Abbott: understanding analysis I'm going through exercises in my book but I can't figure this one out.

How to show that $\mathbb{Q}[\pi^{2^n}]\subsetneqq \mathbb{Q}[\pi^{2^{n+1}}]$? I easly prove that $\mathbb{Q}[\pi^{2^n}]\subset \mathbb{Q}[\pi^{2^{n+1}}]$, now I want to prove that $\pi^{2^n}\not\in\mathbb{Q}[\pi^{2^{n+1}}]$ but I can't do that in an elegant way (my proof need a lot of calculati...

$ \sum_{n=1}^{\infty} -\frac{1}{2^{n}} + \frac{1}{2^{n+1}+1} + \frac{1}{2^{n+1}+2} $ No idea which convergence test use for this one.

Consider a set of cubes $F$, such that each corner $(x,y)$ of any given cube of $F$ satisfies $0\leq x,y \leq n$, and each cube has a corner with coordinates $(0,0)$. What is the maximum number of cubes in $F$ so that no cube of $F$ contains another cube of $F$? So far I have managed to prove t...

Could someone please explain to me the definition of a prime ideal and a proper ideal. I honestly do not understand this concept. If possible please explain your version of the definition in the most brain dead way possible. Imagine you are trying to explain this to a little kid, how you would ...

I have a doubt at the definition of compact spaces. So if you have a topological space X, then X is compact if every open cover of X has a finite subcover. In other words, if X is the union of a family of open sets, there is a finite subfamily whose union is X. So, if {A_i} is a family of open s...

I need a paper which explains all the process to prove that it´s enough to riffle cards 7 times. ¿Someone has a link? If you have it in spanish i will be too greatful.

Please could somebody verify this: $T$ is diagonalizable $\iff$ there exists a basis of $V$ (the vector space ) consisting of eigenvectors of $T$ $\iff$ the algebraic multiplicity of each eigenvalue is equal to its geometric multiplicity. If even one geometric multiplicity is strictly less than ...

Consider the following vectors in R 3 v1 = 1 √ 3   1 1 −1   , v2 = 1 √ 2   1 −1 0   , v3 = 1 √ 6   1 1 2   , (a) Show that they form a basis of R 3 . (hint: compute the inner products v t i · vj ). (b) Work out the coordinates of a vector x =   x1 x2 x3   in the basis {v1 , . . . ...

In quantum mechanics we know that if $q$ corresponds to a complete set of parameters characterizing a quantum system, then the state vectors $|q\rangle$ satisfy the following identity: $$\int |q\rangle\langle q| d\tau_q = \Bbb 1$$ where $|q\rangle\langle q|$ is a projection operator and $\Bbb 1$ ...

his is not Bayes theorem, is it? How to prove this formula?Sorry for this naive question. I'm a first year math student.

$$\sum_{n=0}^{n=\infty}\frac{a^n}{(n-1)!}$$ Assuming that $a>0$

$$\sum_{n=1}^{\infty}{x}a^{x-1}$$ where a is between 0 and 1. can anyone explain me how it can be evaluated to be $${1}\over{1-a}$$

Suppose you have five books in your book bag. Three are novels, one is biography, and one is a poetry book. Today you grab one book out without looking, and return it later. Tomorrow, you do the same thing. What is the probability that you grab a novel both days. I thought it would be $$ 3 / 5 $...

Im working on some problems in Arthur Engel's problem solving strategies book and one of the problems is: On page 132: the question is: Now here is what the back of the book says: So I am trying to find the other solution based on the last sentence. My guess is it would involve modular ari...

Can you find some real numbers $x_1,x_2,...,x_n$ that follow $\sum_{k=1}^n{x_i^q}=q$ for all positive integers $q > 1$? Take $n$ to be some positive integer.

I'm trying to understand one-forms in the context of general relativity. Lee (Introduction to Smooth Manifolds) says that at a point $p$ and with a vector field $X$ we define a covector field $df$, called the differential of $f$, by$$df_{p}\left(X_{p}\right)=X_{p}f.$$ My question is, is th...

I need to find the reduced row echelon form of this matrix -8 6 -2; 6 -7 4; -2 5 -3; I performed the following row operations: r1+r2= -2 -2 2 r2+2(r3)= 2 1 -2 (-1/2)r1= 1 1 -1 r2+r3= 0 5 -5 r3+r2=0 5 -5 r3-r2= 0 0 0 (1/5)r2= 0 1 -1 r1-r2= 1 0 0 ...

let $x_1, x_2, y_1$ be uncorrelated data, $$X = x_1 + px_2$$ $$Y = x_1+ qX$$ then what is the correlation between X and Y? I am guessing its q?

Why do simply connected solvable analytic groups have no nontrivial compact subgroups? I'll appreciate any help on this question.

Derive a generating function for the sequence $a_n = n^2$ I know the power series for $n^2$ is $${x(x+1)}\over{(1-x)^3}$$ However I am struggling to connect the two with the proof. Any help would be greatly appreciated thanks!

How do I factor 2 + 3$\sqrt{2}$ into primes in $Z[\sqrt{2}]$? I know that primes are irreducible in $Z[\sqrt{2}]$ and that units are of the form $\pm(1\pm\sqrt{2})^n$. How are primes and units related, if at all? I understand that all primes of $Z[\sqrt{2}]$ are obtained by factoring rational pr...

I need to find the directional derivative of $f(x,y,z)=xy+xz+yz$ at $P(1,2,3)$ in the direction of $\overrightarrow{v}=\langle 2,1,-1 \rangle$ I think I started this incorrectly and would greatly appreciate any hints! $$z=\frac{-xy-xz}{y}$$ $$f_x=0$$ $$f_y=\frac{xy+xz}{y^2}$$ When I got to ...

I fully understand both the Lemma and proof of Gauss's Lemma on number theory which states $$\bigg(\frac{a}{p}\bigg) \equiv (-1)^n, $$ where $(\frac{a}{p})$ is the Legendre symbol https://proofwiki.org/wiki/Gauss%27s_Lemma_(Number_Theory) My question here is, what was the reason behind this l...

let V={(x_1, x_2...x_n...)}|x_i are real numbers } under normal operations and T((x_1, x_2...x_n...))=(x_1+x_2+,x_2+x_3...(x_n)+(x_n+1)...) Find T((x_1, x_2...x_n...))=lamba((x_1, x_2...x_n...)) so so i start like (x_1+x_2+,x_2+x_3...(x_n)+(x_n+1)...)=(((lamba)x_1, (lamba)x_2...(lamba)x_n......

Find the rectangle of maximum area that can be inscribed in a right triangle with legs of length a=43 and b=44 if the sides of the rectangle x,y are parallel to the legs of the triangle, as in the figure. Okay, I know that I should first draw out an image of everything so that I can visual what ...

Let $f: X \rightarrow Y$ be a continuous map of topological spaces, and $F$ a sheaf of rings on $X$. The direct image sheaf $f_{\ast}F$ on $Y$ is given by the formula $V \mapsto F(f^{-1}V)$. If $x \in X$, is it true in general that $F_x \cong (f_{\ast}F)_{f(x)}$? We have $$(f_{\ast}F)_{f(x)} =...

Let (n1,m1),(n2,m2),. . .,(n9,m9) be integer lattice points in the plane (ie. ni and mi are integers). Show that the midpoint of the line joining some pair of points is also an integer lattice point. I think that I need to use the pigeonhole principle but I'm not sure how to get to that point.

The Question: Let $X_{1}, X_{2}, ..., X_{9} $ be a random sample of size 9 from a normal distribution $N(2,4)$. Let $Y_{1}, Y_{2} , Y_{3}, Y_{4}$ be an independent random sample from a normal distribution $N(1,1)$. Let $\bar{X}$ and $\bar{Y}$ be their sample means respectively. Let's assume that ...

T/F: The set: $$\{(x,y): sin(x^{2012} +y^3) + x^2 + y^4 ≤ 1$$ is a compact set in $R^2$. I think it is false since I don't believe the set is closed nor bounded? Is that correct?

I am confused as to how to turn a fraction into a sum using geometric series. I need to find Laurent series I have $\frac{z+2}{(z-1)(z-4)}=\frac{2}{z-4}+\frac{-1}{z-1}$ I do not know how I turn the last 2 fractions into geometric series and write them as sum. Can someone please help me?

I am trying to prove - Let Fn be the nth Fibonacci number. Then (Fn+1)2 - (Fn+1Fn) - (Fn)2 = (-1)n - I am not sure where to start with this.

I got this question for a finals review: The answer says it has no inflection point, but I got the second derivative to be y''=2-2sinx And when y''=0, x=2kπ+π/2 So how can this be? Thanks. P.S. The homework tag seems to be gone for some reason, so I can't add the tag.

The Thomson's Lamp paradox: A mad scientist owns a desk lamp. It begins in the toggled on position. The scientist toggles the lamp off after one minute, then on after another half-minute. After a quarter-minute the lamp is toggled off, then the scientist waits an eighth-minute and turns the lamp...

I have a problem with the following integral $$\int_{-\infty}^{\infty}\frac {x\sin x}{x^4+1}$$ Can someone please help me with the way the solution goes? I would highly appreciate it Thanks in advance!

Im asked to prove the inequality: $0\leq a<b$ and $x>0$ $$ a^x(b-a)<{b^{x+1}-a^{x+1}\over{x+1}}<b^x(b-a) $$ So far I have seen that obviously: $$a^x(b-a)<b^x(b-a)$$ and that $$b^{x+1}-a^{x+1} = (b-a)(b^x+b^{x-1}a+...+ba^{x-1}+a^x) > a^x(b-a)$$ $$$$ I was thinking it may have to do with $a<{a+b\...

i thought the unit vector of projection of Z on v perp is the vector.but cant show it.

It seems that the following is the basic property of the supremum of a function: $$\sup_{x \in I}f(x) - \inf_{y \in I}f(y) \geq \sup_{x, y \in I}f(x) - f(y)$$ I think This property is so obvious that the textbook just uses this without mentioning why. But I want to know if this can be proven ma...

My professor says that the answer is 1/3 but he never explained to us how he got that. Just hoping for some insight! Thank you!

http://i.stack.imgur.com/a1gnS.png So I used the pythagorean theorem to find the missing leg first. a^2 + b^2 = c^2 a^2 + 6^2 = 10^2 a^2 + 36 = 100 100 - 36 = 64 √64 = 8 Now I'm just lost from here. It was just recently that I started studying tan, cos, sin etc... so any help is appreciated.

I am curious if anyone recognizes the following function as belonging to a particular class of functions (like quadratics) or belonging to any particular academic laws (economic, biological, etc). It is not out of a text book or class, but I believe it could describe personal living expenses as a...

If $f:X\rightarrow\mathbb{R}$ is a semicontinuous function, then its points of discontinuity lie in the union of countably many closed nowhere dense sets. Thanks a lot.

Need help on this proof: Let p be an odd prime. Suppose that p divides a − b and a + b. Prove that p divides a.

Let A be an invertible $n \times n$ matrix with entries from a field F. Prove that if rank$\begin{pmatrix} A &B\\ C&D\end{pmatrix}$=rank(A), then $D=CA^{-1}B$ .

Suppose that Y is a binomial random variable based on n trials with success probability p and consider Y* = n - Y. a. Argue that for y* = 0, 1, ..., n P(Y* = y*) = P(n - Y = y*) = P(Y = n - y*) b. Use the result of part a to show that part b of the exercise Hi. Someone can give me a hi...

Need help in solving this: the prime factorization of all integers n such that φ(n) is a power of 2 Where φ(n) is the Euler's phi function

I have gathered in my reading that there is a difference between the behavior of the wave equation depending on whether the initial position or initial velocity is zero, but my impression is not precise enough to be very clear to me. If we have the one dimensional wave equation $u_{tt} = u_{xx}...

Let $X$ be a non empty set. Let $M$ the set of all sequences $(x_{n})$ of elements of $X$. For $x=(x_{n})$ and $y=(y_{n})$ in $M$, let $k(x,y)$ the smallest integer $n$ such that $x_{n}\neq y_{n}$. Let $d:M\times M\to \mathbb{R}$ $d(x,y)=\dfrac{1}{k(x,y)}$ if $x\neq y$ and $d(x,x)=0$. Show that $...

$f(x)=\left\{ \begin{array}{ll} x & \mbox{if $x \in \mathbb{Q}$};\\ x^2 & \mbox{if $x \in \mathbb{Q}^c$}.\end{array} \right.$ This question asks to find function $g$ and function $h$, which are continuous on [0,2] such that if P is a partition of [0,2] that includes the point 1, then $L(f,P)=L(g...

I want to show that two surfaces are not homeomorphic, say $S^2$ and $\mathbb{R}P^2$, for example. Is it enough to show that their Euler characteristis differ to conclude this?

I'm having difficulties understanding the following proof: I'm used to seeing different notation (used in Erwin Kreyszig's functional analysis) where the canonical map is $C: X \to X''$ with $x \mapsto g_x$. And $g_x: X' \to \mathbb{K}$ where $f \mapsto f(x)$ In the image above having troubl...

I want to prove that: The $\mathfrak{g}$-module $V$ is irreducible if and only if $V\subseteq \mathfrak{g}(v)$ for each nonzero $v\in V$. Since $\mathfrak{g}(v)$ for any non-zero $v\in V$ generates a submodule of $V$, we want these to not be proper. Proof of $\iff$: $(\,\,\Longleftarrow \...

Please help with my exam review question- thank you!! When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it suppresses noise from the audience. Suppose the microphone is placed 2 m from the front on the stage (as in the figure) and the...

I am studying Laplace transforms right now and got stuck at this step that involves a weird partial fraction decomposition. It looks like the instructor skipped a bunch of steps and assigned numerators to a bunch of the fractions without assigning them dummy variables. Any idea how he got to this...

Let p be a prime number, and d: $\mathbb{Z} × \mathbb{Z}$ → [0, +∞) be a function defined by $d_p(x, y) = p^{−max(m∈N : p^m|x−y)}$ . Prove that $d_p$ is a metric on $\mathbb{Z}$ and that $d_p$(x, y) ≤ $max(d_p(x, z), d_p(z, y))$ for every x, y, z ∈ Z What I tried: I tried to use triangle inequal...

Let us say I have a function $f(z)$ please can you give me the steps to show if this has a branch point at infinity and how to determine its order?

Define the $p$-adic logarithm$$\log_p(1 + x) = \sum_{i =1}^\infty (-1)^{i-1}x^i/i.$$How do I see that if $p > 2$ and $|x|_p < 1$, then $|\log_p(1 + x)|_p = |x|_p$?

I have this math problem I'm kind of stuck on. You intercept the message 27284682555982882069237 which was encrypted using a public modulus of 124137798108168664109413 and an encryption exponent 257. The modulus is now too large to be factored by testing successive candidate divisors...

Can someone help me in a step-by-step derivation for the Fourier Transform of a line ? It appears to be simple but still cannot figure out. I know what is the end result but I am unable to figure out the intermediate steps. I tried to use the Shifting and the Similarity theorems but I am not gett...

Let $S$ be the set of 3x3 matrices $A$ such that $A^tA$ = $\begin{bmatrix}1&0&0\\0&0&0\\0&0&0\end{bmatrix}$. How to prove that $A$ is of rank one and $S$ contain a nilpotent matrix? First part : since$~~$ $rank(A) = rank(A^t)= rank(AA^t) =rank(A^tA)$, first part i can solve. How to prove the...

Leaky bucket problem $$\beta = 16 KB$$ $$packet\ size = 1KB$$ $$\rho = 8\ packets/sec$$ What is maximum Burst Size? Taken from here https://www.youtube.com/watch?v=4eMrQXU0DdA&list=PLpherdrLyny-zJw95jcE-uJkcsIAG1MEn&index=103 Video provides example but never solution... I did not understand h...

Prove that for every real numbers $a,b$ and $c$ we have $$(a+b+c)^5\ge 81(a^2+b^2+c^2)abc.$$

I'm a beginner in smooth manifold theory and was just wondering if this was easy to prove.

Consider the following game. P has a fair $6$ sided die and continuously rolls it while keeping track of the sum, $S$, of the top faces. P can bail anytime and leave with $S$ number of dollars. However, if $S$ is ever a perfect square, the game ends and $P$ gets nothing. When is it m...

Where I am doing wrong? if any one can tell!!! I have done this equation as $$y ={x\over 1+(x-x^3+x^7)^3} = x [1+(x-x^3+x^7)^3]^{-1}$$ Using binomial expansion, $$y= x[ 1-(x-x^3+x^7)^3+...]$$ $$y= x[ 1-(x^3+x^9+x^{21}-3x^5+3x^9+3x^7+3x^{13}+3x^{15}-3x^{17})+...] $$ $$y= x[ 1-x^3-x^9-x^{21}+3x^5-...

Here $k$ and $k_{1}$ are constant $\frac{\partial }{\partial t}f\left(s ,t\right)=k{f}^{2}\left(s,t\right)-{k}_{1}f\left(s,t\right)$

Let D be the region in the xy-plane bounded on the left by the line x=2 and on the right by the circle x^2 + y^2 = 16. Evaluate \iint (x^2 + y^2)^(-3/2)dA

$f$ and $g$ are two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing .Then which of the following is true $?$ $A)$ If $g$ is continuous then so is $f\circ g$ counterexample : $g(x)=x$ or $g(x)=1$ and $f(x)={1\over {1-x}}$ $B)$ If $f$ is continuous then so is $f\circ g$ ...

Given y=(a-x)(x+√(x^2+b^2). How to see whether it is max or min and find that value also ?

drawing In the drawing, $ABCDS$ is a pyramid whose base is a parallelogram. $O$ is the intersection of the parallelogram diagonals. The following holds: $\overrightarrow{SF} = k \cdot \overrightarrow{SD}$ $\overrightarrow{SE} = t \cdot \overrightarrow{SO}$ The problem is to express $t$ using...

How can I prove that $R$ is complete using the construction with Cauchy sequences ?

A value x is said to be an integer when floor(x) = x, where x ∈ ℝ or more equivalently floor(x)/x = 1 Since 0 ∈ ℝ, and we assume that 0 is either an integer or non-integer, from the same equation to determine whether or not 0 is an integer we find: floor(0) = 0 Since floor(0) takes the highes...

So I have the initial value problems below. From a previous part, I have already confirmed that v(t) is a solution to the the problem above it. Now I am left to write a MATLAB code that plots log(h) vs log(e(h)) and that verifies that the Euler Method's approximation is first order by computing t...

Given $x_n \sim N(0, \frac{1}{n})$, does $x_n$ almost sure converge or converge in probability or converge in distribution? How to prove it

Is it true that if in $$AX=0$ where $A$ is $n\times n$ matrix over field $\Bbb K$ and $X$ is a length $n$ vector of variables if $rank(A)=n-1$ we will have an unique solution?

Evaluate $\lim\limits_{x\to\infty} \dfrac{[x]+x}{x}$, where $[x]$ denotes greatest integer $\leq x$. We have, if $[x]=k$, then $k\leq x<k+1$. But I don,t know how to evaluate this. Any hints?

Sample $n^2$ integers $a_{11},\dots,a_{nn}$ in $[-d,d]$ uniformly. What is the probability that the resulting matrix $[a_{ij}]$ has rank $r$?

I'm looking for a diagram $D$ (as simple as possible) in the category of semi-simplicial sets (i.e $sSet$ with only monos) such that $R(\text{lim}\,D) \ncong \text{lim}\,R(D)$, where $R$ is the geometric realization on semi-simplicial sets. Moreover, in $sSet$ the equivalence should be true, i.e....

I think I accidentally found a proof of the famous theorem that the sum of the angles of a triangle add up to $180 ^\circ$, but am not sure if it is correct. Here it is: It can be proved that the median in a right triangle is equal to half the hypotenuse. Consider an aritrary triangle $QRS$ , $...

The integration of |e^-(2+j)t |^2 from zero to infinity is 1/4 when I seperate above as |e^-2t|^2 * |e^-2jt|^2 and integrate. |e^-2jt| was taken as 1. But when I integrate the problem taking |e^-(2+j)t|^2 = e^-(4+2j)t the answer for the integral is 2-j /10. Can some one tell what is the correct...

Show that the maximum number of edges in a simple graph with n vertices is {n(n-1)\over 2} ?

I'm new to this site, Can someone explain if and if not Image to the question

Let $f(x)=\frac{x}{1+x}$ and let $g(x)=\frac{rx}{1-x}$ Let $S$ be the set of all real numbers $r$ such that $f(g(x))=g(f(x))$ for infinitely many real number $x$.Find the number of elements in set $S.$ I found out $f(g(x))=\frac{rx}{1-x+rx}$ and $g(f(x))=rx$ But i do not know how to solve it f...

I have this doubt that I cannot solve. ∫|x−1|^a/|x2−y2|^b dxdy in D= circle x^2+y^2<1 If I use polar coordinates I cannot solve anything. Could you help me? Thank you

Let $X\sim G(p)$ Why $\mathbb{P}(X>n)=q^n$ in geomtric distribution, I know that $\mathbb{P}(X=n)=(1-p)^{n-1}p$

How many ways are there to arrange the letters in "engine" so that no letter appears next to itself? Initially, I know that there are 6! possible arrangements of the letters. But we have to divide the 6! by 4 because there are 4 duplicate letters. There are 2 e's and 2 n's, so I must subtract th...

let $f(n)$ be the number if ways to lay down tiles in a formation of size 2 x n using tiles of size: $$ \begin{matrix} 1 \\ 1 \\ \end{matrix} $$ and tiles of size: $$ \begin{matrix} 1 & 1 & 1 \\ 1 & \\ \end{matrix} $$ Turning and ...

I've spent many hours on this and I just can't understand how to do this. Could you please go through this with me? I have a test, and I really need to understand how to do these types of problems. Thank you so much!! "Find vectors x and y with sum-norm of x = 1, max-norm of y = 1, such that 1-n...

I'm designing a tool for students and right now im working with coordinates. I have the plan to shift the north pole in my code to Munich and I would like to do this by working with a spherical triangle and the angle at one point. As result I would like to have a formula which tells me how to cal...

Hello I had this question below on a midterm on Calculus I had proved the series diverge by testing the Absolute value of the series. By using the Root test which found the limit to be: e^2/3> 1 Is it necessary to test if it converge conditionally? The question This is the the prove by symb...

I am having a bit trouble understanding group actions. if I am given a set A = {a,b,c,d} and a group action s: Z mod 4 -> $S_A$, how would one then be able to show if there exists a group action s such that s(2) = (a b). The only thing I seem to be able to get out of this information is that: $s...

Let A be a symmetric n x n matrix. Then A is positive definite if and only if all the principal minors of A are positive.

Just started learning a little bit about group theory. Can anyone give me a hint (or counterexample if the statement isn´t true). Let be $P $ Principal ideal then every Ideal $P^{'}\subset P$ is also a Principal ideal. May be a stupid question to ask but I need to know.

Which number fits into this geometrical symbolic relationship? enter image description here

V is isomorphic to U as vector space. U is banach if and only if V is banach.

Is there a closed form for this?$$\sum_{n=1}^\infty \frac 1 {2^n - 1}$$ Tried searching but couldn't find anything.

Let $A,B,C$ be events, and $B,C$ be pairwise independent. Can I say that $P(A|B,C)=P(A|B)$? The only thing I get is that $P(A|B,C)=P(A|B)P(A|C)$ Any help would be appreciated.

I am getting nowhere with this question. I found a similar question on this site but I need help in a more step by step way. I would be very grateful.

I want to prove $e^{i \cdot \pi \cdot x} = \cos x + i \sin x$. Proof: $e^{i x \pi} = \sum \frac{x^{n} i^{n} \pi ^{n} }{n!} = -i\sum_{odd} (-1)^{n} \frac{x^{2n+1} \pi ^{n} }{(2n+1)!} + \sum_{even} (-1)^{n} \frac{x^{(2n)!} \pi ^{n} }{(2n)!} $ and this should give the taylor serie for $\sin$ an...

I am searching an example of a function $f$ on $[a,b]$ such that $f$ is a bounded function having intermediate value property but is not Riemann Integrable on $[a,b].$ Please give me such type simplest example which can easily be visualized. Thanks in advance.

Show that for every ε>0 there is a probability distribution p so that for every tree B: L(B,p)>H(p)+1−ε.

I know that when applying SVD on a matrix (m * n) I should have these three outputs: S: m × n diagonal matrix with non negative numbers U: m × m orthogonal matrix V: n × n orthogonal matrix but when using R statistical package. I got for S a vector instead of a matrix: look please: this is t...

how I can compute expectation of Dirichlet distribution of... \operatorname{E}[\ln X_i] = \psi(\alpha_i)-\psi(\textstyle\sum_k \alpha_k) can someone explain it to me in detail!? https://en.wikipedia.org/wiki/Dirichlet_distribution

I need to complete the definition of bcount so that bcount(n) returns the total number of odd coefficients n k , 0 ≤ k ≤ n. For instance, the values of n k for n = 6, with odd values highlighted, are: 1, 6, 15, 20, 15, 6, 1, bcount:= proc(n::TYPE) description "Count odd binomial coeffic...

I FOUND THE SOLUTION IN THE WOLFRAM I NEED STEPS SOLUTION

I'm studying on Lang's Algebra the formal definition of polynomial's ring, in particular I'm at the chapter of several variables polynomials. Let $A$ be a commutative ring with unity ($1$), so we have associated at $A$ the ring $A[X_1,X_2,...,X_n]$ Define $X:=(X_1,X_2,...,X_n)$ and $f(X) \in A[X...

I have a Matlab code which solves the Lyapunov equation $AX + XA^T + Q =0$ for a 3-D array of Matrices $A$ using matlab function 'lyap(A,Q)' . My problem is, sometimes the resultant Matrix $X$ is positive definite and sometimes it is not. My question is -- given that the matrix $Q$ is positive de...

How do I solve this question, what are the steps ? Use the method of separation of variables to derive the solution $u(x, t)$ to the equation for a vibrating string $\frac{\partial ^2u}{\partial \:t^2}=9\:\frac{\partial ^2u}{\partial x^2} , (0 < x < 4, t > 0),$ with fixed endpoints $u(0, t) = ...

How to integrate $\displaystyle\int\sqrt{q_m}dG_m $ if $F_n(0)=1-a_n, G_n(0)=1-b_n$ and $q_n=\frac{dF_n}{dG_n}$ It is written that $F_n, G_n$ are ''concentrated'' on $\{0,1\}$ the result should be $\sqrt{(1-a_m)(1-b_m)}+\sqrt{a_m+b_m}$, does it make sense to have an expression like $\int\sqr...

May be this is a stupid question but why ∫√((sin x)^2).dx = ∫|sin x|.dx instead of +-∫sin x.dx ? I think may be because it violates the rule that a function can't have more than 1 output for a single input, which brings me to my next question does the intgrand need to be a function?

Show the equality $\frac{d\mu_n}{d\nu_n}=\prod\limits_{m=1}^{n}q_m$ I don't understand why $\frac{d\mu_n}{d\nu_n}=\prod\limits_{m=1}^{n}q_m$ with the definitions; $F_n=\mu(\xi_n\le x)$, $G_n=\nu(\xi_n\le x)$ and $q_n=\frac{dF_n}{dG_n}$, $\mathcal F_n=\sigma(\xi_m:m\le n)$ and $\mu_n,\nu_n$...

I consider the surface $N^{\epsilon}_t = \text{graph}\{ u/\epsilon - t/\epsilon\}$ for all $t >0$. Let $\nu$ be the outward unit normal to $N^{\epsilon}_t$ and $H$ the mean curvature of the surface. I also introduce a cutoff function $\phi \in C^2(\mathbb{R})$ such that $\int \phi =1$. I have ...

I would be very grateful if somebody could explain me the part in the red rectangle in the proof below. I don't quite get it why the author claims that if $\delta$ is sufficiently small then $p+1\le q-1$. I don't get it why it is the consequence of the inequality $\beta-\alpha \le x_q-x_{p-1}<(q-...

I need .tex file of a template paper of Geometry and Topology http://msp.org/gt/about/journal/submissions.html, can someone send me Thanks

Use mathematical induction to prove that 5^n + 9^n + 2 is divisible by 4, for n is a positive integer.

Look at the result of (-1)^(1/10000000)on the google calculator You should get 1 + 3.14159265 × 10-7 i Why does Pi occur in imaginary number operations that don't include Pi?

So I have -(x - 2)² Do I rewrite it as -(x - 2)*-(x - 2) and distribute the negative to the inside making it (-x + 2)(-x + 2) or add the negative at the end of doing FOIL?

How can I prove the following function is convex? $\psi (x)=(x_1-(\frac{x_1+x_2+x_3}{3})^2+ (x_2-(\frac{x_1+x_2+x_3}{3})^2+ (x_3-(\frac{x_1+x_2+x_3}{3})^2$

What the answer when t goes to infinity for this function -2 * (t*e^-t + e^-t )

I thought this was true since, $f(x)=f(0)+f'(0)x+f''(0) \frac {x^2}{2!} + \dots$ But I am wrong. Where did I make mistake?

Assume that $X$ is a nonempty set and that for every $x\in X$ we have a nested family of nonempty sets: $\{E_{x,t}: t\in [0,\infty)\}$ for which $t_1\leq t_2$ implies $E_{x,t_1}\subseteq E_{x,t_2}$, $E_{x,t}\neq\emptyset$ for all $t$. Nothing more is assumed. Is there always a pseudometric $d$ o...

A function $f(x)$ satisfies the condition,$f(x)=f'(x)+f''(x)+f'''(x)+f''''(x)+......\infty$,where $f(x)$ is a differentiable function indefinitely and dash denotes the order of the derivative.If $f(0)=1$,then find the function $f(x).$ $f(x)=f'(x)+f''(x)+f'''(x)+f''''(x)+......\infty$ $f'(x)=f'...

Hello did a few exercises about supermum and infimum but im not sure if my solutions are correct. The following set are given: (i) {n is element of the whole numbers | n³ > 10} (ii) {n is element of the whole numbers | (3/n) +4} (iii) {x element of real numbers | x² < 5} (iv) {k element of in...

Does $\Bbb E[X|Z]=\Bbb E[Y|Z]$ if $X,Y$ are identically distributed random variable, where $Z$ is a third random variable? Thank you!

I have a problem understanding the meaning of the delta-function on the sense of distributions. E.g. I have the following equation: $(\frac{d}{dt} \theta(t) ) f(t) = \delta(t) f(t)$. What does this mean for the function f(t)? I cannot really understand what the delta function does here in t...

$2xf(x)+x^2f'(x) = \frac{e^x}{x}$. $f(2) = \frac{e^2}{8}$. And how to prove it? What I can get is $f'(2) = 0$ but that's not enough.

I have a series of the following form: \begin{align} \sum_{k=2}^\infty \left( 1 - e^{-ns^{k-1}} \right)^k \end{align} where $0<s<1$. I would like to compute an approximation of this series, for the regime where $n$ is large. One natural thing to try would be to use the approximation $1-e^{-x} \...

I'm looking for an example of a curl free vector field on a non-simply connected region which is still a gradient.

X(t) =e^-t * (cost+jsint) determine x(t) is periodic or nonperiodic and the period of its periodic

I've got this problem: You are given a grid n X n with n rows and n columns. For every row i you are given number r_i and for every column j you are given number c_j . ...

Why is $\Pr(Z_n=k, n\ge N)=0$ if $\Pr(\xi_i^m=1)<1$ and $\mu=1$ for any $k>0$ It is a theorem from Durrett PTE $Z_n$ is a Galton-Watson process with $Z_{n+1}=\xi_1^n+\dots+\xi_{Z_n}^n$ with $\xi_i^m$ iid nonnegative integer valued. In the proof it is stated that: $Z_n$ is a martingale conve...

Question: Let $H,K,N$ be subgroups of a group $G$ such that $H\leq K$, $H\cap N=K\cap N$ and $HN=KN$. Show that $H=K$. Here is my attempt: $H=K\Leftrightarrow$ $H\subseteq K$ and $K\subseteq H$. Clearly, $H\subseteq K$ as $H\leq K$. To show $K\subseteq H$; $HN=KN\Rightarrow k=hn$ where $k...

How can i check which Algorithm is used by GAP for its working. Like ` gap> p_1:=(1, 2, 4, 5);; gap> p_2:=(1, 2, 3);; gap> g_1:=Group(p_1, p_2);; gap> p_3:=(1, 2, 4, 3);; gap> p_4:=(1, 2, 5, 4, 3);; gap> g_2:=Group(p_3, p_4);; gap> g_1 = g_2 True Which Alg...

Problem Suppose $351_7=aca_b$ for positives $a,b,$ such that $b-1=c$. Compute the ordered triple $(a,b,c).$ Here is what I tried: $351_7 = 183 = a*b^2+c*b+a = a(b^2+1)+c*b = a(b^2+1)+(b-1)*b = b^2*a+b^2-b+a \implies b^2*a+b^2-b+a -183 = 0 \implies 1-4(a)(a-183) \geq 0.$ I am not sure if th...

What symbolizes the number and what is the number missing from the relation?enter image description here

Prove that the $\lim_{n \rightarrow \infty} \frac{2^{n} n!}{n^{n}} = 0$ The Eqaution is equal to $\frac{2^{n} n!}{n^{n}} = $ $(\frac{2}{n})^{n} n!$ Its possible to say that $\lim_{n \rightarrow \infty} $$\frac{2}{n}$ is $0$ and because of this reason $\lim_{n \rightarrow \infty} $$(\frac{2}{n})...

As simple as this may sound, I just do not understand what this statement implies. An nxn matrix A is symmetric if and only if: $$\bar{x}.(A\bar{y}) = (A\bar{x}).\bar{y}$$ Why is this true, and what does it even signify?

Find the cardinality of the set of all points in the plane which have one rational and one irrational coordinate. Justify you answer My thoughts so far. We know that $Q$, the set of all rational numbers has a cardinality of $aleph_0$. Also, since the set of irrational numbers is just the reals ...

I have the following system of PDEs for which I have given parameters $\gamma, \tau$ and $\mu$, $$\begin{align} T_t = &\ \gamma\,(L +\tau F-T)\\ F_t = & -F_x-(F-LT)\\ L_t = &\ \mu L_{xx}+(F-LT)\end{align}$$ with no flux boundary conditions at $x=0$ and $x=\infty$ for $F$ and $L$. The initial c...

The question is: Let S be a set, R be a binary relation on S, and x an element of S. Express in English the negation of the statement “For all x in S, xRx”. I was originally thinking since the negation is just the opposite, I would switch S and R to get an expression of "For all x in R, xSx"

I want to appear in$Regional Mathematical Olympiad$Help me to prepare for it successfully so that I can clear that exam in one attempt

i'm tackling project euler problems to learn to code, and this is the problem i'm trying to solve now (number 12): The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be: 1, ...

The d'Alembert functioal equation is: $$f(x+y)+f(x-y)=2f(x)f(y)\tag0$$ This equation plays a central role in determining the sum of two vectors in Euclidean and non-Euclidean geometries. Is there a good characterization of the solutions of this equation?

What two convergent subsequences immediately prove that sequense sin(n) diverge?

I was wondering if definitions in mathematics are "if and only" statements? I know for sure that theorems are not "iff" statements. Thank you in advance for your help.

A two part question. 1 True or False: when working with an equation or inequality, everything that you do is either: a substitution, or an operation performed on each side Note that algebraic or numerical simplifications are substitutions - $2+2=4$, so we're free to substitute $4$ where $2+...

Wich are the "best" books to initiate in algebraic K-theory study? I've got Algebraic K-theory by Bass,H and K-theory by Atiyah. Websites will be apreciated too. Thanks.

Let $M$ be a smooth closed $d$-manifold embedded in $\mathbb{R}^n$, and let $d_M: \mathbb{R}^n\to \mathbb{R}$ be given by $d_M(x) = \min_{m\in M} d(x,m)$. Define $M\oplus \epsilon = \{ x\in \mathbb{R}^n \mid d_M(x) \leq \epsilon\}$. The reach of $M$, ${\rm Reach}(M)$, is the largest $\epsilon$ ...