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...
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 $\Bbb 1$ ...
$$\sum_{n=1}^{\infty}{x}a^{x-1}$$ 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$$ then what is the correlation between X and Y? I am guessing its q?
Derive a generating function for the sequence $a_n = n^2$ I know the power series for $n^2$ is $${x(x+1)}\over{(1-x)^3}$$ However I am struggling to connect the two with the proof. Any help would be greatly appreciated thanks!
How do I factor 2 + 3$\sqrt{2}$ into primes in $Z[\sqrt{2}]$? I know that primes are irreducible in $Z[\sqrt{2}]$ and that units are of the form $\pm(1\pm\sqrt{2})^n$. How are primes and units related, if at all? I understand that all primes of $Z[\sqrt{2}]$ are obtained by factoring rational pr...
I fully understand both the Lemma and proof of Gauss's Lemma on number theory which states $$\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 ...
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.
In my company we have a spring project which has seen many years of development. It is a massive web services project with many many rest end points. The problem is that now when we are growing into multiple agile teams, working on this monolithic project is becoming painful. There are too many ...
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 $$b^{x+1}-a^{x+1} = (b-a)(b^x+b^{x-1}a+...+ba^{x-1}+a^x) > a^x(b-a)$$ $$$$ I was thinking it may have to do with $a<{a+b\...
It seems that the following is the basic property of the supremum of a function: $$\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
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 $...
\dfrac
with \frac
in the title. (autocomment) — Normal Human 21 secs ago$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?
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?
I know there is a maximum cap of 200 for reputation per day for a user. Is there a maximum cap that you can get for answering a question? In theory if I get a 1K votes for an answer that will get converted to 10K reputation, right? Also do I loose reputation if the post gets converted to a commun...
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$$ What is maximum Burst Size? Taken from here https://www.youtube.com/watch?v=4eMrQXU0DdA&list=PLpherdrLyny-zJw95jcE-uJkcsIAG1MEn&index=103 Video provides example but never solution... I did not understand h...
Where I am doing wrong? if any one can tell!!! I have done this equation as $$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 $f\circ g$ ...
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...
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?
\dfrac
with \frac
in the title. (autocomment) — Normal Human 21 secs agoI just created implementation since it seems that many questions could possibly fall into this category and it would make it easier to find similar questions. I started wondering, would it make sens to re-tag some old questions with this tag now? This would bring many old questions on the top of ...
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...
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} $$ Turning and ...
I've spent many hours on this and I just can't understand how to do this. Could you please go through this with me? I have a test, and I really need to understand how to do these types of problems. Thank you so much!! "Find vectors x and y with sum-norm of x = 1, max-norm of y = 1, such that 1-n...
In this question the OP posted a question, without any error messages. After asking about the error messages he provided it, in the comments. Since this error message can be useful in order to answer the question, I suggested an edit, moving his error message to the question. To my surprise, my ...
I'm designing a tool for students and right now im working with coordinates. I have the plan to shift the north pole in my code to Munich and I would like to do this by working with a spherical triangle and the angle at one point. As result I would like to have a formula which tells me how to cal...
Hello I had this question below on a midterm on Calculus I had proved the series diverge by testing the Absolute value of the series. By using the Root test which found the limit to be: e^2/3> 1 Is it necessary to test if it converge conditionally? The question This is the the prove by symb...
I am having a bit trouble understanding group actions. if I am given a set A = {a,b,c,d} and a group action s: Z mod 4 -> $S_A$, how would one then be able to show if there exists a group action s such that s(2) = (a b). The only thing I seem to be able to get out of this information is that: $s...
Let A be a symmetric n x n matrix. Then A is positive definite if and only if all the principal minors of A are positive.
Just started learning a little bit about group theory. Can anyone give me a hint (or counterexample if the statement isn´t true). Let be $P $ Principal ideal then every Ideal $P^{'}\subset P$ is also a Principal ideal. May be a stupid question to ask but I need to know.
Is there a closed form for this?$$\sum_{n=1}^\infty \frac 1 {2^n - 1}$$ Tried searching but couldn't find anything.
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.
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...
Couldn't find this asked already, can a query be made for viewing the average view count for questions in a given time frame? For example, x number of views on average within 1 year, y number for 2 years, et cetera. It should logically be restricted to a per tag category for more a more localize...
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...
\displaystyle
in the title. (from a bot) — Normal Human 20 secs agoMay 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$
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'm looking for an example of a curl free vector field on a non-simply connected region which is still a gradient.
I was just looking around the Meta, when I found this amazing page of the website. It is damn fascinating, though I'm interested in something. Does it count questions, solutions and all the stuff real-time? No need to go into details, if you are not desired, because I'm sure there's an efficient...
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?
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.
I'm learning about functions of bounded variations and need help to understand the proof of this lemma: Lemma. If $f : [a,b] \rightarrow \mathbb{R}$ is of bounded variation, then f is also bounded and satisfies $\left\lVert f \right\rVert_{\infty} = \left|f(a) \right| \leq V_{a}^{b} f$. ...
I have a problem I think it is a little hard or at least has some points need to be considered to solve it. I know that if the transformation function is with the form of fog I should calculate the region with g then claulate the new region with f. But whatever I did, didn't help. The question is...
Prove using the definition and the Cauchy criterium that $x → x3$ is Riemann integrable on $[a,b]$ and that $\int_a^{b}x^2dx= a^3 - b^3 /3$ Thank you.
Can someone help me to prove this equivalence : let $f:E \rightarrow F$ we have $1)$ for all $y\in F$ and for all open $U$ from $E$ such that $f^{-1}(\{y\})\subset U$, there exists a neighborhood $V$ of $y$ such that $f^{-1}(V)\subset U$. 2) $f$ is closed I strated by $1) \Rightarrow 2)$ l...
Please is the following assertion true? Let $\tau\subset\mathbb{R}_+=[0,\infty)$ be a measurable set (the standard Lebesgue measure), $\mu(\mathbb{R}_+\backslash \tau)=0$. Then there exists an increasing sequence $\{x_i\}_{i\in\mathbb{N}}\subset \tau,\quad x_i\to\infty$ such that for any $k,i,\...
I'm a bit confused with this determinant. We have the determinant [picture determinant][1] [1]: http://i.stack.imgur.com/veqk0.png for a matrix $\mathbb{M_n}$ with n$\geq2$. I compute $\delta 2=19$, $\delta 3=65$ Then I would like to find a relation for n$\geq 4$ which links $\delta_n, \delta...
Let p and q be distinct prime numbers and N = pq. G ={natural numbers less than pq that are relatively prime to pq}. if a and b are in G and k is a natural number, then (1) G has (q-1)(p-1) elements. (2) ab mod N is never 0. (3) ak mod N is never 0. (4) ab mod N and ak mod N are each elements...
I am looking for names/examples/references for probability distribution functions which are supported on a closed interval, say $[0,1]$, and increasing there.
What is the answer to the following puzzle: If 3 and 2 = 7 5 and 4 = 23 7 and 6 = 47 9 and 8 = 79 then what is 10 and 9 = ?
I have a positive function $f(x,y)$, where $x\in{\mathbb R}^n$ and $y\in{\mathbb R}$. I know that for $y$ fixed, $g(x)=f(x,y)$ is convex, and that for $x$ fixed, $h(y)=f(x,y)$ has positive second derivative. If this enough to show that $f(x,y)$ is convex?
Let $\mathcal{X}$ be a smooth geometrically irreducible projective curve over $\mathbb{F}_q$. Fix a closed point $\infty\in \mathcal{X}(\bar{\mathbb{F}_q})$. Let $K$ be the function field of $\mathcal{X}$ and $\mathcal{A}$ the ring of functions on $\mathcal{X}$ regular away from $\infty$. Let $\p...
I read somewhere that a line is made up of infinite points. Between any two points on that line, there are another infinite points. and between any two points BETWEEN those 2 points there are another infinite points. Is this true? If so , HOW? Also, i have read in physics that the smallest p...
$(1) ([8], [10]) \in Z_{12} \times Z_{18}$ $(2) ([3], [6], [12], [16]) \in Z_4 \times Z_{12} \times Z_{20} \times Z_{24}$
If I have $$ n < \cfrac{n^2(s-2)-n(s-4)}{2}$$ can I make it so that one side becomes completely known when I give a value to $s$, for example(obv. not true) $$123n^2+456n+789<987s+654$$
How can prove the following optimization model is convex for $n>3$? $\psi (x)= \min \sum_{i=1}^{n} (x_i-\frac{\sum_{j=1}^{n}x_j}{n})^2$
Since math is the most important part for programming I want to implement the gradient tool and I need to understand the math behind it . Can any body also explain the differences between linear , radial ,angle,reflected,diamond gradient? 2 []3 []4
How do I prove that any finite subgroup of SO2 must be cyclic? Also, what are all the finite subgroups of O2?
My question is: Given sets A = {a1, a2} and B = {b1}, let the function f from A to B be given by the following set of ordered pairs, f = { (a1, b1), (a2, b1) }. If f has an inverse function, call it g, and write g as a set of ordered pairs. If f does not have an inverse function, explain why it d...
I asked a question here. As you can see, noone really liked it, and people voted to close it as off-topic. I don't have a problem with this, I only want to understand why. In my question, I ask for the explanation of a comment on php.net. The first comment to my question is that the comment re...
Is there a direct way to find the matrices representing the symmetries for example of a tetrahedron with vertices (1 1 1) (-1 -1 1) (-1 1 -1) (1 -1 -1)
I have known that ∀x [s(0)∗ x = x] is a theorem of Peano Arithmetic. but how to Prove that this formula is not a theorem of Robinson Arithmetic?
I would like to obtain the letter $K$ from $K[1,2]$ in Maple. I tried to use op(K[1, 2]). But I get $[1,2]$. Are there some function $f$ in Maple such that $f(K[1,2]) = K$? Any help with be greatly appreciated!
I’m stuck on the following problem. There are two sources $S_A$ and $S_B$ at the ends of a channel. Both are made up of a white noise component $W_i$ plus an impulsive component $I_i$: $S_A = W_A + I_A$; $S_B = W_B + I_B$. Two sensors are co-located with the sources. The first one records the...
Solve the equation $\frac{dy}{dy}=\frac{x+2y-5}{2x+xy-4} $ I tried substituting $x=X+h$ and $y=Y+k$ but the $xy$ term is creating problem. How to solve it?
$S^1 := \{z\in \mathbb{C}: |z|=1\}$ is the set of roots of unity. We can use the fact discovered by Euler that any complex number $z$ can be represented as $z=re^{\pi i k}$, where $r\in \mathbb{R}$ is the radius of $z$ and $k$ is any real number. Any complex number with radius $1$ can thus be def...
The inverse of covariance matrix can be used to find the conditional independencies among variables. This inverse is more sensitive to changes then the correlation matrix. As a result two samples with very similar correlation matrices and standard deviations, may still point to different conditio...
Let p and q be distinct prime numbers and N =pq. Do multiplication mod N. Define G = {natural numbers less than pq that are relatively prime to pq} and s =(p-1)(q-1) . Then (1) m^s =1 for all m in G. I have no clue with first part of this problem, and don't know how to start with it.
I do understand the concept of Naive Bayesian classification, as it tries to calculate the probability of an outcome of a class given multiple evidences. It comes from the Bayes theorem and it is called naive as it tries to each of piece of evidence as independent. This approach is why this is ca...
prove E(X1+X2|Y)=E(X1|Y)+E(X2|Y) This question is given in the text book as an exercise. It looks trivial but took me hours to think of a proof. I don't know what to do after expand it using definition of expectation...
« first day (39 days earlier) ← previous day next day → last day (535 days later) »