0
votes
2answers
17 views

Conditional probability about card picking.

A card is picked at random from N cards labeled 1,2,3,,,,,N and the number that appears is X. A second card is picked at random from cards numbered 1,2,3,,,X and its number is Y. I am asked to ...
1
vote
1answer
38 views

Evaluating $\int_0^1 \frac {x^3}{\sqrt {4+x^2}}\,dx$

How do I evaluate the definite integral $$\int_0^1 \frac {x^3}{\sqrt {4+x^2}}\,dx ?$$ I used trig substitution, and then a u substitution for $\sec\theta$. I tried doing it and got an answer of: ...
0
votes
2answers
20 views

Points on a straight line (Complex Analysis)

I encouter a problem in complex analysis course : Let $a, b, $ and $c$ be three distinct points on a straight line with $b$ between $a$ and $c$. Show that $\frac{a-b}{c-b} \in \mathbb{R}_{<0}$. ...
1
vote
1answer
20 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
0
votes
1answer
4 views

minimal genus of surface representing a homology class

Can you give an example of a 4-manifold with embedded surfaces of different genera representing the same homology class? (If $i:\Sigma\to X$ is an embedding of a closed surface in a closed smooth ...
2
votes
3answers
39 views

Is it possible to find the area of a shape from its perimeter?

Is it possible to find the area of a free form shape knowing the perimeter? An example would be a clover leaf shape. If the perimeter is 96 how would I know what the area would be?
2
votes
0answers
14 views

Solve $\lim_{n\to\infty} \frac{\sqrt[n^2]{1!2!\cdots n!}}{n^\alpha} = \beta$

Find the ordered pair $(\alpha,\beta)$ with non-infinite $\beta \ne 0$ such that $$\lim_{n\to\infty} \frac{\sqrt[n^2]{1!2!\cdots n!}}{n^\alpha} = \beta$$ My approach: $$\ln (1!2!\cdots n!) = (n)\ln ...
7
votes
2answers
966 views

Is there a solution manual for Royden fourth edition?

I bought the fourth edition of Royden Real Analysis, this book is awesome and is quite different of third edition that has less excersices. I have the solution manual for the third edition. Is there ...
1
vote
1answer
12 views

How to take the z-transform of an alternating serie?

For $x(k) = 1$, when $k$ is even, $0$ for $k$ is odd How do you characterize such a function in order to take the z transform $X(z) =\sum_k x(k) z^{-k}$ Thanks
0
votes
2answers
22 views

General Formula for Principle Square Root of Complex Number

How can I prove that $ \sqrt{z} = \sqrt{|z|} \frac{(z + |z|)}{|z+|z||} $ without using mathematical induction, and if I cannot -- how would I go about using induction in the set of complex numbers ?
0
votes
1answer
11 views

How to calculate the probability of a die with a wild side?

So I have a 6-sided die with 5 different values in 5 of their sides. Its sixth side can be treated as any of the other 5 values. So my question revolves around which is the probability of getting any ...
0
votes
3answers
31 views

What is the integral of $\frac{\sqrt{x^2 +4}}{x}dx$

I use trig substitution then get to this step but then I get stuck: $\int \frac{2\sec ^3\theta}{\tan \theta}d\theta$ anything I do seems to further complicate it. Thanks in advance.
1
vote
2answers
14 views

Determining order class of $T(n) = nT(n-1) + n$ with $T(1) = 1$

I'm trying to solve the following problem: Define $T(n) = n\cdot T(n-1) + n$ with $T(1) = 1$. Is $T(n) \in \mathcal O(2^n)$? I started by finding the time complexity of $T(n) = n\cdot T(n-1) + ...
0
votes
2answers
28 views

If I have a polynomial $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ with a solution at $x = -1$, how do I get the other root

If I have a polynomial $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ with a solution at $x = -1$, then I know I can just take $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ and divide it by $x+1$ to get the other root. In a ...
1
vote
1answer
19 views

Solve the PDE by the method of characteristics.

I am trying to figure out where my solution went wrong. I am off by a factor of two. $$ u_x + u_y + u = e^{x+2y}$$ I first found that the characteristic curves are determined by $$\frac{dy}{dx} = 1 ...
0
votes
0answers
13 views

Setting up a double integral in terms of x and y to find flux

I am presented with the following problem, and it wants me to set up the double integral in terms of x and y, but I have no idea on how to continue solving this one, any ideas? Set up a double ...
3
votes
1answer
40 views

Solving A Certain Diophantine Equation

I am stack on finding the solution of the diophantine equation: $d(2^{k+1}-1)-b^2(2^{k+1}-2)=1$. where $k\geq 1$ and $b^2>d$ for $b$ an odd composite integer. Is there a solution to this ...
0
votes
1answer
12 views

Finding the minimum value of $1/2 (x_{1}^2+x_{2}^2)-x_{2}b_{2}$

I am trying to find the minimum value of the following: $1/2 (x_{1}^2+x_{2}^2)-x_{2}b_{2}$ I know this is equal to: $1/2 ([x_1 x_2] Id [x_1 x_2]^T )-[x_1 x_2] [0 b_2]^T$ To find minimum we have to ...
0
votes
0answers
14 views

Short intervals with all numbers having the same number of prime factors

How to prove that for some $k, n_0$, for all $n \ge n_0$ it is never the case that all integers in $\{n, n+1, \dots, n + \lfloor (\log{n})^k \rfloor\}$ have exactly the same number of prime factors ...
1
vote
1answer
14 views

A prime ideal with the algebraic set reducible

In "Algebraic Curves" by Fulton, section 1.7, page 11, there is the following corollary of the nullstellensatz: Corollary 2: If $I$ is a prime ideal, then $V(I)$ is irreducible. There is a ...
0
votes
0answers
10 views

Form of the steady state solution of the temperature of a rod, if its thermal conductivity is described by $p(x) = x^s$

The steady state temperature distribution of a rod given by: \begin{equation} \frac{\textrm{d}p(x)y'}{\textrm{d}x} - y = 0,\; 0 \leq x \leq 1,\; \text{and} \;y(0) = 0, \end{equation} ...
0
votes
1answer
18 views

How many cases can draw diagonals?

Imagine a n_regular polygon that vertex is named by 1 to n. We know can draw (n)(n+3)/2 diagonals in n_regular polygon,Also know if we want to draw Maximum diagonals that not intersecting each other ...
0
votes
0answers
19 views

Are there fundamental differences between a complex differential equation versus real valued differential equation?

What is the difference between the solution of an ODE in the form: $ay(x) = y'(x)$ versus $ay(z) = y'(z)$
-2
votes
4answers
44 views

What function can I use to represent the next sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3…

What function can I use to represent the next sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3... I am looking for general solution.
0
votes
0answers
9 views

Cauchy Equations and Navier Stokes

I'm attempting to take the Navier Stokes Equation and coming up with an expression that will allow me to numerically determine the velocity of non-Newtonian fluid flow. The text I'm using is Cengel ...
0
votes
1answer
3 views

Random Variable Modeling

I am trying to understand how to model a random variable. So using a biased coin with $P(Head) = q$. If I am to generate a random variable $Y$ that is equally likely to be either a or b depending on ...
1
vote
0answers
19 views

Is $\mathbb{C}[T \times_T T] = (\mathbb{C}[T] \otimes \mathbb{C}[T])^T$?

Let $T$ be an algebraic group. There is a left $T$ action on $T$ given by left multiplication and a right action on $T$ given by right multiplication. Let $T$ acts in the middle of $T \times T$ by the ...
-1
votes
2answers
46 views

Math word problem [on hold]

Laura is 2 inches taller than Rebecca. Max is 4 inches taller than Rebecca. The total height for the three girls is 150 inches. How tall is Rebecca? I am having trouble figuring out Rebecca's height ...
2
votes
1answer
36 views

Finding all the group elements of a certain order of a finite group

Consider a group $G=\langle P, Z, Q \rangle$ generated $P,Z,Q$ where $$Z=\left[ {\begin{array}{ccc} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1\\ \end{array} } \right] ...
0
votes
0answers
7 views

Find the equations that determine minimizing x for the following

I don't quite understand what the question is asking and how to approach it. I am given the following two equations: i. P=1/2 $x^T A^T AX- x^TA^Tb$ ii. E=$||Ax-b||^2$ I could use some pointers ...
0
votes
2answers
10 views

Representing number $X$ in base $r$

In general, let $X = (X_{nāˆ’1}X_{nāˆ’2}...X_0)_r$ be an n-digit number in base r. Give an algorithm or explain in English how to represent $X$ in base $r^2$. I ...
0
votes
0answers
3 views

The Derivation of the Ito-Wentzell Formula

Is there a good derivation of the Ito-Wentzell Formula which is a generalization of the Ito's Lemma?
0
votes
0answers
17 views

A question on quadratic optimization

I am sorry that I haven't taken any course on optimization, could anyone tell me how to solve the following problem? Thank you very much. Minimize $$x^T\left( \begin{array}{rr} 1 & -1 \\ -1 & ...
0
votes
2answers
34 views

Show that if $F = \emptyset$, then the statement $x\in\bigcap F$ will be true no matter what $x$ is

Show that if $F = \emptyset$, then the statement $x\in\bigcap F$ will be true no matter what $x$ is I know that $x\in\bigcap F = \forall A \in F, x\in A$ But how can $x$ be in any set, much less ...
0
votes
1answer
22 views

Flat connection of a vector bundle over a 1 dim. manifold

I'd like to show that a connection of a vector bundle $E$ over a 1 dim. manifold $M$ is flat, or equiv. that its curvature is zero. Let $D$ denote the connection, $\sigma$ a section of $E$ and $v,w$ ...
1
vote
2answers
28 views

Every $p$-norm ($p \in [0,\infty]$) generates the same class of open sets on $\mathbb{R}^n$

The following claim has been made in my multivariable analysis class, and I think I have the idea of the proof but I can't quite seem to get down to the rigorous proof the instructor wants: Every ...
1
vote
1answer
9 views

Infinite dimensional FG-modules

So the way I understand FG-modules is that it is analogous to a vector space defined over a field F with G a basis. However, I encountered a problem given the hypothesis that V is a possibly infinite ...
2
votes
0answers
15 views

Proving a metric on X.

Let $X$ be the collection of all sequences of positive integers. If $x=(n_j)_{j=1}^\infty$ and $y=(m_j)_{j=1}^\infty$ are two elements of $X$, set $$k(x,y)=\inf\{j:n_j\neq m_j\}$$ and $$d(x,y)= ...
-1
votes
0answers
14 views

Which of the Following statements are true? algebra 2

I need help with this problem. Help me find out which one of the statements are true. There can be more than one. I'm positive that one of them is A.
0
votes
0answers
4 views

Random Variable 1

This is my answer: E[X]=1(0.4)+2(0.2)+3(0.1)+4(0.3) =0.4+0.4+0.3+1.2 =2.3 E[W(X)]=E[200-10X] =E[200]-E[10X] =200-10(2.3) =200-23 =177 and i don no how to find ...
2
votes
5answers
77 views

Limit problems and quandaries: finding $\lim_\limits{n\to \infty } {({n^2-n\over n^2+1})^{n+10} }$.

Find $\lim_\limits{n\to \infty } {({n^2-n\over n^2+1})^{n+10} }$. What I did is: $\lim_\limits{n\to \infty }{({n^2-n\over n^2+1})^{n+10}}=\lim_\limits{n\to \infty } {({n^2+1-1-n\over ...
-3
votes
0answers
18 views

analysis/number theory study group (online)

I plan on studying analysis from landau, rudin probably others and am looking for people (hopefully more than 1) where we could solve theorems/problems and ask each other questions. Online ...
1
vote
1answer
33 views

Integration of a function defined by its graph, the union of semi-circles and a line segment

I don't understand how to do this problem and I would someone to help me with it.Please step by step for me. I just started on integration so this problem is a bit too hard for me due to my lack of ...
-2
votes
0answers
24 views

Idele for a rational number q=63/550 .

Wikipedia, in its article "p-adic number", has taken an arbitray number x=63/550 to show the p-adic absolute value with respect to different primes. Obviously, the p-adic absolute value is 1 for ...
17
votes
2answers
439 views

Irrationality of sum of two logarithms

I try to prove that the number $$\log_2 5 +\log_3 5$$ is irrational. But I have no idea how to do it. Any hints are welcome.
4
votes
0answers
18 views

Is there a name for this special, “most parallel” ultraparallel line in hyperbolic geometry?

Suppose you're in the hyperbolic plane, and you have a line L and a point P not through L. There are an infinite number of lines parallel to L that go through P. However, there's one line M which is ...
0
votes
0answers
18 views

a valuable question about differential geometry, the curve$\beta(s)=\alpha(s)-rn(s)$

Let $\alpha(s)$,s$\in [0,l]$ be a closed convex plane curve positively oriented. The curve $$\beta(s)=\alpha(s)-rn(s)$$,where r is a positive constant and n is the normal vector, is called a parallel ...
5
votes
2answers
112 views

Arc length contest! Minimize the arc length of $f(x)$ when given 3 conditions.

Contest: Give an example(s) of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ ...
0
votes
0answers
9 views

Continuity of a function on the ring of formal power series, with a metric defined.

Let $E$ be the ring of formal power series over a field $K$. Consider $S,\ T \in E$. Define a metric $d$ on $E$ by $d(S,T)=0\ $ if $S=T$ and $a^{(-k)}$ for $k=\mathrm{order}(S-T)$, where $a>1$ is ...
0
votes
1answer
15 views

Pattern problem

April is beginning an exercise program. She plans to walk 1 mile for 2 days, 2 miles for 3 days, 3 miles for 4 days, and so on until she is walking 6 miles each day. How many miles has April walked ...

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