1
vote
2answers
39 views

Seeking to prove Continuity of $f(x) =\frac{x}{1+||x||}$

How would I prove that $f:\mathbb{R}^n\rightarrow B(\theta,1)$, where $f(x)=\frac{x}{1+||x||}$, is continuous? For metric spaces, I understand that if $f(x)$ is continuous at a point $p$ in $\...
1
vote
2answers
67 views

$(f_n)$ Borel, $f(x)=\lim_{n\to\infty} f_n(x) \implies f$ Borel

Prove that if $(f_n)$ is a sequence of Borel measurable functions and if $f(x)=\lim_{n\to \infty}f_n(x)$ exists in $\mathbb{R}$, then $f$ is Borel measurable. In fact $f$ is Borel measurable even if ...
2
votes
2answers
80 views

Can $ 2^{3^{4^{.^{.^{.^{n-1}}}}}}\equiv 1 \bmod {n} $ for some $n>7$.

Prove or disprove that there isn't any positive integer $n>7$ such that the linear congruence below is true. $ 2^{3^{4^{.^{.^{.^{n-1}}}}}}\equiv 1 \bmod {n} $
0
votes
1answer
62 views

Tangent space to the tangent bundle

I'm having some trouble understanding the tangent space to a point on the tangent bundle. I have to prove for homework that the tangent bundle to a smooth manifold has a canonical orientation, even if ...
1
vote
1answer
109 views

Finding Fixed Point

If there is a continuous function from the closed unit disk to itself such that it is identity map on boundary, must it admit a fixed point in the interior of the disk?
2
votes
1answer
53 views

Möbius function on a finite poset (X, $\leq$)

I'm having some difficulties with the following problem: Give an example of a finite poset $(X, \leq)$ and elements $a,b \in$ X such that $\mu(a,b)=-4$ where $\mu$ is the Möbius function of $(X, \...
0
votes
1answer
51 views

contour integral to show the sum of a series

Let $Γ_n$ be the cycle that traverses the square with vertices ±(n+ $\frac 12$ )(1±i). Show that there is a constant δ > 0 such that |sin(πw)| > δ for all w ∈ $U_n$ $Γ_n$. By considering the ...
0
votes
2answers
37 views

Prove that there is exactly one pair of reals $a, b$ such that $x^{2} = x\sin x + \cos x$ for $x=a,b$

Some observations. Let $$f(x) := x^{2} - x\sin x - \cos x.$$ Then $$f'(x) = 2x - x\cos x.$$ On setting $f'(x) := 0,$ I obtain $$x = 0$$ (because either $x=0$ or $x \neq 0$ and if $x \neq 0$ then $\...
1
vote
0answers
28 views

What is a “nonparametric function”?

I am looking for a formal definition of the term "nonparametric function." I understand the term and I use nonparametric regressions http://en.wikipedia.org/wiki/Nonparametric_regression but I haven'...
14
votes
6answers
2k views

Prove $1+2\sqrt3$ is not a rational number

How would I go about proving $1+2\sqrt 3$ is not a rational number assuming $\sqrt 3$ is not a rational? Would direct proof be the easiest? Total beginner here, any insight would be appreciated.
0
votes
0answers
10 views

an upper set is of the form $[a, \infty) $ if it has the least element $a$

$(DEF):$ Let $X$ be a totally ordered set. A set $A \subset X$ is called an upper set if $a \in A $ and $x > a $ implies $x \in A $. Suppose $A \subset X$ is an upper set that has the least ...
1
vote
0answers
70 views

Finding square root of $-5-12i$ by formula and by De Moivre's Theorem

I was trying to obtain the square root of $-5-12i$ by the formula for square root (given below) and also by De Moivre's theorem and verify that both give the same result. But the two results are ...
0
votes
1answer
31 views

Best way to prove adjacency of a graph

Assume n is even. Considering a graph where each vertex in $v_1,...,v_n$ is adjacent to the next (ie $v_i \sim v_{i+1}$ for $1\leq i<n$) and where $v_1,v_n$ are each connected to at least $n/2$ ...
0
votes
1answer
17 views

Probability with Combinations

There are $13$ cards dealt. Find the probability of getting at least $1$ heart? I am not sure how to set up the problem. I would appreciate any help!
1
vote
0answers
56 views

Prove $ \lim_\limits{n \to\infty}{(1 + \frac{1}{1!} + \frac{1}{2!} + \cdot\cdot\cdot + \frac{1}{n!})} = e $ [duplicate]

Use that $ \lim_\limits{n \to \infty}{\left(1 + \frac{1}{n}\right)^n} = e $ to show that $ \lim_\limits{n \to \infty}{\left(1 + \frac{1}{1!} + \frac{1}{2!} + \cdot\cdot\cdot + \frac{1}{n!}\right)} = e ...
0
votes
2answers
108 views

If $\tan x=\sin x/\cos x$ then what is $\tan 3x$ equal to?

Would $\tan 3x$ be equal to $\sin 3x/\cos x$? Or perhaps $\sin 3x/\cos 3x$? Regards, Tom
1
vote
3answers
46 views

if x is an integer, then $(x^3+1)\bmod 3 = (x+1)^3 \bmod 3 $

Can anyone help me explain why if $x$ is an integer, then $(x^3+1)\bmod 3 = (x+1)^3 \bmod 3$? I know there are 3 cases. $x=0\bmod3,\ x=1\bmod3,$ and $x=2\bmod3$ totally new to this form of mathematics,...
0
votes
1answer
96 views

Number of ways of putting n indistinguishable balls into k indistinguishable groups.

http://www.campusgate.co.in/2011/10/permutations-balls-and-boxes-related.html Can someone explain how the recurrence table for Case 4 has been obtained ?
5
votes
2answers
75 views

Maximize the prime sums in a table

The entries of a $3×3$ table are integers from $1$ to $9$, and each number appears exactly once. Consider the row, column, and diagonal sums of numbers in the table. Find the maximum number of these ...
0
votes
1answer
109 views

Difference between Double and triple integral?

Hi all I am going to be starting multivariable calc and I am trying to read up but I can't seem to quite grasp this exactly yet. What are the differences between double and triple integrals? I am ...
1
vote
0answers
85 views

Roots of Unity and Primitive roots

For $\mathbb{Z}/13$, I want to find its primitive roots and 4th roots of unity. For $g$ to be a primitive root, we must have that $g^6 \neq 1 \pmod{13}$ and $g^4 \neq 1 \pmod{13}$. $2$ satisfies this....
0
votes
1answer
24 views

Series having strict inequality implies limits having strict equality?

I was wondering if I have two convergent series, say, $\sum_{n=1}^{\infty} s_n = s$ and $\sum_{n=1}^{\infty} t_n$ = t, and for all their partial sums we have that: $s_n > t_n$. Is it then ...
0
votes
1answer
109 views

Differential Geometry-Hodge Star

The Hodge star is given by $$*(dx^{i_1}\wedge dx^{i_2}\wedge....\wedge dx_{i_p})=\frac{1}{(n-p)!}e_{i_1 i_2....i_p i_{p+1}...i_n}dx^{i_{p+1}}\wedge dx^{i_{p+2}}\wedge....\wedge dx^{i_n}$$ The question ...
0
votes
1answer
112 views

Find a sequence of R-integrable functions whose pointwise limit is R-integrable but the limit of $\int (f_n)$ does not equal to $\int f$

There exists a sequence of Riemann integrable functions $(fn)$ on $[0, 1]$ whose pointwise limit $f$ is still Riemann integrable, but for which $$\begin{equation} \lim_{n\to\infty} \int_0^1 f_n(x)\,dx ...
3
votes
1answer
127 views

Continuous function from the closed unit disk to itself being identity on the boundary must be surjective?

If there is a continuous function from the closed unit disk to itself such that it is identity map on boundary, must it be onto?
2
votes
0answers
39 views

Zeros of vector field extended from field lines

Suppose I have a finite number of 1-dimensional curves embedded in a 3-dimensional space. What can I say topologically about vector fields chosen so that these closed curves are field lines? For ...
0
votes
3answers
45 views

Proving statistical assertions?

Are these assertions True or False? I'm not quite sure how to go about proving or disproving them… If $X ∼ \text{Unif}[0, 2]$, then $E(\ln(X)) \le 0$. If $X ∼ \text{Unif}[0, 2]$, and $Y = 1[...
1
vote
1answer
26 views

Random Variables and Probability Density Function

I'm having a little trouble with a homework problem. I will write out what I've figured out and would appreciate some help with what I'm unable to understand. You flip a weighted coin four times. ...
0
votes
0answers
69 views

Matrix of rank r is sum of r column-times-row matrices

Prove that if $F$ is a field and if $A \in M_{m,n}\left(F\right)$ has rank $r$, then there exist $v_1, v_2, \ldots, v_r \in F^m$ and $w_1, w_2, \ldots, w_r \in F^n$ such that $A = \sum\limits_{i=1}^r ...
0
votes
1answer
19 views

Finding critical points of a multivariable

I'm a bit confused on the procedures for finding the zeros of the partial derivatives of these kind of functions. If you could correct what I am doing wrong it would be much appreciated. $$ f(x,y)=8xy-...
0
votes
1answer
37 views

Show that the real projective line, P1, is orientable

I'm asked to show that the real projective line, P1, is orientable. I'm not quite sure how to define orientable to prove this. Thanks.
1
vote
0answers
62 views

What does invariant exactly mean and how does it get the invariant?

I have read many journal about simulation of regularized long wave. In numerical test section,many researcher use invariant of mass,momentum and energy to check accuracy of their method but i found ...
0
votes
2answers
88 views

Unwind the equation

Let $x, y, z, t$ be positive integers. Given that $$68(xyzt+xy+zt+xt+1)=157(yzt+y+t)$$ Find the value of the product $xyzt$. I couldn't even start with the problem. I just know that the expression n ...
1
vote
1answer
99 views

Proving the recursive formula for “Virahanka Numbers”

So apparently Virahanka was an Indian mathematician that, in a way, discovered the Fibonacci sequence 500 years before Fibonacci. He was interested in finding the number of patterns of short syllables ...
0
votes
1answer
21 views

Boundedness in a linearly ordered set

Let $X$ be a linearly ordered set and $A \subset B \subset X $ Can it happen that $A$ is bounded in $X$ but not in $B$ ? Can it happen that $A$ is bounded in $B$ but not in $X$ ? Attempt: I think ...
0
votes
0answers
43 views

Pricing of Binary or Digital Options or Feynman-Kac Equation for $\mathbb E f(X_T)$ with diffusion $X$ and discontinuous function $f$.

I am trying to find references (books, papers, etc.) for calculating $\mathbb E f(X_T)$, where $X_T$ is a diffusion and $f$ is a real function that is not continuous by means of solving a PDE or ...
1
vote
1answer
38 views

$(f_n)$ Borel measurable implies $\sup_n f_n$ and $\inf_n f_n$ Borel measurable

Suppose $(f_n)$ is a sequence of Borel measurable functions. Show that both $\sup_nf_n$ and $\inf_nf_n$ are Borel measurable. Attempt: Suppose $(f_n)$ is a sequence of Borel measurable functions. ...
0
votes
1answer
105 views

Find $\lim_\limits{n\to \infty}\underbrace{{\sin\sin\cdots\sin(x)}}_{n\text{ times}}$. Why am I wrong? [duplicate]

Find $\lim_\limits{n\to \infty}\underbrace{{\sin\sin\cdots\sin(x)}}_{n\text{ times}}$. It is known that after the first sine, we get something in $[-1,1]$. If it is $0$ then it is constant and ...
2
votes
1answer
83 views

Minimizing the product $xy$ subject to a polynomial constraint on $x, y$

Given that $$16y(x^2+1)=25x(y^2+1),$$ where $x,y$ are positive integers, find the smallest possible value of $xy$. I wrote my expression as a quadratic in $x$ and calculated it in form of $y$. Then $...
1
vote
1answer
30 views

Proof on the exterior measure

Let $E \subset \mathbb{R}^n$ be a measurable set and let $\delta > 0$. Show that $m^*(\delta E)=\delta^n m^*(E)$, where $m^*$ is the exterior measure (outer measure). Attempted proof: Let $\{Q_j ...
0
votes
1answer
24 views

Help Constructing an infinitely differentiable function…

Given $a<b$, I need to find a function $\psi\in C^{\infty}(\mathbb{R})$ so that $\psi(x)=0$ when $x\leq a$, $0<\psi(x)<1$ when $a<x<b$, and $\psi(x)=1$ when $x\geq b$. Previously, I ...
5
votes
5answers
372 views

Prove if $n^2$ is even, then $n^2$ is divisible by 4

I am working on this question Prove for every integer n if $n^2$ is even, then $n^2$ is divisible by 4. prove by contradiction Proof: Since there exists an integer $n$ such that $n^2$ is even,...
3
votes
1answer
76 views

Evaluate a rational function of $x,y,z$ given two polynomial equations in $x,y,z$

Let $x, y, z$ be real numbers. Given that $$2x(y^2−1)+2y(x^2−1)=(1+x^2)(1+y^2)$$ and $$4z(1−y^2)+4y(1−z^2)=(1+z^2)(1+y^2)$$ Find the value of the following expression: $$\Bigg(\frac{2x}{1+x^2}−\frac{...
3
votes
1answer
82 views

Order zero maps in matrix algebra

Let $a$ and $b$ are two elements in a $C^*$algebra $A$. We say $a\perp b$ if $ab=ba=a^*b=ab^*=0$. We say a completely positive map $\phi: A \rightarrow B$ is of order zero if for any positive elements ...
0
votes
1answer
54 views

In an equilateral spherical triangle, show that SecA=1+Seca

Q. In an equilateral spherical triangle, show that $SecA=1+Seca$ So A is the vertex or the angle of the triangle and a is the side of the equilateral spherical triangle. I started off the proof by ...
1
vote
2answers
86 views

Equivalent definition of limit of a function (Reference request)

Let $f: \mathbb R \to \mathbb R$, $x_0 \in \mathbb R$. We write $$\lim_{x\to x_0} f(x) = L$$ if for all $\epsilon>0$, there is $\delta >0$ so that $$|f(x) - L | <\epsilon$$ whenever $|...
5
votes
2answers
298 views

Sum of squares in geometric progression

In the geometric progression $b_1, b_2, b_3,\ldots, b_1+b_3+b_5=10$ and $b_2+b_4=5$. Find the sum of the squares of the first five terms. If you solve for the first term and the common ratio, you ...
1
vote
1answer
61 views

Definition of connection on vector bundle

A connection on a vector bundle $E$ is a map $ D:\Gamma(E)\rightarrow \Gamma(T^*(M)\otimes E)$ satisfying 1) For any $s_1,s_2\in \Gamma(E)$, $D(s_1+s_2)=Ds_1+Ds_2$ 2) For $s\in \Gamma(E)$ and $\...
1
vote
1answer
23 views

Showing that $\varepsilon_k$ make a basis

With $\mathbb{C}^n := \{(a_0, \dots, a_{n-1}) : a_j \in \mathbb{C}\}$ and $$\varepsilon_k = (1, \zeta^k, \dots, \zeta^{(n-1)k})/\sqrt{n}$$ where $\zeta = e^{2\pi i / n}$, show that $\varepsilon_0, \...
1
vote
1answer
88 views

Why is the sample Mean a consistent Estimator for the Logistic Distribution?

I think is this a very trivial question, but non the less: How can I show that the $ \hat\theta_n = $ $ \bar x $ is a consistent estimator of $ \theta _0 $. Since $ \theta _o $ is $ \mu $ for the ...

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