Questions about the process of studying mathematics without formal instruction.

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1answer
32 views

On the isomorphisms $(\mathcal{O}_{Z,X})_\mathfrak{p}\cong\mathcal{O}_{Y,X}\text{ and }\mathcal{O}_{Z,X}/\mathfrak{p}\cong\mathcal{O}_{Z,Y}$.

Suppose you have two closed, irreducible subvarieties $Z\subseteq Y$ in some variety $X$. (I'm not sure if it matters, but for ease I'll just assume everything is over an algebraically closed field.) ...
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1answer
47 views

Learning functional analysis and measure theory

I have taken a first course in real analysis and I'm currently studying analysis in $\mathbb{R}^N$ on my own. I want to start functional analysis after this, and I also want to learn measure theory ...
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1answer
29 views

Bound for Outlyingness

Given a sample of $n$ data, $x_1, \dots, x_n$. Define the sample mean $$\bar x := \frac{1}{n}(x_1+\cdots+x_n),$$ and sample variance $$s^2 := \frac{1}{n-1} \sum_{i=1}^n (x_i-\bar x)^2.$$ To measure ...
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0answers
30 views

Continuity of set function on field and relation with continuity in topological space

I am trying to understand how continuity of measures relates to the definition of continuity in topological sets : Every open set in range corresponds to an open set in domain. A real valued set ...
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0answers
30 views

Continued fraction approximation to a function and its derivative

I am recently working on fitting a model with incomplete beta function. In order to put it into my optimization algorithm, I must find out the derivatives of the incomplete beta function $B_p(x,y)$ ...
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1answer
24 views

Convergence of Remainder from Taylor Expansion

For a distribution function $F$ and its variance functional $T(F)$, it can be shown that the Taylor expansion of $T(F)$ at $F$ in the direction of the empirical distribution function $F_n$ gives the ...
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1answer
32 views

Problem of Partial Differential Equations

For this question, I get stuck when I apply the second initial equation. My answer is $θ= Ae^-(kλ^2 t)\cos λx$, where $A$ is a constant. Would anyone mind telling me how to solve it?
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1answer
46 views

Lebesgue-Stieltjes Integral (Several Variables)

Let $\mathcal F$ be a convex set of probability measures or distribution functions and $F, G$ be two elements in $\mathcal F$. Let $T$ be a functional on $\mathcal F$ defined as follows. Note that $h$ ...
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1answer
47 views

Showing that the indicator/characteristic function is not a regulated function

I want to show that the indicator function (aka. the characteristic function) is not a regulated function. \begin{align} \chi : \begin{cases}[a,b] & \longrightarrow \mathbb{R} \\ x & ...
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2answers
125 views

Still forget even if theorem-proof “self-discovered”; Importance of intuition/proficiency of concepts in research work…

It is widely said if we go through concepts/theorems/proof on our own by actively doing instead of passively reading, the idea will be ingrained in mind. I agree with that, it really often helps. ...
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3answers
62 views

Confused about transfinite induction

QUESTION: I seem to be confused about how transfinite induction is carried out. I have looked at several examples and they seem to follow a procedure consisting of grounding the induction, proving the ...
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0answers
26 views

Observed and expected Fisher information of a Bernoulli Random Variable

If $X$ is a Bernoulli random variable with parameter $p$, the probability mass function is given by $$ f(k) = p^k(1-p)^{1-k} $$ and the loglikelihood, $\ell(p)$, is given by $$ \ell(p) = ...
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1answer
54 views

Why is a Hyperplane called a “Hyper”plane?

I just had this curious question. In other fields, the word "hyper" is actually used to refer to something which is "over; beyond; above" as defined by Google. An example of such terms would be ...
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2answers
93 views

How to learn math? [closed]

I am 19 years old and I'm computer programmer and Software Engendering college Student, And I am smart (mean: I am not stupid) and know programing better than other, I think math is like programming. ...
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1answer
37 views

The map $f\colon\mathbb{A}^2_k\to\mathbb{A}^2_k$ given by $f(x,y)=(x,xy)$ is birational?

I'm reading a bit about rational maps, and I'm still trying to get get my head around birational maps. Consider the map $f\colon\mathbb{A}^2_k\to\mathbb{A}^2_k$ on the affine $2$-space over $k$ ...
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0answers
34 views

Videolectures and Spivak's Calculus

I'm reading Spivak's Calculus but I have problems understanding some topics, so I would be glad if someone share with me some Videolectures that will make my self-learning more efficient. Sorry if I ...
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0answers
41 views

Where can I get detailed and comprehensive notes of a functional analysis course taught using the book by Erwine Kryszeg?

Where on the Internet can I find detailed and comprehensive lecture notes of an elementary functional analysis course taught using the book Introductory Functional Analysis with Applications by Erwine ...
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2answers
64 views

A metric that makes $l^\infty$ separable

I know that "The metric space $l^\infty$ is not separable with the metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ as $\sup\limits_{i\in\Bbb{N}}|{a_i-b_i}|$. ...
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0answers
16 views

Help with understanding step in Optimisation Book

I am reading an Optimisation book. My knowledge on multi-variable calculus is minimal. Hence I do not understand the block-quoted step. We take $ \underline w = \underline x^* + t(\underline x - ...
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1answer
85 views

In war with exercise, any future for me?

I love theory with theorems, definitions & proofs, but i don't like exercise, I need more context around it. Is there a different way of practicing theory except given exercises, maybe some ...
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6answers
81 views

What is the value of $a+b+c$?

What is the value of $a+b+c$? if $$a^4+b^4+c^4=32$$ $$a^5+b^5+c^5=186$$ $$a^6+b^6+c^6=803$$ How to approach this kind of problem. Any help. UPDATE: Thank you all for answers. Now I ...
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1answer
52 views

Properties of a differentiable and strictly convex $f:(a,b) \to \mathbb{R}$

Let $f:(a,b) \to \mathbb{R}$ be a differentiable and strictly convex function I tried to explore some of the properties of such a function. For all $x,y \in (a,b)$ with $x \neq y$ I could apply ...
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1answer
34 views

Expectation of multinomial distribution

Three fair dice are cast. In 10 independent casts, let X be the number of times all three faces are alike and let Y be the number of times only two faces are alike. Find the joint pdf of X and Y and ...
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1answer
32 views

Probability of unbiased die

One of the numbers 1,2,...,6 is to be chosen by casting an unbiased die.Let this random experiment be repeated five independent times.Let this random variable $X_1$ be the number of termination in the ...
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1answer
19 views

How do I compute the normalisation of $A=k[X,Y]/(Y^3 - X^5)$?

I'm trying to solve exercise 4.7 in Reid's UCA: "Find the normalisation of $A=k[X,Y]/(Y^3 - X^5)$." I can easily show $A$ is not normal: let $x$ and $y$ denote the images of $X$ and $Y$ in $A$. Thus ...
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2answers
51 views

Prove that $A \subset B$ if and only if $A \setminus B = \emptyset$

Prove that $A \subset B$ if and only if $A \setminus B = \emptyset$. What is the correct and mathematically strict way to prove the above? (slightly different than Prove that if $A \setminus B = ...
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1answer
40 views

How to show that $\lim_{x\to \infty}f'(x)=0$

Let $f$ be a real-valued, bounded, twice differentiable function defined on $(0,\infty)$ with $f'(x)\ge 0$ and $f''(x)\le 0$. Show that $$\lim_{x\to \infty}f'(x)=0$$ I understand $f: (0,\infty) ...
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15answers
2k views

Nobody told me that self teaching could be so damaging…

Even though I've been teaching myself math for a couple of years now I only just started (a month ago) at the university. My experience is rather mixed. For starters, I'd like to mention that I'm 21 ...
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1answer
24 views

Tightness of random variales

If $\{X_n\}$ is a tight family of positive r.v.s. can we say something about $\{f(X_n)\}$ where $f$ is a continuous function?
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1answer
18 views

Expected winnings from a game

A game of lucky dip consists of drawing 2 chips without replacement from a box containing 1 green chip, 2 blue chips and 40 white chips. You are paid according to the type of chips drawn. The green ...
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2answers
46 views

Probability of scratch and win card

A game of “scratch-and-win” is played as follows. You scratch 2 out of 3 covered circular tabs on a game coupon • • • to reveal 2 images. The coupons are of types (A), (B), (C) with images ♥ (heart), ...
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0answers
17 views

Encode A11B modulo 37

Encode the word A11B modulo 37 using the encoding 0=0, 1=1, . . . , 9=9, A=10, B=11, . . . , Z=35, blank space=36. I took the weighted sum: 5(10) + 4(1) + 3(1) + 2(11) + 1(c) ≡ 0 mod 37 Solving, i ...
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1answer
59 views

Sequence in a hilbert space.

$M$ is a closed subspace of the Hilbert space $H$, and x $\in H$. Call $d = \inf_{y \in M} ||x - y||^2$ Show that there exist a sequence of elements $y_n$ of M such that $||y_n - x ||^2 \rightarrow ...
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0answers
16 views

How to check $H_0$ hypothesis using Pearson's criteria?

How to check hypothesis by using Pearson's criteria ( $\chi^2$ test), that $H_0:$ random variable $X$ is normally distributed given that $k=7$ (count of intervals) and $\alpha=0.1 $ (significance ...
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0answers
70 views

Geometry textbook question

I have just started the textbook Geometry: its elements and structure by Alfred Posamentier. The first set of questions refers to the following diagram: The very first question is "What is the ...
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1answer
27 views

Uniformly Converging Sequence but Not Normly Converging

I want to find a sequence $(f_n) \subset \mathcal L^1(\mathbb R)$ of integrable functions which uniformly converges to $0$ but with $\int f_n \nrightarrow 0$. I came up with the following example: the ...
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0answers
29 views

Computation of large powers

How do I check if $2^{123456789}$ is divisible by 9? I tried using modular exponentiation but it is way too tedious. Is there an easier or faster way to solve it? Thanks!
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3answers
37 views

On inversive geometry

I am given the following problem set: Observe the circle $K$ with center $0$ and radius $r$ in the complex plane $ \mathbb{C} \simeq \mathbb{R}^2$. Show that the inversion on $K$ is given by the ...
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0answers
24 views

Deriving the normal distance from the origin to the decision surface

While studying discriminant functions for linear classification, I encountered the following: .. if $\textbf{x}$ is a point on the decision surface, then $y(\textbf{x}) = 0$, and so the ...
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1answer
34 views

Surjective Operators

Let $X$ be a separable Banach space. Show that there is a surjective linear operator $P: L^1(\mathbb N) \to X$ and a subspace $S \subset L^1(\mathbb N)$ such that $$\|P(x)\| = \inf_{y \in S} ...
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1answer
32 views

How to see this is an isometry

Let $X$ be a separable Banach space, and $(f_n)$ be a countable dense subset. Recall that for each $f_n$ there exists a linear functional $l_n \in X^*$ such that $\|l_n\| = 1$ and ...
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1answer
44 views

Cartesian Product of Borel Sets is Borel Again

Let $E$ and $F$ be Borel measurable subsets of $\mathbb R^{d_1}$ and $\mathbb R^{d_2}$, respectively. Then $E \times F$ is also Borel measurable in $\mathbb R^{d_1 + d_2}$. I suppose it is ...
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1answer
44 views

Find a linear operator such that $\|T\| > \sup_{\|f\| \leq 1} \left|\langle Tf, f \rangle\right|$

Find a linear operator $T: \mathbb C^2 \to \mathbb C^2$ such that $$\|T\| > \sup_{\|f\| \leq 1} \left|\langle Tf, f \rangle\right|,$$ where $\langle \cdot, \cdot \rangle$ is the standard inner ...
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0answers
55 views

Adjoint Operator of a Compact Operator

In the proof of the fact that the ad-joint operator $T^*$ of a compact operator $T$ defined on a separable, infinite dimensional Hilbert space $\mathcal H$ is also compact, I read that "$\|P_nT - T\| ...
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1answer
23 views

Union of Two Rectangles is the Disjoint Union of at most $6$ Rectangles

Let $X = X_1 \times X_2$ and suppose that $(X_1, \mathcal M_1, \mu_1)$ and $(X_2, \mathcal M_2, \mu_2)$ are two measure spaces. Consider the set of all rectangles, i.e., sets of the form $A \times B$, ...
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1answer
17 views

Probability Question involving dices

Two fair dice are thrown. Given that the total score obtained is even, find the probability of throwing a double. So I got that the sample space is all the possible outcomes and, |S|= 21. A = Event ...
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1answer
231 views

Is this study plan sufficiently general, or overly specialized? [closed]

My current study plan is in order below. I will be completing these textbooks in this order one at a time. I have been told that I don't have textbooks in my plan that approach topology in a general ...
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2answers
40 views

Probabilty question

You have a bunch of n keys of which only one one opens the door of a storeroom, You wish to get into the storeroom. You choose one key at random and try it. If it does not work, you discard and try ...
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0answers
62 views

Convergence of Expectations (cont'd)

The question is related to this question. Suppose $\{X_n\}$ is a sequence of indep. random variables with zero expectation. Consider their sum $S_n$ which have the following properties: $S_n$ ...
2
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1answer
133 views

Fundamental Matrix

Determine $\phi(x,0)$ for $A(x)=\begin{pmatrix} -1 & \cos(x) \\ 0 & -1\end{pmatrix}$, where $\phi(x,0)t_{0}$ is a solution of $\frac{d}{dx}t(x)=A(x)t(x)$. I am not entirely sure as to ...