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 fact that you're self-studying is what your question is _about_.

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32 views

Permutation of word

Question: Find the permutation of letters of the word EXERCISES in which vowels are together. My Efforts: I have rearranged the word in such a way that all the vowel come together. EEEI XRCSS Now ...
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0answers
47 views

Problem showing a double summation equality.

I'm trying to show that $$G(L) = \sigma^2(\sum_{j=0}^\infty \psi_{j}^2 + \sum_{h=1}^\infty\sum_{j=0}^\infty \psi_j \psi_{j+h}(L^h-L^{-h}))$$ is equal to: ...
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3answers
74 views

Given $o(a)=5$, prove $C(a)=C(a^{3})$

Given $o(a)=5$, prove $C(a)=C(a^{3})$ At this point I would like a hint rather than a full solution. I know we are given $a^{5}=e$ and that we wish to prove this implies that $C(a) =\{ x \in ...
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1answer
397 views

Is the Gamma Function a jointly sufficient statistic?

A random sample $X_{1},...,X_{n}$ are pulled from a gamma distribution. Are there jointly sufficient statistics based on these observations for the two unknown parameters? The definition of a gamma ...
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23k views

Good abstract algebra books for self study

Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, ...
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1answer
29 views

$Var(\bar{X})$ for a random sample from Bernoulli Distribution

Let $X_1,...,X_n$ be a random sample of size $n$ from a Bernoulli distribution with parameter $p$ where $0< p< 1$ is unkown. (a) Find $\theta^2=Var(\bar{X}).$ (b) Find the value of $c$ so that ...
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0answers
77 views

Periodic curve on unit sphere and torsion

Define $S^2 \subset \mathbb{R^3}$ be the unit sphere. Suppose that $\alpha :\mathbb{R} \to S^2$ is a differentiable curve parametrized by arc-length. a) Show that $\kappa(s)$, the curvature of ...
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0answers
40 views

Characteristics of Singularity Points

Determine the character of the singularity at $z=0$ for each of the following functions: \begin{align} &\frac{1/z^{7}}{e^{z}-1} \tag{1} \\ &\frac{}{} \\ ...
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2answers
143 views

Expectation of Truncated Random Variables

Let $X_n$ be a sequence of $iid$ random variables with zero mean, finite variance $\sigma^2$ and partial sum $S_n:=\sum_{k=1}^n X_k$. Let $0<\delta<0.5$ and $\epsilon >0$ and define ...
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1answer
188 views

Soft question: How does basic differential geometry “fit together”?

I'm self-studying diff geom from Lee's Introduction to Smooth Manifolds. He warns the reader that there's a lot of machinery to construct, which is fine, and he explains things with wonderful clarity. ...
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1answer
227 views

Problem about coordinate curves and quadrilateral formed?

Below is a problem which states a fact about "Tchebyshef net". I don't understand meaning of bolded part. Please help me. The coordinate curves of a parametrization $x(u, v)$ constitute a ...
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1answer
45 views

Follow-up question on mathematical induction with arbitrary base case

Note: This question has already been answered here Proving mathematical induction with arbitrary base using (weak) induction. I was trying to 'reconstruct' at least one proof given in this question ...
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1answer
55 views

Poles and Zeros

Determine the number of zeros and poles inside (counted with multiplicity) of the function \begin{equation} f(z)=\frac{z^{6}\sin(\pi z)}{(1-z^{2})(2-z^{2})(3-z^{7})} \end{equation} inside the ...
1
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1answer
34 views

Kolmogorov Exponential Bounds (Upper)

This is one version of Kolmogorov exponential bound from Allan Gut's Probability: A Graduate Course (2005, p385-386). Let $Y_k$ be an independent sequence of random variables with zero mean and ...
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4answers
72 views

Existence of the square root in $\mathbb{C}$

I am stuck on the following Proposition: Proposition: Show that for every $z \in \mathbb{C} \setminus (- \infty, 0]$ there exists exactly one $w \in \mathbb{C}$ such that $w^2=z$ and Re$(w)>0$ ...
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2answers
230 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
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3answers
64 views

Applying Rouché's Theorem

Determine how many zeros of the following polynomial lie inside the circle $|z|=2$ \begin{equation} z^{5}+2z^{4}+z^{3}+20z^{2}+3z-1=0\end{equation} My Reasoning If we put $f(z)=z^{5}+2z^{4}$ and ...
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0answers
24 views

Order Type Stratification

Is there some sort of interesting way of organizing certain order types that aren't ordinals? When I say certain order types, some examples include but are not limited to: order types of dense ...
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1answer
38 views

Residue Calculus (Computing an Improper Integral)

Use residue calculus to compute the integral $\int_{-\infty}^{\infty}\frac{1}{(z^{2}+25)(z^{2}+16)}dz$ My solution If we add to the interval $I_{R}=[-R,R]$ add the semicircle $\gamma_{R}$ in the ...
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2answers
41 views

Property of a system of two inequalities

I have this system $$\begin{cases} a+b>1 \\ a-b>1 \end{cases}$$ can I sum the second inequality to the first getting $a>1$? Or this property can be used only equations?
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2answers
219 views

Number of possible eight digit number divisible by 9

An eight digit number divisible by 9 is o be formed by using 8 digits out of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 without repetition. Find the number of ways in which it can be done. I know divisible rule of ...
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1answer
31 views

Writing The Derivative Of $f(x)$ With Respect To $g(x)$ In Limit Form

What would be the proper way to represent this derivative in the limit form? $$\frac{\mathrm{d} }{\mathrm{d} g(x)}[f(g(x))]$$ In my attempt to solve this I've tried to word out the derivative: The ...
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1answer
29 views

On the equivalency of two indefinite integrals using u substitution.

I am reading the Separation of variables page on wikipedia, at a certain point it states that the following equation Is equal to (1) because of the substitution rule of integrals. The ...
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1answer
46 views

Largest possible number of student passed in all three subject

Question: From $50$ students taking examination in Mathematics, Physics and Chemistry, $37$ passed in mathematics, $24$ passed in physics and $43$ passed in chemistry. At most $19$ students passed in ...
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1answer
50 views

How is Fubini Theorem used here?

Let $\mu$ be a $\sigma$-finite translation invariant measure defined on the Borel subsets of $\mathbb R^d$ and $\lambda$ be the usual Lebesge measure. My question is how the Fubini theorem is used in ...
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1answer
33 views

Ratio of hazards in Proportional Odds model

In the proportional odds model we have the the odds of survival in 1 group are proportional to the odds of survival in another group $$\dfrac{ S_1(t)}{1-S_1(t)} = \psi \dfrac{S_0(t)}{1-S_0(t)}$$ ...
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2answers
89 views

How to gain a mentor for self study

I am trying to independently learn mathematics at an upper-level undergraduate and first year grad student level. It's mostly linear algebra and statistics. The textbooks that I'm reading are quickly ...
3
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3answers
721 views

Derivative of a function with respect to another function.

I want to calculate the derivative of a function with respect to, not a variable, but respect to another function. For example: $$g(x)=2f(x)+x+\log[f(x)]$$ I want to compute $$\frac{\mathrm ...
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0answers
32 views

Joint Limiting Distribution of Min and Max

Let $X_1,\ldots,X_n$ be iid from the uniform distribution $U(a,b)$. Let $X_{(1)}< cdots< X_{(n)}$ be the order statistics. Find the joint limiting distribution of $(n(X_{(1)}-a),n(b-X_{(n)}))$ ...
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1answer
123 views

Uncountable Dense Linear Orders

Is there an example of two uncountable equipollent dense linear orders without endpoints that don't satisfy the same first order properties? Or is it true that two uncountable equipollent dense linear ...
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0answers
60 views

On what value of $f(X)$ minimizes $E[(Y-f(X))^2|X]$

$X$ and $Y$ are random variables. The question is: what value of $f(X)$ minimizes $E[(Y-f(X))^2|X]$. I am pretty sure I have found the solution to this problem by writing: $$E[(Y-f(X)-E[X|Y] +E[X|Y] ...
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2answers
100 views

Path to 3d Mathematics programming, where to start?

This might read like duplicate of this question https://math.stackexchange.com/search?q=where+to+start However since that one wasn't answered, and I have a more specific problem in regards to ...
1
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1answer
49 views

Normal Distribution and Iterated Logarithm

Let $X_n$ be independent $N(0, \sigma^2)$-distributed random variables with partial sum $S_n := \sum_{k=1}^n X_k$, $n \geq 1$. Then I read the following results. $$ \sum_{k = 1}^n \mathbb P (S_n > ...
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0answers
86 views

How to select good exercises?

I'm studying on Rudin "Principles of of Mathematical Analysis" which I begin to find as a good and complete reference. I wonder how many exercises shall I do at the end of each chapter ? In case of ...
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2answers
78 views

Determining final and initial object in a certain category

I am reading Paolo Aluffi's greatly entertaining book "Algebra: Chapter $0$" and I got stuck on some excercises dealing with universal properties. Let $C$ be a category, and let $A$ and $B$ be two ...
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3answers
35 views

Limit with terms very similar to those that should give an exponential function

I have been trying to solve the following limit but am completely stuck. $$\lim_{\alpha \rightarrow \infty} 1-\left( \frac{y+\alpha}{\alpha-1} \right)^{-\alpha}$$ I have tried inverting the ratio ...
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1answer
467 views

Posterior Probability

a) Your initial belief is that a defendant in a court case is guilty with probability 0.5. A witness comes forward claiming he saw the defendant committed the crime. You know the witness is not ...
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5answers
3k views

Are there any memorization techniques that exist for math students?

I just watched this video on Ted.com entitled: Joshua Foer: Feats of memory anyone can do and it got me thinking about memory from a programmers perspective, and since programming and mathematics ...
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1answer
36 views

Confusion about the associative property and the mechanics of Parenthesis

This is a follow up question on my earlier post (Updated): Showing that a set $M$ with two elements classifies as a field. I feel this post is necessary because I realize that what confuses me is how ...
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1answer
39 views

(Updated): Showing that a set $M$ with two elements classifies as a field

My question is more conceptual, so I will come straight to the exercise: Exercise: Let $M= \lbrace g,u \rbrace $ be a Set. On $M$ the Addition and the Multiplication is given by: \begin{align} ...
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1answer
37 views

Simple probability question

Question: In class of 125 students, in examination 70 students passed in mathematics and 55 students passed in statistics and 30 passed in both the subject. Find the probability of the event where ...
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4answers
128 views

Constructive proof for existence of integer part of real number

I try to prove de following exercise of my analysis textbook. Show that for every real number $x$ there is exactly one integer $N$ such that $N \le x < N + 1$. I have been finding a ...
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3answers
41 views

Is this derivative correct?

I'm newbie at Calculus, so I'm doing some exercises of derivates, I know by the formula: $f(x) = \sqrt u$ $\frac {df(x)}{dx} = \frac{u'}{2 \sqrt u}$ that the derivate of the next function is: ...
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1answer
66 views

Question about Lebesgue Covering Dimension

Suppose we have a metric space equipped with two different metrics: $(X,d), (X, d')$. What must be true of the metrics: $d, d'$ in order for $X$ to have the same Lebesgue covering dimension? A ...
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1answer
44 views

Want to check measure theory proof

I need to show that sigma-finiteness implies semifinite. Does the following proof work? Let $(X,m,\mu)$ be a $\sigma-$finite measure. Let, $E\in m$, and $\mu(E)=\infty$. By $\sigma-$finiteness, ...
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1answer
44 views

Proof on existence of the natural numbers, crucial step.

I am trying to understand/reconstruct the proof given by my Professor addressing the existence of natural numbers. However there is one step in particular I don't understand and the more I think about ...
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1answer
215 views

Without solving the equation, determine the nature of its roots: $x^2 + ax + a^2 = 0$

I'm working through the book Core Maths for Advanced Level on my own, and, after solving the above problem, I'm not getting the same answer as the book. So, given: $$x^2 + ax + a^2 = 0$$ Using the ...
3
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3answers
507 views

Probably of 2 six in 5 dice rolls

What is the probability of obtaining exatcly 2 six when rolling a dice 5 times? In order to obtain this probability, I will need to devide the number of favorable events by the number of possible ...
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1answer
69 views

An exercise on first order logic formulas, terms and Polish notation

This is part of my homework (not mandatory and not accredited). Please comment/answer if my reasoning for the exercises is correct, because I'd like to see if I understand the material. I will start ...
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1answer
474 views

Am I reading Bott - Tu right?

Summary: I'm finding Bott - Tu to be too brief and terse. I constantly have to look elsewhere to fill in details. This is not time-efficient. Am I missing something? If not - what other books do ...