# All Questions

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### about definition of a cardinal

Definition: the cardinality of a set $A, |A|$ is the least ordinal s.t. $A \sim \alpha$ Definition: We define a cardinal to be an ordinal $\alpha$ s.t. $\alpha = |\alpha|.$ i.e, an ordinal s.th. ...
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### Nonexistence of the limit $\lim_{(x,y)\rightarrow (0,0)} \frac{x^2y^2}{x^3+y^3}$

How can we prove that the limit $\lim_{(x,y)\rightarrow (0,0)} \dfrac{x^2y^2}{x^3+y^3}$ doesn't exist? I have tried a lot of different paths and all of them lead to zero. I have only tried paths ...
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### Holomorphic functional Calculus in Dunford and Schwartz

I am currently studying the spectral theory for bounded operators as described in the book "Linear Operators" by Dunford and Schwartz because I would like to obtain a better understanding of the ...
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### Uniqueness of the group with order 5

I need to prove the uniqueness of the group G of order 5 up to isomorphism (i.e. G = $\{e,\ a,\ b,\ c,\ d\}$). So far I have shown that $\forall a\in\ G\ ,$ $\ a^2\neq e$ where $e$ is the identity ...
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### Solving for x - Trig

Someone mind helping on this? I think have done the question correct but the system isnt accepting my answer.
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### Real analysis: convergence question

I found the closed formula for the sum of 1/( k^2+3k+2) from 1 to infinity which is 1/2 - 1/(n+2). Could you first check whether this is right. If so, how to find the sum of this series? Is it just ...
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### Prove: $\liminf_{n \to \infty} s_n \le \limsup_{n \to \infty} s_n$

I am looking over examples and the definitions for this section but I am still not familiar with all the tricks. I appreciate any help with proving this (from hints to maybe a solution. It is only a ...
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### Tate Circles in Motivic Homotopy Theory

First off, this is a vague question about a survey which is, I guess, meant to be vague. So bear with me In Morel's "Motivic Homotopy Theory" survey he mentioned the following fact in motivating the ...
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### Weird system of equations

X : 2 = 7 Y : 2 = 6 X + Y = 15 Find X and Y. I think maybe this is some unpositional number system. I've tried positional, and it works for basis 21 (if we take X=D, and Y=C), but professor told me ...
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### if the improper integral $\int^\infty_a f(x)\,dx$ converges, then $\lim_{x→∞}f(x)=0$ [closed]

I need to prove that: $$\lim_{x→∞}f(x)=0$$ if $$\displaystyle∫^∞_af(x)\,dx$$ converges. I need a proof or an specific, and if possible simple, counterexample. Would really appreciate your help! ...
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### Fundamental Polygon of Real Projective Plane

Wikipedia gives the following fundamental polygon for the real projective plane $\mathbb{R}\mathrm{P}^2$ The problem here is that the corners aren't identified to a single point (like in the ...
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### Abstract Linear Algebra for Statistics

Is abstract linear algebra required for a deep understanding of statistics? I'm a computer science major deciding between a linear algebra for applications class versus a very theoretical proof based ...
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### How to construct point finite covering in collectionwise normal spaces

I am actually looking for a related reference (and ideally if anyone knows the answer) on the following construction problem: Let X= $\prod_{i=1,..,n} X_{i}$ be a collectionwise normal and Hausdorff ...
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### On a Congruence Relation Between Polynomials

Problem: If $f \in \mathbb{Z}[X]$ and $f(a) ≡ 0 \pmod n$ for some $a \in \mathbb{Z}$, then there exists a $g \in \mathbb{Z}[X]$ such that $f(X) ≡ (X − a) g(X) \pmod n$. I think that ...
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### The product of limit point compact Hausdorff spaces is not limit point compact

Let $X, Y$ be limit point compact Hausdorff spaces (to be clear, a space is said to be limit point compact if every infinite subset of it has a limit point). Is it true that $X \times Y$ is limit ...
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### Set of 4 anticommutative matrices

How would you go about showing that there cannot be a set of four 2 by 2 matrices that satisfy the anticommutative relation $AB + BA = 0$ or $2I$ if $A=B$? i.e minimum order has to be 4. I know that ...
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### Unit length vectors that sum to zero

Let's say we have a collection of $n$ vectors in $\mathbb{R}^2$ where $n$ is odd. Suppose each vector has unit length and that the sum of the vectors is zero. Is it necessarily true that the vectors ...
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### Show that is $X_ \alpha$ Hausdorff, regular and normal

Show that if $\displaystyle \Pi_{\alpha \in J}X_\alpha$ is Hausdorff, or regular, or normal, then so is $X_ \alpha$.
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### Having trouble understanding Series and Sequences

So all I could get from my teachers thick accent in class today is that: A sequence is finite and converges when bounded by x? and A series is infinite and diverges because no matter how small the ...
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### Determining the length of a space curve?

What does the author integrate the magnitude of r'(t) with respect to u when the definition right above it says to integrate with respect to t?
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### When to do u-substitution and when to integrate by parts

I'm in my first semester of calculus, so the problems I'm facing are about as hard as those on KhanAcademy calculus playlist. I'm currently doing integration, a somewhat diffucult part of the course. ...
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### Probability question cards

This is a probability question. There are 6 cards with letters a, c, e, i, m, n in a box. Somebody picks cards in a random order. What is the probability of getting the word “cinema”? Don't solve it ...
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### Example of sheaf hom not commuting with stalk

I came up with the following example and I wanted to know if it is correct. Let $\mathcal F$ and $\mathcal G$ be sheaves of smooth functions on $\mathbb R$. Consider the smooth function $f$ ...
Let $X,Z$ be two correlated variables and $Y,Z\sim N(0,1)$ where Y is independent of $X,Z$. Consider the expectation: $$E[f(X,Y)Z].$$ If $f(X,Y)$ and $Z$ are independent then clearly ...
### Why $p$ factor in $p \log p$ in the entropy formula?
It is explained that we have log for additivity of information in the entropy formula. But, why is the $p$ factor? It is redundant, since we already have it in the $\log p$!