-4
votes
1answer
149 views

Proof that $0 = 1$ or that $\frac1\infty = 0$ [closed]

Proof that $0.9\ldots = 1$: Let $x = 0.999\ldots$ $$10x = 9.999\ldots\\ 9x = 9.999\ldots - 0.999\\ x = 1$$ Proof $0 = 1$: $$1 - \frac{1}\infty = 0.999\ldots\\ 1 - \frac{1}\infty = 1\\ ...
6
votes
2answers
293 views

Show that $\lim\limits_{n\to\infty} x_n$ exists for $0 \le x_{n+1} \le x_n + \frac1{n^2}$

Let $x_1, x_2,\ldots$ be a sequence of non-negative real numbers such that $$ x_{n+1} ≤ x_n + \frac 1{n^2}\text{ for }1≤n. $$ Show that $\lim\limits_{n\to\infty} x_n$ exists. Help please...
3
votes
2answers
321 views

Proof of Hilbert's Basis Theorem: won't $\deg (f_{i})$ be a strictly decreasing sequence?

Say we have an ideal $I\subset R[X]$. We select a set of polynomials $f_{1},f_{2},f_{3},\dots$ such that $f_{i+1}$ has minimal degree in $I\setminus (f_{1},f_{2},f_{3},\dots f_{i})$. Can't $\deg ...
1
vote
0answers
105 views

Entropy Rate of a sequence of Random Variables with Distributions related to Primes

Let us consider a stochastic process $\mathcal{X}=\{X_i\}_{i \in \mathbb{N} }$ where $X_i$'s are independent and $X_i$ is distributed as $$X_i=p_k \ \mbox{w. p.}\frac{p_k}{\sum_{l=1}^{i}p_l},\ 1\leq ...
1
vote
1answer
84 views

Complement of invariant subspace

Assuming that I have a given vector space $V$ and a subspace $U$, which is invariant under an endomorphism $A\in End(V)$. I want to prove that $U^\perp$ is also invariant on $A$ .
1
vote
3answers
64 views

Finding limit function

\begin{align} f(x) &= \lim_{n \rightarrow \infty} n ((x^2 +x + 1)^{1/n} -1) \\&= \lim_{n \rightarrow \infty} n ((\infty)^{1/n} -1) \\&= \lim_{n \rightarrow \infty} n (1 -1)\\& = ...
0
votes
3answers
113 views

Binary operation

Let binary operation $ \circ $ on set $X$ be function $\circ : X \times X \rightarrow X$. Binary operation on set X is : unitary if for some element $1 \in X$ and any $x \in X$ we've got $(1 ...
5
votes
2answers
90 views

How can i solve this System of first-order differential Equations?

My Problem is this given System of differential Equations: $$\dot{x}=8x+18y$$ $$\dot{y}=-3x-7y$$ I am looking for a gerenal solution. My Approach was: i can see this is a System of linear and ...
6
votes
2answers
10k views

How to determine the arc length of ellipse?

I want to determine the arc length of a ellipse. So what data should I know ? And what law should I use ? For example I have this ellipse on picture below: How can I determine the $d$ length of ...
1
vote
3answers
3k views

How do I remove a number from the numerator of a fraction so that I am left with the variable in the denominator in this equation?

The question is this: $$\frac{1}{R^p} = \frac{1}{4.5\times 10^2} + \frac{1}{9.4\times 10^2}$$ I calculated the equation so that it simplifyed to: $$\frac{1}{R^p} = 0.003286$$ But now I am stuck... ...
0
votes
1answer
188 views

Block matrix and invariant subspaces

I was wondering what the exact relationship between invariant subspaces and a block matrix is? Is it correct to say: Each diagonal block matrix "creates a vector space decomposition" and vice versa? ...
1
vote
0answers
58 views

Unique continuity property

Can someone told me what is :"the unique continuity property" in the following paragraph ? and what is the meaning of : .... and either $v\in E(k)$ or $v\in E(k+1)$ Please help me Thank you .
2
votes
0answers
35 views

The dimension of birkoff polytope

Let $P_m$ be a subset for R^mxm be the polytope given by: $x_i,_j \ge 0$ $x_i,_1 + ... + x_i,_m \le 1$ $x_1,_j + ... + x_m,_j \le 1$ $\sum_{1 \le i,j \le m } \ x_i,_j \ge m-1$ Contruct a ...
0
votes
1answer
69 views

Show that the equation of a line can be given as ℑm(αz+β)=0

I've just started a non-Euclidean Geometry course and the book we are using has a very brief (and not-so-helpful) section on complex numbers that we sort of went over in class. One of the questions ...
1
vote
3answers
735 views

Definite integral of a product of normal pdf and cdf

Denote the pdf of the standard normal distribution as $\phi(x)$ and cdf as $\Phi(x)$. Does anyone know how to calculate $\int_{-\infty}^y \phi(x)\Phi(\frac{x−b}{a})dx$? Notice that this question is ...
1
vote
2answers
59 views

Random walk confusion

If a ransom walk is binomial (1/2 probability of going forward, 1/2 backward) why isn;t the variance a) $\sigma=(\frac{n}{4})^.5$ b) instead of $\sigma=(n)^.5$ these sources seem to give ...
0
votes
2answers
47 views

Graphing on a region with integral

Can someone help me with this I am very lost in this
1
vote
2answers
100 views

Property of Banach algebra with involution

Let $\mathcal{B}$ be a Banach algebra with involution *. Is it always true that $\forall A \in \mathcal{B}: \| A \|^2 \geq \| A^* A \| $? (motivation: I read a proof that bounded linear operators on ...
3
votes
2answers
120 views

Show that $\int_{(0,1)\times (0,1)} \frac{1}{1-xy} dxdy = \sum_{n=1}^{\infty} \frac{1}{n^2}$

I'm looking for a clever way to show that $$ \int\limits_{(0,1)\times (0,1)} \frac{1}{1-xy} dxdy = \sum_{n=1}^{\infty} \frac{1}{n^2}.$$ All suggestions will be appreciated!
3
votes
1answer
42 views

ve that the perpendiculars to the sides at these points meet in common point if and only if $ BP^2 + CQ^2 + AR^2 = PC^2 + QA^2 + RB^2 $

$P, Q, R$ are points on the sides $BC,CA,AB $ of triangle $ABC$. Prove that the perpendiculars to the sides at these points meet in common point if and only if $ BP^2 + CQ^2 + AR^2 = PC^2 + QA^2 + ...
2
votes
0answers
35 views

If $E_1, E_2$ are connected and $A_1\subseteq E_1$ and $A_2\subseteq E_2$ then $(E_1\times E_2)-(A_1\times A_2)$ is connected [duplicate]

Let $E_1$, $E_2$ connected metric spaces. Let $A_1\subseteq E_1$ and $A_2\subseteq E_2$ proper subsets. Show that the complement of $A_1\times A_2$ $$(E_1\times E_2)-(A_1\times A_2)$$ is connected. I ...
3
votes
2answers
59 views

Can $\Phi^{-1}(x)$ be written in terms of $\operatorname{erf}^{-1}(x)$?

Can the inverse CDF of a standard normal variable $\Phi^{-1}(x)$ be written in terms of the inverse error function $\operatorname{erf}^{-1}(x)$, and, if so, how? This seems like an easy question, but ...
3
votes
2answers
218 views

If $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not ring-isomorphic to $n\mathbb{Z}$

Show that if $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not ring-isomorphic to $n\mathbb{Z}$. Can I get some help to solve this problem
1
vote
1answer
47 views

Justify conditional

If there is an interpretation $I$ in wich $A=\forall x (P(x)\rightarrow Q(x))$ is true, then $(P(x)\rightarrow Q(x))$ is not true ands is not false in $I$. I need to justifiy this is false. So, if A ...
3
votes
2answers
137 views

Set of permutation matrices

I'm stuck in this problem. Prove the set $P$ of $n×n$ permutation matrices spans a subspace of dimension $(n−1)^2+1$
1
vote
2answers
92 views

Number of combinations when you can choose none or multiple options for a questions.

I would like to know both the formula and the math name for such combination. Simple example: 1 questions with 3 options, you can choose none, one or multiple options. How can I calculate the number ...
1
vote
0answers
55 views

One partial differential equation

Where can I find information about equation $$\frac{\partial u(x,t)}{\partial t}-\operatorname{div}\left(A(x)\nabla u(x,t)\right)=f(x,t),\text{where } A(x) \text{ is a matrix 2x2} ?$$ I would be ...
4
votes
2answers
200 views

Can any finite group be realized as the automorphism group of a directed acyclic graph?

We are given a finite group $G$ and wish to find a DAG (directed acyclic graph) $(V,E)$ whose automorphism group is exactly G (a graph automorphism of a graph is a bijective function $f:V\to V$ such ...
2
votes
1answer
395 views

The smallest example of a Carmichael number

A composite integer n is a Carmichael number if the only Fermat witnesses for $n$ are those $a \in \mathbb Z_n^+$ which are not coprime with $n$. The smallest example of such a number is $561 = ...
2
votes
2answers
118 views

Are $p$-groups Engel groups?

A group $G$ is termed Engel if whenever $x, y \in G$, there exists an integer $n$ (depending on $x$ and $y$ such that $[x,y,\dots,y]=1$, where $y$ occurs $n$ times. Is it true that every infinite ...
8
votes
2answers
885 views

Proving the equivalency of Principle of Mathematical Induction and Well Ordering Principle

I would like to know how from the very basic I can teach some one the above title statement. Here is my plan. $\textbf{First}$ I will state WOP: Every non-empty set of positive integers contains a ...
2
votes
0answers
172 views

Exact and Closed forms on Manifolds with Boundary

Let $\bar{M}$ be a manifold with boundary and let $M$ be its interior. Is this statement correct? A smooth k-form $\alpha$ on $\bar{M}$ is closed (exact) if and only if its restriction to $M$, i.e. ...
2
votes
3answers
145 views

Show that $|z-z_1|^2 + |z-z_2|^2 = |z_1 - z_2|^2$

The problem is : if $z$ lies on a circle with diameter having endpoints $z_1$ and $z_2$ then show that $|z-z_1|^2 + |z-z_2|^2 = |z_1 - z_2|^2$ where $z, z_1, z_2 \in \mathbb{C}$. The angle subtended ...
5
votes
1answer
439 views

Proof Complex positive definite => self-adjoint

I am looking for a proof of the theorem that says: A is a complex positive definite endomorphism and therefore is A self-adjoint. Does anybody of you know how to do this?
0
votes
1answer
69 views

Angle limit problem

I have been trying to interpret orientation angle data retrieved from a sensor device. It returns the angle in Radian units towards North that the device is measuring at the moment. The problem I am ...
2
votes
3answers
3k views

Odds of being correct X times in a row

Is there a simple way to know what the chances are of being correct for a given number of opportunities? To keep this simple: I am either right or wrong with a 50/50 chance. What are the odds that ...
7
votes
11answers
961 views

limit question: $\lim\limits_{n\to \infty } \frac{n}{2^n}=0$

$$ \lim_{n\to\infty}\frac n{2^n}=0. $$ I know how to prove it by using the trick, $2^n=(1+1)^n=1+n+\frac{n(n-1)}{2}+\text{...}$ But how to prove it without using this?
0
votes
2answers
795 views

When does convergence in distribution imply convergence in probability?

I was looking at the proof for the Delta Method (http://en.wikipedia.org/wiki/Delta_method#Proof_in_the_univariate_case) and there is something I am quite confused about. It gives $\sqrt(n)[X_n - ...
3
votes
2answers
80 views

How do we know $p/q$ can be expressed as a terminating fraction in base $B$ only if prime factors of $q$ are prime factors of $B$?

On cs.stackexchange I asked a math question: How to demonstrate only 4 numbers between two integers are multiples of .01 and also writable as binary. Yuval Filmus answered with a explanation ...
1
vote
1answer
29 views

Finitely many non-convergent ultrafilters

I am trying to prove that if a space $X$ has finitely many non-convergent ultrafilters, then every non-convergent ultrafilter $\mathcal U$ contains a set $A$ that is not contained in any of other ...
8
votes
3answers
193 views

For real numbers $x$ and $y$, show that $\frac{x^2 + y^2}{4} < e^{x+y-2} $

Show that for $x$, $y$ real numbers, $0<x$ , $0<y$ $$\left(\frac{x^2 + y^2}{4}\right) < e^{x+y-2}. $$ Someone can help me with this please...
0
votes
1answer
62 views

Applied Probability with Four Disjoint Subsets.

I have a rather longwinded question, so please bear with me! A number of employees conduct duties at company X. Each type of employee meets by department 4 times yearly. Occasionally, a ...
-1
votes
2answers
696 views

Families of curves, differential equation problem [closed]

Obtain the D.E. having a solution as the equation representing all circles whose radius 1 and center's on the line $y=x$.
5
votes
2answers
117 views

Functions whose derivatives can be written as a function of themself?

What kinds of function $f: \mathbb{R} \to \mathbb{R}$ can be written as some function of itself? I.e. $f'(x) = g(f(x))$ for some function $g$? If $f$ is given, can $g$ be solved in terms of the ...
3
votes
2answers
121 views

In how many ways can $3$ balls be tossed into $3$ boxes?

$3$ balls are tossed into $3$ boxes. In how many ways can that be done? Well, I did in the following way. We have $5$ objects: $3$ balls and $2$ walls of boxes. So a configuration, for example: ...
2
votes
1answer
100 views

Reference request for the law of the stopping time in the gambler's ruin problem

Suppose we have a sequence of independent and identically distributed random variables $(X_n)_{n\ge 1}$ such that $$ P(X_n=1)=p,\quad P(X_n=0)=r,\quad P(X_n=-1)=q $$ with $p,q,r\in[0,1]$, $p+q+r=1$, ...
1
vote
1answer
69 views

Is this function involving matrices convex?

Let $X\in \mathbb{R}^{n \times n}$. Then, is the function $$ \text{Tr}\left( (X^T X )^{-1} \right)$$ convex in $X$? ($\text{Tr}$ denotes the trace operator)
1
vote
1answer
419 views

Two questions on Lebesgue Decomposition of an increasing function?

I come up with the question in doing Stein's Real analysis, Chap3. Ex. 24, which assert that any increasing function $f$ on $[a,b]$ can be decomposed as $$F=F_A+F_C+F_J,$$ with $F_A$ is absolutely ...
2
votes
3answers
1k views

How do we take second order of total differential?

This is the total differential $$df=dx\frac {\partial f}{\partial x}+dy\frac {\partial f}{\partial y}.$$ How do we take higher orders of total differential, $d^2 f=$? Suppose I have $f(x,y)$ and I ...
8
votes
1answer
100 views

Curve over ring, covered by two affines

The reference is http://www.math.columbia.edu/~masdeu/files/notes/FallSeminar.pdf, page 9: Let now $C/R$ be a curve over a noetherian ring $R$; this means that $C$ is smooth, connected, integral, ...

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