The random vector $(X,Y)$ has the following joint distribution $P(X=m, Y=n)$ = $m\choose n$$\frac{1}{2^m}\frac{m}{15}$ where $m=1,..,5$ and $n=0,...,m$. Compute $E(Y|X=m)$

It is unclear to me how I can manipulate an expression like $2sin^2(2x)$ using identities.

A step function can be defined to be a linear combination of a sequence of brick functions. My question is - Are step functions always Lebesgue integrable ?

The (probably) famous Gauss-Lucas theorem states that the roots of the derivative $P'(z)$ are contained in the convex hull of the roots of $P(z)$, where $P(z)$ is complex variable polynomial. I am interested here in could it be the case that we always have some polynomial of any degree (except $...

Is this factorization true? $$(x^n - y^n) = (x+y)^{n-1}(x-y)^{n-1}$$ I am trying to use it in my computation of the determinant of a Vandermonde matrix. Thanks,

I am not a mathematics wizard. Like a N.log(N) function belongs to quasilinear family of functions, what is the function family that N/log(N) function belongs to ?

Have 5 yards of ribbon; need 50 equal pieces. 180 inches in 5 yards. So closes I get is 50 pieces...3 inches each ?

Just... hwy? Why is that the case? Why? What is the answer? I have no idea. I don't know hwy there are quality Czechs on the problem. 85 down vote accepted Why am I getting this message? All new questions are subjected to a "minimum quality" filter that checks for characteristics of extremely ...

As a follow-up from this question: Groups of order 42 I realize that this would probably mean that $G$ is cyclic, could someone walk me through the logic behind this?

I am really trying to solve this problem but I am not following the the solution book. I have used integration by parts. Yet at that point I become stuck. Help me understand. Is the considered a constant? Can I move it outside the indefinite integral? Am I on the right track? It seems as if I a...

Let Y be a supspace of a vector space X. Let x=a+Y=b+Y and y=c+Y=d+Y. How can I show that a+c+Y=a+d+Y=b+c+Y=b+d+Y. I have tried somethings but I couldn't. How should I approach to this proof. Thank you..

A group G with orden 15 is simple? There are a theorem for realize it? Thanks for all you help!

Are there any major areas of modern research that deals with the interactions of algebra and geometry/topology besides the obvious ones such as algebraic topology and algebraic geometry?

Given that: xs^2+yt^2=1 and x^2s+y^2t=xy-4 where x=x(s,t) and y=y(s,t) find ds/dx, ds/dy, dt/dx and dt/dy. (where these are all partial derivatives) at the point (x,y,s,t)=(1,-3,2,-1)

I have this function f(x,y)=5x^3-7y , 0<=x , y<=1 Find the values of the global min and max So, I got partial derivatives fx=15x^2 , and fy=-7 But then I run into confusion on how to proceed. Thanks for help

How can I differentiate a complex number by a complex number? How can I differentiate a complex matrix by a complex matrix?

So, there's something that I can use to show that $f(x,y) = \frac{x^2 - y^2}{(x^2+y^2)^2}$ is or is not Lebesgue integrable in [0,1]^2?

Given $a\equiv x \bmod r_1$ and $b\equiv x\bmod r_2$, we can construct $x$ from $$x=a r_2 [r_2^{-1}]_{r_1} + b r_1 [r_1^{-1}]_{r_2}$$ where $(r_1,r_2)=1$. Suppose $(r_1,r_2)=r$ and $a\equiv x \bmod r_1$ and $b\equiv x\bmod r_2$ and $a\equiv b \bmod r$ holds then what is the constructive formula ...

This is the problem: Design a rectangular milk carton box of width w, length l, and height h which holds 520 cm^3 of milk. The sides of the box cost 2 cent/cm^2 and the top and bottom cost 3 cent/cm^2. Find the dimensions of the box that minimize the total cost of materials used. My approach was...

Suppose $f: D \to R$ and $g:D \to R$, $x_0$ is an accumulation point of $D$, and $f$ and $g$ have limits at $x_0$. If $f(x) \le g(x)$ for all $x \in D$, then $\lim \limits_{x \to x_0}f(x) \le \lim \limits_{x \to x_0}g(x)$ Assume that $\forall x \in D$, $f(x)$ and $g(x)$ both have limits at s...

How many subgroups of index $p$, where $p$ is a prime, are there in the additive group $\mathbb{Z}^2$?

Can a complete hyperbolic surface be isometrically embedded into flat R^4?

Let $\;y_1\left(x,\,\lambda\right),\: y_2\left(x,\,\lambda\right)\;$ be respective solutions of eigenvalue problem \begin{align}y\,'' &= \lambda \,y, & x&\in\big(\,0,\,\infty\,\big)\end{align} with boundary conditions \begin{align} &\begin{cases} y_{1}\big(0,\,\lambda\big)=1\\y_{1}'\big(0,\,\lamb...

Assume the existence of a $\xi \in Z^+$. A $\beta \in Z^+ ; 1\le\beta\le \xi$ is said to be $\xi$-adhering if $$\big\lfloor{\xi \over\lfloor{\xi/\beta}\rfloor}\big\rfloor=\beta$$ Let $f(\xi)$ be the number of $\xi$-adhering numbers for each integer $\xi\ge 1$. Question: Compute $f(1)+f(2) \c...

Certain pieces of antique furniture increased very rapidly in price in the 1970s and 1980s. For example, the value of a particular rocking chair is well approximated by V = 105 (1.75)^t, where V is in dollars and t is the number of years since 1975. Find the rate, in dollars per year, at which...

Given the assumption, I am asked to prove the property of floor and ceiling functions. Is my approach correct for using that assumption by showing that the assumption's pre-conditions are valid and the rest?

I've been self studying algebra for some time alone and a common problem I bump into that always throws me off is something like this (Complete idiots guide to algebra, if you know a better book please tell.) simplify 5-(-3)-(+2)+(-7) Now the double signs are removed so the non brackets become ...

How this sum can be performed if not with the use of usual formula? (1/6)(SUM from -2 to n)(0.5*e^(-jpik/3))^n

problem A cat chases a rat. For every $5$ leaps of the rat , the cat takes $3$ leaps, but the $2$ leaps of the cat are the same as $3$ leaps of the rat. Compare the speeds of the cat and the rat a)$4:5$ b)$12:13$ c)$16:15$ d)none of these

In how many ways can 2 different prizes be given to 10 students if one student may receive both prizes?

Don't get me wrong I don't really want to but I need to in order to have experience. I want Thomas Staggs position as chief executive officer of the entire corporation. Thanks

Use the Cauchy-Schwarz inequality to show that (acos(θ)+bsin(θ))$^2$ $\leq$ a$^2$ +b$^2$ for all a,b,θ ∈ $\mathbb{R}$ What I was trying to do was to take the smaller of either a or b and prove that (acos(θ)+bsin(θ))$^2$ $\leq$ a$^2$ if a were smaller than b. Then I created two vectors $\vec{v}$,...

$$\text{If} \ S=\displaystyle\sum_{n=1}^{\infty}\dfrac{\sin (n)}{n}, \ \text{then what is} \ 2S+1$$ I know that $\sum \frac{\sin(n)}{n}$ converges. But now what do I do?

Let $g=e^{-ix^2}, x\in \mathbb{R}$. Let $T$ be an operator defined as $T(f)=f*g$. Show that $T$ cannot satisfies $p-p$ inequality unless $p=2$. Note: We say an operator $T$ satisfies $p-p$ inequality if $||Tf||_p\le C||f||_p$ whenever the expression on the right side exists finitely. An...

Write an equation to model the generation of Ecoli if the doubling time is 20 minutes. I can't for the life of me figure this out.

Prove that if $a$ is a quadratic residue modulo $p$ and modulo $q$, then $a$ is a quadratic residue modulo $pq$.

The total area is 4046 sq meter. The 3 side perimeter are 126.636 meter, 31.762 meter & 127.275 meter. What is the length of the 4th side?

Find the equations of the locus of the point P (x,y) that is equidistant from the lines 4x+3y-2=0 and 12x-5y+6=0. Do I use simultaneous equations? I cant remember:(

I'm currently working on a problem for an assignment, so forgive me if my question is a little clumsy or vague. I'm trying to get myself headed in the right direction and offer a meaningful question here without just being handed the answer. I'm trying to prove that $\forall$$a$ $\epsilon$ $\mat...

Given. (1 + x^2005 + x^2006 + x^2007)^2008 = A0 +A1*X +A2*X^2 + .... An*X^n We are required to calculate A0 - A1/3-A2/3 + A3 -A4/4 -A5/5 + A6 ...... I tried approaching the problem by setting x to 1 and ω in two different cases. For x=1, we get sum of all coefficients as 4^2008 For x=ω , w...

i wanted just to understantd how this equality is found, by change of cordinnates \int_{0}^{1}{1/{x^{p-1}}\int_{0}^{1/x}|ln{t}|^p/|1-t|^p dtdx= \int_{0}^{1}|ln{t}|^p/|1-t|^p \int_{0}^{1}{1/{x^{p-1}}dxdt + \int_{0}^{+\infty}|ln{t}|^p/|1-t|^p \int_{0}^{1/t}{1/{x^{p-1}}dxdt thanks

It is given that $\text{Im}(f'(z))=6x(2y-1)$ and $f(0)=3-2i$, $f(1)=6-5i$ then what is the value of $f(1+i)=?$. I tried by assuming $f(z)=u+iv$ and $z=x+iy$ so $\text{Im}f'(z)=v'=6x(2y-1)$ but couldn't reach to a profitable conclusion. Any help or hint please.

So I understand the construction of this semi-direct product and it's properties. However I am confused to why $K^n$ is normal in Aff(n,K). My main confusion comes from this. I believe understand how T$_{A,b}$ $\in$Aff(n,K) acts on x$\in$K$^n$. It would be T$_{A,b}$(x)=Ax+b right? But how d...

Let ABC be a triangle in which AB = AC and let I be its in-centre. Suppose BC = AB + AI. Find ∠BAC.

I am new to axiomatic probability and am self-learning this topic. This question is from Grimmet's book. A conventional knock-out tournament(such as that at the Wimbledon) begins with >$2^n$ competitors and has $n$ rounds. There are no play-offs for the positions >$2,3,...,{2^n}-1$, and the ...

I have this question from higher algebra by Hall and Knight: if $ \frac{y}{x-z} = \frac{x+y}{z} = \frac{x}{y} $ then find the ratio of x,y and z? There are two answers given for this question , the first is $\frac x4 =\frac y2 =\frac z3$ and the second is $\frac x1 =\frac y{-1} =\frac z0 $.Now i...

{(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)} this set had no relation. Is it true and why?

This is yet another problem, where I have run into trouble. Let ${F}$ be a ${\sigma}$-field of the subsets of $\Omega$ and suppose that $B\in{F}$. Show that $G=\{A\cap{B}:A\in{F}\}$ is a $\sigma$-field of the subsets of $B$. I know a $\sigma$-field is, and that it is closed under complement...

I was wondering how do I solve the summation? I found it in this link on power towers http://mathworld.wolfram.com/PowerTower.html

How to solve the following simultaneous equations? x + y + z = 0 x^2 +y^2 +z^2 = 1 x^3 +y^3 +z^3 = 0 Thank you!

This might sound like an easy question, but I need to prove: $A$ is a set. $A\cap A=A$ $A\cup A=A$ $A\setminus A=\emptyset$ I have tried; let $x\in A \therefore x\in A$.

If {$f_k $} is a sequence of Lebesgue integrable functions, then {$f_k$} is said to "converge in norm" to an integrable function $f$ if $\int | f_k - f | $ converges to zero . Can someone explain to me the concept behind convergence in norm ?

I was reading Fredholm Operators from the book "A course in Functinal Analysis " by J.B Conway. There I got stuck with the following problem. Let $A\in B(\mathcal H)$. Show that $A(\mathcal M)$ is closed for every closed subspace $\mathcal M$ of $\mathcal H$ iff $A$ has finite rank or $A$ is le...

Show that two nonzero vectors $\vec{v_1}$,$\vec{v_2}$ ∈ $\mathbb{R_3}$ are orthogonal if and only if their direction angles satisfy cos$α_1$ cos$α_2$ +cos$β_1$ cos$β_2$ +cos$γ_1$ cos$γ_2$ =0. Note: I tried to turn all of the cos$[angle]_2$ to sin[angle] and then convert all of the cos[angle] ter...

We have equation $$c=\frac{y-1}{xy}$$ Find the value of c using initial value condition $y(0)=1$ i.e. when $x=0 \implies y=1$

Let N be a positive integer And N=$(a^p)(b^q)$ Then prove that sum of the divisors or factor of N is =$$({a^p+1}-1)({b^q-1}-1)/(a-1)(b-1)$$

Let x and y be positive integers such that arctan(1/x)+arctan(1/y)<(pi/2). Show that there exists positive integers a1,a2,...,ak none of which equals x or y such that arctan(1/x)+arctan(1/y)+arctan(1/a1)+arctan(1/a2)+...+arctan(1/ak)=(pi/2). I simplified the given inequality to arctan((1/x+1/y...

Show that two nonzero vectors $\vec{v_1}$,$\vec{v_2}$ ∈ $\mathbb{R_3}$ are orthogonal if and only if their direction angles satisfy cos$α_1$ cos$α_2$ +cos$β_1$ cos$β_2$ +cos$γ_1$ cos$γ_2$ =0. Note: I tried to turn all of the cos$[angle]_2$ to sin[angle] and then convert all of the cos[angle] ter...

Is there any criterion (theorem or algorithm) to find out a given polynomial $f(x)$ is irreducible over the field $\mathbb{F}_2$?

I'm becoming confused by this. Say I have the following model: $$ y_t = c+\phi y_{t-1} +\epsilon_t \,, \epsilon_t|\Omega_{t-1} \tilde{} WN(0,\sigma_t^2 ) $$ $$ \sigma_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2 $$ $$ |\phi|<1 \,, \alpha_1<1, \alpha_0 \ge 0, \alpha_1>0 \,. $$ I know that an AR(1) i...

For a standard ellipse, a chord subtends an angle of 90 degrees with the centre (0,0). To find the locus of the foot of perpendicular to this chord from the centre of the ellipse, I wrote the equation of chord with parameters A, B: xcos(A+B/2)/a + ysin(A+B/2)/b = cos(A-B/2) Since an angle of 90 i...

Is there any (fast) algorithm that produces irreducible polynomials over $\mathbb{F}_2$?

Is thermal conductivity(thermal conductivity) of nano sized particles and bulk particles are same ? If no whats the difference and is there any relation between property and size ?

I have used the pythagorean identity to show that tan x = 4/3. Then I used the unit circle to see that the angle lies in the third and/or fourth quadrant because they are between pi and 2pi. This means that tan in those quadrants is either positive (in the third quadrant) or negative (in the fou...

I came across a definition of NP problems: Definition. A decision problem $X ∈ NP$, if there exists a polynomial time verifier $V$ such that For every yes instance $x ∈ X$, there exists a polysized proof y such that $V (x, y) = yes$. For every no instance $x ∈ X$, and for every proof y, we ha...

I am trying to study the series $$ \sum_{n=1}^{\infty} \frac{ (-1)^k (\log n)^k}{n^2} $$ Where $k \in \mathbb{N}$. What test would work in this case?

I was just thinking lately that how do we know that literally every number can be expressed in binary? And that too, with a unique representation?

We all have seen the basic tally mark sign: So anyone has a better notation, which includes more objects in one grouping, more informative and easily readable? (As compared to this)

I am going through a textbook on Matrix Algebra and I am not able to understand the difference between the two!

How do I calculate the time between frames on Android? I'm using this for the thread: public class GameThread extends Thread { private int FPS = 30; private double averageFPS; private SurfaceHolder surfaceHolder; private Game gamePanel; public boolean running; public ...

Which is the derivative of $f: \mathbb{R} \to \mathbb{R}, f(x)=x^c, c \in \mathbb{R}$ ? Is the same as in the case where $ c \in \mathbb{N}$

Chapter 4, exercise 2.3, problem 5 I can't understand the first part of the problem, the inequality seems holds for some functions at some points. Is the problem wrong?

Does exist a complete, correct but no recursive deductive axiom for second order logic?

I was informed in my last question that the Inverse function theorem: $$(f^{-1})^{\prime}(f(a))=\cfrac{1}{f^{\prime}(a)}\tag{I*}$$ was needed to show that $$\rho_x (x)=\rho_\alpha(\alpha)\left|\frac{\mathrm{d}x}{\mathrm{d}\alpha}\right|^{-1}\tag{A}$$ is the same formula as $$\rho_y(y)=\rh...

just some basic concept questions to clear up some definitions: Does Ax = 0 have a non-trivial solution? Why? My answer to this is no, because to have a rank 5 that means there is a pivot position in every column and therefore can't have a non trivial solution. Is this right? I feel as though ...

$$f(x) = \prod_{k=0}^{n} (1+kx)^{ (-1)^k \binom {n} {k} }$$ How to prove that Taylor expansion of this function at zero stars from $x^n$ (all lower terms are zero)?

Which type of function can I use to repeat one number? For example: number 2 repeat 3 times and get 222?

'There are n sweets in a bag. Six of the sweets are orange. The rest of the sweets are yellow. 'Hannah takes a sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. 'The probability that Hannah eats two orange sweets is 1/3. Show tha...

Let S :P3(F)--> P2(F) be diferentiation twice. Then the matrix M(S,(1,x,$x^2$,$x^3$),(1,x,$x^2$) of S with respect to the bases (1,x,$x^2$,$x^3$) and (1,x,$x^2$) of P3(F) and P2(F) respectively is f 'x(1)= 0.1+0.x+0.$x^2$+0.$x^3$ f 'x(x)=1.1+0.x+0.$x^2$+0.$x^3$ f ' x($x^2$)= 0.1+2.x+0.$x^2$+0.$...

Is f(x,y)=ln(x^2+y^2)/(1-x^2-y^2) uniformly continuous on {(x,y):x^2+y^2<1}: I know the definition of uniform continuity and Since f=g∘r is the composition of UC functions, it is also UC I need a hint, please help.

I know that b_{n} is an even function, due to the function being an even function, however I am struggling to compute a_{n}

$\int{x(ln(x))^3}{dx}$ Hi, I was hoping you could help me solve this with changing the variable. Thanks

I'm completely clueless on this one. I can easily calculate the limit using L'Hopital's rule, but proving that the series is converging to 0 is far more tricky. $$a_n= {n \over 2^n}$$ Any help?

Example 1 on this page: http://mathinsight.org/parametrized_curve_tangent_line_examples Why do they use $(t-1)$ in the last step ($l(t)=c(1)+(t−1)c′(t_0)$)? Why not just use $t$?

Why define $(\nabla^2F)(X,Y)=\nabla_X(\nabla_YF)-(\nabla_{\nabla_XY}F)$? I can't find the motivation of this define.I don't know the purpose of defining so.

hints on solving lim x->+inf of lan(x^2 - x +1)/lan(x^10 + x +1) would be appreciated. i tried multiplying each polynomial with the inverse of highest power but it didnt work out. no lhospital

I need to find the 4th root of -64. I have the answers but I don't know how to do it. This has to be done without the use of De Moivre's theorem as we haven't been taught it. Thank you very much for any and all help :)

Linear map from R2 to R4 1 1 2 2 3 3 4 4 Now I know the null-space is everything in the form (x,-x) so this is one dimensional. Apparently the range is also one dimensional however if I apply my matrix to say (x,y) I get (x+y,2(x+y),3(x+y),4(x+y)) transposed. The answer is that the range is...

I am currently looking through a proof in a book and am failing to understand one particular step. The step is this: $\sum_{i=0}^{\infty} x(\frac{d}{dx})x^n= x(\frac{d}{dx})\sum_{i=0}^{\infty}x^n$ How is it that you can treat the: "$x(\frac{d}{dx})$" term as a constant here? Does the derivative...

Prove that the fraction(21n+4)/(14n+3) is irreducible for every natural number n. I know the solution already. Just for sharing new idea.

Please, help me to find limit of this sequence: $\lim_{n\to \infty} \left(\frac{n^2 + n}{n^2 + n + 2}\right)^n$

Suppose you are given with the first three terms of the Maclaurin Series expansion of the function f(x)=1-e^-x Then show, with reasoning, that f(x)>x-x^2/2 for 0 Now, I began by deriving the expansion and it was like this f(x)=1+x/1-x^2/2+.... But I don't understand the logic behind the i...

I don't know why ? $$\int_{ln4}^{ln9} e^\frac{x}{2} dx = 2$$ i calculated 1.

Suppose $X$ is a Banach space and $f:X \rightarrow \mathbb{R}$ is a continuous function. Is it true that $$ \left| \lim_{t \rightarrow 0}{\dfrac{f(x+ty) - f(x)}{t}} \right| \leq \lim_{t \rightarrow 0}{ \dfrac{|f(x+ty) - f(x)|}{|t|}}$$ where $t \in \mathbb{R}, x,y \in X$ ? I think it is true wh...

So I started reading Conjecture and Proof by Miklos Laczkovich and one of the first proofs he provides is that of the irrationality of the square root of two. I am aware there are alternative proofs (one of which is geometric and another that uses the fundamental theorem of arithmetic) but I have...

How are these two equal? My teacher said it is obvious, am I missing something? $$\prod_{k=1}^{n}(\cos{kx}+i\sin{kx})=\cos{(1+2+...+n)x}+i\sin{(1+2+...+n)x}$$

Please, help to proof that: $$\lim_{n\to \infty} \left(\frac{(\sqrt[n]{a} + \sqrt[n]{b})^n}{2 ^ n}\right) = \sqrt{ab}$$

What is the distribution of the position of the maximum of a Brownian bridge?

I need a book or lecture notes for the course functional analysis, which I took this semester, the lecturer mentioned some book at the beginning and currently I'm also reading the book of Vitali Milman but some chapters, we treated in the lecture, are not contained in the book (the starred chapte...

The question is like: Prove that the equation ln(x)=1/x for x>0 has a unique solution and explain why. When it asks about the "unique solution" I try to find the exact value. Is it possible to find it, how would I solve it? Thanks.

Is the spaces $\mathcal{C}(\Omega),\quad \mathcal{C}^1(\Omega),\;\Omega \;\mbox{open set in} \;\mathbb{R}$ hilbert ? If so, how to prouve that ?

I have this question is this integral converge always $$\int_{0}^{1}x^\alpha|ln(x)|^\beta dx$$ where $$\alpha >0 ,, \beta >0$$ thanks

What do you obtain when you apply the projection P2,3,5 to the 5-tuple (a, b, e, d, e)?

How do I solve this second order PDE? $\frac{\partial^2 X}{\partial t^2}+X= \cos(\omega t)+\frac{1}{4}(1-exp(-\tau +F(\sigma))^\frac{-3}{2} [\sin(6t-6\psi)+3\sin(4t-4\psi)+3\sin(2t-2\psi)]$ $\omega$ is a constant. $F(\sigma)$ is just a general function of $\sigma$. $\tau$ and $\sigma$ are slow...

can anyone please explain how we can find number of distinct solutions of the following equation : (5/4)cos^2(2x) + cos^4(x) + sin^4(x) + cos^6(x) + sin^6(x) = 2 between the interval [0,2pi] thanks for your answers in advance

So the title says it all. I assume that polygons have straight line segments as their edges and that they have finite number of edges. The number $n$ of pieces is, of course, $n>1$, to avoid triviality that every polygon tiles itself.

A Frobenius group $G$ is a transitive permutation group which is not regular, but in which only the identity has more than one fixed point. Let $K=\{x\in G \vert x=1 \ or \ x \ has \ no \ fixed \ point \}$. Then $K$ is called the Frobenius kernel of $G$. It is known that if $G$ is finite, the...

Let ABC and DBC be two equilateral triangle on the same base BC,a point P is taken on the circle with centre D,radius BD. Show that PA,PB,PC are the sides of a right triangle.

Let S be a set with n elements and let a and b be distinct elements of S. How many relations are there on S such that: a) (a, b) ∈ S? b) (a, b) ∉ S? c) there are no ordered pairs in the relation that have “a” as their first element? d) there is at least on ordered pair in the relation that has “...

Let f,g  be non-zero entire functions having growth orders o(f),o(g)   respectively. (a) Find the growth of order of f+g  . (b) Find the growth of order of fg.

I am trying to solve a question where you are given two points, (2, 0) and (3, 5), asked to find the equation - seems simple enough, I get $y = 5x - 10$, which my plotting on a graph seems to confirm is correct. Then, given the point (-1, -1), it asks to find the equation of the perpendicular li...

I got a bit confused in the below question of probability. Please help me to clarify...thanks! May will read either one chapter of history book or one chapter of comics. If the number of misprints in a chapter of history book and comics are Poisson distributed with mean 2 and 5 respectively, t...

Without the 2, the answer is easily 3/5. But with 2, how do you visualize the new triangle in order to get the result?

I came across a binomial distribution question which states that X follows B(20, 0.75). Is it possible to find the median with only this much information?

$$u_t = c^2u_{xx} \mid (x,t) \in (0,L) \times (0, \infty)$$ $$u_x(0,t) = u_x(L,t) = 0 \mid t > 0$$ $$u(x,0) = 0 \mid x \in [0,L]$$ I tried energy method based on Pinchover and Rubinstein (*): $$\text{Let } E(t) = \frac{1}{2} \int_{0}^{L} u^2 dx$$ $$\to E'(t) = \frac{1}{2} \int_{0}^{L} 2uu_t ...

In the weak formulation of the Poisson equation $\nabla^2u = g$ with boundary conditions $u = \bar{u}$ on $\Gamma_e$ and $\frac{\partial{u}}{\partial{n}} = \bar{q}$ on $\Gamma_n$, why is the integration of the weighted residual expressed as $$I = \int_\Omega w(\nabla^2u-g) d\Omega - \int_{\Gamma...

Hell! I'm writing the prorgram on python that can approximate time series by sin waves. The program uses DFT to find sin waves, after that it chooses sin waves with biggest amplitudes. Here's my code: __author__ = 'FATVVS' import math # Wave - (amplitude,frequency,phase) # This class was cre...

So we all know the Monty Hall problem where there's you're on a gameshow and you have 3 doors, one of which has a car behind it and the other 2 a goat. I understand that if you pick one door, there is a 1/3 chance of winning and that the other two doors combined have a 2/3 chance of having a car...

I asked this earlier but since have been asked to be more specific. I'm trying to find the curve of a cantilever beam y(x), which is described by a fourth order Euler-Bernoulli beam equation: d^4y(x)/dx^4 = f(x)/EI. f(x) is the force per unit length (uniformly distributed load), E is Youngs m...

The definition of convergence states that the nth Partial Sum converges. Suppose the Sn (Partial Sum) converges to 0. Will that be considered as convergence or not?

How I solve this? (Jensen's inequality?)

Write down the first three terms in the binomial expansion of (1 – x)^15. By substituting x = 0.01, find an approximate value for 0.99^15 I understand the first part however how do I substitute x? The first three values are: -x^15 + 15x^14 - 105x^13

Given a null sequence $(a_n)_{n\in\mathbb{N}}$ and $a_n\in\mathbb{R}^{>0},\forall n\in\mathbb{N}$ I need to prove that $\forall\epsilon >0,\exists (a_{n_k})_{k\in\mathbb{N}}$ such that $(\sum_{k=1}^{\infty}a_{n_k})<\epsilon$. So basically, I think I need to construct a (geometric?) series whic...

In the below two images you see one with three lines leaving point A each 120 degrees apart and a umbilic torus made from twisting a triangle around a circle. I have tried to construct an umbilic torus with, rather than a triangle, the shape below. I am doing this with the intent of investigati...

$f(x)$ is a fifth degree polynomial. It is given that $f(x)+1$ is divisible by $(x-1)^3$ and $f(x)-1$ is divisible by $(x+1)^3$. Find $f(x)$.

I have a rather trivial question, when the book states: "Assume that the function of n variables f is of class $\mathsf{C^2} $" I know that $\mathsf{C}$ is the complex set, but how about squared? What can I conclude from that. Sorry for my silly question.

If $f(x)$ is a real function such that $f'(x) → 3$ as $x → 0$, does that mean $f'(0)$ exists? I ask this because of problem 9 on page 115 of Rudin's Principles of Mathematical Analysis. The actual question is slightly different: Let $f$ be a continuous real function on $ℝ^1$, of which it is k...

Today I study about oscillatory integral: let $q$ be a nondegenerate real quadratic form on $\mathbb{R}^n$, $a\in A^m$ and $\psi\in C_0^\infty$ such that $\psi=1$ if $|x|\leq 1$ and $\psi=0$ if $|x|\geq 2$, so we have $$\lim_{n\to \infty} \int{e^{i q(x)} a(x) \psi(2^{-n} x)}dx$$ exist, any by def...

At my multivariable calculus class we gave this definition for the limit of a function: Definition: Let $ \mathbb{R}^n \supset A $ be a open set , let $f:A \to\mathbb{R}^m $ be a function, let ${\bf x_0}$ be a point of $A$ and ${\bf P}$ a point of $\mathbb{R}^m$. To say that $f$ ...

If X follows a poisson distribution where P(X = 1) = 0.4, P(X = 2) = 0.6, what is the value of P(X = 0)? How will I solve this question?

How to prove that, for every real $t$, one has $$\frac{1}{3}e^{2t} + \frac{2}{3}e^{-t}\leq e^{2t^2}?$$

Here is the question i'm particularly interested in : Let $f$ be a modular form, suppose we know the $a_p(f)$ for all but finitely many prime $p$. Is this enough to know the modular forms i.e. to know all the $a_n$ ? If it is true can you give a proof or reference ? I'm also interested (but it'...

I'm confused about the correct method for finding Taylor/Laurent series for complex functions. For example, I'm given this function $$ f(z) = \frac{z^2-1}{(z+2)(z+3)} $$ and I need to find the Taylor/Laurent series centered at $0$ for all $z \in \mathbb{C}, |z| \neq 2, 3$. From my understanding, ...

If a^2 + b^2 not equal to c^2 +d^2 and if a not equal to b and c not equal to d and if a b c d are greater than or equal to 0 AND NOW IF L GREATER THAN OR EQUAL TO 1 THEN TO PROVE (a+L)^2 + (b +L)^2 not equal to (c + L)^2 + (d +L)^2

If $f(x)$ is continuous over [0,1] and $f$ only takes rational values, if $f( {1 \over 2})={1\over2}$, how do I prove $f(x)={1\over2}$ everywhere on [0,1]?

Given $(a_n)_{N\in\mathbb{N}}$ such that $a_n \geq 0$, and $a_n\to a \geq 0$. how can i prove that $\sqrt[n]{b_n}\to 1$ when $n \to \infty$?

I'm currently studying Calculus from Stewart's book, and for the The Fundamental Theorem of Calculus Pt. 1, he defined a function $g(x) = \int_0^x f(t) dt$ which represented the area under $f(x)$ from $0$ up to $x$ and proved that $g(x)$ is the antiderivative of $f(x)$ and, in this case, if I plu...

Here user Mario Corneiro formulated statement which I find really interesting and useful and I proved it and I would like to know is my proof correct? If sequence $\{a_n\}_{n=1}^{\infty}$ implies the following conditions: 1) $a_1<a_2<\dots<a_{n}<\dots$ 2) $a_n\to \infty$ as $n\to...

If I have a three-dim. vector $$ (x_1,x_2,x_3)\in\mathbb{R}^3, $$ what then is its projection onto the $(x_1,x_3)$-space? I am not sure.

I'm trying to show that {1, x, x^2,...,x^n} is a linearly independent set (in P_n) without being circular; so without using either the Fundamental Theorem of Algebra or the fact that this is the standard basis for P_n. I understand if a,b,c are all distinct positive integers and t1, t2 arbitrar...

For #1 I am able to prove it but I am not sure what kind of example can be used and how I would prove that it would fail. For #2, I am not sure how to go about it. 1) Let {$A_n$} be an infinite sequence of measurable sets increasing to A. Prove that m(A)=$\lim_{n\to \infty}$ m($A_n$). Give an ...

Given $\lim_{n\to\infty} a_n = a \neq0$. Need to prove that $\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}=1$. So i know that if $\lim_{n\to\infty} a_n = a$ so $\lim_{n\to\infty} a_{n+1} = a$ for the getting $1$, but how do i prove it?

i am confused on how to go about solving for this. perhaps i am not understanding how to solve congruence problems. any help will be appreciated.

Please help to prove this, using mathematical induction: $$(x_1 + x_2 + ... +x_n)(\frac{1}{x_1} + \frac1{x_2} + ... \frac{1}{x_n}) \ge n^2, $$ $$ x_i > 0, $$ $$i = 1, 2, ... n$$

$\sum_{k=1}^n k2^k = 2^{n+1}(n-1)+2$ How can I justify that using difference and shift operators?

Find a bijection form Z into (0,1) ^ Q can anyone help me with this

For some exercise I need to compute the generators of $E(\mathbb{Q})/2E(\mathbb{Q})$, where $E: y^2 = x(x^2 + 3x + 5)$ I did this by the algorithm from Cassels book 'Lectures on Elliptic Curves' and got the answer $< (0,0), (1,3) >$ which seems reasonable. A friend of mine did the same computati...

$$\lim _{ x\rightarrow \infty }{ \frac { x+8 }{ x+3 } =1 } $$ Proof: Let $\epsilon > 0$ $$\left| \frac { x+8 }{ x+3 } -1 \right| <\epsilon $$ $$\Longrightarrow \left| \frac { x+8 }{ x+3 } -\frac { x+3 }{ x+3 } \right| <\epsilon $$ $$\Longrightarrow \left| \frac { 5 }{ x+3 } \right| <\epsil...

Consider IVP = ln(t) dy/dt +2y = tan(t) and y(1.5) = -2 . Find the longest open interval where the unique solution to IVP is certain to exist.Can you help me out ?

Let $R$ be a ring without unity (not necessarily commutative). Suppose that $char R=n$. Does there must exists an element $r\in R$ with $|r|=n$?

this is what i got but im not sure if it is correct. 5x≡3mod3 3*5x≡3*mod3 x≡15x≡6mod3 x≡0mod3 x=3k k belongs to integers

Here's an interesting differential equation: $$f''(x) = \frac{x(f'(x))^2+f(x)^2}{f(x+1)}$$ It was created by a someone I know, and I am wondering if anyone can solve it. I mainly want to see what other people do on it before I reveal my own work.

What does $\Delta^{-1}$ mean? I have seen it in a question such as "justify that $\Delta^{-1}k^{(n)} = {k^{n+1}\over{n+1}}$". Thanks for your help.

The original question is: given A={0,1} B = {{0,1},{1}} What is the result of P(B{A})

If I have a nonsingular matrix M, How can I prove that M*M is positive definite? M* is the transpose matrix of M. Thank You!!

i have 3x mod 10=1 and i split to 3x mod 5 = 1 3x mod 2 = 1 now how to solve that?

I am trying to prove that {1, x, x^2, ..., x^n} is a linearly independent set with no hand waving and without using the fact that it is a basis for P_n or the Fundamental Theorem of Algebra. Suppose (a_0) + (a_1)x + ... + (a_n)x^n = 0. How do I show a_i = 0 for 1 <= i <= n?

$\frac{2}{\pi}\int_{0}^{\pi}\cos(\frac{x}{2}).\cos(nx) dx$ please show step by step solution

If $p$ is prime and $\gcd(m,p) = 1$ show that $\gcd(m,p^k) = 1$ where $k\geq1 $. I think I have come up with a solution: Suppose $m$ has prime factorization $m = p_1^{a_1}...p_n^{a_n}$ where $p_i$ is prime. Since $\gcd(m,p) = 1$ none of $p_i = p$. Now if $\gcd(m,p^k) \neq 1$ then the $\gcd$ mus...

I'm required to generate random number which follows Shadowed Rician distribution as enter image description here

I have a field of view on something I am working on of 41.5933 (is correct) however the equation that was supplied to me that was supposedly used to calculate it for some reason does not add up to the 41.5933 I get 4.07958271 when I try it in unless I am not doing it correct. Any help to determin...

This problem has been asked and discussed in following posts: Logic problem: Identifying poisoned wines out of a sample, minimizing test subjects with constraints, Finding 2 poisoned bottles of wine out of a 1000 Identifying poisoned wines I know that there are better optimized solutions to...

I have some doubts with this problem because The concept of Limit is confusing for me: "If one category $J$ is a disjoint union (coproduct) $\coprod_{k}F_{k}$ of categories $J_{k}$ for index $k$ in some set $K$, with $I_{k}:J_{k}\rightarrow J$ the injection of the coproduct, then each functor $...

Hi I am stuck on a problem about dual spaces which I've spent hours on but I just cant grasp the idea of functions of functions- The problem in mind uses the vector space V of polynomials of degree $\leq 2$ over the field $\mathbb{R}$ and I have been given 3 linear maps from V to R- e.g a map $\p...

As tangent is a line passing through a curve from one point secant from two points so is sine also something like this?

Consider a one-dimensional classification problem with X = R and Y = {-1, +1}: Visualize the marginal distribution p(x) and the conditional distributions p(y = -􀀀1 | x) and p(y = +1 | x). Guess from the visualization of p(y = 􀀀-1 | x) and p(y = +1 | x) what the Bayes-optimal classifier is li...

How can I use base case of n=1 to prove the following? $$\forall z \in Z: \forall n \in N: n \gt 0 \Rightarrow \lceil z/n \rceil = \lfloor (z+n-1)/n \rfloor.$$ By assuming: $$\forall l \in Z: \forall m \in N: m \gt 0 \Rightarrow [\exists q \in Z: \exists n \in N: l = qm+n \wedge n < m]$$ My wor...

Let $X$ be a Hausdorff topological space such that every closed subset has finitely many connected component. How can I verify that $X$ is finite?

Show $\int_0^\infty 2y^2e^{-y^2} dy = .886$ $2\int_0^\infty y^2e^{-y^2} dy$ Then by parts; $f=y^2$ $dg=e^{-y^2}$ $df=2y$ $g=-2ye^{-y^2}$ $2\int_0^\infty y^2e^{-y^2} dy$ = $-2y^3e^{-y^2} -\int_0^\infty -4y^2e^{-y^2}$=$-2y^3e^{-y^2} +4\int_0^\infty y^2e^{-y^2}$ equivalently; $-2\int_...

Given a matrix size $n$, I want to produce a matrix $A$ with $\rho(A)<1$, which is not diagonal. Is there a way to do so? Thanks.

I'm having trouble proving the following statement: "There exists an elementary matrix E_1 such that (E_1)^2 = I" I'm thinking about how the inverse of E_1 is qua; to E_1 (E_1^-1 = E_1) but I'm not sure how to show the product of it, if that is even the right step towards the proof. Could some...

(A) Show that dy/dx=y/2y-x, (B) Find all points (x,y) on the curve where the line tangent has a slope of 1/2, (C) Show that there are no points (x,y) on the curve where the line tangent to the curve is horizontal, (D) Let x and y be functions of time t that are related by the equation y^2=2+xy. A...

Do you know if there is some kind of solutions manual to Rosenlicht's "Introduction to Analysis"? I would be very grateful for a link (if there is one). Thank you.

The text is from courant calculus volume 1. First few pages. So obviously I don't understand this part. It says we divide each interval on the number line by q parts. Thus each subdivision length will be 1/q. I understand this. Then it says every point P is p/q, or lies between p/q and ...

Among 5 people, what is the probability that exactly 2 of them are born in the same month.

If the events A and B are independent, and the events A and C are independent, is it true that the events A and B − C are also independent? Prove, or give a counterexample.

Let T:P3→P3 be the linear transformation such that T(−2x^2)= −2x^2 − 2x, T(0.5x + 2)=3x^2 + 4x−2, and T(2x^2 − 1)= 2x + 1. Find T(1), T(x), T(x2), and T(ax^2 + bx + c), where a, b, and c are arbitrary real numbers. I understand how to find T(x^2) where you just divide the given T(-2x^2) by -2 t...

Is there an easy way to graph tuppers self referential formula between $$k<y<k+17 $$for a value of K? Like any sites or programs? or would i have to do it the long way of dividing by 17, turn it into binary, and draw it out myself?

find the order of $\overline {15}+ <\overline 4>$ in $\Bbb Z_{45}/<\overline 4>$ how can I start? is it $\Bbb Z_{45}/<\overline 4>\cong \Bbb Z$

I have the differential equation dR(t)/dt=S(t)R(t), I know that S is a skew symmetric matrix and that R(0) is a rotation matrix. How do I prove that R is then a rotation matrix at any time t?

Let X be an rv with pmf $p_k= (nCk) p^k (1-p)^(n-k)$ [binomial]. If F is the corresponding df, find the distribution of F(X). I know for certain that I need to get the sum of k binomial rv's, however I'm having a hard time in doing it. I read from books that it will be needing of mathematical in...

Let T be a linear operator on a n-dimensional vector space V.Suppose that T commutes with every diagonalizable linear operator on V .Prove that T is a scalar multiple of identity operator.

Let $p:X\to B$ be a covering and $g:I\to B$ be a path on $B$ with $p(x_0)=g(0)=b_0$. Then there exists unique lifting path $f:I\to X$ s.t. $g=pf$ and $f(0)=x_0$. I didn't get fully the proof from the book, so I've tried to prove it by myself, but I'm not sure about its correctness. Proof: Since ...