathbb{Q} \exists x \in \mathbb{R}\setminus \mathbb{Q}\left(a \lt x \lt b\right)$

I want to prove that $\forall a,b \in \mathbb{Q} \exists x \in \mathbb{R}\setminus \mathbb{Q}\left(a \lt x \lt b\right)$. Would the following proof be correct? Proof: Let $ I = \mathbb{R} \setminus \...
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0 votes
0 answers
9 views

Why isn't the non-differentiablity of a path at some finite number of points $a_1<\cdots<a_m$ points a problem when calculating the contour integral?

My question is quite elementary, but I can't for the life of me remember/find the appropriate result addressing it. Suppose that $\gamma(t), a \leq t \leq b$ is a continuously differentiable path on $(...
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0 answers
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Is the upper triangular matrix algebraic over the center of the division ring?

Let $D$ be a division ring and let $n>1$ be a positive integer. Now, we consider $$A=\begin{pmatrix} 1&\ast&\ast&\cdots&\ast\\ 0&1&\ast&\cdots&\ast\\ \cdots&\...
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0 answers
24 views

Choose to stop or roll a dice 3 times. What is wrong with my reasoning?

Problem: "Suppose we play a game. I roll a die up to three times. Each time I roll, you can either take the number showing as dollars, or roll again. What is your expected winnings?" My ...
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  • 525
3 votes
1 answer
45 views

$2019f'(x)+2020f(x)\geq2021$

Find all continuous function $f:[0,1]\rightarrow\mathbb{R}$ which is differentiable on $(0,1)$ and $$f(0)=f(1)=\frac{2021}{2020}\textrm{ while }2019f'(x)+2020f(x)\geq2021,\forall x\in(0,1).$$ The ...
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0 votes
0 answers
24 views

Is there a transformation that satisfies $f(Ax+By)=x+y$ where x, y are vectors and A, B are matrices?

Given $x, y$ $\in \mathbb{R}^{n}$, and A, B $\in \mathbb{R}^{n\times m}$, is there a transformation $f :\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ that satisfies $$f(Ax+By)=x+y$$ ?
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  • 695
0 votes
1 answer
42 views

Can anyone provide me List of prime numbers which are sum of the cubes of three natural numbers?

Pierre de Fermat discovered that if $p$ is a prime and congruent to 1 mod 4, then it can be written as the sum of the square of two natural numbers. Similarly, I was trying to find the list of those ...
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0 votes
0 answers
11 views

What am I missing in this proof using the projection lemma (LMIs)?

I am trying to apply the projection lemma to combine a specific pair of linear matrix inequalities (LMIs) into one LMI. I thought I had found also a specific (desirable) structure in the slack ...
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0 votes
0 answers
5 views

Quadratic Programming Problem to Second-Order Cone Program

Currently I'm trying to convert Quadratic Programming Problem to Second-Order Cone Program in Matlab and I found this link https://mathworks.com/help/optim/ug/convert-qp-to-socp.html, but I don't ...
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0 answers
12 views

Number of solutions to a diophantine equation with general terms

Let $\gcd(a,n)=1=\gcd(b,m)$. Find the number of pairs $(a,b)\in\mathbb{R}^2$ such that $am-bn=k$ for some integer $k$, where $m,n>1$ are fixed integers. My idea: Define $f(k)$ to be the number of ...
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0 votes
0 answers
11 views

How to intuitively understand cross-product

I have learned about the cross-product of vectors in both mathematical studies and physical ones but I am still curious as to how one can obtain an intuitive understanding of how two vectors create a ...
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