Questions about studying mathematics without formal instruction.

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7 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
22 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
11 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
53 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
12 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
58 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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0answers
16 views

Proving convergence of a martingale in $L^2$ [on hold]

I'm stuck with the following problem: Let $X$ a positive martingale bounded in $L^2$. Show that $\lim_{n\to \infty} X_n = X$ a.s. and in $L^2$.
3
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2answers
35 views

Simple everyday math

I am looking for a book that contains (instead of crossword puzzles) math problems beginning with simple addition, subtraction, multiplication and division and progresses to more difficult area ...
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1answer
25 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
25 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
33 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
17 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
32 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} ...
3
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1answer
29 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 ...
0
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1answer
25 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
34 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\| ...
2
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1answer
20 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
16 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 ...
4
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1answer
192 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 ...
2
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2answers
36 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 ...
2
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0answers
58 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
votes
2answers
120 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 ...
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1answer
18 views

Find the volume of the region by triple integral

What is triple integral? How can I sketch region $D$ as well as evaluate it's volume? I get stuck.
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0answers
43 views

Finding dominating integrable function

Hi everyone I'm not completely familiar with this kind of argument and I'd appreciate if someone can help me to see if the argument is correct and also any suggestion to improve it. Thanks in advance. ...
2
votes
2answers
54 views

On sums and identities

I am given the following problem set: (a) the Riemann $\zeta$-function for $s > 1$ is defined through the convergent sum: $$\zeta(s) := \sum_{n = 1}^{\infty} \frac{1}{n^s}$$ show the identity ...
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0answers
18 views

Continuous Strong-Strong Implies Continuous Weak-Weak

Let $X$ and $Y$ be two Banach spaces and let $T$ be a linear map between $X$ and $Y$. Show that $T$ is continuous strong-strong if and only if $T$ is continuous weak-weak. I can see that $T$ ...
10
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0answers
135 views

How beginner researcher or ph. d student should learn new concepts?

I hope this question is o.k for S.E. M. When some one is reading book(monograph); possibly helpful in his/her research, and content is large enough; At the first glance it seems very frustrating and ...
0
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1answer
21 views

Finding an specific counterexample of the interchanging of the limit and integral.

Hi I found the following exercise where I'm stuck, I'd appreciate any help. Thanks in advances Let $0\le f_n\to f$ and $\int f_n\to c>0$ pointwise. Show that $\int \lim f_n d\mu\in [0,c]$ and ...
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1answer
18 views

About continuous functions and continuous continuations and their uniqueness

How would you access the following problem: (a) Show that for every $s \in \mathbb{Q}$ the function $$f: \mathbb{C}^* \rightarrow \mathbb{C}$$ $$ f(z) := \frac{\overline z}{\vert z \vert ^s}$$ is ...
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1answer
20 views

On functions and their linear independence

How would you access the following problem: Show that the set of functions $$ \phi_n : \mathbb{R}_{>0} \rightarrow \mathbb{R}$$$$\phi_n(x) = \frac{1}{n+x}$$for $n \in \mathbb{Z}^{\ge 0}$ is ...
2
votes
1answer
42 views

Conservative vector field, potentail function and work done

For (i), is that I have to show $curl F = 0$ ? For (ii) and (iii), what should I do in order to find the potential function and work done? Also, is the answer $4$ for (iii)?
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2answers
37 views

Convergence of Expectations

Suppose $\{X_n\}$ is a sequence of non-negative random variables such that $$EX_n<\infty, \text{ }\lim_{n\rightarrow \infty}EX_n =\infty$$ and $\lim_{n\rightarrow \infty}X_n$ exists a.s. May I ...
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1answer
31 views

Multiple integrals: Double integrals

For this question, how to evaluate the integral by changing the order of integration? Also, how to sketch the region of integration? I really get stuck.
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1answer
28 views

Finding partial derivatives from a given function

For i), is the answer $df/du = (a)df/dx + (2cu)df/dy, df/dv = (b)df/dx + (2dv)df/dy$ For ii), is the answer $d^2 f/du^2 = (a^2) d^2 f/dx^2 + (4acu) d^2 f/dxdy + (4c^2 u^2) d^2 f/dy^2$ , $d^2 ...
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0answers
16 views

On a summation manipulation

I have $R_t= \frac{1}{h} \sum_{j=0}^{h-1} E_tr_{t+j} + \theta_t$ where $E_tr_{t+j} = E[r_{t+j}| I_t]$. By subtracting $r_t$ from both sides and after some manipulations I should get: $$R_t - r_t= ...
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2answers
33 views

How to do this Integration (in Orthonormal Family for Continuous Functions)

Here is a common example in the discussion on orthonormal family. Let $\mathcal L = C[a, b]$. For $k \in \mathbb Z$, let $e_k \in \mathcal L$ be defined by $$e_k(\xi) := \frac{1}{\sqrt{b-a}} ...
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0answers
33 views

Learning pipeline for developing own optical flow algorithms

I am really sorry if this question is outside of this resource or too silly I am bachelor of computer science and a programmer in small company. And i am faced with the task of developing own custom ...
1
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1answer
19 views

Convolution Operator and Integration Operator

I have some questions about the following two operators. A convolution operator $T$. If $k \in \mathcal L^1(\mathbb R)$, then $$f(x) \mapsto \int_{-\infty}^\infty k(x-y)f(y) dy: \mathcal L^2(\mathbb ...
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1answer
44 views

Closed Subspaces of Hilbert Spaces

I read the following statements. But I do not know how to show it or any example to support it. Could anyone provide some explanation and examples, please? Thank you! The subspace $C^\infty$ ...
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0answers
22 views

Basis for Finite Dimensional Hilbert Spaces

Verify that a Hilbert space orthonormal basis in a finite dimensional Hilbert space is the same as an orthonormal basis in the sense of linear algebra. Here is what I know. Hilbert space ...
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0answers
20 views

Maximum of squared point to plane distance?

I am looking at the squared distance from a hyperplane that passes throught the point $\mathbf{a}$ to the point $t$ which I understand is given by the formula $$\frac{\left[\mathbf{k} \left( ...
7
votes
3answers
58 views

$f$ an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$ implies constant

Let $f$ be an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$, i.e. takes values in the complement of the nonpositive part of the real axis. Show that $f$ is ...
0
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1answer
27 views

Eigenvalue of Compact Operators

To prove that the set of eigenvectors of a compact linear operator on a normed space $X$ is countable, I read "it suffices to show that for every real $k > 0$ the set of all eigenvalues whose ...
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1answer
50 views

A Question on Compact Operators on Hilbert Space

I read this question which I have no idea how to start. Could anyone provide me with some detailed answer, please? Thanks. Suppose that a linear operator $F$ from a Hilbert space $\mathcal H$ to ...
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2answers
39 views

To discuss differentiability of function at origin and my attempt

P1: $F = |x| + |y| when x,y is not equal to 0, = 0 when x = y = 0 P2 :Discuss the differentiability at origin of $F = y sin(1/x)$ :
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0answers
18 views

Finite Sums and Riemann-Stieltjes integral

Show that every finite sum $\sum_{k=1}^{n}a_k$ can be written as a Riemann-Stieltjes integral. My thoughts As far as I understand step functions provide a bridging link between Riemann-Stieltjes ...
2
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0answers
45 views

Can any one recommend a way to “quickly” learn a subject?

I would love to read a well written book on a subject - provided that I have the time. But sometimes we do not need to become experts on a particular field but still need the basics. For example, a ...
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0answers
21 views

Sum converging a.s. (cont'd)

Let $X_k$ be independent random variables with zero expectation s.t. $\sum_{k=1}^nX_k\rightarrow_{a.s.} X$ and $X$ has a nondegenerate distribution. Is there a way to estimate $E[(\sum_{k=1}^\infty ...
2
votes
3answers
69 views

limits multivariable calculus. where am i wrong with my attempt?

P : $\lim_{(x,y) \to (0,0)} f(x,y)$ where $$f(x,y) = y\sin\frac1x + \frac{xy}{x^{2} + y^{2}}$$ Text book says Limit doesnot exist . So where i am wrong with my proof below ? EDITED ATTEMPT : Or ...