2
votes
1answer
42 views

Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$

Problem : Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$ Then which of the following is always true $(a) f(2) = f(7)$ $(b) f(4) = f(10) $ $(c) f(2) =f(4) $ ...
113
votes
6answers
7k views

In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...
1
vote
1answer
61 views

Finding the definite integral $\int_{0}^{2\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx$

$$\int_0^{2\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx$$ My try: $$I=\int_0^\pi \frac{e^{\sin x}\cos(x)}{1+e^{\tan x}} dx+\int_\pi^{2\pi} \frac{e^{-\sin x}\cos(x)}{1+e^{\tan x}} dx$$ also ...
1
vote
1answer
11 views

Linear regression relationships

Velocity $= X$, distance to stop $= Y$ $\beta_0= -17.5791$, $\hat{\operatorname{se}}(\beta_0)=6.7584$ $\beta_1 = 3.9324$, $\hat{\operatorname{se}}\beta_1 = 0.41.55$ degrees of freedom $=48$ (a) is ...
1
vote
1answer
995 views

What is wrong with my solution of finding remainder of $50^{(51^{52})}$ when divided by 11?

I used the following method using remainder theorem. (I used method from here: Find the remainder of $128^{1000}/153$.) $$\begin{align} (50^{{51}^{52}})/11 & = (50^{2652})/11 \implies \\ ...
1
vote
2answers
46 views

What is the remainder when 50^51^52 divided by 11?

To find the remainder: $$50^{51^{52}} \mod 11$$ I have solved till: $$6^{51^{52}} \mod 11$$ But not able to proceed further. Help please.
1
vote
1answer
43 views

All integer solutions to diophantine equation: $x^2+p y^2=z^2$?

I would like to find all integer solutions to the Diophantine equation $$ x^2+p y^2=z^2 $$ where $p\ge2$ is a given prime number. Also prove that my (probably parametric similar to Pythagorean triples ...
0
votes
3answers
46 views

Continuous but not uniformly continuous example

Let $f(x) = \frac{1}{x}$ for $x > 0$ and take our set at which the function act on $(0,1]$. This function is continuous but not uniformly continuous on $A$. To prove this consider $\epsilon = ...
1
vote
2answers
27 views

Is it true that $x_k\rightarrow x$ iff. $\exists N \in \Bbb{N}$ st. $k>N$ implies $|x_k-x|<a_k$

My question is, Is it true that $x_k\rightarrow x$ iff. $\exists N \in \Bbb{N}$ st. $k>N$ implies $|x_k-x|<a_k$ for some $a_k$ where $a_k>0$ and $a_k \rightarrow 0$ as $k \rightarrow 0$. ...
0
votes
2answers
31 views

Trigonometric Form of Complex Numbers question.

What is the following quotient expressed in polar form: $$\frac{10(\cos(35^{\circ})+ isin(35^{\circ}))}{5(\cos(100^{\circ}) +i\sin(100^{\circ}))}?$$ Please enter your answer in cis notation and ...
0
votes
2answers
20 views

Binary expansion, finding the greatest power of $2$ less than a given number

I'm looking to better understand binary for a CS50 problem set. I'm not understanding transferring decimal notation to binary. For example, use 237. How to find the largest power of $2$ less than ...
13
votes
0answers
63 views

Curtains and groups

This picture is a copy of the pattern on my curtains. The points of a hexagonal lattice are each coloured with one of four possible colours. It has translational symmetry in two directions: a ...
1
vote
1answer
35 views

Curl of a vector field cross itself

How we can use the property that $$A×(B×C) = B(A.C)- C(A.B)$$ to prove the relation: $$a×(∇×a) = ∇ (a^2/2) -(a.∇)a.$$ When I use it, the result directly appear to be $$∇(|a|^2 )-(a.∇)a$$ instead of ...
0
votes
3answers
49 views

$\mathbb{N}$- a complete metric space with $d(x,y)=|x-y|$

$\mathbb{N}$- a complete metric space with $d(x,y)=|x-y|.$ This seems quite intuitively correct, but I do not know how to prove this formally, does anyone know how they would go about this?
0
votes
1answer
23 views

Solving differential equation using Laplace transform

Can this DE be solved using Laplace transform? $\frac{\mathrm{d} y}{\mathrm{d} x}\cos x=y\sin x+\cos ^{2}x$
1
vote
0answers
21 views

Example of $S[1/a] \cong S[1/b]$ as rings via $\phi$, where $S$ is a UFD, $a, b \in S$, and $\phi(U(S)) \neq U(S)$.

Above, $U(S)$ refers to the units of $S$. This problem stems from reading a paper titled "Translates of Polynomials" from 2000, where a fact about a ring isomorphism between $S[1/a]$ and $S[1/b]$ is ...
3
votes
4answers
260 views

How to analyze risk vs. reward for spending on research and development work?

Imagine I have a company that makes widgets, where each widget costs me A dollars to make. Each month I can allocate money toward research and development with the aim of finding a new process that ...
0
votes
1answer
66 views

If $g$ is a permutation, then what does $g(12)$ mean?

In Martin Lieback's book 'A Concise Introduction to Pure Mathematics', he posts an exercise(page 177,Q5): Prove that exactly half of the $n!$ permutations in $S_n$ are even. (Hint: Show that ...
1
vote
2answers
40 views

Does construction of infinite product measure require axiom of choice?

I am learning about infinite (countable) product measure, which the exact statement of the theorem I write below. I was wondering if the theorem requires axiom of choice or not. I would appreciate any ...
1
vote
2answers
109 views

Compact Operators: Trace

Given a Hilbert space $\mathcal{H}$. Consider a bounded operator: $$A:\mathcal{H}\to\mathcal{H}:\quad\|A\|<\infty$$ Regard ONB's: ...
0
votes
0answers
30 views

Finding sequences in $\pi$ [duplicate]

I recently came across some programs which were able to calculate exactly, when a particular sequence of digits appears the first time in the decimal expansion of $\pi$. This made me wondering, if it ...
2
votes
0answers
36 views
+50

Let $f(x)$ be defined over all rationals $x$ in $[0,1]$ and let $F(n) = \sum_{i=1}^n f(\frac in)$

also define $$F^*(n) = \sum_{i=1\,\,(i,n)=1}^n f(\frac in)$$ then prove that $$F^* = \mu * F$$ where $\mu$ is the Möebius function and the $*$ means the Dirichlet convolution. I tried the Bell series ...
0
votes
1answer
26 views

Is linear independence preserved through column transformations?

Let's say you have a $m\times n$ matrix $A$, and the $n$ column vectors are linearly independent. And let's say you have a transformation $T$. You perform the transformation on each column of $A$, ...
1
vote
1answer
327 views

Hilbert cube is compact

Let $\{u_n\}_{n\in \mathbb N}$ be an orthonormal set in $H$ (Hilbert space). How prove that the set $\displaystyle Q=\{x\in H :\ x=\sum_{i=1}^{\infty}{c_nu_n}, \ \mbox{where} |c_n|\leq\frac{1}{n} \}$ ...
1
vote
0answers
9 views

Basic examples of functions in Hörmander class

The Hörmander class $S_{\rho,\delta}^m$ (with $\rho,\delta\in[0,1]$) consists of smooth functions $p(x,\xi)$ with $$|D_x^\beta D_\xi^\alpha p(x,\xi)|\leq ...
0
votes
2answers
2k views

Matrix Norm Inequality

So I'm trying to prove that $\lVert A\rVert_\infty \leq \sqrt{n} \lVert A\rVert_2$. I've written the right hand side in terms of rows, but this method doesn't seem to be getting me anywhere. Where ...
2
votes
1answer
23 views

Minimizing long equation with hyperbolic functions

In physics book that I am reading it is said that minimizing the expression $$\phi = - N T k \log (2 \cosh(H \beta)) - \frac{J N}{2} z \tanh^2(H \beta) + H N \tanh(H \beta) $$ with respect to $H$ ...
1
vote
2answers
50 views

A doubt on Krull's Principal Ideal Theorem Proof

Sorry if it is a dumb question but i'm studying the proof of Krull's PIT from this pdf and i don't understand why the author uses in his proof the ideals $P^{(n)}=P^nR_P\cap R$ instead of the simpler ...
1
vote
2answers
525 views

Rotating the gradient

Suppose I have a triangle T in 3dimensional space and i want to rotate it in arbitrary ways. The coordinates for T are given by $f: T_R \in \mathbb{R}^2 \rightarrow T \in \mathbb{R}^3 $ where $T_R$ is ...
0
votes
0answers
16 views

Proof of x = 0 modulo 3 only if the sum of its digits 0 modulo 3 [duplicate]

Okey, lets beggin from a helpfull proposition I've already proved: $$$$ if $a_i\equiv b_i\:\forall 0\le i\le m$ then to any $m$ numbers: $p_1,p_2,...,p_m\in \mathbb{Z}$ $$\sum ...
1
vote
2answers
31 views

Show $f: S^1 - {N} \to \mathbb{R} $ $f(x_1,x_2) = \frac{x_1}{1-x_2}$ is Homeomorphism

$S^1$ is a unit circle and $N := \{ (0,1) \in S^1\}$. The question hints that the for any $(x_1,x_2) \in S^1- {N}$, line joining $N$ and $( x_ 1 , x_ 2 )$ meets the $x$ -axis at ($f ( x_ 1 ;x_ 2 ) , 0 ...
0
votes
1answer
39 views

Why is the discriminant of the discriminant negative?

On this link is a question about functions. My question is, in that question itself, a pivotal part of the solution is to realise that the discriminant of the (positive) discriminant is negative. ...
0
votes
1answer
18 views

Polynomial with arithmetic values

Can I find a polynomial in a second degree in two variables from the values of which can be found an infinite arithmetic progression? Thank you!
1
vote
0answers
31 views

Why is the coproduct $\Delta$ of the quantum double $D(H)$ an algebra homomorphism?

If $H$ is a a finite dimensional Hopf algebra, $H^\ast\otimes H$ has a Hopf algebra structure with multiplication $$ (\phi\otimes h)(\psi\otimes g)=\sum \psi_2\phi\otimes h_2g\langle ...
0
votes
1answer
35 views

Asymptotic direction

Me and my classmates are interested in a visual description of an asymptotic direction at a point of a surface. The normal curvature in an asymptotic direction at a point is zero. And a curve on a ...
2
votes
1answer
22 views

Direct Limit of finitely generated groups

Is every group the direct limit of its finitely generated subgroups? This is true for abelian groups, I have not seen this statement for nonabelian groups, so i am wondering if this is true. Seems ...
0
votes
2answers
63 views

How to differentiate the following interesting vector product?

How do we differentiate the following vector product with respect to $\boldsymbol r$. \begin{equation} \frac{d}{d\boldsymbol r}\bigg[(\boldsymbol \omega \times\boldsymbol r)\cdot (\boldsymbol \omega ...
0
votes
1answer
30 views

Prove that if events $A,B$ independent of C then $P(A\cap B\cap C)= P(A\cap B)P(C)$

I am trying to prove why the intersection of two events $A, B$ that are independent of C is also independent of C so that the following equality holds: $$P(A\cap B\cap C)= P(A\cap B)P(C)$$ ...
1
vote
2answers
19 views

Can we replace the limit of a sequence with that of a function?

Let $f$ be a function defined in $[1,\infty]$. If $\lim_{x\to\infty}f(x) = L$ and $a_n = f(n)$ for integer $n\ge 1$ then $\lim_{n\to\infty}a_n = L$. Found this theorem in many references, but ...
1
vote
1answer
39 views

Logarithm Propr

I'm having a bit of trouble proving the following property: Theorem If $Re(z)>0 $ and $ Re(w)≥0$, then $\log(zw)=\log(z)+\log(w)$, where log is the principal branch. I know that $\log (zw) = ...
0
votes
1answer
45 views

Direct Sum Proof

I am reading Axler's Linear Algebra Done Right Book and I am on the part about the direct sum. He gives the following proposition: When he assumes that $a$ and $b$ hold to prove that the proof gives ...
2
votes
2answers
41 views

Prove line connecting intersection of tangents and opposite vertex bisects segment containing intersection of tangents and a vertex

Let $\triangle ABC$ be an isosceles triangle with $AB=BC$. Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let the tangents at $A$ and $B$ intersect at $D$, and let $DC\cap\Gamma=E\neq C$. Prove ...
3
votes
3answers
46 views

Laplace Transform of a Heaviside function

Find the Laplace transform. $$g(t)= (t-1) u_1(t) - 2(t-2) u_2(t) + (t-3) u_3(t)$$ I understand that the $\mathcal{L}\{u_c(t) f(t-c)\} = e^{-cs}*F(s)$ Finding $F(s)$ is the hard part for me. My ...
4
votes
3answers
530 views

Expected Value of Local Maxima and Local Minima

Recently I came across this question: Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima ...
1
vote
2answers
26 views

Best way to answer average question with large range of data

I have expense data for 30+ departments. I want to figure the best way to answer the question 'what is the average expense?' The problem is that each department has a different range of data and size. ...
1
vote
1answer
23 views

Sizes of Quotient Rings of DVRs with Finite Residue Field

If $R$ is a discrete valuation ring (DVR) with maximal ideal $\mathfrak{m}$ such that $R/\mathfrak{m}$ is finite, then all quotient rings of $R,$ namely $R/\mathfrak{m}^n$ for $n \in \mathbb{N},$ are ...
-1
votes
0answers
21 views

What does the Graph of ax0 + bx1+cx2+dx3… look like? [on hold]

What does the Graph of ax0 + bx1+cx2+dx3... look like? How do I plot it in mathematica? I don t have numbers for a,b,c.
1
vote
2answers
55 views

The bisector of $\angle BAC$ of triangle $\Delta ABC$ cuts $BC$ at $D$

The bisector of $\angle BAC$ oF triangle $\Delta ABC$ cuts $BC$ at $D$ and circumcircle of triangle at $E$. if $$AD=5 \text{ cm} ,\ DE=3 \text{ cm},\ AC=4 \text{ cm}, $$ then what is the length of ...
2
votes
2answers
28 views

Homomorphism with intersection of all Sylow p-subgroups as kernel?

Does anyone know of a homomorphism from a group $G$ to another group with kernel as the intersection of all Sylow $p$-subgroups? I was trying to prove that the intersection of Sylow subgroups is ...
1
vote
3answers
121 views

Prove that for every planar graph, there is a partition $V = V_1 \cup V_2 \cup V_3$ such that the graphs with those are acyclic

Prove that for every planar graph $G = (V,E)$ with $|V| \geq 3$ there is a partition of V to $V = V_1 \cup V_2 \cup V_3$ such that $V_1 \cap V_2, V_1 \cap V_3, V_2 \cap V_3 = \emptyset$, where for ...

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