# All Questions

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### Is there exists unbounded collections rather than be restricted into the world “class”?

In set theory,there is a question states that whether the class which contains all the classes exists? The answer is "NO" obviously.Otherwise it will lead to paradox.The reason is that there is a ...
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### Are $X$ and $Y-E(Y|X)$ independent?

The question is quite simple: if $X$ and $Y$ are two random variables (with unknown distribution, possibly independent), are $X$ and $Z=Y-E(Y|X)$ independent? Thinking about the conditional ...
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### Corollary of Schur's Lemma - why abelian

Corollary (of Schur's Lemma): Every irreducible complex representation of a finite abelian group G is one-dimensional. My question is now, why has the group to be abelian? As far as I know, we want ...
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### Mathematical notation/representation of a sum-product expression

I can write a general term, say $m$th term $T_m$, as a sum of $2^{m-1}$ terms in which each term contains two type of variables, i.e., $a_{i,j}$ and $b_i$ where $i,j\in 0,\cdots,m$. However, I do not ...
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### Please explain solution for this problem.

A shopkeeper sells a badminton racket, whose marked price is Rs. 30, at a discount of 15% and gives a shuttle cock costing Rs. 1.50 free with each racket. Even then he makes a profit of 20%. His cost ...
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### $f=g$ or $f=\bar{g}$?

Let $R$ be a ring with identity and $\mathbb{C}$ the ring of complex numbers. Suppose $f,g:R\rightarrow \mathbb{C}$ are two ring homomorphisms such that for every $r$ in $R$, $|f(r)|=|g(r)|$. Prove ...
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### Probability (boxes)

A contest consists in choosing one of three boxes that are covered, inside of which there are envelopes and only one of these envelopes contains the prize. Box 1 contains 8 envelopes, box 2 contains 5 ...
1answer
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### 3 card monte carlo variation

A friend wants to play a betting game with you. There are 3 upside-down cards on the table 2 black and 1 red. Your job is to find the red card. For every dollar you bet he will give you 2 to 1 odds (i....
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### Prove there is only finitely many Fourier coefficients of f that are non-zero

Can you give me a proof of this?
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### Leonhard Euler's Books in the Analysis and Algebra

I am an aspiring mathematician who is deeply interested in the analysis, topology, and their applications to the microbiology. Recently, I started to become very curious about why concepts and ...
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### What is the summation of $1^1 + 2 ^ 2 + 3 ^ 3 + …+ n ^ n$?

I am looking for the formula with proof, At least the approach for proving it would be helpful. Thanks in advance
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### Existence of an analytic function, isolated essential singularity

From the characterization of essential singularities, the Laurent series expansion at the points z=n has infinitely many negative degree terms. Which tests guarantee the existence of an analytic ...
1answer
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### can mickey mouse divide by 7?

In the figure displayed in the link below : divisibility by 7 To find the remainder on dividing a number by 7, start at node 0, for each digit D of the number, move along D black arrows (for digit 0 ...
1answer
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### Dugundji topology textbook. Is this an erratum?

3.4. Theorem Let $W$ be well-ordered, and $\Sigma\subset I(W)$ any family with the following properties: (a). Each union of members of $\Sigma$ belongs to $\Sigma$. (b). If $W(a)\in\Sigma$, then ...
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### Eliminating $x=-3,5$ when we solve the system $y=|3-x|, 4y-(x^2-9)=-24$

$$y=|3-x|$$ $$4y-(x^2-9)=-24$$ I use two methods to solve it, Method 1: Squaring both sides and eventually get a equation like this$$(x-5)(x-7)(x+3)(x+9)=0$$, $x=-9,-3,5,7$. Method 2:Using $$y=3-x$$ ...
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### Intermediate step in proving Cauchy's Integral Formula

I'm trying to understand the proof of the Cauchy's Integral Formula from the J. Conway, Complex Integration book. He states that However, I don't know how to solve what he left as exercise 1, at ...
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### How to minimize $f(x)$ with the constraint that $x$ is an integer?

I would like to find the integer x that minimizes a function. That is: $$x_{min} = \min_{x \in \mathbb{Z}}{(n - e^x)^2}$$ The goal is to write a program that ...
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1answer
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### Is the singular locus of a variety (as a variety itself) a smooth variety?

A general fact about the singular locus $Sing(X)$ of a variety $X$ (analytic or projective) is that they form a subvariety of the oringinal variety $X$. And we know that the boundary of a manifold ...
1answer
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### If A is 5 by 3 and B is 3 by 5 (with dependent columns), Is $AB = I$ impossible?

Let me first introduce the problem. This is part of the quiz problem from MIT's 18.06 course (Spring 2012 semester, quiz 1, problem 3). My question is related to (b) but (a) is mentioned in the ...
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### Fractionalth Dimension?

Is it possible for an object with a fractional number of axes to have calculable properties (like for an example a 1.5 dimensional vector)? If so, then what would be a good wikipedia page(s) for it. ...
2answers
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### Determine a set on which a sequence of holomorphic functions converges

Determine the set in $C$ on which $$\sum^{\infty}_{n=0}\left(\dfrac{1-e^{z}}{1+e^{z}}\right)^{n}$$ converges. My thought was to solve the inequality $\left|\dfrac{1-e^{z}}{1+e^{z}}\right|<1$. I ...
1answer
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### How many 3-digit numbers can be formed using the digits 2, 3, 4 and 5 as often as desired?

The method I tried: 5 x 4 x 3 = 60 different numbers I solved it this way because if a number needs to be formed with a certain number of digits (with no restrictions - which I think 'as often as ...
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Let $A\in M(m,n,\{0,1\}$) (i.e. $m \times n$ matrices with entries in $\{0,1\}$) and $x,y\in \{0,1\}^n$. We will denote the $i$-th row of $A$ as $row_i(A)\in \{0,1\}^n$. Define, $z_i = \begin{... 0answers 27 views ### Determining all functions$f(x+c)=-1/(f(x)+1)$I've noticed in my free time when the functional mapping$f(x+c)=-1/(f(x)+1)$is iterated twice, it yields the original function$f(x)$(i.e.$f(x+3c)=f(x)$). So I thought to study it as a periodic ... 0answers 31 views ### Elements of$\text{Spec}(\mathbb{C}[x_1, …, x_n])$I'm just curious as to what the elements of$\text{Spec}(R)$are when$R = \mathbb{C}[x_1,..., x_n]$. I'm aware that$\text{MaxSpec}(R) = \mathbb{C}^n$. 3answers 45 views ### Proving that$\lim_{(x,y) \to (0,0)} (x^2 +y^2 -x^3 y^3)/(x^2 +y^2) =1$How can I go about proving that $$\lim_{(x,y) \to (0,0)} \frac{x^2 +y^2 -x^3 y^3}{x^2 +y^2} = 1 ?$$ I checked some lines along$x, y$and$x=y$and it all gave$1\$

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