Integrate \int (e^-xy)*sin(x) step-by-step Found solution for e^x*sin(x). It used integration by parts. Tried it here but keep getting too many Ys.

Related to this similar problem. I am attempting to solve the following: Find a natural cubic spline function whose knots are $-1$, $0$, and $1$ and that takes on the values $S(-1)=5$, $S(0)=7$, and $S(1)=9$. Neither my textbook nor my class notes have been particularly helpful in showing ...

I am trying to implement an algorithm described in a paper. At one point, it says "Let $\phi_0,\phi_1,\ldots$ be a collection of functions from $\mathbb{R}^d$ to $\mathbb{R}$ such that $\phi_0\equiv1$ and $\{\phi_i(X_t)\}$ form a basis for $L^2(\Omega,\sigma(X_t),\mathbb{Q})$ for all $t=1,\ldot...

Pick a sequence of 10 balls from a sack containing red, blue, green, yellow, white and black balls. Each time a ball is picked, it is replaced in the sack before the next ball is picked. a) How many sequences have exactly 3 balls? b) At least 4 blue balls? c) exactly 2 black balls AND at ...

Are there rules for higher powers? It seems like even and odd is preserved by powers, but how do I prove that?

I'm trying to solve this with two methods. P(Min of 3 Dice Rolls < 3) = P(D1 < 3 U D2 < 3 U D3 < 3) = 1 - P(D1 >= 3 n D2 >=3 n D3 >=3) For the union probability I'm getting .7 repeating, while for the intersection probability I'm getting .703 repeating. Can someone show me both methods are ...

Problem: Show that the set of polynomials with rational coefficients is countable. Idea: We know that the set of rational numbers is denumerable. This implies that the set of rational numbers is countable. We also know that the degree that each polynomial can be is a natural number (I think, an...

If i'm given 3 distinct points (a, b, and c) how would i describe the set of all points (d) in which abcd is a cyclic quadrilateral? I'm thinking that it is any point on the circle which contains triangle abc with exception of the vertices of said triangle. Is this correct?

Problem: Suppose that there is a function mapping S onto T. Prove that Card(S) is greater than or equal to Card(T). Issue: I can't seem to find a reason why this follows. If S is onto T then this guarantees that for any element in T there is at least one element in S. Therefore, I keep coming u...

A 10 foot high tank shaped like an inverted pyramid with a square base measuring 16ft by 16ft is filling with water at a rate of 2.5 cubic feet per a second. How fast is the water rising when the water is 8 feet high.

John is building a rectangular pen against his garage using wooden fencing. He wants to divide it further into 4 smaller pens all of equal size by placing 2 perpendicular units of fencing in the middle. John has 100 feet of fencing and wants to make the pen as big as possible. Find both the dimen...

Let $K$ be a number field, $\mathcal{O}_k$ its ring of integers, $\operatorname{Cl}(K)$ its class group and $h_K = \lvert \operatorname{Cl}(K)\rvert$ its class number. Let $X = \operatorname{Spec}(\mathcal{O}_K)$ and let $U\subseteq X$ be an open subset of $X$. Apparently (ex.3.19 Liu), using the...

I tried to prove this by contradiction and what I did is. I used contradiction proof that is n is odd then 5n^2 - 3 is even. But my Professor said this is wrong you need to prove that 5n^2 - 3 is even then n is odd. Can someone help me please?

Sample deviation is calculated as population standard deviation/sqrt(sample size) σ/√(n). Then why in an example like this below, is the answer 5.086? I understand that it comes from 4.2+1.96*1.5/sqrt(11), but why are we dividing with sqrt(11) when going from sample to population, when according ...

Let f(x)=x^2+1/x^2-9 So to find the y intercept I take f(0) correct? So when I substitute 0 for x I got -1/9 so is the y intercept (0,-1/9) Also to find the x intercept I set the numerator equal to 0. So then I got x^2+1=0 but wouldn't that make x^2=-1 which is imaginary? I'm a little confused ...

I would like to stress the kind of reference I am looking for... In statistics there are lots of motivating (and sometimes unexpected) examples that are interested for everyone such as Birthday Problem, Simpson Paradox, secretary problem, St Petersburg Paradox that easily motivate people from dif...

Let be $C_n$ the rectangle, positively oriented, which sides are in the lines $$x=\pm(N+\dfrac{1}{2})\pi~~~y=\pm(N+\dfrac{1}{2})\pi$$ with $N\in\mathbb{N}$. Prove that $$ \displaystyle\int_{C_N} \dfrac{dz}{z^2\sin(z)}=2\pi i\left[ \frac{1}{6}+2\sum_{n=0}^\infty \dfrac{(-1)^n}{n^2\pi^2}\ri...

How to prove the Schwartz function in $\mathbb{R}^n$ is dense in the space $W^{2,p}(\mathbb{R}^n)?$

How I calculate this integral using contour integration? $\int_0^\pi \frac{3cos(n\theta)}{5+4cos(n\theta)}d\theta$ I know I can start by using that $cos(n\theta) = Re (e^{in\theta})$, but I get bogged down in the computing.

Let E/F be a field extension. (a) If (E : F) = 2, is E/F a Galois extension? Justify your answer. (b) If (E : F) = 3, is E/F a Galois extension? Justify your answer.

I found the following in Baby Rudin: Theorem If $x,y\in \Bbb R, x>0$ then there exists a natural number $n$ such that $$nx>y$$ After this, he proves another theorem using the following Let $x<y$, we have $y-x>0$ then the theorem above yields that a natural number $n$ exists such ...

Prove that for all positive $x,y,z$, $(2e^x+\dfrac{2}{e^x})(2e^y+\dfrac{2}{e^y})(2e^z+\dfrac{2}{e^z}) \geq 64$ I dont have that much experience with inequalities but I know I can rewrite $64 $ as $4^3$ So here is my approach, if I can maybe prove that $(2e^x+\dfrac{2}{e^x}) \geq4$ as well for $...

Instinctively I think yes, since a = k + b for all a and b, but then I think about, eg. 4 congruent to 7 (modulo 1) and I'm confused about what the least residue of that would be...

I am trying to show that $$\sum_{i=1}^{100}(n+i)^{n+i}\equiv 0 mod 100$$ Any tips?

I don't know weak derivative of 1/r^n where r is norm of x. (i.e. distance of from origin to x) If n=1, it's easy by logarithm and integration by parts because log(x) is locally integrable. but if n>1, how to approach to it?

Double Integral Problem I just don't understand how to do the last step and calculate the actual value with the integral with respect to y. Any help is appreciated. Thank you so much!

What is the expected number of tosses of fair coin to get $THH$? Here's my approach: Let $N := $ number of tosses to get $THH$. Then by partitioning the sample space, $$ E(N) = E(N; H) + E(N; T). $$ If we get a $H$ on the first toss (with probability $\frac{1}{2}$) then we basically start over...

THis is an exercise on Martin Braun's Differential Equations and Their Applications p.544 Exercise 5. Let $P_n(x)$ be the Legendre polynomial of degree n. (a) Show that $P_n'(x)$ satisfies a selfadjoint equation with $\lambda = n(n+1) - 2$ (b) Show that $\int_{-1}^1 P_n'(x) P_m'(x) (1-x^2)dx ...

I'm trying to think of functions that are injective but not surjective. I can think of f(x) = e^x for f: R --> R (since the range should be positive R for this to be surjective, but R+ is a subset of R) and f(a,b) = a^b for f: ZxZ --> R (the range should be Q for this to be surjective, but Q is a...

A is a 2x5 matrix with two pivot positions. Does the equation Ax=b have at least one solution for every possible b? If the answer is yes, then I understand why. If the answer is no, please explain.

The problem said: Let X and Y be identically distributed independent random variables such that the moment generating function of X + Y is M(t) = 0.09e 2t + 0.24e t + 0.34 + 0.24et + 0.09e2t, −1 < t < 1 Calculate P(X <=0). I found the path to have the correct answer: We must just b...

I want to prove the following: Let $K$ be a field, let $g(X) \in K[X]$ be an irreducible polynomial, and let $L$ be a splitting field for $g(X)$ over $K$. Prove that $g(X)$ has at least one repeated root in $L$ if and only if char($K$)$ = p$ for some $p > 0$ and $g(X) = h(X^p)$ for some $h(X) \i...

Let $f$ be a given continuous function on $[0,1]$. Prove that there is a unique continuous function $g$ on $[0,1]$ satisfying $$g(x) = \frac{1}{2}g\left(\frac{x+1}{2}\right) + f(x)$$ for all $x$ in $[0,1]$.

$X_1,X_2,...,X_n\;i.i.d\;$with bernoulli distribution(p.m.f is $\theta^{x}(1-\theta)^{1-x})$. I want to gain distribution of $\bar X$. Then I calculated moment generating function of $\bar X$(by calculating $E[e^{\frac t n (X_1+...+X_n)}]$. And I gained M.G.F of $\bar X$. Result is $(1-\theta...

A paperweight has a slanted top described by x + y + z = 2. Its edges are orthogonal to the xy–plane, and the bottom of the paperweight is formed by the triangle with vertices (1, 0, 0),(0, −1, 0) and (0, 1, 0). Use a triple integral to find the volume of the paperweigh. From the base of the gra...

isomorphic copies of $V$, where $V = k^n$. I was able to show that R has a unique simple module V, but I'm stuck on the rest of the exercise. Any help is appreciated.

Sketch the graph of the function y=x(4-x)-83lnx. Indicate the transition points (local extrema and points of inflection). Local minimum x= Local maximum x= inflection at x= I understand how to solve this problem mathematically, however, is there any way of analyzing the question to get a better...

Hi I was checking my answers with my professor and I think the way he solved it was wrong can you please check? Question: Let R be the region in the xy-plane bounded by the axes and the line x + y = 1. Use the change-of-variable transformation T(u, v) = (u + v,u v) to evaluate, ZZ R(x y)5(x + y)...

If $u_k\rightharpoonup u$ weakly in $W^{1,q}(U)$,how to show $$ \sup\limits_{k}||u_k||_{W^{1.q}(U)}<\infty $$

Consider the functions f(x)=(x^2-14x+49) and g(x)=(x-7) and f(x)/g(x)=(x^2-14x+49)/(x-7), notice how f is g^2 because you can factor (x^2-14x+49)=(x-7)^2 so f/g = (x-7)^2/(x-7) = (x-7). So since f/g = (x-7) which is just a line the domain should just be (-infinity, infinity) right? But here is wh...

I would like a verification of a proof for the following statement. Let $S$ be a multiplicatively closed subset of a ring $R$. If $R$ is a PID, then $S^{-1}R$ is a PID. Let $I = \left<r_1/s_1, r_2/s_2\ldots\right>$ be an ideal of $S^{-1}R$. Since $1/s_i$ is a unit in $S^{-1}R$, we have $\left<r_...

$$\int\int_D 6x\sqrt{y^2-x^2}dA, D=\{(x,y)|0\leq y\leq 2, 0 \leq x \leq y\}$$ I tried: $$\int_0^2 \int_0^y 6x\sqrt{y^2-x^2}dxdy$$ But that is incorrect.

Let $Au_i=u_{i+1}-(2-\beta)u_i+u_{i-1}$ whith $u\in \ell^2=\{(u_i)_{i\in Z}:\sum_{i\in Z}u_i<+\infty\}$. How to compute $||A^{-1}||$ or estimate it?? Is it a sectorial operator of type $(\omega, \theta)$ with $\omega <0$????

How come in u-substitution if u is to the -1 power it becomes equivalent to the ln of u? I don't have a specific problem where that is the case , however I do recall this being a rule of thumb. Can someone explain this to me?

Let $f : A \rightarrow B$. Prove that if $X \subseteq A, Y \subseteq B$, and $f$ is a bijection, then $f(X) = Y$ if and only if $f^{-1}(Y) = X$.

C=a∨b∨c∨d∨e is a clause in SAT D= (a∨b∨x)∧ (¯x∨c∨y)∧ (¯y∨d∨e) is the another form of C to make sure every clause has only threeliterals Is D true when C is true and false when C is false? Why? I think when a,x,y=true and b,c,d,e=false, then C is true but D is false.

As the title says I'm looking for Math textbooks whose copyright has expired or have an open source / creative commons compatible copyright.

Consider $$\begin{bmatrix} C_{11}&C_{12}&\dots&C_{1n}\\ C_{21}&C_{22}&\dots&C_{2n}\\ \vdots\\ C_{n1}&C_{n2}&\dots&C_{nn}\\ D_{11}&D_{12}&\dots&D_{1n}\\ D_{21}&D_{22}&\dots&D_{2n}\\ \vdots\\ D_{n1}&D_{n2}&\dots&D_{nn}\\ \end{bmatrix}$$ where each $C_{ij}$ and $D_{ij}$ is $1\times n$ submatrix. $...

Picture Link Here If someone could please explain in English what the question is actually asking, that'd be great. An explanation would be nice too, but no worries. (I know the answer is e, it's not homework)

The problem is Kreyszig 10ed international edition : 16.2 #9. What is the order of pole at $z=\pi$ of the function $f(z)$ below? $$f(z)=\frac{\sin z}{z-\pi}$$ I thought that it will be simple pole because $\sin z$ is analytic for all $z$$\in$$\Bbb C$ and $z-\pi$ has a 1st order zero at $z=\pi$....

if $f$ is a unitary matrix representation, then $f$ is a equivalent to a direct sum of irreducible unitary representation via an equivalence implemented by a unitary matrix $T$.

If $$\frac{1}{\sigma_\widehat{e}^2}=\sum_i\frac{1}{\sigma_i^2}\tag{1}$$ Pick anyone of the $\sigma_j$ and multiply both sides of $(1)$ by $\sigma_j^2$ $$\implies\frac{\sigma_j^2}{\sigma_\widehat e^2}=\color{red}{\fbox{$\sum_i\frac{\sigma_j^2}{\sigma_i^2}=\color{blue}{1}+\sum_{i\ne j}\frac{\sigm...

I was learning (practicing to solve) simplifying the rational expressions. I know how to simplify the rational expressions... but I can't understand some part of the questions. The question that I can't understand If you look at this image, you could see sentence "First, let's set the denominat...

Let A={x∈R, x not equal 1} B={x∈R, x not equal 2} f(x)=2x/(x-1) A -> B How to find the bijective of f(x)?

What are the differences between a so called mathematical law and axiom? Please define both concepts in your answer.

Are there better alternative definitions than $\exp(x) = {\large\sum\limits_{k=0}^\infty} \dfrac{x^k}{k!} , \ln(x) = {\large\int_1^x} \dfrac1t\ dt$. that can be used for derivation of their identities e.g. $\exp (x+y)=\exp(x)\exp(y)$, $\ln (xy) =\ln (x)\ln(y)$ ( and by extension sin and cos rela...

If E/F is a field extension and (E:F)=2, is it Galois? What if (E:F)=3? I don't see why the degree of the field extension matter if it is finite. Thanks!

Consider the following informal definition for a function calc(x,y) = (0*y) + (1*y) + … + (x*y) For example, we have that calc(2,5) = (0*5) + (1*5) + (2*5) Give a recursive definition for the function calc. Give a trace to show each step involved in calculating calc(3,4) using your def...

How to solve this differential equation:$$Ri(t)+ L\frac{di}{dt}= A\cos(\omega t+ \alpha)\;?$$ Any hint would be appreciated.

Given an optimization as follows: \begin{align} \text{minimize}\quad &c^Tx \\ \text{subject to}\quad &Ax = 0 \\ & \|x\|_2^2 \leq 1 \end{align} where $A \in \Re^{m\times n}$ and $c \in \Re^n$. Let $x^*$ be the optimal solution to this problem. Because $c^Tx$, $\|x\|_2^2 - 1$ are convex and...

If I'm given 3 vertices, say (5,3), (9,3), and (9,7), how can I draw this triangle in a 2 dimensional way along 2 axises and still represent it as a poincare triangle?

$T\begin{bmatrix}a\\b\\c\\d\end{bmatrix} = 2a-3b+c-2d$ $v_1=\begin{bmatrix}1\\2\\3\\4\end{bmatrix}$ $v_2=\begin{bmatrix}1&2&3&4\end{bmatrix}$ then $T(v_1) = -9$ or $\begin{bmatrix}2\\-6\\3\\-8\end{bmatrix}$? then would $T(v_2) = -9$ or $\begin{bmatrix}2&-6&3&-8\end{bmatrix}$ or what?

Few days back one of my friend and I were discussing about Galois life and his ideas. Though we are not trained in Galois theory, but I am recently started to learn it by myself and hope to take up research on it some times soon. So I am wondering about the open problems in Galois theory? Is ther...

Is it true that the equation $27x^2+1=7^3y^2$ has infinitely many solutions in positive integers $x,y$ ?

$\lim_{x\rightarrow 0 } \dfrac{1}{x} \int_{x}^{2x} e^{-t^2} dx$ I dont have any idea to solve this integral ?

The two ends of the frastum are circles with radii of 3cm and 9cm. Find the volume of the frastum. How to do the question?

Brian Lara, the famous batsman, scored 6,000 runs in certain number of innings. In the next five innings he was out of form and hence, could make only a total of 90 runs, as a result of which his average fell by 2 runs. How many innings did he play in all, if he gets out in all the innings? (a)...

I'm trying to prove $\Box A\rightarrow \Box\Box A$ from $KG_r$ where $KG_r$ is the axiom $$\Box[ \Box A \rightarrow A ] \rightarrow \Box A$$ I'm given the hint that $$ A \rightarrow ((\Box\Box A\land \Box A) \rightarrow (\Box A \land A))$$ is a theorem. I know that in any normal system you...

Theorem: Let A and B be sets such that A$\cup$B≠B and B\A={}. Then we have that B$\subset$A Theorem is True. A and B are not identical sets. A contains all of B's elements, therefore B is a subset of A. Is this correct?

The population of an ant colony doubles every day and a half. A biologist predicts the ant colony will eventually overtake a flower bed. What is the number of hours between the time the flower bed is half full and the time the flowerbed is completely full? Explain.

What is the value of $\dfrac{1}{1!} + \dfrac{1+2}{2!} + \dfrac{1+2+3}{3!} + \dots $ How to proceed with it ?

G(x,t) =\begin{cases} a+ b(logt) & \text{if $0<x<t$ } \\[2ex] c+ d(logt) & \text{if $t<x<1$ } \end{cases} is a Greens function for $xy''+y'=0$ subject to y being bounded as x tends to $0$ and $y(1)=y'(1)$ if a) a=b=c=d=1 b)a=c=1,b=d=0 c)a=c=0,b=d=1 d)a=b=c=d=0 I don't know how to tackle th...

We are given initially some $k,c,d\in\mathbb{N}$. How many solutions $(x_1, x_2, ..., x_k)$ does the equation $\sum_{i=1}^{k}{x_i}=c$ have, where $x_i\in\mathbb{Z}$ and $0\leq x_i\leq d$?

I am a Calc I student having trouble understanding why exactly L'Hopital's rule works and when exactly it can be used. I know this question is general, I just want to gain more of an understanding of the rule...thank you for your time.

There is an expression given for a piece-wise linear function, fn(Phi). fn is being a function of Phi: u = u0*cos(w*t-y) = u0*cos(Phi) where Phi = w*t-y Assume u0, w, t are positive Assume u0 > Fy/kf fn(Phi) = Fy + kf * (u - u0) for 0 <= Phi < Theta fn(Phi) = -Fy for Theta <= Phi < pi fn(Phi...

How to find an integer $x$ such that $140x \equiv 133 \pmod{301}$? There is a hint that $gcd(140,301)=7$

I have 3 circles: $C_1$ centered at $(0,0)$ with radius 1 $C_2$ centered at $(a,0)$ with radius $a+1$ $C_3$ centered at $(-a,0)$ with radius $a+1$ (so $C_1$ is internaly tangent to both $C_2$ and $C_3$ ) Question: What are the centres of circle $C_4$ and $C_5$ that are tangent to all three ...

What is the value of $\lim_{n\to \infty}(\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}...+\frac{1}{2n})$? Please give some hints to proceed .

for checking it $$lim_{n \to \infty}cos(1/n)=lim_{n \to \infty} \frac{e^{i/n}+e^{-i/n}}{2}=(1+1)/2=1$$ this means that the series diverges is it right thought.

What is the origin and context of the phrase 'solvable in polynomial time' in computer science? Are they related to the notion of 'polynomials' in mathematics?

I am trying to solve a system of ODE's using the backward euler's method in MATLAB. The system has 3 equations with the initial conditions given below. x' = λ - ρx - βxz; y' = βxz - δy; z' = py - cz; x0=43100; y0 = 0, z0 = 0.0033, λ = 388, ρ = 0.009 δ = 0.18, p = 50000, c = 23, β=3.61e-8 Can s...

Let $A=\{\sum _{i=1}^\infty \dfrac{a_i}{5^i}:a_i=0,1,2,3,4\}\subseteq\Bbb R$. Then is $A$ a finite set? the series $\sum _{i=1}^\infty \dfrac{a_i}{5^i}$ always converges to $0$ .So $A=\{0\}$ Is it correct?

Suppose that X is uniformly distributed on (0,2). A) Find E[X] I got that to equal to 1 B) Find E[e^X] I got that to equal to 3.1946 I am wondering if these are right and if not could you please show me how to do it the correct way.

I am trying to compute the expected value of a function V^2, given that V is a normal random variable with mean = 0 and variance = sigma squared?

Do there exist two dense sub-spaces of a Hilbert space whose intersection is {0}?

A great friend of mine recently sat for an interview. He was asked a question which has fascinated me since some days now. It is Consider you have 100 balls which look the same but one out of them is either heavier or lighter than the rest . You are given a beam balance , then find out in mi...

I am trying to find the expected value of the function Z = mV^2/2, in terms of sigma and m, where m is a constant greater than 0.

Intuitively it seems that function $\sin(1/x)$ should be bounded, but analysis shows it is not the case. The function takes arbitrarily large values around zero. Why it is the case?

I have troubles to solve this kind of exercises. For example, let $G_1=<x,y |x^3=y^4=1> $ and $G_2=<x,y |x^6=y^6=(xy)^3=1> $. I want to check that 1) $G_1$ is infinite and nonabelian and 2)$xy^2x \neq 1$ in $G_2$. For the first part, I have seen that it is useful to define a group homomorphism...

Can we say that $\sum{{a_n}{b_n}}$ converges / converges absolutely?

What is the total no. of positive integral solutions of: X1 + X2 + ... + Xk-1 < Xkwhere Xi < Xi+1 and each of Xi <= N

Let $H$ be a non-separable Hilbert space. Assume $E$ is an orthonormal basis in $H$. Let $E_0=\{e_n\}$ be a countable subset of $E$ and let $\{\zeta_n\}$ be a bounded sequence in $H$. Let $E_1$ be a countable susbet of $E$ and denote $p$ by the projection onto $\overline{span\{e: e\in E_1\}}$. ...

I am working through Parzen and I came across a problem that has completely stumped me. I have an urn which has M black balls and N white balls. Each turn, I randomly reach in and choose one ball without replacement. If the ball is black, I add on white ball to the urn (and no not replace the dra...

I am considering the following optimization problem: $\underset{u}{\text{minimize }}\quad \underset{v}{\max} f(u,v)$ where for a fixed $u$, the function $f(u, .)$ is piecewise constant. Do you know if there exists any simple approach to solve this kind of problem? Which one should I use? Than...

$\int e^{x^x} $. taking u = $x^x$ so du= $x^x$[logx +1] dx -> $\frac {du}{[u(log(x) +1)]}$=dx Now to remove log(x) from denominator I took log(u)= x log(x) -> log(x) = $\frac {log(u)}{x}$ Again If I use that log(u) to remove x I get log(x). How to proceed from here, Or is there another me...

$2^{100} \equiv 1 \ (\text{mod}\ 125)$ and is divisible by $8$. Why then is $2^{100} \equiv 376 \ (\text{mod}\ 1000)$?

In English, I would generally read the mathematical notation $\bar{A}$ as "a bar" or "big a bar." What would this be in Russian? Thanks.

Let $X$ and $Y$ be an independent random variables with exponential distribution $Exp(1)$. Compute $E[\min{\{X,Y\}}|Y^{2}]$. My attempt: I. If we want to compute $E[\min{\{X,Y\}}|Y]$ may be sufficient to use the following formula: $$E[f(X,Y)|Y]=\int\limits_{-\infty}^{+\infty}f(x,Y)f_{X}(x)dx$...

I want to find the degree of the splitting field for $x^3-3x-1$ over $\Bbb Q$ and $\Bbb F_5$. My attempt is contained below.

Let X1,X2,X3,... be an i.i.d. sequence of Bernoulli random variables where Xi=0 with probability 0.5 and Xi=1 with probability 0.5. For a fixed value of N, define Xbar = X1+X2+X3+...+XN/N A) Find u(subscript Xbar) = E[Xbar] and Sigma^2(subscript Xbar)= Var[Xbar]

How can I find the PDF of the function Z=mV^2/2, in terms of sigma and m, where m is a constant greater than 0?

Let $f:V\times V \rightarrow C$ be a bilinear form in a finite inner product space. Will there always be a single linear transformation $T:V\rightarrow V$ for which $f(v,u) = <Tv,u>$ for each $v,u\in V$. If not - what's an example of this not happening? (I can prove this for bilinear forms $f:V\...

There are several books and articles on topos theory which mention the internal language, but I can't manage to see the big picture from any of them. I would like a soft explanation of how the entities in the definition of an elementary topos come to together and give rise to this internal logic....

Please solve the following integral.$$\int x^2 e^{-\frac{x^2}{2}}dx.$$

Why does $3f_3 + 4f_4 + 5f_5 + ...= 2E$? Is there a specific proof for this or is it just due to the fact that each edge makes up parts of 2 different faces. I have looked all over the internet and I cannot find a proof online, everyone just seems to 'assume' that it does. Please help. Thanks

Let $X$ be obtained by taking two disjoint copies of the interval $[0,2]$ (with the Euclidean topology) and gluing each $t$ in the first copy with the corresponding $t$ in the second copy, for all $t \in [0,2]$ different from the middle point. Explicitely, one may take the space $$ Y=[0,2]\times...

How are simplicial complexes more helpful than CW complexes with respect to smooth manifolds? What are some of the benefits of one vs. the other?

Considering statements (1) , (2) , (3) & (4) to be true , $$ \sum_{j=1}^B (w_kj + y_kj) + (e_k - b_k)a \leq f_l$$ $$ \sum_{j=1}^B (w_kj + y_kj) \leq f_l - (e_k - b_k)a $$ $$ \sum_{j=1}^B (w_kj + y_kj) \leq f_l + (b_k - e_k)a $$ Summing up in terms of $k$ , $$ \sum_{k=1}^{4}\sum_{j=1}^2 (w_...

If $x=\pi^{3}$ and $y=3^{\pi}$ then what is the relation between x and y? I have no idea of how to solve this problem. Please help.

Looking at sets (in ZF, Set Theory of Kunen) I could not escape from looking at logic as well. A formal language is presented containing basic symbols ($\wedge,\neg,\exists,(,),\in,=,$ and $v_{i}$ for $i=0,1,\dots$). Also things are said about free variables and bound variables. Formulas are d...

I have the following problem. Let $H_t$ be an adapted process with trajectories a.s. of class $C^1$ on $\mathbb{R}_{+}$. Compute using simple process $\int_o^t H_s d B_s$. My idea is to firstly set $U_n(s):=\sum_{i=0}^{m(n)-1} H_{t_i^n} \mathbb{1}_{(t_i^n-t_{i+1}^n]}$ (where of course I consider...

I know the definition of an outer measure. Let X be a set. An outer measure on X is a function $\theta : \mathcal{P}X \to [0,\infty]$ such that (i) $\theta \phi=0$ (ii) if $ A \subseteq B \subseteq X$ then $\theta A \leq \theta B$ (iii) for every sequence $<A_n>_{n \in \mathbb{N}}$ of subsets...

Can anyone prove the angle theta (smaller angle) between diagonals of a parallelogram is given by the equation cos(\theta)=\frac{a-c}{a+c} where "2*a" is the base length of parallelogram; "2*c" is the horizontal distance to the edge when we draw a vertical line as seen in the picture below. ...

Given $$F(x)=\frac{1}{2x}\int_{-x}^xf(t)dt,$$ where $f:\mathbb{R}\to\mathbb{R}$ is continuous and $f'(0)$ exists, how can I prove that $F'(0)$ also exists?

Let $H$ be a Hilbert space. Some books define an orthonormal basis as one that is in the title. I am using the standard linear algebra definition, that is, an orthonormal basis is one that is a basis and orthonomal in the linear algebra (finite-dim) sense (every element in $H$ is a linear (poss...

Let $A=\{\sum_{i=1}^{\infty} \frac{a_i}{5^{i}}:a_i=0,1,2,3$ or $4 \} \subset \mathbb{R}$. Then which of the following are true: a. $A$ is a finite set. b. $A$ is countably infinite. c. $A$ is uncountable but does not contain an open interval. d. $A$ contains an open interval. Each such seri...

Let triangle $\triangle{ABC}$ have incenter $I$. Prove that the Euler lines of $\triangle{AIB}, \triangle{BIC}, \triangle{CIA}, \triangle{ABC}$ are concurrent.

The theorem is: Let X be a set and $\theta$ an outer measure on X. Set $\sum= \{E: E \subseteq X, \theta A=\theta(A \cap E)+ \theta(A \backslash\ E)$ for every $A \subseteq X\}$ What is the intuition behind this definition? and what does the Caratheodory method try to achieve?

I've got what I think is a proof, am wondering if I've made a mistake. Proof by contradiction: Suppose sqrt(n)=a/b, with a and b being integers and coprime meaning a/b is rational. Square it, so n=a^2/b^2 and take the b^2 over: b^2 n = a^2 The way I see it, since a^2 is taking the prime factor...

I am looking at the following: Could you explain to me what a constant field is?

enter image description here How to integrate (cos x)^n.(sin x)^m with reduction formula. the actual steps please in terms of (n-2,m) and in terms of (n, m-2) like in the image shown I'm stuck halfway Thanks https://en.wikipedia.org/wiki/Integration_by_reduction_formulae

Find the value of the series $\sum_{i=1}^ \infty \dfrac{n}{2^n}$ The series on expanding is coming as $\dfrac{1}{2}+\dfrac{2}{2^2}+..$ I tried using the form of $(1+x)^n=1+nx+\dfrac{n(n-1)}{2}x^2+..$ and then differentiating it but still it is not coming .What shall I do with this?

I cannot seem to solve what I think should be a straightforward intersection theory question in Ravi Vakil's algebraic geometry notes. Let $X$ be a scheme and $Y$ a closed subscheme of dimension less than or equal to $n$, and $\mathscr{L}_1,\ldots,\mathscr{L}_n$ be a collection of line bundles o...

I really need help on this question. diagram: http://media.apexlearning.com/Images/200702/13/1ef91461-c2f5-4d1a-9188-5c5fe1fd849a.gif

The question reads : A box (with no top) is to be constructed from a piece of cardboard of sides A and B by cutting out squares of length h from the corners and folding up the sides as in the figure below: Suppose that the box height is h = 3 in. and that it is constructed using 134 in.^2 of c...

Show that there is no perfect square whose last three digits are 341. Solution: Let x be any integer. Note that if $x^2$ ended with the digits 341, then we would have $$x^2 = 1000k+341$$ for some integer k, so $$x^2 = 1000k + 341 = 341 = 5 (mod 8)$$. We make a table of squares modulo 8. $$ x: ...

An equation of motion when a particle moves in a resting medium is given by dv/dt=-(kV+bt) where k and b are constants.Given that v=u,when t=0..show that v=b/k^2-bt/k+(u-b/k^2)e^-kt

I have a professor who made a very good point about the data titanic analysis during a lecture this week. I am still however trying to better understand. He argued that it is also possible to have a model of titanic data (GLM) using Poisson rather than the typical Binomial approach. I am curious ...

So, first of all I'm gonna start with stating that I'm a complete noob in mathematics that started doing this as a hobby. I encounter problems that I can't figure out on a regular basis, like a few times per page but I usually find someone to help me. This time those same people were of no help s...

I have just started revising number theory and I am getting stuck on a lot of the "prove that" questions. Any tips and advice would be much appreciated!

I am in calculus class right now and I have no idea. I'm sorry for my ignorance.

I'm totally stuck at this problem and any help would be appreciated: Suppose $R_\rho \in M_2$ denotes a reflection across a line which is through the origin and at a anticlockwise angle of $\rho$ with the x-axis. The question asks that for any $n \in \mathbb{N}$, does there exist a subgroup of $M...

Over 2,300 years ago Euclid proved that the number of primes is infinite, so two possible questions come to mind: How many primes are there less than the number x? There are infinitely many primes, but how big of an infinity? This document will focus on the first question. The seco...

If h(t) represents the height of an object above ground level at time t and h(t) is given by h(t)=-16t^2+13t+1 find the height of the object at the time when the speed is zero. Suppose h(t)=t^2+14t+7 . Find the instantaneous rate of change of h(t) with respect to t at t=2 . Suppose G(x)=6x^2+x+...

Hi can anyone help me on this problem? In have the function 2^n*(2n+1)-1 and I need to prove that the function has an inverse function. So I did: f(x,y)= 2^n*(2n+1)-1 f(x,y)=t, where t=59 so: 2^n*(2n+1)-1 =59 2^n*(2n+1)=60 (Factorize 60 it becomes) 2^n*(2n+1)=2*2*3*5 after that I am los...

$$x''=-2\beta x'-x+\gamma\cos(\omega t)$$ with $\beta,\gamma,\omega$ positive constants. I started by finding the homogeneous part of the solution by solving $x''+2\beta x'+x=0$ and found $$x_{hom}=c_1e^{\lambda_+ t}+c_2e^{\lambda_- t}$$ with $\lambda_{\pm}=-\beta\pm\sqrt{\beta^2-1}$. Now to find...

The problem statement is: Let D be the domain consisting of all points satisfying Im $z>0$ and (Re $z)^2$+(Im $z−1)^2>1$. Map $D$ onto the upper half plane. So the domain is everything on in the upper half plane, minus the circle of radius 1, centered at $(0,i)$. I first applied the logarit...

Let $G$ be an abstract group (say, finitely generated). Endow $G$ with the profinite topology. I would like to get comfortable with computing the closure of subgroups of $G$. More specifically, I would like to understand how to use the profinite completion of $G$ to do that. If $G$ is residual...

I am not relieved of the following problem and trip. Let $\varepsilon>0$ be a small parameter, $a>0$ be a given constant, $x_{\varepsilon}\in(0,a)$ be a given sequence such that $x_{\varepsilon}\to a$ as $\varepsilon\to0$ and $f:[0,a)\times[0,a)\to\mathbb{R}$ be a continuous function such that $...

In an exam there are 10 questions. If you answer correctly to a question, you get $1$ point. If you answer incorrectly to a question, you get $-1$ point, or lose a point. If you don't answer to a question, you get $0$ point. You pass the exam if you get at least $7$ points. A pupil read the ques...

Show that the number of ways to stack coins with n coins in the bottom row is denoted by Catalan number Dn = (2nCn)/n+1. I have tried in the way described in the picture below

Found this on a textbook and I know I am missing out something really simple as it's been a while for me with vectors so here it goes:If $ \Bbb P=mi-2j+2k$ (where i,j,k are unit vectors) and $ \Bbb q=2i-nj+k $ are parallel to each other, then find $m$ and $n$

integral from -1 to 1 f(x) dx = (5/9)f(-sqrt(3/5))+(8/9)f(0)+(5/9)f(sqrt(3/5)) verify your answer by continuing the method of undetermined coefficients until an equation is not satisfied. Now im stuck, i dont really understand Gaussian quadrature and can some one help me use the method of undet...

If $EX^2<\infty$ and $E(X|\mathcal G)$ is $\mathcal F$-measurable then $E(X|\mathcal G)=EX$ There is one step in the proof which I don't understand, set $Y=E(X|\mathcal G)$ and then why is $E((X-Y)Y)=0$, from a theorem I know that $Y\in\mathcal F$ is such that $E(X-Y)^2$ is minimal

I try to prove that $[0,1]$ is conex.I have to follow the next steps: $R$ is complete using the construction of $R$ with Cauchy sequences $\Rightarrow$ $[0,1]$ is complete ,$[0,1]$ compact$\Rightarrow$ $[0,1]$ is conex.How can I prove that?

Suppose that the variable of a mono-dimensional wave equation is an angle: $$\frac{\partial^2 f(\phi)}{\partial \phi^2} + k_{\phi}^2 f(\phi) = 0$$ This equation is derived from a more complex (and separable) equation in cylindrical coordinates and $\phi$ is one of these coordinates. According t...

I want to find the Galois group of $x^6-2$ over $\Bbb Q$. I have posted my attempt in an answer below. Is there a better way? Alternative proofs are greatly appreciated.

Poisson distribution with a mean of 2 books bought per hour. (a) What is a probability that no one will buy a book in a given day (24 hours)? (b) The probability that there is at least one book bought in the next minute?

In a book I'm reading, the following statement is written: Let $\varepsilon>0$ and $\delta>0$ be parameters with $\varepsilon<\delta$, $\{x_{\varepsilon}\}_{\varepsilon>0}\subset(0,1)$ be a sequence such that $x_{\varepsilon}\to a\in(0,1)$ and $f:(0,1)\to\mathbb{R}$ be a continuous function. ...

Given a convex quadrilateral in which we know the angles of the corners and of the one diagonal, can we find the angle of the second diagonal? quadrilateral

A. has exactly one solution B. has no solution C. has a countably infinite number of solutions D. has uncountably many solutions I choose option C as the answer. This is because $\frac {|f(x)-f(y)|}{|x-y|} \ge 1 \forall x,y \in \Bbb R \Rightarrow |f'(x)| \ge 1 \forall x \in \...

It's theorem from PMA Rudin. But I think that we can conclude more stronger conlusion: that $\sum |a_n|$ converges i.e. $\sum a_n$ converges absolutely. Am i Right?

Let $\mu,\alpha_n:\mathbb R^+\to \mathbb R$ continuous function with $\mu$ bounded function. Let $N^{(n)}$ the trajectory of a Poisson process with intensity $(\alpha_n \mu)(t)$. Let $0=T_0^{(n)}<T_1^{(n)}<..$ jumps of $N^{(n)}$ Let $M_n(t)=\sum_{i=1}^{N_t^{(n)}} \frac {1} {\alpha_n (T_i^{(n)})}...

Attached is my proof. Is there any flaw in my proof? Thanks in advance.

Use double integral to determine the area bounded by the curve y=x^2 and line y=2x+3

For the closed ball $ {x\in \Bbb{R}^{3} ; || x || \le 1} $ I want to draw Eulidean norm, $L_1 $ norm and $L_{\infty}$ norm by using matlab.

I read the answers to these questions: Proof of Lagrange theorem - Order of a subgroup divides order of the group Lagrange's theorem How do I know that the cosets of H have the same number of elements?(That equivalence classes form a partition of G.)

I have a set of descriptions about the Lebesgue measure on $\mathbb{R}$. (i) A half open interval in $\mathbb{R}$ is a set of the form $[a,b[=\{x:a \leq x<b\}$ where $a,b \in\mathbb{R}$ We allow $b \leq a$ and in this case why does $[a,b[=\phi$? (ii) If $I \subseteq \mathbb{R}$ is a half open ...

I am try to simplifying this boolean expression (A+C+D)(A+C+D)(A+C+D)(A+B)`into four literals since from four days. Can anybody please help me out.

Any continous function on $\mathbb R$ such that $f(n) = 0 $ for all $n \in \mathbb Z$ . Then Which one of the following is true 1) Image of $f$ is closed. 2) Image of $f$ is open. 3) $f$ is uniformly continous. I am unable to solve ,Please give me hint how to solve. Thank you

I cant get my head round how to prove this, I tried using the euclidean algorithm but that didn't help me much.

If $\phi : G \to Perm(G/H)$ where $\phi$ is the group action on $G/H$ by $G$. $\phi := g(g'H) = (gg')H$ Why is the kernel of $\phi$ equal to $\cap_{x\in G} xHx^{-1}$ I thought the kernel is $\ker \phi = \{g | gx =x \}$ (so ker is a stabilizer).So in our case it is $g(g'H) = g'H$ . So all forms...

Please how to prove by using two methods that Irrational numbers are limits of a rational sequence After that deduce that $\mathbb{Q}$ is not closed in $(\mathbb{R},|.|)$ Thank you

Prove that if an integer n is not divisible by 6, then $ (n-2)^2 $ is not divisible by 4. Prove: $ 4 \nmid (n-2)^2 \Rightarrow 6 \nmid n $ Proof by contrapositive: $ 6 \mid n \Rightarrow 4 \mid (n-2)^2 $ n=6k,$ \\\ k \in \Bbb Z $ $((6k)-2)^2 = 36k^2 - 24k+4 = 4(9k^2 + 6k+1), (n-2)^2=4c \\\ w...

Non-standard analysis offers very convenient tools to prove facts about continuity or differentiability. I am looking for such tool in infinite-dimensional calculus. To be more precise, let $X$ and $Y$ be Banach spaces and let ${}^*X, {}^*Y$ be their non-standard extensions. Suppose we have fun...

I want to find two different fields $K_1,K_2$ such that $\Bbb Q\subset K_i \subset \Bbb Q(\alpha,\zeta)$ such that $K_i /\Bbb Q$ are Galois. A few things: $\alpha$ is the real $6^{th}$ roof of $2$, $\zeta$ is a primitive $6^{th}$ root of unity. Let $L= \Bbb Q(\alpha,\zeta)$, I have shown here ...