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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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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30 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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3answers
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Group isomorphisms and a possible trivial statement?

I have the following set $G=\lbrace a,b,e \rbrace$ and I successfully computed the following Cayley-Table \begin{align} \begin{array}{|c|c|c|c|} \hline \circ& a & b & e \\ \hline a& ...
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1answer
62 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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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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91 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
29 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
33 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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1answer
100 views

Bernoulli Related Probability Distribution

Find the probability of having $4$ or more girls in a family of $6$ children. Find also the probability that among $5$ families, each with $6$ children, at least $3$ of the families have $4$ or more ...
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How to use math textbooks

I'm a higher schooler who was recently gifted a book by my teacher (Schaum's outline of advanced calculus) which is really awesome and I've started working my way through it. I have run into a ...
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1answer
309 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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1answer
35 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
39 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
99 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
50 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
68 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
32 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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3answers
727 views

Book that is more accessible than Shoenfield

My logic course is based on my Computer Science education and on some random Internet pages (mostly Wiki). I want to make my knowledge of logic more coherent and fill in missing gaps. Thus I started ...
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3answers
353 views

I want to study higher mathematics. Where do I start?

Since last year, I've been interested in higher mathematics and don't really know where to start and don't know what knowledge I still need to obtain, before being able to understand the concepts. ...
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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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2answers
820 views

Vladimir Zorich vs Rudin/Pugh/Abbott

There have seen various comparisons between books on Analysis. I was surprised to find out that Zorich's book on Analysis was not compared anywhere. Can anyone give a comparison between Zorich and ...
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2answers
54 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
71 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: $S_n$ ...
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0answers
146 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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53 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. ...
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1answer
21 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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2answers
61 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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53 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 $T$ ...
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1answer
37 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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2answers
37 views

On subgroups of isometries and their respective fixed-points

I am working on the following problemset: Let $G < \DeclareMathOperator{\Iso}{Iso}\Iso(E)$ be a finite subgroup of the isometries in the euclidean plane. Denote: $$G = \{g_1, \ldots , g_n \} ...
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1answer
25 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
22 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 ...
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2answers
51 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
40 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
31 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
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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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1answer
83 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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1answer
2k views

Which is better strategy to learn and read books, traditionally one by one OR re-read carefully on perfect books

(Just focus on how to learn and master the stuff pretty well, not involve the aspect of courses or exam) Because recently I always feel that the time and energy are pretty limited, I want to try ...
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1answer
93 views

Proving Euler Summation by Parts Without Using Integration by Parts

Assume $f$ has continuous derivative $f'$ on [a,b]. Prove the following summation formula, without using partial integration: \begin{equation} \sum_{a< x \le ...
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2answers
38 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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1answer
82 views

How to avoid losing the woods for the trees in daily study/lecture time

When facing to some new material in mathematics, I feel easily to be overwhelmed by lots of details with losing the woods for the trees. So is there some good strategy to study the materials ...
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1answer
39 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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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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1answer
44 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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3answers
88 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 ...
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1answer
68 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
52 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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1answer
89 views

Nash's Axiomatic Bargaining: Source of problems sets and practice questions.

From where can I practice questions related to the following topic: Nash's Axiomatic Bargaining. Any form of book reference or a link to some online problem set would be highly appreciated.
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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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1answer
153 views

Norm of the Resolvent of a Self-Adjoint Operator

Let $\mathcal H$ be a Hilbert space and $\mathcal L$ is a self- adjoint operator with a discrete spectrum $\{\lambda_{j}\}$. I read that it is well known that for, $\lambda \notin \sigma(\mathcal ...