I figured out that $$\int_0^x \floor t = x*\left(x-1\right)/2 when x=\floor x$$ But what would the equation look like when $x \neq \floor x$ ?

I'm working with put/call options for a finance class, and am having just a little bit of confusion with the formulae. For call options, I know that the formula to determine price (C(0)) is equivalent to C(t) = $\left(\frac{s^u - p_c}{s^u-s^d}\right)$ * s(t) - $s^d$ $\left(\frac{B(t)}{B(1)}\right...

Problem: Let Y denote the amount of milk (in gallons) remaining in a 1-gallon container on its expiration date. Suppose that the density function for Y is f(y)=2y for 0 \le y \le 1. Suppose that a random sample of 5 gallons of milk is taken and the amount of milk remaining on the expiration date ...

I arrived at the following problem when using separation of variables to solve a PDE on $\mathbb{R}^2$ using polar coordinates. In that case I needed to impose the condition that $u(r,0)=u(r,2\pi)$ and $u_\theta(r,0)=u_\theta(r,2\pi)$ in order to have a continuous function. This led to the foll...

I found this equation in a book (m0 × v0^2) + (m1 × v1^2) = (m0 × v0Final^2) + (m1 × v1Final^2) It says that Notice that you have a different equation with the same two unknown variables: v0Final and v1Final. You can now factor these out and come up with a single equation for each unkno...

I'm trying to find the moment of inertia for a sphere of radius 1, with density of $1-\rho^2$ at a point distance $\rho$ from the center. I already found the mass which is $8\pi/15$, but now I'm trying to find the moment of inertia if rotated around the z axis. Any help is appreciated, although...

Can anyone help me with this? Do anyone know the expansion of (n+1)^n but not using Taylor expansion?

The function $f: \mathbb R \rightarrow \mathbb R$ is defined by $$\begin{equation} f(x)=\begin{cases} 2^n, & x=n \in \mathbb Z \\ 0, & \text{otherwise} \end{cases} \end{equation}$$ Is $f$ borel measurable? Is $f$ Lebesque measurable? Please help on this. I want to understand why... Trying to se...

I was helping a high school student with some fundamental concepts of planes and lines, when I realized I am rusty on some definitions myself. I found some minimal coverage in his high school math and precalculus textbook, and I also found a section on my vector calculus text by Marsden. However,...

I have a test on Tuesday and I'm trying to review a section I wasn't here for during class and I'm really confused. I know how to find taylor and maclaurin polybomials but there is a question that asks me to approximate sin 4 degrees to five decimal places. I know I need to turn it into radia...

$X_1$ and $X_2 $are iid mixed type random variables with $f_{x_1}(0)=$ $f_{x_2}(0)=0.5$ and constant density it the interval (0,2] determine of df of$X_1+X_2 $ thanks

I am asked to show $\int_{-\pi}^{\pi}log|1-e^{i\theta}|d\theta=0$. So I start with noting that $\int_{-\pi}^{\pi}log|1-e^{i\theta}|d\theta=lim_{\epsilon->0} [\int_{-\pi}^{-\epsilon}log|1-e^{i\theta}|d\theta + \int_{\epsilon}^{\pi}log|1-e^{i\theta}|d\theta]=0$ So now I create a contour which is ...

You are given that $E[(S-30)_+]=8$ and $E[(S-20)_+]=12$ the only possible aggregate claim in (20,39] is 22 with $f_s(22)=0.1$ calculate$ F_s(20)$

I have to find the values of t for which the system has a unique solution x+y+tz=1 x+ty+z=1 tx+y+z=-2 I'm not sure if I have to compute the determinant of the associate matrix and then find the roots. Would that be enough? Does the -2 en the third equation affects in any way?

Toss a fair coin 20 times. Find out how many outcomes have 7 H's and 13 T's.

http://i.imgur.com/vc9byVK.png Find a parametrization of the lemniscate (x^2+y^2)^2 = xy by using t = \frac yx as a parameter. Then use Area enclosed by C = \frac 12∮_C_2 (x dy - y dx)to find the area of one loop of the lemniscate.

My textbook gives the following answer: sin(4x) + 2sin(2x) = 0 2sin(2x)cos(2x) + 2sin(2x) = 0 2sin2x (cos(2x) + 1) = 0 2sin2x = 0 sin2x = 0 2x = πk x = kπ/2 So in the interval [0,2π) you have the solutions 0,π/2,π and 3π/2. The book then shows the other solutions from (cos(2x...

I understand everything until the last part where they get 2/5 sqrt(5)??? i know the ||[2 -4]|| means sqrt(2^2 + -4^2) Photo of the question because i cant embed

My last problem is let U be a set of points from $R^3$ and d:RxR $\to$ $R_{\geq0}$ an euclidean distance. For every partition of U with k classes, ($S_1$,...$S_k$), we define a quality of it as the shortest distance between 2 points from different classes. The below algorithm determines the parti...

so I have 1/2 + 1/4 + 1/8 + ... + 1/2^n = 1 - 2^(-n) to prove by mathematical induction. I did all the steps and I end up with: 1-2^(-n)+2^(-n-1)=1-2^(-n-1) I tried this out in Wolfram and another "calculator" and it says that it's true. How? How do I transform this so it's equal?

Let $\Sigma$ be $\sigma$-algebra of subsets of $\mathbb R ^2$. Set $$A= \{ S \subseteq \mathbb R: S \times \mathbb R \in \Sigma \}$$ Prove that $A$ is $\sigma$-algebra of subsets $\mathbb R$. Please can someone show me how to do this.

I'm not entirely sure how to go about proving this so hopefully someone can point me in the right direction. The definition I have for a limit point is "$a$ will be a limit point if for a sequence $x_n$ there exists a subsequence $(x_{n_k})$ such that $\lim_{k\to\infty} (x_{n_k})=a$". In the fo...

I need help showing a couple of things. First one , how can I describe which kind of singularity (removable, essential,finite order pole) I have in the origin if the given function is $\frac{z^4}{(cotz-1)^2} $ since $(cotz-1)^2$ is zero when $z=0$ and obviously $z^4$=0 when $z=0$ seems like deriv...

I am having some issues with the following multivariable limit: $$\lim_{x,y\to0,0} \frac{x^2+y^2}{x+y}$$ I am trying to show whether it exists and is equal to 0, or whether it does not exist. What I tried to do was convert it to polar coordinates and then show the limit was zero from there, how...

I know that $\mathbb{Z}[x]$ is all the possible polynomials in $x$ with integer coefficients: $\mathbb{Z}[x]=\{a_0+a_1 x + a_2 x^2 + \cdots:a_n\in\mathbb{Z}\}$. Where $a_n$ are not all necessarily distinct. Modding out by $x$ basically cuts the polynomials off at the first power of $x$, as I u...

I came across the following power series while looking at a problem. $$-1+a_1x+a_2x^2+a_3x^3+.......$$ where $a_0=-1$ and the choice of $a_1$ is arbitrary.The other coefficients are dependent on $a_1$ in the following way : $$a_n= (-1)^{(n+1)} \frac{n^{n-1}}{(n-1)!}.a_1^{n}$$ For $a_1=0$,such a...

A question I have asks if $K(x)$ satisfies $K(1)=0$ and $K'(x)={1\over{x}}$ then show: If $f(x)=K(10x)$ then $f'(x)={1\over{x}}$ So Im not sure if I have to prove it or do something else but this is what I did. $$ f'(x)=K(10x)^0\cdot K'(10x) \cdot 10 \\ f'(x)={1\over{10x}}\cdot 10={1\over{x}} $...

I am asked to solve the following Let G be a finite group with $p^r$ elements, where p is prime. if G acts on the finite set S with N elements and $(p,N)=1$, prove that there exists $s\in S$ such that $g*s=s,\forall g\in G$ I am able to get that $\sum_{g\in G}I(g)>0$ but I don't see how to conv...

Is ther an example of a module that has an infinite number of composition series? I would think not, if there is it would have to have an infinite number of submodules.

In the interest of housekeeping, I recently took a look at what what polylogarithm integrals are still in the unanswered questions list. Some of those questions have probably languished there because the solutions methods are presumably too tedious and too similar to previously answered questions...

Let X and Y be independent random variables with X being uniformly distributed in [0, 1] and Y being exponential with parameter λ. How do I find the pdf of X+Y and X/Y?

I've been thinking about this for a while and I couldn't understand how we could apply this to a bigger case. There's a graph with x vertices and y edges. The vertices are randomly put into set A or set B, with a 1/2 chance for getting put in A, and a 1/2 chance for getting put in B. The cut is a...

For $p>0$, does $\int_1^\infty x^{-p/x}$ diverge? I've tried the root test, the comparison test, and the limit comparison test without success. Any assistance would be appreciated.

I came across this statement in hypothesis testing chapter and I am confusing myself how the answer came. 'What is the probability of finding 20 defectives out of 100 samples, if, in fact p=0.10? • The answer is ~0.002.' Can you please give me an idea how 0.002 came ? Using sampling distributio...

i have done some work to get the problem to look like this (not sure if its correct): 2a2 + a0 + Σ ((n+1)(n+2)a(n+2)+nan+an)xn=0 then making the coefficients =0 i get a2=-1/2a0 also getting an+2=-an/(n+2) Now i am not sure where to go from here....

Explain the flaw in the following induction argument which shows all of Lucas’ toys are the same colour. Proof: We will show by induction that: for every integer n ≥ 1, in any group of n of Lucas’ toys, all the toys in this group are the same colour. Basis: If Lucas had only one toy, then clearl...

I am just curious if there is some website or book comprehensively covering theme of counting limitis of functions.

Under a probability space, given a set of events $B_i, i\in I$ where $I$ is an index set. If $\forall J\subset I, k\notin J$ we have $\Bbb P(B_k|\bigcap_{j\in J}B_j) \ge \Bbb P(B_k)$, then we say the events $B_i, i\in I$ are positively correlated. The problem is Is $\Bbb P(B_k|\bigcap_{j\in J...

Not sure where to start.. would someone please point me to the significance of invertability to the result of $gfgf = gf$?

I know that to prove a single-parameter onto function, you just find x in terms of y. How do you do it when the function has parameters such as "f(r, s) = r + 3pi*s"?

I tried to solve following second order ODE, But I am almost stuck : $$\frac{\partial ^2\sigma}{\partial x^2}-h(x) \sigma=0$$ where $h(x)$ define as: $$h(x)=G_o+\left(G_{\infty }-G_o\right)\left(1+\frac{2}{\pi }\sum _{m=1}^{\infty } e^{-\frac{\pi ^2 \mathcal{D} m^2 t}{L^2}}\frac {\cos (\pi m)-...

Prove $\forall a\in \mathbb Z, \forall b\in \mathbb Z, \forall c\in \mathbb Z, (a | b \land a\nmid c) \rightarrow a\nmid(b + c)$. Maybe a gentle nudge in the right direction

My Question First I let u = y' and employed the Chain Rule to obtain du/dx = du/dy * u But I am not sure where to go from there. Any tips, suggestions, or solutions to the problem would be much appreciated!

describe the Fast Fourier Transition F(f) for a 1-periodic function f(t) given at eight points t=0,1/8,2/8,3/8,4/8,5/8,6/8,7/8.

Find the number of positive integers such that logarithm of whose reciprocals to the base 10 has the characteristic $-2$. Let $x$ be a positive integer. Now the characteristic of $\log_{10}(\frac{1}{x})$ is $-2$ I dont know how to solve further.How to count number of positive integers?Please h...

How do I find representation of -3, -1/5, and 1/5 as 2-adic integer? Is 2-adic representation same as binary expression: for example, 1/5 = .00110011... ?

Let $B_2$ be the $\sigma$-algebra of Borel subsets of $\mathbb R^2$, i.e. $B_2=\sigma (\mathbb G_2)$, where $\mathbb G_2$ is the collection of all open subsets of $\mathbb R^2$. Prove $B_2= \sigma (R_2)$ where $$R_2=\{ (a,b) \times (c,d): a<b \, \, \text{and} \, \, c<d\, \, \text{are real number...

How do I include the matlab "engine.h" file in my C project (Visual Studio 2012) #include "engine.h" I've done what is shown above and added engine.h to source files, but when I try to compile I receive error "Cannot open include file: "engine.h"

I'm reading the book "Elements of the representation theory of associative algebras volume 1".And I can't understand the proof of the proposition 3.11 on padge 124. The place where marked green "because P is injective,u factors through P",how to get this result?

Let $p(t), q(t) ∈ C[t]$ be relatively prime, $A ∈ M_n(\mathbb{C})$. Show that $rank(p(A))+rank(q(A)) ≥ n$. I have been stumped on this question for quite awhile. Could someone please enlighten me in regards to a fitting theorem? I'm assuming this is related to Bilinear and Quadratic forms but I ...

I am completely lost. So do I prove the two directions (starting from the first one and starting from the last one)? and how would I prove this?

Prove that if $B \subseteq \mathbb R$ is a Borel set, then $B \times \mathbb R$ is a Borel in $\mathbb R^2$, i.e. $B \times \mathbb R \in B_2 $ I'm still trying to understand the definition properly. Please can someone show this. It would help me so much. Thanks in advance!

I am trying to show that the following relation holds: \begin{equation} \log(1+ax) = log(x) + o(log(x)) \end{equation} as $x\rightarrow \infty$, where $a$ a positive number. I tried using Taylor expansion but I could not come to the results. Any hints would be really helpful! Thanks!

I thought about doing (.25^5)+(.75^5), not sure if this indicates that he makes 5 in row and then misses 5 in a row or if it is correct.

H(n) = { \begin{array}{lr} 0 & n\leq 0\\ 1 & n = 1 \textrm{ or } n = 2\\ H(n-1) + H(n-2) - H(n-3) & n>2\\ \end{array} Prove that $\forall n\in \mathbb N, n\geq 1 \rightarrow H(2n) = H(2n-1) = n$. Maybe I'm just an idiot but I approached this question by drawing a graph. Assuming H= n. But thi...

Prove that Petersen graph is class 2 ( class 2 = $\Delta(G) + 1$. I try to draw the petersen graph and i find only one method to coloring the outer cycle ( outer cycle is colored by $1,2,1,2,3$) and we know that the edge connecting the outer cycle with tne inner cycle. Because of that I need 4 ...

Let $A = \cup_{i \ge 1} A_i$, $i = 1, 2, \cdots$. This is union of countably infinite sets. Also, $A_i \subsetneq B$ for all $i$, i.e. there exists at least one element in $B$ that is not in $A_i$ for every $A_i$. Then is it true that $A \subsetneq B$? Intuitively, it seems true because for eve...

Fourier cosine series expansion of $f(x)=1 for 0<x<\pi$ Hint iss "thought is better than calculation".

[enter image description here][1] I created a video poker hand evaluator and I am not getting the optimal hand and I know my mistake is because of the way I am calculating the expected payout. Above is a picture from the web version of the game, how this "2.510638" got calculated? This is a ja...

Quick question, mostly just for my knowledge, but I'm working on a problem: Determine whether the indicated set $A$ is an ideal in the indicated ring $R$: $A = \{0,2,4,6,8\}$ in $R = \mathbb{Z}/10\mathbb{Z}$. For short hand, can I denote $A$ as $2\mathbb{Z}/10\mathbb{Z}$?

Here's my situation: I understand that to prove the "onto" ness of a function it is necessary to show that X can be expressed in terms of Y by isolating it using standard mathematics. However, I am confused as to how one can prove the trait of being onto in a function that has multiple parameter...

If |A| = |B| = 5, how many functions f: A → B are invertible? I'm a bit lost on how to start this problem, any help would be much appreciated.

After expansion, we have $$ (x_1+x_2+\dots+x_n)^m=a_1x_1^m+a_2x_1^{m-1}x_2+\dots $$ where $x_{()}$ is the variable and constant indeces $n>m$. What is the expressions of all these possible coefficients $a_{()}$? Thanks in advance.

A is the event that Red rolls 1, 2, or 3; B is the event that Red rolls 2, 4, or 6; and C is the event that the sum of the two rolls is 5. (a) Find p(A|B), p(B|C), and p(C|A) (b) Find p(A|B ∩ C), p(B|C ∩ A), p(C|A ∩ B) (c) Are the three events pairwise independent? Mutually independent? My at...

I drew the graph of root x but I cannot see the link to the strict inequality. Any hints would be welcome. enter image description here

Define the sequence $a_k$ recursively by $\displaystyle\sum_{d|k}a_d=2^k$ with $d>0$. Prove that $a_k$ is a multiple of $k$.

A partial quote from a book I am currently reading: "For example let X be a Riemann surface of genus 0 and $\mathbb{P}^1$ be the standard projective line. We choose a point $p \in X$. Because $\deg \left[ p \right] = 1 \geqslant 2g - 1$ ..." The question I have is with the notation $\left[ p \r...

Find the equation of a plane perpendicular to each of the two planes x - y + z = 0 and 2x+y-4z-5=0 and containing the point (4, 0, -2) Need help! Thank you. ^_^

Find the locus of the midpoint of the line joining the focus (ae,o) to any point on the ellipse x^2/a^2 + y^2/b^2 = 1

Find the value of $\sum_{n=1}^{50}\arctan(\frac{2n}{n^4-n^2+1})$ $\sum_{n=1}^{50}\arctan(\frac{2n}{n^4-n^2+1})$ $\frac{2n}{n^4-n^2+1}=\frac{2n}{1-n^2(1-n^2)}$ I am not able to split it into sum or difference of two $\arctan$s.Please help me.

new to the site, hopefully you guys will be able to help me. I need to recursively define asub(n) = 6asub(n-1)-9asub(n-2), and prove asub(n)=(5-n)*3^n Sorry for the weird format, I am not familiar with the format you use on here. I've already done the basis case for this induction problem, but...

Prove the following inequality, $$\sqrt{n} \le (n!)^\frac{1}{n} \le \frac{n+1}{2} \ \ \ \ \forall \ n\in \mathbb{N} $$

If $a, b$ are non-square whole numbers, and $c$ is an positive whole number, prove there exists no solution to the following equation: $$\sqrt{a}+\sqrt{b}=c$$

So I have the following which I must prove : $\sum_{(n1,n2,n3):n1+n2+n3=n)} \binom{n}{n1, n2, n3}$ = $3^n$ I'm not sure where I must begin. This is btw a multinomial.

While calculating electric field due to charged sphere i came across this integral Which i have no idea how to solve. Please help me out. $$\int\frac{(z-Ru)(du)}{{(R^2+z^2-2Rzu)}^{3/2}}$$ It is given as hint that it can be solved by partial fractions.

I am having trouble in handling such sequences and don't know how to proceed. Please help me. Thanks in advance.

A man has an umbrella, and he commutes from his house to work and back. If it is raining, and he has an umbrella, he takes his umbrella. If it is not raining, or he does not have an umbrella, he does not take it. I am trying to establish the transition probabilities for the Markov chain. I unders...

Given $f(x)$, how would I find $(f^{-1})'(x)$? As an example how would I find that for this problem: $f(x) = 4x^3 + 5x + 2$

I'd like to be able to prove the following inequality: $\frac{{{H_{n, - r}}}}{{{n^r}\left( {n + 1} \right)}} \le \frac{{{H_{n - 1, - r}}}}{{n{{\left( {n - 1} \right)}^r}}}$. It's clear that as $n \to \infty$ we get equality, the limit on each side is $1/(r+1)$, and it also seems clear that this ...

$a=\sqrt{57+40\sqrt2}-\sqrt{57-40\sqrt2}$ and $b=\sqrt{25^{\frac{1}{\log_85}}+49^{\frac{1}{\log_67}}}$ and $c$ is the value of $x^3+3x-14$ where $x=\sqrt[3]{7+5\sqrt2}-\frac{1}{\sqrt[3]{7+5\sqrt2}}$.Find the value of $a+b+c$. I tried to solve and simplify this problem but no luck.Please help m...

How would I express -3 in 2-adic representation? Is it just revercimal calculation of binary expression of -3? like: -3 = -11 in binary, so using revercimal, -11. in binary = 01. ?

I desperately need help on these three natural deduction problems. If anyone can help, it'd be greatly appreciated.

A farmer harvested $1$ section (which is $640\, acres$) of wheat and $2$ sections of barley. The total yield of grain for both areas was $99,840\, bushels$. The wheat sold for $6.35\, /bushel$ The barley sold for $2.70\, / bushel$. The farmer received $363,008$ for both crops. What was the yiel...

Definition. Let $A$ be an $r \times s$ integer valued matrix. $A$ is "special" if there exists an integer $k$ such that $a_{ij}=0$ unless $i=j$ and $i \leq k$ and $a_{ij} \neq 0$ if $i=j \leq k$. Show that any matrix $A$ can be reduced to a "special" matrix by elementary row and column operat...

"The value that the sampling distribution of the estimator be will be centered at"

I've found a reference to "ancient time" from Google. It's mentioned in e.g. the book Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture, by Qi S. Zhang. What's "ancient time" and why is the name such?

I'm an engineering student who is doing some self study in analysis. The book I'm using is Maxwell Rosenlicht's "Introduction to Analysis". My background includes multiple variable calculus, ODEs, and some linear algebra (not proof based). As you likely know, all the exercises in the book involve...

Problem: Two players play the following game. Initially, X=0. The players take turns adding any number between 1 and 10 (inclusive) to X. The game ends when X reaches 100. The player who reaches 100 wins. Find a winning strategy for one of the players. This is my solution, which hopefully you c...

Find the value of the expression $\tan\frac{\pi}{16}+\tan\frac{5\pi}{16}+\tan\frac{9\pi}{16}+\tan\frac{13\pi}{16}$ I identified that $\frac{\pi}{16}+\frac{13\pi}{16}=\frac{5\pi}{16}+\frac{9\pi}{16}=\frac{14\pi}{16}$ $\tan(\frac{\pi}{16}+\frac{13\pi}{16})=\tan(\frac{5\pi}{16}+\frac{9\pi}{16})$ ...

I have to prove that the function $f\colon [0,\infty)\to R$, defined by $f(x)=\lbrack\!\lbrack \sqrt{x}\rbrack\!\rbrack$ $\forall$ $x\in [0,\infty)$, is continuous at $c=2$. The function $f(x)=\lbrack\!\lbrack \sqrt{x}\rbrack\!\rbrack$ takes a number and rounds it down to the closest integer if a...

While we were introduced to integration, we were told about some basic concepts that, as we were told, could not be proved based on our level of sophistication. They are as follows: $\int_a^b \! f(x) \, \mathrm{d}x=\phi(b)-\phi(a)$, where $\phi$ is a primitive of $f$ in $[a,b]$ $\int_b^a \! f(x...

I want to plot the partial sums of the reciprocals of the squares between 1 and 10. So far, I've got the following code. It's something along these lines but I can't quite manage it. I need to sum the terms so far for each point between 1 and 10 but all I know how to do is sum all the terms. f...

I saw this in the text. F = ~D~H~L + D~H~L + D~HL and it said this is equivalent to F = D~H + ~D~H~L Anyone know what is the process to this equivalence. I try to solved it but I'm not able to.

As the question says How to prove $$\tanh ^{-1} (\sin \theta)=\cosh^{-1} (\sec \theta)$$ I have tried to solve it The end result that got for LHS $$=\log \frac{1+\tan\frac{\theta}{2}}{1-\tan \frac{\theta}{2}}$$ I am stuck here Please help

I wrote some short alternative proofs (sketches mostly) to my book, can someone tell me if they are okay. The unity of a subfield is the unity of the whole field. Let $H \subset F$ with $F$ being the field. Let $1_H=hh^{-1}$ for every $h$, and let $1_F = ff^{-1}$ for all $f \in F$. Let $f =...

How do I prove that every meromorphic function on the extended plane is a rational function?

Let $\gamma:[0,1]\to \Bbb C$ be a closed rectifiable curve and consider $\gamma^{-1}:[0,1]\to \Bbb C$ given by $\gamma^{-1}=\gamma(1-t)$. Show that trace($\gamma$)=trace($\gamma^{-1}$). I tried to show that $\gamma$ and $\gamma^{-1}$ is equivalent, since equivalent paths has the same trace. ...

Forgive me for asking this question. I am deriving the Wigner function, $$ W\left(x_{1},p_{1},x_{2},p_{2}\right)=\frac{1}{4\pi^{2}}\int dx'_{1}dx'_{2}e^{-ip_{1}x'_{1}-ip_{2}x'_{2}}\Psi\left(x_{1}+\frac{x'_{1}}{2},x_{2}+\frac{x'_{2}}{2}\right)\Psi^{\ast}\left(x_{1}-\frac{x'_{1}}{2},x_{2}-\frac{x'_...

My teacher told me to write a program to solve this problem, but I thought it would be more fun to solve this without brute force of my PC. To explain this problem, for instance, 0000001001101111 has six 00s, three 01s, two 10s, four 11s.I tried to solve this by using recurrences.Define the numbe...

We have a permutation of $\pi$ numbers $\{1,2,...,10\}$. Let $A_1$ be $\pi(1)>1$ and $A_2$ be $\pi(1)>2$. (number on position 1 or 2 must be greater thatn 1 or 2, respectively). What is the probability of $A_1$ and $A_2$? Are $A_1$ and $A_2$ independent? I don't know how to approach this problem...

Suppose we model a physical phenomenon with a 2nd order linear differential equation: $a_2$(t)y''+$a_1$(t)y'+$a_0$(t)y=$f(t)$, where 't' stands for time. In choosing an appropriate driving function f(t), suppose we want just the segment of a certain function $g(t)$ for 5≤t<10 only. Further, suppo...

I have a formular for the convolution theorem, and read several chapters in several scripts about it. This is the formula: $(f*g)(x)=\int_{\mathbb{R}^d}f(x-y)g(y)dy$ However much I read, I cannot figure out where exactly the y comes from. So, if I had an excercise where I have to convolute to ...

I encountered this problem from Conics and Cubics by Bix. Please help me answer this. Let $C$ be a nonsingular, irreducible cubic with a flex $O$. Add points (commutative) of $C$ with respect to $O$ as base point. Let $P$ be a point on $C$ of order 6. Prove that $2P$ is a flex of $C$ collinear w...

https://drive.google.com/file/d/0B1YZD9uzvB5TVXdneEcyeVhsM3c/edit I can do most of them fine, but I don't feel ready for calculus. I'd love to start learning calculus but I'm not sure if I am ready and want a second opinion to see if these are enough for me to start calculus. I am skeptical beca...

A straightforward corollary to Menger's Theorem states that if we pick two non-adjacent u,v∈V then the maximum number of internal vertex-disjoint u−v paths is equal to the minimum size of an u−v vertex-cut. Let's put in place a bound for the length these path have. Denote A_n(u,v) the maximum nu...

This question came up in a math competition a few weeks ago. My reasoning for (a) was that if we took away the 9 smallest numbers (1-9), the smallest 9 numbers that we would then be able to choose from then would be 10, 11, 12, 13, 14, 15, 16, 17, 18. The sum of these numbers is already more ...

Given an alphabet $\Sigma$ and 3 words $u,v,w \in \Sigma^*$, and 3 claims about $u,v,w$: 1. $w^5=v^3$ 2. $uvw=wuv$ 3. $wu=uw$ Which claim implies the other and which not? When I first saw the claims, it seems like $2\implies 3$ and $not(1\implies 3)$. That is because intuitively, if we remove $v$...

Let us consider surfaces $P_t = \partial\{v<t\} $ moving under inverse mean curvature flow where $v$ is a function defined on $\mathbb{R}^n \setminus \{0\}$. I need to show that the eccentricity $ \Theta(P_t)$ is decreasing. I want to use expanding spheres as barriers and I think that since w...

I am asked to prove that order of a Boolean Algebra cannot be prime greater than 2. I have a dificulty to show this in an appriopriate way. I know the definition of Boolean Algebra. The definition I have seen is as follows: A structure (B,meet,join) is called a Boolean Algebra if B is distribu...

Is Heisenberg group solvable? Does solvability depends on the topology?

Find the solution : sin 5theta + cos 4theta=2 I am preparing for IIT and just stumbled upon this question.

By my course, all manifold of dimension 1 is isomorphic to $(0,1), (0,1],[0,1)$ or $\mathbb S^1=\{x^2+y^2=1\mid x,y\in\mathbb R\}$. I was thinking of a curve in the plan, with a knot. (See picture) I agree that the gluing (in pink) is possible in $\mathbb R^3$, but if we consider this curve in ...

The question is general, but I'll first give a simple example. Suppose you have a candy machine with $N$ candies. The machine is weird, when you give it a quarter it gives you $1$ to $N$ candies (all numbers equally probable), this is called one "buy". You may assume you have more than $N$ quart...

I saw a comment at the OEIS website for the sequence of entry points, of Fibonacci factors. https://oeis.org/A001177 It referenced a paper by Mark Renault in 1996, with the quote from OEIS: http://webspace.ship.edu/msrenault/fibonacci/FibThesis.pdf If m has prime factorization m=p1^e1 * p2^e2 *...

http://s15.postimg.org/kjkxd1nxn/Screenshot_2015_11_29_23_08_38.jpg Ques no. 1 &2 . It is very interesting.

I have the question about the non-square matrix. If A is m by n matrix, the m is not equal n. x is a vector which the dimension is n by 1. Is it correct that ||Ax|| <= ||A|| ||x|| If A is m by n matrix and B is n by m matrix, is it correct that ||AB|| <= ||A|| ||B||

A question from the theory of bounded analytic functions. Let $f$ be analytic in the circle $D: |z|<1$ and bounded in $D$ by absolute value by a constant $M>0$. We assume that $N$ derivatives of $f$ at $z=0$ are bounded by absolute values by known constants $\epsilon_i$, $i=0\dots N$ (small one...

I don't understand how to do this. The tip I have for the question is to first find a bijection between (0,1) and (1,∞).

Solve the equation,$\sqrt{\log(-x)}=\log{\sqrt{x^2}}$ $\sqrt{\log(-x)}=\log{\sqrt{x^2}}$ $\sqrt{\log(-x)}=\log{|x|}$ Now two cases arise,when $x>0$ and when $x<0$ When $x<0$, $\sqrt{\log(-x)}=\log(-x)$ I found $x=-1,-10$ When $x>0$ $\sqrt{\log(-x)}=\log x$ $\log(-x)=\log x\times\log x$ I could...

im stuck with a basic probability problem: n balls --> n-1 extraction. Only one black ball. No replacement. example: 7 balls(6 white, 1 black). 6 extractions. i know that the probablity of 6 whites is: 6/7 · 5/6 · 4/5 · 3/4 · 2/3 · 1/2 = 0.14 aprox, so the prob of get the black ball is: 1 - 0...

$f(t)$=$g$$(t-10)$$U$$(t-15)$$-$$g$$(t-10)$$U$$(t-20)$ The above $f(t)$ contains terms of the form $f$$(t-a)$$U$$(t-b)$, where $a$ doesn't equal $b$. Describe the form that $L${$f$$(t-a)$$U$$(t-b)$} takes. [Hint: The formula for L{g(t)U(t-a)}$=$$e$^${-as}$L{g(t+a)}

I am trying to show that $d(a \wedge da)=0$ if $k$, the degree of k-form $a$ is even. I have said: $=da \wedge da + (-1)^k a \wedge d^2a$ I believe the first term is zero due to repeated indices and the second is zero since $d^2=0$. However, I need it not to always be zero if k is odd, so where...

I came across this problem: An insurance agent offers an insurance policy to 2n costumers. Each costumer accepts the offer with probability $0<\alpha<1$ and rejects it with probability $1-\alpha$. A costumer who accepted the offer choose randomly one of 3n service centers of the insurance compan...

The Question: A full conical water tank of height 25m and diameter 50m drains in 100 minutes after the bottom plug (at the vertex) is removed. a) After the tank is drained, water is pumped into the tank at a rate of r=2m^3/min. Find the highest level to which the water will rise: Now I was able...

I want to run Metropolis-Hastings on a problem which involves two parameters that are not independent. I.e. I want to estimate both of these parameters. At the moment I'm trying to understand if this is possible and if so how or what to watch out for... I considered implementing a single-compone...

Prove that: $ \forall x \in R_{+}{*}, arctanx> \frac{x}{1+x^2} $. I can't prove that. Help me please..

How do I find the first four terms of the Taylor series expansion of $$f(z)=\frac{1}{(z^2+ 1)}$$ around $z=e^{i \pi/4}$

What is the shape of the graph $|z-1|+|z+i|=2$ in the complex plane? $(A)\text{two points}\hspace{1cm}(B)\text{a line}\hspace{1cm}(C)\text{a parabola}\hspace{1cm}(D)\text{an ellipse}$ Let us take $z=x+iy$ $|(x-1)+iy|+|x+i(y+1)|=2$ $\sqrt{(x-1)^2+y^2}+\sqrt{x^2+(y+1)^2}=2$ Upon simplifying,$3x^2+3...

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I am working at some Fuzzy-Logic and I am having my problems with the inferece. While using the generalised modus ponens you are using this formula μB'(y) := sup{min(μA'(x),min(μA(x),μB(y))) | x∈X} for y∈Y My Question is, where is the Difference between the min/max Operators and the sup/in...

I have the following sum that I wish to evaluate: $$ \sum_{n=0}^{\infty} \left(\int_0^1 du\,u^{1+n+\epsilon}\right) \frac{\Gamma(n+2) \Gamma(1+\epsilon)}{\Gamma(n+3+\epsilon)} {}_2F_1\left(1, n+2, n+3+\epsilon, -\frac{a}{b}\right) = \sum_{n=0}^{\infty} \frac{1}{n+2+\epsilon} \frac{\Gamma(n+2) \Ga...

I have two constants $k,s$ and would like to find the maximum $t$ such that $$k^2 \frac{t(t-1)}{2}\left ( 1 - \frac{2}{(t+1)t} \right )^{st} < 1$$.

Find the value of A,B and C given: 3x 2 + 4 ≅ A ( x + 2)2 + B (x + 2) + C I've managed to expand the brackets, however i am still confused on what to do next. Please help and explain the process.

We have observations ($y_i$,$x_i$) (i=1,...,n). Suppose that give $x_i$, $y_i$ is a discrete random variable with distribution $$ P(y_i=y|x_i=x) = \frac{e^{-x\beta}(x\beta)^y}{y!}, y=0,1,2,...$$ Assume that $x_i>0$ for all i. Compute the maximum likelihood estimator of $\beta$ and its exact dist...

I think it is trivial, but maybe a vector be linear dependence in one vector space but not in another vector space?

Is there any general formulation to find Green's functions of coupled ODEs or a system of ODEs? What is 'Green's function matrix'? Do Green's functions of coupled ODEs form a matrix which is called Green's function matrix? Please refer to a good text in this regard.

I have a question from my professor and I need help on one part. The question reads: Suppose I have a $2n \times 2n$ grid of unit squares where $n>1$. Suppose we partition this grid into $2 \times 2$ contiguous grids of unit squares. We have a valid grouping of these squares if: At le...

So far, I understand that we need to show that you can form a path, but I am very confused on how to go about it. Can anyone provide hints please?

There is prize in a box. The prize has a value of a positive integer between 1 and N and you are given N. To win the prize, you have to guess its value. Your goal is to do it in as few guesses as possible; however, among those guesses, you may only make at most g guesses that are too high. The v...

I am aware that this is a continuous function however I am having trouble proceeding with a valid argument. Please help

We can find complex integration of a function over a closed contour by residue theorem if there are only finite many singularity inside the contour. But my question how to find the integration if there are infinite many singularity inside the contour? Please help me solve this type of problem me...

Problem: If $x\in [0,1)$ write $a_1 = [10x], a_2=[100(x - \frac{a_1}{10})], a_3 = [1000(x - \frac{a_1}{10} - \frac{a_2}{100})], . . .$. Prove that $0 \leq a_k \leq 9$ for each $k$ and that $\sum_{n=1}^{\infty}\frac{a_n}{10^n}$ converges. Prove that $x = \sum_{n=1}^{\infty} \frac{a_n}{10^n}.$ Pro...

In a hand of 13 playing cards from a deck of 52 whats the probability of drawing exactly one king. My approach would be $${4 \choose 1}*{48 \choose 12}/{52 \choose 13}*{2}$$ I divided by 2 because I felt I had ordered the king and the other 12 cards chosen but this is wrong. Can someone please ...

$\vDash \forall x (\alpha \lor \beta) \implies ( \forall x \alpha \lor \forall x \beta) $. I am not able to start this question, any help would be appreciated. Thanks in advance!

What is/are the possible value(s) of 'a' such that the system of linear equestion in x,y and z has a unique solution? x-2y+az=0 y+3z=-1 -2x+3y+z=-2

Let c be an irrational number and $f:~\mathbb{R}\to {{S}^{1}}\times {{S}^{1}}$         $t\to ({{e}^{2\pi it}},{{e}^{2\pi ict}}) $ Show that the image set $f(\mathbb{R})$ is dense in ${{S}^{1}}\times {{S}^{1}}$

Consider $x \in \Bbb{Z}_p$. Then I want to find the valuation of $(1+p)^x-1$. I think that $val_p((1+p)^x-1)=1+val_p(x)$. Is this right?

when the joint distribution of a set of items is different from the distributions of the individual items in the set. this is the context: Our discussion so far has focused on item-based top-N recommendation algo- rithms in which the recommendations were computed by taking into account rela...

In a triangle $ABC$, let $M$ be the midpoint of side $BC$ and $N$ be the midpoint of median $AM$. Let $O$ be the circumcentre of triangle $ABM$. If the circumcircle of triangle $BOM$ cuts the side $AB$ at $P$(leaving the point B), prove that the points $P,O,N$ are collinear. I am quite weak at g...

Let $A$ be the matrix $\begin{bmatrix}3 &1 \\ 1&2\end{bmatrix}$. What is the maximum value of $x^T Ax$ where the maximum is taken over all $x$ that are the unit eigenvectors of $A$? $5$ $\frac{(5 + \sqrt{5})}{2}$ $3$ $\frac{(5 - \sqrt{5})}{2}$ Eigenvalues of $A$ are $\frac{(5 \pm \sqrt{5})...

Show using residues that, for all $b > 0$ and $-1 < a < 0$, $$\int_0^{\infty}\frac{x^a \log x}{x+b}dx = \frac{\pi b^a}{\sin^2 \pi a}(\pi \cos \pi a - (\log b)(\sin \pi a))$$ I'm really confused on how many integrals in answer includes. I'm not sure if I need one for when b>0 and when a>-1 and wh...

X ~ norm(15, 1.25). I want to find a and b, such that P(a < X < b) = .95. I've attempted this multiple times and spent like two hours and all my answers are wrong. Please help!!!!

Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis for $H$. Let $E_0$ be a countable subset of $E$ and $\{\delta_i\}_1^{\infty}$ be a bounded sequence of $(0,\infty)$. For given an arbitrary sequence $\{h_i\}_1^{\infty}$ of unit vectors in $H$, I am looking for an operator...

If we take the Cantor set and instead of removing the interval [1/3, 2/3], we remove the open interval [x,1-x], with 0

I'm trying to understand how LP formulaton for shortest path problem. However I'm having trouble understanding constrains. Why this formulation work? http://ie.bilkent.edu.tr/~ie400/Lecture8.pdf I'm having trouble understanding how the constraints work at pages 15 and 17. I got the main idea a...

I have L={Contains 'a'} and Alphabet(E)={a,b} Can i create a NFA Like this

I was completely lost when handed this at a math competition a couple of weeks ago. Please help. I am trying to understand how to do these questions so that I am ready for the next competition.

Let $G$ be a group of order n=pq, where $p$ and $q$ are prime numbers and let $x$ $\in$ $G$. My question is how hard is to compute $x^{-1}$ in $G$ ?

Since $L^{p,0}=L^p$ and $L^1$ is not reflective, thus in general Morrey space is not reflective, but how about for $L^{p,\lambda}$ with $1<p<+\infty$ and $0<\lambda<n$, where $n$ is the dimension of domain. What's more, it seems that the dual space for Morrey spaces are not clear so far?

Is $$\prod_{i=1}^{n}\frac{x_i}{\theta}=\frac{\prod_{i=1}^{n}x_i}{\theta}$$ or $$\prod_{i=1}^{n}\frac{x_i}{\theta}=\frac{\prod_{i=1}^{n}x_i}{\prod_{i=1}^{n}\theta}=\frac{\prod_{i=1}^{n}x_i}{\theta^n}$$ ???

How to prove that there exist a fractal with similarity dimension D = x, where x is between 0 and 1?

How would I set this up? I don't know what to do.

Well I know that de axiomatization is important to establish certain laws, etc. But what other arguments do exist?

I'm in the market for a mathematical (or otherwise) term to describe a slice of a hypercube. Tensor is out of the running as that's the name of the object I am slicing. The second I could use a hand with is a term to describe an index (or access point) that spans more than a single point in eac...

Given there are 5 pairs of siblings, answer the question: What is the probability that 2 ppl randomly chosen are siblings? My answer: 10/10 + 1/9 What is the probability that all 4 people randomly chosen are all not related? i.e. no siblings among the 4 chosen. My answer: 1/10 + 1/9 + 1/8 + 1...

Change the order of integration in x varies from 0 to 2a and y varies from 0 to sqrt(2ax-x^2);integrand is (x^3 + y^2.x)/(sqrt(4.a^2.x^2 - (x^2 + y^2 )^2) dx dy and evaluate it. After changing the order, i found out the limits of integration as y varies from 0 to a and x varies from a-sqrt(a^2 ...

What is the solution for the following functional equation? g(x).g(z) = g(x+z) + g(x-z) The solution given is: g(z) = 2cos(z). In the derivation of the result (using Taylor expansion), there is a step that is like this: g"(x) = b.g(x), where g"(x) is the second derivative of g(x) and b is a ...

Use Monotone Convergence Theorem to prove that $$f(x) = [(2x-4)^{-1/2}]1_{(2,4]} + 01_{\{2\}}$$ is Lebesgue integrable on $[2,4]$. According to notes from a different class So I just replace $x$ w/ $f(x)$?

an anyone please help me with the following proof: Let $$F(x)=\ln{x}\ln{(1-x)},0<x\le\dfrac{1}{2}$$ show that $$F'(x)>0$$ because $$F'(x)=\dfrac{(1-x)\ln{(1-x)}-x\ln{x}}{x(1-x)}$$ It suffices to show that $$G(x)=(1-x)\ln{(1-x)}-x\ln{x}>0,0<x\le\dfrac{1}{2}$$

Give the NFA accepting the language over the alphabet {O, 1} that have the set of strings which contain 01 as substring. I have created the following.Im i right?

The rank of a matrix A is the dimension of the vector space generated by its columns. It shows the dependent and independent column of matrix A. Suppose I have a nxn matrix A with rank n/2. As a specific example, let A =[3 1 3 1;1 4 1 4;3 1 3 1;1 4 1 4] with the two eigenvalues 4.7639, 9.23361 ...

Let p and q be positive numbers satisfying 1/p+1/q= 1; and let f ∶ [0,+∞) → R be the function f(x) =1/p x^p− x +1/q Show that f has an absolute minimum at x = 1 and hence deduce the inequality ab ≤1/p a^p + 1/q b^q for any positive a and b. Hint. What can you say on the magnitude of p and q? I ...

Let $X=(0,1)$ endowed with the Euclidean topology. Is the sequence $x_n=\frac{1}{n}$ convergent in the topological space X? I don't know how to start these kind of questions. Anybody who can give me an extensive answer? Thank you very much!

Find $sup_{x\in(0,+\infty)}(2^{-x}+2^{\frac{-1}{x}})$. Help me please, I can't find it. I tried but it was difficult for me.

Let $X$ be a topological space having the property that every $ x \in X$ has some open neighborhood which is homeomorphic to some open subset of $\Bbb{R}^2$ . Give some justification or a counterexample for each of the following questions. (i) Is $X$ a first countable space? (ii) Is $X$ a second...

I define $P_t =\partial \{v<t\}$ a compact subset of $\mathbb{R}^n\setminus\{0\}$ where $v$ is a function defined on $\mathbb{R}^n\setminus\{0\}$ . I am looking at the evolution of the eccentricity $\Theta$ of these surfaces $P_t$. I want to justify why $\sup_t\Theta(P_t) < \infty$. Is this ob...