0
votes
0answers
2 views

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 ...
0
votes
0answers
4 views

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 ...
0
votes
0answers
5 views

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 ...
0
votes
0answers
8 views

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 ...
-1
votes
2answers
18 views

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 ...
-3
votes
1answer
20 views

$f=g$ 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 ...
0
votes
0answers
13 views

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 ...
0
votes
1answer
8 views

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....
0
votes
0answers
16 views
2
votes
0answers
16 views

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 ...
1
vote
0answers
37 views

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
0
votes
0answers
11 views

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 ...
3
votes
1answer
46 views

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 ...
1
vote
1answer
31 views

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 ...
1
vote
3answers
29 views

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 $$y=3-x$$ ...
2
votes
0answers
27 views

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 ...
0
votes
1answer
14 views

Comparing variance for two games

I saw this question here Compare two coin tossing games and wanted to figure it out myself, and I had some trouble with variance. I can see easily that both games have expected value $\$500$. But my ...
0
votes
0answers
2 views

What is the difference between deterministic vcs, probabilistic vcs and random grid vcs

I am confused about "deterministic visual cryptographic scheme( dvcs)", "probabilistic vcs", and " random grid vcs". Can we say every "random grid vcs" is "probabilistic vcs"? If yes why. What is the ...
1
vote
0answers
13 views

Bisections of the unit line inequality

Let $x_1 \in (0, 1)$. Iteratively define intervals $I_1,I_2,...$ and points $x_2,x_3,...$ by: $I_k$ is the longest sub-interval of $(0, 1)$ not containing any of the points $x_i , 1 \leq i \leq k$, (...
1
vote
1answer
27 views

Why use Kronecker product

I have found many references on Kronecker product but I did not see any reference talking about why this way of multiplication exist and whats the intuitive use of this particular product. Appreciate ...
1
vote
3answers
56 views

What is Absurdity and Contradiction?

I want to know the meaning of these mathematical terms. What do they mean in mathematical logic? Do they refer to same thing or are they different. I am trying to learn "Proof by contradiction" Please ...
9
votes
1answer
71 views

A trivial combinatorics result I found, is my proof correct?

I have just finished highschool and have started learning on my own some combinatorics and how to do proofs, and while messing around with sums and Pascal's triangle I found an interesting yet trivial ...
1
vote
3answers
23 views

How many different 3-digit numbers can be formed if the number can consist of the digits from 0 to 7, cannot start with 0 and must be even?

Original question: A 3-digit number is made up using the digits 0, 1, 2, 3, 4, 5, 6 and 7 at most once each. The number cannot start with 0. How many different numbers can be formed if the number ...
2
votes
1answer
21 views

A compact subset of Holomorphic function space with uniform convergence topology

Let $F=\left\{f:f\left(z\right)=\sum^{\infty}_{n=0}a_{n}z^{n},\left|a_{n}\right|\leq n,n=0,1,...\right\}$ (a).Prove that every $f\in F$ defines a holomorphic function on $D$. (b).Prove that $F$ is a ...
0
votes
1answer
40 views

What type of function is $y=x^{-1}$?

What type of function is $y=x^{-1}$? It's not a polynomial I think because they should be positive exponents, so what are these functions called?
0
votes
0answers
13 views

Express in terms of the spectral decomposition of $A$ the set of $x, y$ for which an inequality is satisfied

I'm confused by this problem: Let $A \in \mathbb{C}^{n \times n}$ be diagonalizable with eigenvalues $0 \leq \lambda_1 \leq \cdots \leq \lambda_n$. Express in terms of the spectral decomposition ...
0
votes
2answers
26 views

Compare two coin tossing games

Compare the following two games: You have a fair coin. After one toss, you will get 1 dollar if you get a head, and 0 dollars if you get a tail. How much will you be willing to pay to play this game ...
-4
votes
0answers
32 views

Request for counter example in group theory [on hold]

Is the converse of the statement that,every subgroup of an abelian group normal, true? If yes, give proof. If no, give counter example.
2
votes
4answers
56 views

If $4x/3y = 7/2$, what is the value of $y/x$?

If $4x/3y = 7/2$, what is the value of $y/x$? This is a multiple choice question, and the choices are as follows: A. $3/14$ B. $8/21$ C. $21/8$ D. $14/3$ I started off answering this by cross ...
0
votes
1answer
22 views

Using holomorphic functions to approximate some meromorphic functions

Let $A=\left\{z:\dfrac {3}{4}<\left|z\right|<1\right\}$ ,$f_{1}\left(z\right)=\dfrac{1}{2z-1}$ and $f_{2}\left(z\right)=\dfrac{1}{2z-3}$. Is it possible to uniformly approximate $f_{1}$ or $f_{...
0
votes
0answers
26 views

Definition of theory

I'm studying model theory and mathematical logic by myself and found two different definitions of a theory: (1) by a formalized theory $\mathscr{T}$ we shall understand any triple $(\mathcal{L},C,\...
-3
votes
1answer
19 views

Setting Standard Deviation [on hold]

Suppose that the lifetimes of tires of a certain brand are normally distributed at a mean of 74000 miles and a standard deviation of X. These tires come with a 65,000-mile warranty. The manufacturer ...
2
votes
0answers
22 views

Finding the boundaries on a triple integral

Solve: $$\iiint yz \,dV$$ Over the tetrahedron with vertices on the points $$A(0,0,0), B(1,1,0), C(1,0,0), D(1,0,1)$$ Well, I proceeded to find a the equation of a plane which contained B, C and D. ...
0
votes
1answer
21 views

Complex Numbers, Square, Conics

On the Argand diagram, $P$ represents the complex number $z$, and $R$ the number $\frac{1}{z}$ A square $PQRS$ is drawn in the plane with $PR$ as a diagonal If $P$ lies on the circle $|z| = 2$, (I) ...
1
vote
0answers
50 views

map $x+y \le 1, x,y >0$ to $R^2$?

Is there a bijective continuous function which can map $x+y \leq 1, x,y >0$ to $R^2$? I appreciate any idea and comment.
0
votes
1answer
11 views

Intuition Behind this Theorem About Brownian Motion

I am having a hard time with the intuition behind some of the representation theorems dealing with Brownian Motion. I think if someone can simply explain the intuition behind this theorem then ...
1
vote
0answers
12 views

calculate area of a region defined by holomorphic function

Let $f$ be a holomorphic function on unit disk $D$. (a). express the Jacobian of the map $f$ in terms of $f$ or $f'$ (b).Give a formula for the area of $f(D)$ in terms of the Taylor Coefficients of $...
1
vote
2answers
30 views

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 ...
1
vote
0answers
14 views

For what sets $E \subset \mathbb R^n$ is there a $C^m$ field of polynomials on $E$ with vanishing constant term?

First some definitions, since the notation is somewhat non-standard: let $\mathcal P(m,n)$ denote the vector space of polynomials with real coefficients of degree at most $m$ in $n$ variables. Let $E \...
1
vote
1answer
21 views

Clarification on Limit Comparison Test

For one of my classes we are using Manfred Stoll's, $\textit{Introduction to Real Analysis}$, and I had a question regarding the Limit Comparison Test. Here is the definition that they have: ...
1
vote
0answers
31 views

When is this tensor symmetric?

Let $M$ be a riemannian manifold with metric $\langle \cdot, \cdot \rangle$ and riemannian connection $\nabla$. For a fixed vector field $V \in \mathfrak{X}(M)$, define the tensor $\beta_V = \beta : \...
2
votes
1answer
17 views

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 ...
1
vote
1answer
61 views

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 ...
0
votes
0answers
21 views

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. ...
1
vote
2answers
26 views

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 ...
0
votes
1answer
24 views

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 ...
0
votes
0answers
11 views

Solutions to Binary Equations

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{...
0
votes
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 ...
1
vote
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 $\text{MaxSpec}(R) = \mathbb{C}^n$.
1
vote
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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