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Puzzled in Parametric Linear Regression Matrix Solution

If Ynx1 = Xnxpθpx1, then θ = (XTX)-1XTY. But this doesn't work IRL (e.g. X = [1,2,3]T, Y = [2,4,5]T). I cannot think of why (it should be obvious though). Any help on this part? Thank you!
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Let $2,1+\frac{1}{2},3,1+\frac{1}{3},4+\frac {1}{4},...$ be a sequence does $a_n$ converge/diverge? is there a $sup$ or $inf$?

Let $2,1+\frac{1}{2},3,1+\frac{1}{3},1+\frac {1}{4},...$ be a sequence then which of the statements is true? $a_n$ coverges to a finitie limit or diverges to infinity $\lim \limits_{n \to \infty} sup ...
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A Lipschitz condition

Can we find for any given $\varepsilon>0$ an open subset $A\subseteq[0,1]^2$ with measure $>\frac{1}{100}$ such that, for any smooth curve $\gamma:[0,1]\to\mathbb{R}^2$ of length $1$, the set $\...
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Mutual Information between $v_1$ and $v_2$ coming from the same Inverse-Wishart distribution?

Say that $\left(\begin{matrix} v_1 & c\\ c & v_2 \end{matrix}\right)$ is a bivariate covariance matrix that comes from an Inverse-Wishart distribution $W^{-1}(\Psi, \nu)$. Then what is the ...
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Proof the Moreau Envelope of a proper convex function h is minimised at x iff h is minimised at x

Let h:$\mathbb{R} ^n \rightarrow \mathbb{R}\cup {+\infty}$ be proper convex (i.e convex and has at least 1 finite element in it's range) and differentiable at $prox_{\lambda h}(x)$ (the minimiser of $...
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6 views

Straight line in projective plane is plane in vector space (Road to Reality , page-343)

How do we construct an n-dimensional projective space $P^n$? The most immediate way is to take an $(n + 1)$-dimensional vector space $V^{n+1}$, and regard our space $P^n$ as the space of the 1-...
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Show that the vector field always admits a potential function.

Let $r = ||X||$. Let $g$ be a differentiable function of one variable. Now to show that the vector field defined by $$F(X) = \frac{g'(r)}{r} X$$ in the domain $X \neq 0$ always admits a potential ...
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riemann integrability: If P is a partition of [a,b], Is there always a choice of points q, for P such that $ \sigma(f,P,q) = \underline{s}(f,P)$

If P is a partition of [a,b], Is there always a choice of points q, for P such that $$ \sigma(f,P,q) = \underline{s}(f,P) $$ where $ \underline{s}(f, P) $ is the lower sum by riemann integrability
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What is range of values that the word 'nearby' supposed to represent in this informal definition of continuity.

In my book it gave two informal explanation for the concept of continuity. I had doubt in the second explanation but I cleared it by asking it here. The explanation is , Suppose a function f has the ...
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Durrett's Probability: Theorem 6.2.6

I am having some difficulty understanding the concept of measure preserving, invariance, and ergodic. Here is a proof from Theorem 6.2.6 in Durrett's Probability: Theory and Examples, which states ...
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Mobius inversion formula proof

This is a part of the proof for mobius inversion formula I don't understand how do we get the second and third line. Please explain \begin{aligned} \sum_{d|n} \mu(d) F(n/d) &=& \sum_{d|n}\mu(d)...
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Muntz-Szasz theorem: assumption measure is concentrated in (0,1]

I'm studying the proof of the Müntz-Szasz theorem of Rudin's book.. They define the function $$f(z)= \int_{I} t^z d\mu(t)$$ and we may assume that the measure is concentrated in (0,1]. But why is this ...
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$ E(Var(Y|X)) \leq (1- \rho^2)Var(Y) $

Let $\rho$ be the Pearson correlation coefficient of the random variables X,Y , prove that $$ (*) E[Var(Y|X)] \leq (1- \rho^2)Var(Y) $$ Since $\rho$ exists we know that $Var(X) \neq 0 \neq Var(Y)$ ...
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Autocorrelation Function of Bessel Function

I want to ask that after I take ACF of a Bessel Function and plot it in MATLAB, the shape of the new plot will still be that of a Bessel Function with just one modification that now the graph will be ...
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An ellipse tangent to two circles

A circle is tangent to the circles x 2 + y 2 = 4 and x 2 + y 2 = 12y + 64. Find the locus of its center. I tried to solve this using the system of equations but I cannot seem to arrive to the book'...
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Locally compact and perfect map

Suppose $f:X \rightarrow Y$ is a perfect map between topological spaces $X$ and $Y$, i.e. $f$ is continuous, surjective and closed. Also suppose that the fibers of $f$ are compact. If $Y$ is locally ...
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11 views

How to visualize the graph of multivariable functions, namely functions from R^2 to R?

I'd like to get a graphical approach to Analysis in higher dimension vector spaces, such as \mathbb{R^n}. To make this easier, my goal is to be able to visualize the graph of functions from \mathbb{R^...
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27 views

References for me to understand current approaches to settle $P$ vs $NP$

I am an undergrad student that likes to study approaches to settle $P$ vs $NP$. I know that there is GCT method, and another way is to attack it by logic equivalent. I am a double major student in CS ...
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1 vote
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24 views

On the eigenvalue problem

A standard approach to the eigenvalue problem is as follows. We are looking for solutions $\omega$ of the following: $\Omega\vec v=\omega\vec v \tag{1},$ where $\vec v$ is some element of our vector ...
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  • 599
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Proving Tensor identities using Penrose diagrammatic notation (Page-322, Penrose, Road to Reality)

I want to show that for an anti-symmetric two upper index tensor $S^{ab}$, that: $$Q^{bcd}=S^{a[b} \nabla_a S^{cd]}=0$$ I'm trying to figure out how to show prove the hint equality is true using the ...
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Fractional powers of the Laplacian

I have recently started to study fractional fractional powers of the Laplacian. In the books I've read, fractional powers are defined only for $$-\Delta =-\sum_{j=1}^n\frac{\partial^2}{\partial x_j^2}....
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Linear Algebra SVD [closed]

Can you please help me with proving this? enter image description here Thanks in advance.
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28 views

Show: $\text{lcm}(a,a+p)=\text{lcm}(b,b+p), p \;\text{prime}\implies a=b$

(Romania Mathematical Olympiad). Let $a,b$ be positive integers such that exists a prime $p$ with the property $lcm(a,a+p)=lcm(b,b+p)$. Prove that $a=b$. What I could do: WLOG $p|a, p \nmid b \...
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Shorthand set builder notation?

I was reading some papers and they had the following notation: $$ \big\{k\big\}_{k=0}^n $$ I assume this implies that $$ \big\{k\big\}_{k=0}^n = \{k \in \mathbb{WHAT} : 0 \leq k \leq n\} $$ What is $k$...
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  • 513
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Is $f(x) = 1 , when \ 0 ≤ x ≤ 1 \ and \ 0, when \ 1 < x ≤ 2$ Riemann integrable?

Let $$f(x) = \begin{cases}1 , & 0 ≤ x ≤ 1 \\ 0, & 1 < x ≤ 2.\end{cases}$$ I know that it is Riemann integrable, but don't know how to prove it. $U(f, P) - L(f, P) < \epsilon$ I have ...
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Solve $a^{3}+(a+1)^{3}+\ldots+(a+6)^{3}=b^{4}+(b+1)^{4}$

I tried using mod of $6$ on $6$ consecutive integers as they would have $0,1,2,\dots,5$ as residues and tried checking for residue of $b^{4}+(b+1)^{4}$ mod $6$ but wasn't able to get any satisfactory ...
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Creating non-intersecting linear Diophantine equations

I have a problem involving linear Diophantine equations which I know should be fairly simple, but I've forgotten the math. Let $r(t)$ and $w(t)$ be two linear, partial Diophantine equations: $$ r(t) = ...
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1 vote
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Formula connecting Fourier transforms of function and it's derivative

Suppose $f: \mathbb{R} \to \mathbb{R}$ is continuous, differentiable and suppose $f$ and $f'$ are both summable. Then $$\mathcal{F}(f')(x) = -ix \mathcal{F}(f)(x)$$ (Here $\mathcal{F}$ represents ...
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-3 votes
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15 views

Statistics with expected values

Two teams meet in a game of soccer, team A is the favorite are in the lead of the league, on average they score two goals per game and let in one goal. Team B is in the middle of the league. How to ...
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Non singular point of a projective plane curve

In problem 5.1 of Fulton's Algebraic curves, we're asked to show that a point $P\in\mathbb P^2$, $P=[P_1:P_2:P_3]$ is multiple iff $F(P)=F_X(P)=F_Y(P)=F_Z(P)=0$, I guess it's supposed to be easy but I'...
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3 votes
3 answers
26 views

Let $a_n$ be a positive sequence prove that if $\lim \limits_{n \to \infty} \sqrt[n]{a_n}>1$ then $\lim \limits_{n \to \infty} a_n = \infty$

Let $a_n$ be a positive sequence Which of the following statements is true? if $\lim \limits_{n \to \infty} \sqrt[n]{a_n}>1$ then $\lim \limits_{n \to \infty} a_n = \infty$ if $a_n$ converges or ...
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A polynomial f(x) of degree 10, has all its roots in A.P. with 1 being the smallest root and common diffenrence 2. then [closed]

A polynomial f(x) of degree 10, has all its roots in A.P. with 1 being the smallest root and common difference 2. then
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Show that $\langle a\rangle$ is a normal subgroup of $\langle a,b\rangle$ where $a$ and $b$ are permutations

I'm studying for an exam and in a previous year we had the question: Let $a=(12345678)$, and $b=(26)(48)$ and $G=\langle a,b \rangle$. Show that $\langle a \rangle$ is a normal subgroup of $G$ and by ...
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1 answer
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For $f : \mathbb{R} \to \mathbb{R}$, what is $\sup\limits_{x \in A} f(x)$ if $A=\emptyset$? [duplicate]

Let $f : \mathbb{R} \to \mathbb{R}$ and $A$ be an empty set. What is $\sup\limits_{x \in A} f(x)$ equal to? I reckon it should just be undefined or is there an alternative convention?
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1 vote
1 answer
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Line Integral. Vector field. Parametrization

Let´s say we want to find the circumferences of the plane $C$ that make the line integral $\int_C y^2dx + x^2dy$ worth zero. My attempt: The vector field given is: $\mathbf{F}(x,y)=(y^2,x^2)$. The ...
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  • 15
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Decompose a positive contraction in a continuous masa in $L(H)$

Let $A$ be a continuous masa in $L(H)$ and $T$ be a positive contraction in $A$. Then we can assume that $0<\|Th\|<1$ for all unit vectors $h\in H$. Otherwise decompose $T$ as $P+T_0$ for some ...
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1 vote
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Gauss Bonnet for locally outermost closed minimal surface in Hawkings Theorem

So I'm trying to understand the proof for the following theorem from Lan-Hsuan Huang: "Trapped Surfaces, Topology of Black Holes, and the Positive Mass Theorem": Any orientable locally ...
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Are stochastic processes defined on product of probability spaces?

The following question has confused me lately. Suppose that you have a sequence of random variables $\{\xi_n(\omega)\}$ defined on some probability space $\left( \Omega,\mathcal{F},\mathbb{P} \right)$,...
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1 answer
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A is noetherian and every prime ideal of A is maximal then...

This proof was part of my lecture notes in Commutative algebra and I am having trouble following it. So, I am asking the question here. Statement: If A is noetherian and every prime ideal of A is ...
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  • 1,496
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Finding a subset of the powerset whose union is a subset

Let $P_3(S)$ denote the powerset of $S$ where all elements of length greater than $3$ are removed. Given a subset of $S$, $E$; how many subsets of $P_3(S)$ $R$ are there such that the union of the ...
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15 views

Causal discovery for pairwise independent joint dependent variables

Consider the standard example for variables that are pairwise independent but joint dependent. $$ (x,y,z)= \begin{cases} (0,0,0) & \text{probability 1/4} \\ (1,1,0) & \text{probability 1/4} \\ ...
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1 vote
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Semi-direct product in Tao's proof of Gromov's theorem

Terrence Tao provided an elementary proof of Gromov's theorem (https://terrytao.wordpress.com/2010/02/18/a-proof-of-gromovs-theorem/). I have been working my way through the proof and am stuck at the ...
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1 answer
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Question in the proof that any Artinian ring is noetherian

This theorem is from my lecture notes of Commutative Algebra and I am struck on 2 points of the proof. Statement: Any artinian ring is noetherian. Proof: Let A be an artinian ring. Let $M_1 ,...,...
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  • 1,496
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In this informal definition of continuity (at $x=p$), what does "regardless of the manner in which $x$ approaches $p$" mean?

In my book it gave the informal definition of continuity as If we let $x$ move toward $p$, we want the corresponding function values $f(x)$ to become arbitrarily close to $f(p)$, regardless of the ...
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1 answer
18 views

Matrix entry notation and plane coordinate notation

Let $A$ be an $m \times n$ matrix, and $a_{ij}$ a matrix entry. Usually the index $i$ indicates the row and $j$ the column of the matrix. So changing the index $i$ makes you move in the $vertical$ ...
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2 answers
35 views

Notation: function returning the element of a partition containing $x$

Suppose $Y=\{y_1,\ldots,y_m\}$ partitions the set $X=\{x_1,\ldots,x_n\}$. I would like to define a function $y: X \to Y$ which returns $y \in Y$ if and only if $x \in y$. Is there a way to write this ...
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1 vote
1 answer
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Show that the definition of Scott topology is a topology

To verify the definition of a Scott topology is a topology, I still need to show that it's closed under intersection. Can someone help? Definition 1 (Scott topology) Let $(D,\leq)$ be a complete ...
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  • 19
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Give an equation for a line that contains no lattice points.

Give an equation for a line that contains no lattice points. Explain how you know it contains no lattice points. This problem is taken from section 5.1 exercise Q.9. of book: Number theory, by: James ...
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  • 3,413
3 votes
1 answer
41 views

Lebesgue Dominated Convergence Theorem, $\sin(\frac{x}{n})$ example

We have to evaluate the limit of a sequence $\{f_n\}$ which is $$\lim_{n\rightarrow \infty }\int \frac{n\sin(x/n)}{x(1+x^2)}\,dx$$ using Lebesgue Dominated Convergence Theorem. The hint is: $\int \...
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3 votes
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Are there tensor structures other than a metric which could be defined on a manifold which imply a connection through compatibility criterion?

If we say our connection is torsion free, then the metric compatibility condition completely determines it. While this is geometrically intuitive way to do it, are there other interesting tensor ...
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