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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5 views
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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0answers
20 views
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 $ ...
1
vote
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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0answers
8 views
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(\...
1
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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},$$
where $\oslash$ is the symbol used for the Hadamard division (or element-wise division) and A is a square ...
1
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0answers
52 views
How do I learn undergraduate mathematics well? [closed]
I am Quasar. A quick two-liner about my background - worked as programmer for five to six years, landed my first international assignment as a quantitative analyst in an investment bank, currently ...
4
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1answer
70 views
How do you manage your study time at university? [closed]
I'm studying 3rd year maths at university and havent done that well in my first semester. I'll be starting 2nd semester next week on 4 maths subjects. I want to study through everything in my 3 maths ...
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2answers
48 views
Antiderivative of $g(x)dg(x)$
I am reading a book by Shreve "Stochastic Calculus for Finance II" and after computing a stochastic integral $\int_{0}^{T}W(t)dW(t)$ where $W(t)$ is a Brownian motion he compares it to the integral
$$\...
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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 ...
13
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5answers
1k views
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 ...
1
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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 ...
6
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0answers
52 views
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 ...
1
vote
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 ...
2
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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, ...
3
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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 (...
2
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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+\...
0
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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:
...
0
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0answers
6 views
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,...
10
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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 ...
1
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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 $\...
5
votes
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
$...
-2
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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 ...
3
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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\...
2
votes
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 ...
2
votes
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 ...
1
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0answers
17 views
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 ...
0
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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 ...
1
vote
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{\...
1
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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 $T$. ...
0
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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 ...
0
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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 ...
0
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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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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
$$
\begin{vmatrix}
A ...
2
votes
1answer
32 views
Closed subset of $W^{1,2}([a, b], \mathbb{R})$
In the context of weak solutions of boundary value problems, I want to show that the set $$\{u \in W^{1, 2}([a, b], \mathbb{R}) \; : \; u(a) = 0 = u(b) \}$$ is closed in $W^{1,2}([a, b], \mathbb{R})$. ...
0
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0answers
24 views
What does this notation ($Y = \mathrm dx$) in probability mean?
I'm reading this research article and can't seem to understand what this notation mean,
$P(T = i) = \int P(X>x)P(Y = \mathrm dx) $ where $T$, $X$, $Y$ are all obviously random variables. ...
5
votes
1answer
112 views
Having problem with tom Dieck's algebraic topology text
(An online PDF of the text Algebraic Topology by Tammo tom Dieck can be found here.)
This question is really soft. I'm having problem reading this text. Let me elaborate.
I found this book too ...
2
votes
0answers
25 views
How is this equality established? (binomial/factorials)
In Rick Durrett's book, in a proof for the asymptotic behaviour of Poisson to normal, he uses the following identity:
$$\frac{n!n^m}{(n+m)!} = \left(\prod_{k=1}^m 1 + k/n \right)^{-1}$$
I'm just ...
0
votes
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 ...
0
votes
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 $\sigma^2$, ...
