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Questions tagged [self-learning]

The process of studying mathematics without formal instruction. Don't use this tag just because you were self-studying when you came across the mathematical question you're asking; it is only for when the fact that you're self-studying is what your question is about.

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Show the properties of a spherically symmetric function.

Let $$K^s (\mathbf{x})=c_{k,d}k\{(\mathbf{x}^T\mathbf{x})^{1/2}\}$$ where $c_{k,d}^{-1}=\int_{\mathbb{R^d}}k\{(\mathbf{x}^T\mathbf{x})^{1/2}\}d\mathbf{x}$, $\mathbf{x}=(x_1,\cdots,x_d)^T\in \mathbb{R^...
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Matrix integration problem $\int g(Ax)dx=|A| \int g(y)dy$

In p. 96 of Wand & Jones' (1995) book they asserted that the following equation is valid for linear changes of variables $$\int_{\mathbb{R^d}} g(A\mathbf{x})d\mathbf{x}=|A| \int_{\mathbb{R^d}} g(\...
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0answers
13 views

Least squares estimate of $\beta_k-1$ in multiple linear regression model

Question Let $y_i=\Sigma^k_{j=0} x_{ij} \beta_j+\epsilon_i$ $\epsilon_i$ is $NID(0,\sigma^2)$ and $x_{ij}, i=1,...,n, j=0,...,k$ is the $(i,j)^{th}$ elelement of the $n \times (k+1)$ matrix $X$, ...
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2answers
27 views

Proof that W is a subspace of V vectorial space

Let $ V = \mathbb{R}^3 $, considering $ W = \{ (x,y,z) \in V: x \leq y \leq z \} $. Is W a subspace of V? To proof that I have to demonstrate: 1) W is nonempty, that means $W \ne \emptyset$ 2) if $ ...
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1answer
36 views

Silly doubt regarding onto function

Let function $f:(0,\infty)\to(0,\infty)$ be defined as $f(x)=|1-\frac{1}{x}|$. Then is it onto function? My doubt is that here the codomain doesn't include $0$ but here in the function $0$ is there ...
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1answer
23 views

Combinatorics, number theory, and graph theory, in what order should they be learnt?

Combinatorics, number theory, and graph theory, in what order should they be learnt? I would like to learn discrete math; these are the three main branches that compose it. I was wondering if there's ...
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1answer
30 views

Second derivative Test becomes zero

Consider the domain $D = \{ (𝑥,𝑦) ∈ ℝ^2:𝑥 ≤ 𝑦 \}$ and the function $ℎ: 𝐷 → ℝ$ defined by $ℎ((𝑥,𝑦)) = (𝑥 −2)^4 +(𝑦−1)^4$, $(𝑥,𝑦) ∈ 𝐷$. Find the minimum value of $h$ in the domain $D$: a) $...
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Bregman projection

Given a convex body $K$ and a point $y$ outside the convex body (in the ambient space), the Bregman projection of $y$ , with respect to the regularizer $R$, is defined as $x=\rm{argmin}\{B_{R}\left(\...
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1answer
37 views

Derivative of a trace with Hadamard division

I am trying to solve the derivative: $$\frac{\partial Tr\,[AX'(X \oslash B)]}{\partial X},$$ where $\oslash$ is the symbol used for the Hadamard division (or element-wise division) and A is a square ...
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How do I learn undergraduate mathematics well? [closed]

I am Quasar. A quick two-liner about my background - worked as programmer for five to six years, landed my first international assignment as a quantitative analyst in an investment bank, currently ...
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1answer
70 views

How do you manage your study time at university? [closed]

I'm studying 3rd year maths at university and havent done that well in my first semester. I'll be starting 2nd semester next week on 4 maths subjects. I want to study through everything in my 3 maths ...
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2answers
48 views

Antiderivative of $g(x)dg(x)$

I am reading a book by Shreve "Stochastic Calculus for Finance II" and after computing a stochastic integral $\int_{0}^{T}W(t)dW(t)$ where $W(t)$ is a Brownian motion he compares it to the integral $$\...
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0answers
42 views

Jacobian and area differential

A transformation T (u, v) is said to be a conformal transformation if its Jacobian matrix preserves angles between tangent vectors. Consider that the vector $\langle 1,0\rangle$ is parallel to the ...
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5answers
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How to catch proof errors during self study?

I completed a Bachelor's in Mathematics May 2018 with a 3.6 major GPA. I had trouble with real analysis, scoring B-, B, B, B+ in the four courses I took on the subject despite significant effort and ...
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1answer
28 views

Number of functions between two sets, with a constraint on said functions

Let $A=\{1,2,3,4,5,6\}$ and $B=\{a,b,c,d,e\}$. How many functions $f: A$ to $B$ are there such that for every $x$ belonging to $A$, there exists one and only one $y$ in $A$ such that $x$ is not equal ...
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How should I go about reading mathematics papers and textbooks as a PhD [migrated]

It shames me to admit it, but I feel like I still haven’t figured out the right way to efficiently read papers and textbooks. I am a second year PhD student in pure mathematics, and I struggle a lot ...
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1answer
20 views

Finding base of $\mathbb{R}^3 / U$ (U being a known subspace of $\mathbb{R}^3$)

If $U$ is a subspace of $\mathbb{R}^3$ spanned by the vectors $\{(2,0,-1),(1,2,0),(0,4,1)\}$, how can I find the basis for $\mathbb{R}^3 / U$? I'm having trouble understanding how one can define a ...
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1answer
87 views

Book on stable homotopy theory?

Currently I know nothing about stable homotopy theory other than that it originated from the Freudenthal suspension theorem. But I believe that the following are studied in this field: spectrum, ...
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1answer
36 views

Common Doubt: What did I do wrong here?-Number of ways to arrange green and blue bottles…

Number of ways in which $7$ green bottles and $8$ blue bottles can be arranged in a row if exactly $1$ pair of green bottles is side by side, is ______ Note-Assume all bottles to be alike except ...
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0answers
59 views

Does an isomorphism induce a diffeomorphism?

Consider two isomorphic real vector spaces, $V$ and $W$. Suppose that $V$ gives a (local) chart for a manifold $M$, and that $W$ does the same for a manifold $N$. Does that mean that $M$ and $N$ are (...
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2answers
98 views

Integration problem solving without contour integration

Can the following question be solved without using contour integration. $F:(0,\infty)\times (0,\infty)\to \Bbb R$ be given by $F(\alpha,\beta)=\displaystyle\int_0^\infty\frac{\cos(\alpha x)}{x^4+\...
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1answer
29 views

Steps used to solve this riccati differential equation?

I am reading through examples of linear filtering problems for SDE's, and the process first requires solving a (deterministic) riccati differential equation. In one of the examples, this is given by: ...
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0answers
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Bregman divergence and its property

Suppose, $S$ is a convex set in $\mathbb{R^n}$and $ri(S)$ denotes the relative interior set of $S$.For, $x_1\in S$ and $x_2,x_3 \in ri(S)$ the following property holds \begin{equation} d_{\phi}(x_1,...
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1answer
174 views

Frustration when trying to learn a new topic deeply

When trying to learn a new topic in mathematics I consistently find myself extremely frustrated because I find that far too often, definitions are given without providing context for why they are ...
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1answer
32 views

Grimmett & Stirzaker 7.11.31: Log-likelihood estimation of Markov transition matrix

I have trouble understanding the notation in exercise 7.11.31: I don't understand what $\lambda(\textbf{P})$ is. Lets take the factor $f_{X_0}$. My current understanding is as follows. We know that $\...
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4answers
469 views

Proving the following quadratic inequality? [duplicate]

Apologies if this has been asked before - I could not find a question with this exact inequality. Basically the inequality is $$(a+b+c)^2 \leq 3 a^2 + 3 b^2 + 3 c^2$$ Expanding it out we see that $...
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1answer
44 views

Euler Equations having double roots solutions

If the Euler equation have the double roots as a solution, second solution will be $y_2(x)=x^r\ln{x}$. What is its proof? or how it can be derived? To find a second solution,we will use the fact ...
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3answers
79 views

The number of integral solution of $\alpha+\beta+\gamma+\delta$=18 such that..

Question The number of integral solution of the equation $$\alpha+\beta+\gamma+\delta=18$$, with the conditions: $1\leq\alpha\leq5$; ${-2}\leq\beta\leq4$; $0\leq\gamma\leq5$ and $3\leq\delta\...
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1answer
59 views

Self-Teaching from Big Rudin

first post here so sorry if this question is answered somewhere, but I couldn't find it. I'll start with some background before the question: I've recently completed an undergraduate math degree, ...
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2answers
53 views

How to approach on this - finding minimum distance of point on the ellipse from the centre of it.

Question The minimum distance of any point on the ellipse $$x^2+3y^2+4xy=4$$ from its centre is ______. Attempt Converted the given expression into $$(x+2y)^2-y^2=4$$. But, this becomes equation of ...
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1answer
45 views

The range of values of $a$ such that…

Question The range of values of 'a' for which the common tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ and the parabola $y^2=4x$ and their chord of contact can form an equilateral ...
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2answers
32 views

Completeness of a metric on measurable functions into a complete separable metric space

So this problem comes from Achim Klenke's Probability Theory: A comprehensive course that I am going through on my own. It appears as Exercise 6.2.1 in a chapter on convergence theorems. I have tried ...
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0answers
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Meaning of $L^2 (\mathcal{F}_T, P)$?

The space $L^2(\mathcal{F}_T, P)$ is used in my textbook, but does not seem to be defined anywhere that I can find (and is ommited in the symbols table at the end of the book). Here, $P$ is a ...
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0answers
9 views

Overall Type 1 Error For Multiple Regression Models In Study

The concept of Type 1 error when a study is investigating multiple questions (i.e. multiple models) is something that eludes me, and I can’t seem to find much information on it. I know that the set ...
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1answer
104 views

Logistic Regression: Asymptotic confidence interval for the lethal dose

For the logistic model: $$\log \Big( \frac{\pi(x)}{1-\pi(x)}\Big) = b_0 +b_1x$$ I want to construct a asymptotic confidence interval for the ratio of the m.l.e's of $b_0$, $b_1$: $LD50 = -\frac{\...
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1answer
63 views

How to represent characteristic polynomial in terms of those of invariant direct sum subspaces? [closed]

Suppose $V$ is a complex vector space and $V_1,...,V_m$ are nonzero subspaces of $V$ such that $V = V_1 \oplus ... \oplus V_m$. Suppose $T \in \mathcal{L}(V)$ and each $V_j$ is invariant under $T$. ...
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1answer
17 views

Interchange of differentiation and summation

I came across an example about interchange of differentiation and summation. Can anyone show me how to prove the equation in the picture? Thank you!
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1answer
58 views

Finding sum of digits of $m$ [closed]

If the sequence of 5 positive integers (a,b,c,d,e) satisfy: $$abcde\leq {a+b+c+d+e} \leq 10m$$ then find the sum of digits of m. I don't know how to approach this question. I know it's not a good ...
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1answer
24 views

Representing a martingale as the conditional expectation w.r.t filtration?

Let $M_t$ be an $\mathcal{F}_t$ martingale such that $\sup_t E|M_t|^p < \infty$ for some $p>1$. Show there exists a $Y\in L^1(P)$ such that $M_t = E[Y|\mathcal{F}_t]$. I am not sure how to ...
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1answer
55 views

Finding supporting planes and bounds for function

I am working with this function: $$f(x)=x_1^2+2x_2^2+3x_3^2$$ With the $\min f(x)$ in $-10<=x_i<=10, i=1,2,3$. I was given two points: $p_1=(1,1,1)$ and $p_2=(-1,2,1)$. Using these points I ...
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0answers
81 views

Self Study Real Analysis from this series of books by Jacob & Evans?

Once again a question about self-studying real analysis by an amateur graces these forums, but I am fairly certain these authors have not been mentioned here so far and I would appreciate your ...
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1answer
32 views

Every finite $\sigma$-algebra is of the form…?

Let $\mathcal{F}$ be a finite $\sigma$-algebra. The problem asks to show there exists a partition $G = \{ G_1,\dots,G_n \}$ of $\Omega$ such that for all $A \in \mathcal{F}$, $A$ is the union of all ...
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2answers
64 views

The PDF of $X^3$ where $ X \sim \text{Normal}(0, 1)$

I have been stumped for a few days on this...It would be great if anyone can point me to enlightenment :) Here's what I have tried. Let $Y = X^3$, where X is a standard normal distribution with mean ...
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1answer
27 views

Symmetric matrices property

Reading "Mathematical Physics: Classical Mechanics" by A. Knauf, I found the following statement: The positive symmetric matrices with determinant 1 can be written as $$ \begin{vmatrix} A ...
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1answer
32 views

Closed subset of $W^{1,2}([a, b], \mathbb{R})$

In the context of weak solutions of boundary value problems, I want to show that the set $$\{u \in W^{1, 2}([a, b], \mathbb{R}) \; : \; u(a) = 0 = u(b) \}$$ is closed in $W^{1,2}([a, b], \mathbb{R})$. ...
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0answers
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What does this notation ($Y = \mathrm dx$) in probability mean?

I'm reading this research article and can't seem to understand what this notation mean, $P(T = i) = \int P(X>x)P(Y = \mathrm dx) $ where $T$, $X$, $Y$ are all obviously random variables. ...
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1answer
112 views

Having problem with tom Dieck's algebraic topology text

(An online PDF of the text Algebraic Topology by Tammo tom Dieck can be found here.) This question is really soft. I'm having problem reading this text. Let me elaborate. I found this book too ...
2
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0answers
25 views

How is this equality established? (binomial/factorials)

In Rick Durrett's book, in a proof for the asymptotic behaviour of Poisson to normal, he uses the following identity: $$\frac{n!n^m}{(n+m)!} = \left(\prod_{k=1}^m 1 + k/n \right)^{-1}$$ I'm just ...
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1answer
58 views

What is the best way to learn mathematics? [closed]

As the title, I concern about what is the best way to learn mathematics. I've asked some of the professors in my university. Some said that I should intuitively understand mathematical concepts and ...
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1answer
28 views

Error when trying to derive variance of sample mean

Assume that $X$ is a random variable with mean $E[X]$ variance $\sigma^2$. Let $\mu = \frac{1}{N}\sum_{n=1}^N X_n$, where $X_n$ is i.i.d with respect to $n$ having mean $E[X]$ and variance $\sigma^2$, ...