0
votes
0answers
11 views

rewriting one expression as other expression

Can anyone explain as how we can rewrite the first expression as second ? I am not able to pick the step done to change from 1 to 2. ...
0
votes
0answers
7 views

I need help to verify 5 order relations

Which the following relations are order relations on the set $\Bbb M$? $$\Bbb M: \{1, 2, 3\}$$ $$R:\{(1, 1),(3, 3),(1, 2),(2, 3),(1, 3)\}$$ It is not an order relation because $(2,2) \not \in R$ => ...
0
votes
0answers
2 views

Developed a function optimization strategy - need opinions

I've developed a function optimization strategy which is close to evolutionary optimization strategies. It works fine for various functions, but cannot be used with thorough success for functions with ...
0
votes
1answer
6 views

Considering the complex number $z = m+i$ for which values of $m$ do we have $ \left|\overline{z}+\frac{2}{z}\right| \ge 1 $

Good evening to everyone. I have the following problem that I tried to solve but my mathematical instinct tells me that I didn't solve it right: Considering the complex number $z = m+i$ for which ...
1
vote
3answers
16 views

Why $\sup\limits_{n\ge 1} \sum_{k=1}^n a_k = \lim\limits_{n\to\infty} \sum_{k=1}^n a_k$?

If $\{a_n\}$ is a positive numbers sequence, then $\sup\limits_{n\ge 1} \sum_{k=1}^n a_k = \lim\limits_{n\to\infty} \sum_{k=1}^n a_k$. Is this wrong or right? And why?
0
votes
2answers
26 views

Russel's paradox: what is the contradiction with $R \not\in R$?

Let the Russel's Set be: $$R = \{S | S \notin S\}$$ Where $S$ is a set Suppose $R \in R$, but by definition $R \not\in R$, contradiction. Suppose $R \not\in R$... (I am not sure what should be the ...
-4
votes
3answers
24 views

Proving that if $A\subseteq B$ and $B\subseteq C$ then $A\subseteq C$.

. If $A\subseteq B$ and $B\subseteq C$ Prove that $$A\subseteq C$$
2
votes
0answers
12 views

Is there a method that determines an unknown permutation better than $(n+1)!/2^n$ steps on average?

Suppose I have a random permutation $s \in S_n$ that is unknown to me. However, suppose I can make a query where when I ask if $i$ is in the $j$th position in the permutation, I receive a yes or no ...
1
vote
2answers
20 views

Number of true relations of the form $A\subseteq B$ where $A,B\in\mathcal{P}(\{1,2,\ldots,n\})$

I just started "Introduction to Topology and Modern Analysis" by G.F. Simmons and came across this problem in the exercises. Q. Let $U=\{1,2,\ldots,n\}$ for an arbitrary positive integer $n$. If $...
1
vote
0answers
20 views

Is this metric induced by a norm?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
1
vote
1answer
12 views

Question about arc-connected property in a continuum

Suppose $X$ is metric, compact, connected, and $p\in X$. An arc is a copy of $[0,1]$. Is it possible that every two points in $X\setminus \{p\}$ can be joined by an arc, but there is no arc in $X$ ...
-1
votes
1answer
19 views

Conditional probability using set notation

Got this wrong on a quiz and i don't have the answers. Need to figure this out for a test coming up. \begin{align} P(A) &= 0.75 \\ P(B\mid A) &= 0.9 \\ P(B\mid A^c) &= 0.8 \\ P(C\mid A\...
1
vote
0answers
11 views

'2nd order' Picard Iteration

I'm self-studying differential equations using MIT's publicly available materials. One of the problem set exercises deals with what I'm calling a second order Picard Iteration. To be explicit, we ...
6
votes
0answers
22 views

A tricky integral - $\int_0^1 \sqrt{\frac{1}{(1-t^2)^2}-\frac{(n+1)^2t^{2n}}{(1-t^{2n+2})^2}}dt $

Evaluate: $$\int_0^1 \sqrt{\frac{1}{(1-t^2)^2}-\frac{(n+1)^2t^{2n}}{(1-t^{2n+2})^2}}dt $$ Where $n$ is any positive integer. Introduction: This integral came up while studying the distribution ...
0
votes
0answers
4 views

Are there any examples of higher order ireducible linear differential operators?

Given a monic, linear differential operator $L = D^n + f_{n-1}(x)D^{n-1} + \dots + f_1(x) + f_0(x)$, say $f_0, \dots, f_{n-1}$ analytic for simplicity's sake, we say that $L$ is irreducible if there ...
0
votes
0answers
15 views

Is this a type of recurrence relation

Consider a series of integers $a$ defined by: $$\begin{cases} a_n & = c_n & \text{if $0 \le n \le 2$} \\ a_{2n} & = f(a_n, a_{n+1}) & \text{if $n > 1$} \quad \...
0
votes
2answers
19 views

What are the steps to do to solve this Algebraic problem?

A mixture of 12 ounces of vinegar and oil is 40 percent vinegar,where all of the measurements are by weight. How many ounces of oil must be added to the mixture to produce a new mixture that is only ...
0
votes
1answer
7 views

Image of $\mathbb{P}^1 \times \mathbb{P}^1$ by Segre embedding is a hyperboloid

Let $f: \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^3$ be the Segre embedding given by $((x_0:x_1),(y_0:y_1))\mapsto(x_0y_0,x_0y_1,x_1y_0,x_1y_1)$. Gathmann's notes claim that the real points of ...
2
votes
2answers
31 views

I need help with a limit without using L'Hopital

I need to do the following limit without using L'Hopital and I have not been able, please help $$\lim\limits_{x \to 3} \left(\frac{x-1}{2x-4}\right)^{\frac{1}{x-3}}$$
1
vote
1answer
19 views

If $(V,\left<.,.\right>)$ is hilbert, show that $\sum_{k=1}^\infty \left<v,e_i\right>e_i=v$

Let $(V,\left<.,.\right>)$ a Hilbert space of infinite dimension and let $\{e_i\}_{i=1}^\infty $ an orthonormal basis. Show that $\left(\sum_{k=1}^n \left<v,e_i\right>e_i\right)_{n\in\...
3
votes
3answers
54 views

What are the odds of flipping a coin 100 times and seeing HHHHT?

What are the odds of flipping a coin 100 times and seeing exactly four consecutive heads? Any more than four heads in a row, such as "HHHHH" would not be considered a string of four consecutive heads. ...
0
votes
1answer
14 views

A word problem in Vector Calc

I have been asked to find parametric equations for the tangent like to the cruve of intersection of the surfaces $ x^2 + y^2 + z^2 = 4 $ and $ z^2 = x^2 + y^2 $ at $ (1,1,-sqrt(2) $ My solution; I ...
-1
votes
2answers
20 views

Find $a$ and $b$ in the given equation below

$$a\leq \frac{\sec^{2} \theta–\tan \theta}{\sec^{2} \theta+\tan \theta} \leq b$$ I'm confused. How do I do this? Please help.
-1
votes
1answer
10 views

Suppose X and Y have joint density f (x, y) = 2 for 0 < y < x < 1. Find P (X − Y > z).

Suppose X and Y have joint density f (x, y) = 2 for 0 < y < x < 1. Find P (X − Y > z). Solution is (1-z)^2
0
votes
1answer
21 views

If $(\sqrt{x^2-5x+6} + \sqrt{x^2-5x+4})^{x/2} + (\sqrt{x^2-5x+6} - \sqrt{x^2-5x+4})^{x/2}=2^{\frac{x+4}{4}}$, find $x$.

The main question is : If $(\sqrt{x^2-5x+6} + \sqrt{x^2-5x+4})^{x/2} + (\sqrt{x^2-5x+6} - \sqrt{x^2-5x+4})^{x/2}=2^{\frac{x+4}{4}}$, find $x$. My method : I first began by substituting $x^2-5x+5$ as ...
-4
votes
2answers
40 views

How can we write $x\in \mathbb{R}^n$ in handwriting? [on hold]

Consider the set $\mathbb{R}^n$. For $\bf{x} \in \mathbb{R}^n$ How can we write this $\bf{x}$ in note book. I mean in written handwriting what is the notation for $\bf{x}$.
2
votes
2answers
18 views

A box contains 5 rods whose lengths make triangles.

A box contains five rods whose lengths are 1", 3", 6", 10", 15". How many different obtuse triangles can be made using only three rods at a time. I determined that the answer is 1 because the ...
0
votes
3answers
25 views

Subgroup that generates Z

For reference, the example in question is taken from Contemporary Abstract Algebra (Gallian). $in~Z, <8, 13> = Z$ My first question is to confirm that this is saying $8a+13b=n, where~a,b,n \...
1
vote
0answers
14 views

Why are geodesically convex sets diffeomorphic to $\Bbb R^n$?

In the construction of a good cover for a manifold $M^n$, Bott & Tu use the fact that each point in $M$ is contained in a geodesically convex set (after picking a Riemannian metric). They then ...
0
votes
1answer
27 views

Is $R=\{(a,b)\in \Bbb N\times \Bbb N\,:\,(a - b)$ is an odd number $ \}$ an equivalence relation?

$R \subset \Bbb N \times \Bbb N$ Is this an equivalence relation? $R=\{(a,b)\in \Bbb N\times \Bbb N\,:\,(a - b)$ is an odd number $ \}$ I say it is not because $(a, a)$ is always $0$ which is ...
4
votes
1answer
39 views

An inequality in positive real continuous function

I proposed my conjecture as follows: Let $f(x)$ is a positive real continuous function that is convex on $[m, M]$, let $m \le x_i \le M$, for $i=1,2,...,n$ then show that $$\frac{f(x_1)+f(x_2)+.....+...
-4
votes
1answer
37 views

Evaluate the following:

$$ 1) \cos 2 \theta + \cos 2 \phi $$ and $$ 2) \sin(\theta + \phi) $$ If $$ \sin\theta + \sin\phi = a $$ and $$ \cos \theta + \cos\phi = b $$ please provide a detailed solution. I could not ...
0
votes
0answers
16 views

The Jeep Problem and Nash's Friends

The classical jeep problem is the following. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is ...
0
votes
0answers
8 views

Solve for and Plot the Relationship Between Mean and Standard Deviation of a Normal Distribution Conditional on Satisfaction of A System of Equations

I am trying to use Mathematica, R, or Matlab to solve for (since it cannot seem to be solved analytically) and plot the relationship between mean and standard deviation of a normal distribution ...
3
votes
2answers
34 views

Is the relation $R$ on $\Bbb N$ given by $(a,b)\in R\iff a\mid b$ an equivalence relation?

$R \subset \Bbb N \times \Bbb N$ Is this an equivalence relation? $$R=\{(a,b)\in \Bbb N\times \Bbb N\,:\,a\mid b\}$$ I would argue that it is reflexive because $a\mid a$, but it is not symmetric ...
-1
votes
2answers
27 views

For which values of t does a matrix not have eigenvalues

I need help solving this problem "For which values of real parameter t does the matrix: \begin{bmatrix} π^2t^2 & 36\\ -36 & 0 \\ \end{bmatrix} NOT have real eigenvalues. Thank you.
0
votes
2answers
65 views

What is the definition of real numbers? [duplicate]

I only know that the rational and irrational are together called real numbers. Rational numbers can be written in the form $\frac {p}{ q}$ ,where $p \in \mathbb Z$ and $q \in \mathbb Z$. Are there ...
2
votes
2answers
12 views

Homolorphic vector bundles on amlost complex manifolds

Let $M$ be a real manifold with complex structure $J$, making $M$ into an almost complex manifold. I know that the complexification $T_{\textbf{C}}M = TM\otimes \textbf{C}$ of the tangent bundle $TM$ ...
0
votes
0answers
8 views

Minimum vertex cover exact algorithm analysis

An exact algorithm to find a minimum vertex cover in a simple, undirected graph would be based on the following recursive idea: "either a vertex v is in the minimum cover, or all of its neighbors are"....
2
votes
1answer
28 views

Is $C[a,b]$ isomorphic to $C([a,b]\cup[c,d])$

It is very simple to prove that $C[a,b]$ is isomorphic to subspace of $C([a,b]\cup[c,d])$ and vice versa $C([a,b]\cup[c,d])$ is isomorphic to subspace of $C[a,b]$. Is it true for $\bigl(C[a,b], C([a,...
2
votes
1answer
16 views

Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem: THEOREM. Let $M$ be a ...
0
votes
0answers
11 views

distribution and density of maximum minus element

I am a bit rusty in probability, and for a project I am studying the random variable $Z = max(X_1, \ldots, X_n) - X_i, i = 1, \ldots, n$ where the $X_i$ are positive independent random variables. In ...
0
votes
0answers
5 views

Dual of the Banach space of $k$-times continuously differentiable functions.

Let $C^k([0,1])$ denote the Banach space of $k$-times continuously differentiable functions $f:[0,1]\to \mathbb R$ with norm $$\|f\|_{C^k}:=\max_{i=0,\dots,k}\sup_{x\in [0,1]}|f^{i}(x)|.$$ I'm trying ...
1
vote
2answers
22 views

Why does every irreducible component have codimension one?

I'm a bit confused on the following lemma. Here $\mathbb{P}(V)$ is the set of one dimensional subspaces of $V$. With the choice of a basis for $V$, there is a natural bijection $\mathbb{P}(V) \...
0
votes
1answer
28 views

Prove the following relation

$$ (ax)^\frac{2}{3} + (by)^ \frac{2}{3} =(a^2 - b^2)^ \frac{2}{3} $$ if, $$\frac{ax}{\cos\theta} + \frac{by}{\sin\theta} = a^2 - b^2$$ and $$\frac{ax \sin\theta}{\cos^2 \theta} - \frac{by \cos\theta}{\...
1
vote
1answer
24 views

If $f \otimes \text{id}_{\Bbb Q}$ and $f \otimes \text{id}_{\Bbb{F}_p}$ are isomorphisms, is $f$ an isomorphism?

I would like to know the following "local-global" principle holds (all the tensors are taken over $\Bbb Z$): Let $A,B$ be two abelian groups. Assume that $f \otimes \text{id}_{\Bbb Q}$ and $f \...
0
votes
0answers
4 views

Cluster algebra of finite type

It is proved in the paper that a cluster algebra is of finite type if its Cartan counter part of the principal part of its seeds is a Cartan matrix of finite type. If the initial quiver of a cluster ...
0
votes
0answers
6 views

Matrix Calculation Significance and Multivariate Bayesian Methods

Suppose I have the matrix given by: $$X = \begin{bmatrix}1 & 0 & 0\\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$$ This matrix actually represents whether a user interacted with a ...
0
votes
0answers
16 views

Products of Laurent Series

I'm trying to find the Laurent expansion for $$\frac{e^{1/z^2}}{z - 1}$$ about $z_0 = 0$. Writing the series for $e^{1/z^2}$ and $1/(z-1)$ individually gives $$\frac{e^{1/z^2}}{z - 1} = -\left(\...
-3
votes
1answer
25 views

How to prove a given subset is a subgroup

If $A, B$ are additive subgroups of a ring $(R,+,\cdot)$, then prove that the set $AB=\{r\in R:r=\sum_{i=1}^n a_i b_i \textrm{ for }a_i\in A,b_i\in B\}$ is an additive subgroup of $R$

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