0
votes
0answers
4 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: ...
0
votes
0answers
8 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 : \...
0
votes
0answers
2 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 ...
0
votes
1answer
19 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
12 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. ...
0
votes
0answers
8 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
17 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
9 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
13 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 ...
2
votes
0answers
17 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
1answer
12 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 $1$
-1
votes
0answers
6 views

Can any tell me how to simplify this statistical thing by using maple code [duplicate]

I have a problem similar to the problem given in the link Addition of two Binomial Distribution Mr. Robert found some kind of solution using maple but I do not know how to use maple. Can any one ...
0
votes
1answer
10 views

Commutativity and associativity of isomorphic binary structures

Assume that $(S,*)$ and $(T,\circ)$ are isomorphic binary structures. (a) Show that $(S,*)$ is commutative if and only if $(T,\circ)$ is commutative. (b) Show that $(S,*)$ is associative if ...
0
votes
0answers
14 views

good book for algebra elementary

For us,the prescribed text book was algebra vol 1 by manicavasagom pillay. But I found lot of error in text and the book has only formula and problem ,there is no theory in it. So I think that it is ...
0
votes
1answer
13 views

Deriving equation of ellipse from expanded form?

The equation of an ellipse centered around the origin is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ The expanded form is $Ax^2 + By^2 + Cx + Dy + E = 0$ How do I derive the second from the first? I have ...
0
votes
1answer
22 views

If $H$ is a $p$ dimensional subspace of $\mathbb{R}^n$ and $G$ is a $p$.. [on hold]

If $H$ is a $p$ dimensional subspace of $\mathbb{R}^n$ and $G$ is a $p$ dimensional subspace of $\mathbb{R}^n$ that's contained in $H$, show that $G = H$ I know that a subspace of $\mathbb{R}^n$ is ...
1
vote
0answers
12 views

Class Number of $\mathbb{Q}(\sqrt[3]{19})$ and Hilbert class field

Finding the class number of $\mathbb{Q}(\sqrt[3]{19})$ is an exercise from Marcus 'Number Field'. This question was uploaded by some other user, but it was removed by now. I have worked on details and ...
0
votes
2answers
30 views

Infinitely many positive integers $m$

Let $a_n = \left[\sqrt{(n+1)^2+n^2}\right], n = 1,2,\ldots,$ where $[x]$ denotes the integer part of $x$. Prove that $\quad$ (a) there are infinitely many positive integers $m$ such that $a_{m+1}...
1
vote
1answer
36 views

Prove that $\lim_{x\to 0}\frac{x+a}{1-3x}=a$, $a>0$, by $\epsilon$-$\delta$ arguments

We have \begin{align} \left| \frac{x+a}{1-3x} - a \right| &= \left| \frac{x+a-a+3ax}{1-3x} \right| \\[1em] &= \left| \frac{x(1+3a)}{1-3x}\right|\\[1em] &= \frac{|x|(1+3a)}{|1-3x|} <\...
0
votes
2answers
16 views

Intersection point of two moving objects

Suppose we have 2 moving objects along a linear path: The first object moves at 5 metres/second. The second object accelerates at 1.5 metres/second. How would one calculate the point (in time) ...
0
votes
0answers
19 views

Does the following series converge? If so, is it transcendental?

Let $RF_n(x)$ be a recursive factorial function. ie, $RF_n(1) = 1 \space \forall n \in \mathbb{N}$ $RF_n(2) = 2 \space \forall n \in \mathbb{N}$ $RF_0(3) = 3$ $RF_1(3) = 3! = 6$ $RF_2(3) =(3!)!...
0
votes
0answers
11 views

Ramification in Quartic Field

Let $L = \mathbb{Q}(\sqrt{-1},\sqrt{-5})$ and $K = \mathbb{Q}(\sqrt{-5})$. I am trying to show that $\mathcal{O}_L$ is unramified over $\mathcal{O}_K$. Suppose there exists $\mathfrak{m}\in\mathcal{...
1
vote
1answer
28 views

how to determine the nature of roots of a polynomial equation with degree higher than 30?

the question says $P(x)=x^{32} -x^{25} +x^{18} -x^{11} +x^4 -x^3 +1$.how many possible imaginery and real roots does $p(x)=0$ has. how to determine the nature of roots for such equations of higher ...
0
votes
0answers
8 views

$X = S^{1} \vee S^{1}$ so $\pi_{1}(X) = F\{a,b\}$. Given homo. $\varphi: \pi_{1}(X) \rightarrow \mathbb{Z}/3$, draw associated cover of ker$\varphi$.

Let $X=S^{1} \vee S^{1}$ and so that $\pi_{1}(X)=F\{a,b\}$, the free group on two generators. Let $\varphi:\pi_{1}(X) \rightarrow \mathbb{Z}/3$ be the homomorphism induced by $\varphi(a)=1$ and $\...
0
votes
1answer
29 views

F(x) = 0 for all points except c. Show F is integrable.

Suppose $c$ is a point in the closed $[a,b]$ and that $F(x) = 0$ for all $x$ in $[a,b]$ except for $c$ and that $F(c) = 1$. Show that $F$ is integrable on $[a,b]$ and that $\int_a^bF(x)dx = 0$. By ...
-1
votes
0answers
22 views

What is the Inverse Fourier Transform of this function $ \frac{e^ \frac{1}{2}t(-w^2 +2 \sqrt{2 \pi} u \delta''(w)) }{ \sqrt{2 \pi}} $?

As the title says, what is the Inverse Fourier Transform of this function: $$ \frac{e^ \frac{1}{2}t(-w^2 +2 \sqrt{2 \pi} u \delta''(w)) }{ \sqrt{2 \pi}} $$ The inverse should be taken with ...
1
vote
0answers
10 views

$T_{prod} \subseteq T_{unif} \subseteq T_{box} $

I want to show the opposite containment of basis, i.e. $B_{box} \subseteq B_{unif} \subseteq B_{prod} $. Starting with $B_{unif} \subseteq B_{prod} $, the basic open set of $B_{unif} = B_{\epsilon}(...
2
votes
0answers
22 views

A Problem on an Infinite Series of Functions.

Let $\left\{a_n(t)\right\}_{n\ge1}$ be a sequence of functions from $\mathbb{R^+}$ to $\mathbb{R}$ such that $a_1(t)=t$ and also $$a_{n+1}(t)=a_n(t)-1+e^{-a_n(t)}$$. Let $g(t)=\sum\limits_{n=1}^{\...
0
votes
0answers
6 views

Conjugate function

I am looking for a geometric and intuitive explanation of the conjugate function and how it maps to the below analytical formula. $$ f^*(y)= sup_{x \in dom f } (y^Tx-f(x))$$
0
votes
1answer
20 views

Proof about binary structures and binary operations

Suppose that $*$ is an associative and commutative binary operation on a set S. Show that the subset $$T=\{a \in S \mid a*a=a \}$$ of $S$ is closed under $*$. Proof: We need to show that if $a\in T$...
0
votes
1answer
18 views

Surjective continuous function

I am trying to learn how can construct onto continuous function from rational(irrational) numbers to integers. I believe, I have an example helpfully it is true example let $Q$ denoted the rational ...
-2
votes
1answer
75 views

solve $\cos^n x - \sin^nx=1$

I already post a question on the solution of \begin{align} \cos^n x + \sin^nx=1 \end{align} but it's just a mistake My real question is \begin{align} \cos^n x - \sin^nx=1 \end{align}
2
votes
0answers
21 views

Infinitesimal generator of Brownian motion with additional jumps

A compound Poisson process is a jump process with two parameters, the rate of the jumps $\lambda$ and the distribution of the jumps $\mu$ ($\mu$ is a probability measure on $\mathbb{R}$). The ...
2
votes
0answers
29 views

In what sense is metric space completion universal?

The completion of a metric space is unique up to metric monomorphism (usually called isometry). It is also the "obvious" way to make all Cauchy sequences convergent. Structures which are unique up ...
0
votes
0answers
12 views

I have a problem I would like some advice with. I have to discretize bellow integration over volume in 2D.

I know what is discrete form of $$\int_\Omega\nabla\phi dV$$ in which $\phi$ is vector in 2D and $\Omega$ is volume of cell in CFD field. The result is: $$\frac{1}{\text{Volume}}\sum_{\text{face}} n\...
0
votes
0answers
7 views

Computationally Efficient Way to Partition N-Dimensional Space Around Distinct Values

Sorry if the title isn't super helpful, I'm really just looking for someone to point me in the right direction or let me know if there is a standard way of doing this. What I am wondering is, if I ...
0
votes
0answers
9 views

Ranking: How to adjust for underlying variability (risk)

63 stores. Each day a store sells x number of products. An observation is defined as the percentile rank of a store for a given day. In terms of volume of products sold, the store that sells the least ...
0
votes
2answers
26 views

Simplifying this fraction in a different base

Note: I would appreciate a solution that DOES NOT convert back to base 10. How would one simplify $\frac{43}{70}_8$? I assume, like in decimal, I must recognize a common factor and divide by that ...
0
votes
1answer
10 views

Cokernel of a module homomorphism

Let $A$ a $K$-algebra. Let $M$, $N$ $A$-modules and $f:M\rightarrow N$ a module homomorphism. The cokernel of $f$ is $Cokerf=N/Imf$ I define a homomorphism $\rho:N\rightarrow N/Imf$ by $\rho(n)=n+Imf$....
1
vote
2answers
51 views

Does not exist cover of $\mathbb{R}^n$ by disjoint closed balls

Does not exist cover of $\mathbb{R}^n$ by disjoint closed balls with positive radius. My attempt: Suppose that exists, we can write: $\mathbb{R}^n=\displaystyle\bigcup_{i=1}^{\infty} B_{i}$. Let $C$...
0
votes
0answers
6 views

Does the derivative by extension of Holder continuity coincides original derivative?

Let $B$ be an open bounded convex domain in $\mathbb{R}^{n}$ and $F:\mathbb{R}^{n}-> \mathbb{R}$ be a function that is differentiable at any $x_{0}\in\overline{B}$. Assume further that $F\in C^{1,1}...
0
votes
0answers
3 views

Marginal stability with non-simple poles on imaginary axis

It is known that a system marginally stable if and only if the real part of every pole in the system's transfer-function is non-positive, one or more poles have zero real part, and all poles with zero ...
0
votes
1answer
24 views

Set of points near zero

The subset $A$ of the positive segment of real line has $0$ as a limit point (that is has points of distance less than $\epsilon$ to zero for every positive $\epsilon$). Let $I(x)$ be an interval ...
-1
votes
1answer
32 views

Math Explanation Induction [on hold]

How would you explain to this person why it is that induction actually does work. In my follow up responses to you, I may ask you questions that your friend might as in response to your explanation. ...
1
vote
1answer
14 views

Universal Cover of wedges $S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2}$ and $\mathbb{R}P^{2} \vee \mathbb{R}P^{2}$.

We are asked to find the universal cover of the wedges $S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2}$ and $\mathbb{R}P^{2} \vee \mathbb{R}P^{2}$. I am second guessing myself on this problem because I ...
0
votes
4answers
125 views

Solve $\cos^n x + \sin^n x =1 $

the solutions of this equation as a function of the value of $n$?? \begin{align} \cos^n x + \sin^n x =1 \end{align} I already found the solution if n is odd,
2
votes
1answer
24 views

Walking to infinity stepping on randomly selected lattice points

Suppose you randomly fill the infinite non-negative quadrant of $\mathbb{Z}^2$ with $1$'s and $0$'s, with $1$ occurring with probability $p$ (and $0$ with probability $1-p$). The lowerleft corner of ...
0
votes
2answers
22 views

Elementary Set Theory proof regarding infinite and finite sets

Suppose $X$ is an infinite set and $Y$ is a finite set. Show that exists a surjective function $f:X\rightarrow Y$ and an injective function $g:Y\rightarrow X$.
1
vote
1answer
35 views

Integer solutions of $a^{(b^c)}=b^{(a^c)}=c^{(b^a)}$

Let $a,b,c$ be positive integers and let $$a^{(b^c)}=b^{(a^c)}=c^{(b^a)}.$$ Are there any nontrivial positive integer solutions to this set of equations? This is a question of my own musings. I know ...
0
votes
1answer
15 views

Restriction of sheaf of modules

Let $\mathcal{F}$ be an $\mathcal{O}_x$-module. Let $V \subset U$ open subsets of the scheme $X$. Then the restriction map $\mathcal{F}(U) \to \mathcal{F}(V)$ is compatible with the restriction map $\...

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