I am confusing about the differences of the property of natural log in complex analysis and in real region. This question might be a bit stupid, but any answers or explanations of the log properties will be appreciated. The question is: $log(e^{2\pi i})=log|e^{2\pi i}|+iarg(e^{2\pi i})=log|cos(...

I has a discussion with a friend about the boundary of the boundary, and we do not agree, that's why I ask you the question. For exemple, if $A=[0,1]$, what is $Bd(Bd(A))$ ? To me it's $\{0,1\}$ since $\{0,1\}$ $$Bd(\{0,1\})=Cl(\{0,1\})\backslash Int(\{0,1\})=\{0,1\},$$ but my friend think that i...

I am writing a paper on optimization and was wondering if it is acceptable to say, for example, "There are many local minimums" or should it be phrased to say "There are many local minima"? Are minima and minimums exchangeable vocabulary? Or is there a difference?

I am trying to draw the Young diagram for $S_5$. I know the following pieces of information about $S_5$. The order of the group is $120$. The number of conjugacy classes and so partitions is $7$. Degrees of irreducible representations $1,1,4,4,5,5,6$. The partition is $1 + 10 + 15 + 20 + 20 +...

I have been stuck on this problem for a couple of days, I don't want the answer, but I would appreciate some help in finding it! Thanks in advance! Consider a Symmetric Square Matrix $M(n,k)$ such that there is $n$ lines and $n$ columns, in which there is $-1$ everywhere expect on the diagonal, ...

The moment of inertia is given by equation: $I_L = \lim_{n\to \infty}\sum_{k=1}^n\Delta I_k = lim_{n\to \infty}\sum_{k=1}^nr^2(x_k,y_k,z_k)\delta(x_k,y_k,z_k)\Delta V_k = \iiint_{D}r^2\delta dV.$ What does this equation mean, and what does it signify? Ostensibly, it appears that $\iiint_...

enter image description here Usually the laplace transforms on piecewise functions are only really defined on one interval or zero on all other intervals, but if it's defined on multiple intervals that means there are two different transforms with two unique answers respective to their intervals...

had to prove the above mentioned iff, but I think I have something missing. would appreciate your advice.

My reference for learning abstract algebra says that in an abelian group $G$ the product of two elements (say $a$ with order $n$ and $b$ with order $m$) has order $mn$ if $gcd(n,m) = 1$. What I don't understand is where the $gcd(m,n)=1$ comes into use in proving this fact? For example, $ a^{mn}...

Can someone help me understand this? I feel very wrong asking this simple question, but please help me understand this step. Let $f,g$ be real-valued functions on closed interval $I$ on $E^n$. Show that if $f$ and $g$ are integrable on $I$ then so is $fg$. Solution: Notice that $fg = 1/4(f + g)...

I need to prove Richardson's Method and the first part of the proof is: Consider the linear system $Ax = b$ where the eigenvalues of $A$ are real and positive. Let $G_{\omega } = I - \omega A$, where $\omega$ is a scalar. Let the iteration be $$x^{n+1} = (I-\omega A)x^{n} +\omega b$$ Show all ...

Question on bijection function This is a question on a Final review, and I am having trouble figuring out what to do. Thanks!

So I have to figure out if $1.23456$ is constructible. I think that it's not constructible since: I know that this is $\frac{123456}{100000}$ so this goes into $\frac{2^6*3*643}{2^55^5}$ and the Fermat Primes are repeated in the denominator so then it's not constructible. Am I overthinking this...

Give me an example of how to detect "double peaks" in stocks price movements. Using regular expressions. Is this possible? http://stockcharts.com/school/doku.php?id=chart_school:chart_analysis:chart_patterns:double_top_reversal

if f(x)=(x-1)/(x+1) what is the value, in simplest form of f of f(2015); Pretty intresting problem. But attempted a lot but couldnt solve it! Would love your guys help!!

I was asked to find the number of necklaces that can be made using 4 red beads and 6 blue beads. I understand the steps needed to be taken in order to solve this type of problem but I don't quite understand the logic behind it. I know this type of question is essentially asking you for the number...

For any non-decreasing $f: \mathbb{R} \to \mathbb{R}$ and $S \subseteq \mathbb{R}$, The Lebesgue-Stieltjes outer measure associated with $f$ is $$\lambda^*_f(S) = \inf\left\{\sum_{j=0}^\infty (\lim_{x \uparrow b_j}f(x) - \lim_{x \uparrow a_j}f(x)) \mid a_j, b_j \in \mathbb{R} \land ...

Consider a scalar problem of the form $$y(t)' = \mu^2 y(t).$$ Derive stability conditions for given problem.

I am trying to understand the partitions of $S_5$ created by it's conjugacy classes but two sources have two different partitions. Source 1: Source 2: So, for example, in the first table, the partition for cycle structure ()()()()() i.e. $5$ $1$-cycle is $5+0+0+0+0$ but in the second table i...

Show that $\mathcal{M} = \mathbb{Z}[X] \backslash(X^{2} - X, 4X +2)$ is finitely generated abelian group. What is the general procedure for handling these types?

Let $ P$ be the vector space of all polynomials over $\mathbb R$.Let T and S be two linear maps from $ P$ to itself such that TOS is the identity map.Then which are correct? SOT may not be the identity map. SOT must be identity map, but S and T need not be the identity map. S and T must be the...

I'm using R, and I just wanted to know that how do I calculate residuals for this model? And when I use lsfit for this, why don't I get a value for gamma?

Inside the sphere x^2 + y^2 + z^2 = 25 and outside the cylinder x^2 + y^2 = 4 I've so far converted the formula to z = r^2-5r outside the cylinder r=4 For the internal integral, I have the definite integral going from 0 to 4, but the external integral is from 0 to 2pi. I'm confused as to where ...

I tried to label this with as many soft question labels as possible. To give you a gauge of where I am math-wise, I'm a senior in high school who has gone through all the basic subjects (geometry, algebra, trigonometry, basic statistics/probability, calculus I). In addition, I've explored a few ...

we are given that Re[(z+5)/(z-5)]=0. What is the locus of these points? is it a circle or y axis? my professor told me that it is a circle.

Let f:[a,b] -> ℝ be a continuous function and let a<x<y<b. Then f is continuous in [x,y] and differentiable on (x,y). Can anyone tell me which theorem states this?

There is a card in Magic the Gathering called Lim-Dul's Vault. While it is slightly more complicated than presented, the question I would like to consider is this: Pay 1 life. Look at the top 5 cards of your deck. You may rearrange these cards in any order. You may repeat step 1. If you do...

Approximate $3+ \displaystyle \sum_{i = 2}^{999}\dfrac{3(1000-x)}{1000+x}$. It may help to know that $\ln 2 = 0.69$. I was thinking of doing the integral test to approximate this but I am unsure if this would work.

I have a question asking me to evaluate $\iint_\Sigma \mathbf{F} \cdot \mathbf{n}~dS$, where $\Sigma$ is the lower half of the ellipsoid $z = -2 \sqrt{1 - x^2 - y^2}$ with $\mathbf{n}$ directed upwards. However, I'm having trouble deriving the parametric equations for $\Sigma$. I know the general...

I'm having issues with this problem: Evaluate by changing to polar coordinates ∫∫15xdydx The regions of integration are outer: 1/2 to 0 Inner: sqrt(1-x^2) to sqrt(3x) I know that r=1 because solving y=sqrt(1-x^2) gives a circle with radius 1 as the top limit of integration and the lower being ...

How do I go about showing that this following statement is true?

I was reading the famous "Calculus" by Spivak and at the beginning of the chapter on infinite series, he states: "It's an easy exercise to prove that if both $\displaystyle \sum_{k=1}^{\infty}a_n$ and $\displaystyle \sum_{k=1}^{\infty}b_n$ exist, then $\displaystyle \sum_{k=1}^{\infty}a_n+b_n$=$\...

Following 4 logical propositions are all true : 1. (A → B) ∧ (A → ¬B) 2. ¬A → B 3. ¬(B ∧ D) 4. C ∨ ¬A Truth value of proposition A,B,C,D:

I have created the following FA Im i correct?

How would I come about this? This is very hard. This question is on my practice final. Any help would be helpful. Also, any tips on proving things?

I tried to solve this division of complex numbers and reached to below answer is it true?if not please leave the true answer. $ \dfrac{(-1+i)^{10}}{(-\sqrt{2}-i\sqrt{2})^{15}} $ my solution (first solve the denominator) : $(-\sqrt{2}-i\sqrt{2}):$ $r=\sqrt{2+2}=2$ $\tan(\theta)=\dfrac{-\sqrt{2...

Is there a way to derive the fact: $\langle cA, C\rangle = c\langle A, C \rangle$ from $\langle cA + B, C \rangle = c \langle A, C \rangle + \langle B, C \rangle$? $\langle \cdot, \cdot \rangle$ is an inner product and $A, B, C$ are vectors and $c$ is a scalar. I tried this: plugging $B = 0$, ...

Use the renewal equation to show that the renewal function of a Poisson process with rate λ > 0 is m(t) = λt.

Suppose that $f_n\to f$ in $L^p$. Then, is it true that $\int |f_n|^p \to \int |f|^p$? When $p=1$, the result is clearly true since $\int ||f_n|-|f|| \leq \int |f_n - f|$, but how do you show/disprove the result when $p > 1$?

As far as I understand, with some real number $a$, the function $$f(x) = \frac{\sin(ax)}{ax}$$ is called 'sinc' function. Is there a name for a function like the following one? $$g(x) = \frac{\cos(ax)-1}{ax}$$

This from "Basic Topology" by Armstrong. I can't figure out what $f(x)=-x$ is doing when mapping from unit circles. Is this the antipodal map?

This problem might be simple. How to expand $|(1-H(f))|^2$? I just see one answer in the following: (A typo: $S_X(f)$ should be outside the $\{\}$). My answer is $1+|H(f)|^2 + 2|H(f)|$. However, "$2\text{Re}[H(f)]\neq2|H(f)|$". (One is the lenght of the vector, the other the length o...

Integrate ( (sin 2x)^3-x(cos x)^3 )/(cos x)^2 dx. Help me im quite blur and how should i start.a little show on working are gladly appreciate

I am trying to analyse the following. Assume $S_{3}$ acts on a non empty set $T$, and that is has $3$ orbits. What can we say about the possible cardinalities for the set T? My thoughts: If $G=S_{3}$ then $|G|=3!=6$ and we are consider $G \times T \to T$ , $(g,t) \to g \star t$ I also know t...

Okay I found this one on a test and I am still struggling to solve this one don't know where to start..here it goes $ 1 \le \frac{(3x^2 -7x+8)}{x^2 +1} \le2 $. How do I find the range of x?

A certain cellular device is advertised as being able to last for 7.0 to 9.0 hours of screen-on time. In experiments carried out to test this claim, the time in hours, X, was measured on a random sample of 250 occasions, and the data obtained is summarized by \sum \left ( x-7.6 \right ) = 68.3 an...

Let $C, C_1, C_2 \subset \mathbb{R}^2$ be circles. Let $f: C \rightarrow C_1 \cup C_2$ be a map such that $f(x_1) \in C_1$ and $f(x_2) \in C_2$. Does it follow that $f$ is not continuous?

I tried to solve this question. It is a multiple choice question. First I took the derivative of tan(ax+b) which is sec^2(ax+b) but If I take the derivative with respect to x it will be a sec^2(ax+b). what is find confusing is d/d tan(ax+b) = sec^2 (ax+b) d/d tan(ax+b) *(ax+b) I can't really u...

Let $C_1, C_2 \subset \mathbb{R}^2$ be concentric circles in the plane. Suppose that $C_1$ bounds $C_2$. Let $f: C_1 \rightarrow C_2$ be a map such that for some $y \in C_2$, $f(x) = y$ for all $x \in C_1$ and that $x$ and $y$ must be connected by a path that is entirely contained in the interior...

I'm not sure if this question is coherent. But here goes. (1) Let M is a model of ZFC. For which axioms, A, of ZFC is there a model, M', extending M in which A no longer holds? (For instance, I take it that the empty set axiom fails to meet this condition.) (2) For the conjunction of axiom...

If $f:X\to Y$ takes Cauchy sequence to Cauchy sequence then prove that $f$ is a continuous function. Let $x_n$ be a sequence in $X$ such that $x_n\to x\implies x_n$ is Cauchy $\implies f(x_n)$ is Cauchy but that does not guarantee that $f(x_n) \to f(x)$ . So how is the above result true.Pleas...

As we know, if $A$ and $C$ are not singular, the Woodbury identity is : $(A+UCV)^{−1}=A^{−1}−A^{−1}U(C^{−1}+VA^{−1}U)^{−1}VA^{−1}$. What if $A$ is singular and $C$ is not singular, is there identity (like the Woodbury identity) to calculate the inverse of $A+UCV$ ? Thanks.

Prove that the upper and lower integrals, noted below in R^d, do not depend on the choice of rectangle R as long as it contains E

Older SE question lies here. So I will change the question such that you can understand the question better: $$\sum_{c\space\subset \Bbb{Co}}\frac{1}{c}\le k$$ -where your goal is to get to $k$ with the desired specific composites -You must add up the specific composites (group $\Bbb{Co}$) su...

I am trying to understand how to calculate x^y where y is a decimal number, (2^2.3) According to wikipedia, the 'solution' would be x^y = exp( ln(x) * y ) But if we break it down further, z = ln(x) * y x^y = exp( z ) But in that case, z is almost certainly not going to be an round number ...

$\begin{bmatrix}4 & -5 & 1 \\ 1 & 0 & -1\ \\ 0 & 1 & -1\end{bmatrix}$ I usually set 0 equal to $det(A- (lambda * I))$ to find the eigenvalues, but the book says they are 2, 1, and 0, which isn't what I got. Am I missing a step?

If I know the sides of right triangle, how can I know its angle? without using calculator

Image link : http://tinypic.com/r/30wph1j/9 For the above 8 faced geometrical shape where each edge is perpendicular to the adjacent one. How to get the perimeter of the whole face? I tried to draw different line to make triangle and rectangle to get the unknown edge's length. Is it possible...

Let $V$ be a real normed vector space. Suppose that A is an open set from $V$ Show that the set $\frac12A$ = {$\frac12x: x \in A$} is also open. Let V be a complex vector space. A norm on V is a function || · || : V → R that satisfies the following three conditions: (i) ||v|| ≥ 0, ∀ v ∈ V, and ...

Is this number $\big[log(n) + \sum_{j=1}^n (-log(n+1) - j(log(n)) + 1)\big] \in O(log(n))$?

Find all bifurcation values of the function : x' = u + cos(x) + cos(2x) How do I find all bifurcation values of u? The solution: u < -2, u = -2, -2 < u < 0, and u = 0. How basically, I need to find the values for which u does not exist. Then I use linearization find the stability. But how?? B...

How would I prove that Z × N is countable? The hint given was to follow to indicated order. Thanks!

x = t^2 + 2t y = 4(t+1)^2 Determine the cartesian equation of the locus? What is the domain of the locus?

A body is performing SHM in a straight line OPQ. Velocity at mid point of P & Q is zero. At P & Q are V. Find time period. OP=X, OQ= Y

I have to find the Laurent series for $$ csc(z) \qquad |z|>0 $$ but I really don't know how to start. Please, guys.

I know that every finite dimensional normed space is reflexive, so this would imply X is reflexive. However, the author writes that this also follows directly from the fact that the second dual of $X$ is the space itself. I'm failing to relate how $X = X''$ and reflexivity of are one and the s...

I think the best way to prove this is by contradiction, but I'm struggling with the concept of how t write it properly. $$ A\setminus B = \emptyset \leftrightarrow A \subseteq B$$ Thanks in advance.

Let $0$ be the sequence of real numbers with all the components equal to $0$ and, for each $n \in \Bbb N$, let $\delta_n$ be the sequence of real numbers whose n-th component equals $1$ and all other components are $0$. Show that $0 \in l^2$ and that for any $a \in \Bbb R$, we have that $a\delta...

Suppose given the following: 1. $Y(t)=X(t)\text{cos}(2\pi f_0t+\theta)$ with $\theta \sim\text{unif}[-\pi,\pi]$, and $X(t)$ is a random process. My problem is: If not specified, can we say $X(t)$ and $\text{cos}(2\pi f_0t+\theta)$ uncorrelated? If I want to prove "yes", could I do the fol...

I'm trying to understand queueing systems and I found some notes online. They define $\lambda$ as the mean arrival rate, and $\mu$ as the mean service rate (the average number of customers who can be served by a single service station per unit time). The example they give is Customers arrive...

to prove that e−i|x|e−i|x| is not a characteristic function e−i|x|e−i|x| =cos|x|-i sin|x| Its conjugate will be cos|x|+i sin|x| which is not equal to fi(-x) Is my solution correct ??

Please help finding z-coordinate for constant positive and negative Gauss curvature in Mongé form : $$ x= \sqrt{R^2 + T^2} + R \cos u ,\, y= R \sin u ,\, z= .. $$ (Earlier seen Cassinian Ovals projection used to build up such forms, this may have been done before.)

Why can't Maple simplify this differential equation further? I want to obtain a recursion formula for $c_{n+1}$. I have tried using collect( ..., c[n+1] ), isolate( ..., c[n+1] ) and solve( ..., c[n+1] ) but it doesn't work. First term on left hand side can be written as $$ \begin{split} 3 \...

Given an undirected, unweighted graph $G$, let $A_G$ denote the graph's adjacency matrix. I want to understand in what cases there exists negative eigenvalue for $A_G$, for example, any claim of the form "$G(n,p)$ has at most $x$ negative eigenvalues" would be great for me. If there is any spec...

The question is as follows: Solve ((dy/dx)²+1)(x-y)²=(x+ydy/dx)² . Can we reduce it to Clairaut's form by taking appropriate substitution? Please help.

Can every monad give rise to a monad transformer? In the paper Calculating monad transformers with category theory by Oleksandr Manzyuk, one finds a construction of monad transformers as translating monads along adjunctions. In particular, considering the Eilenberg-Moore construction, we c...

Prove the following property: "If $x_n<=$y_n for every n>=n_0,then: the limit inferior of $x_n is <= than the limit inferior of $y_n." I'm lost on that.Please,could you give me any help?

Land is an indestructible capital asset that offers a perpetual revenue flow. You own a piece of land that returns $20,000 per year forever. The relevant real interest rate to consider is 4%. What is the present value of this land at time 0?

Let's say I have solved an ODE with Euler's forward method, and also solved it using RK4. Is there any way to look at the graph and "see" the order of accuracy of the methods?

I tried to solve this task,but I couldn't do anyway...Please,If you know help me!Thank you! Use integration by parts to evaluate each function ∫𝑥3√(1+𝑥2)𝑑𝑥

I was calculating one physical problem and I stopped at one thing. By definition Dirac delta is given by that expression: $$ \int_{-\infty}^{\infty}\delta(x)dx = 1 $$ and additionaly there is one property which is correct (I sure that, because I've proven it 2 years ago but I can't find a book)....

How can I decide when to use $\frac{\pi}{2}-x$ or $\frac{\pi}{2}+x$ and ${\pi}+x$ or $\pi-x$ while solving integration involving trigonometric function?

Assume that $\underset{x\rightarrow b}{lim}f(x)$ exists $\forall a,b\in\mathbb{R}$ and $a\neq0$ $ \Rightarrow$ $ \underset{x\rightarrow0}{lim}f(ax+b)=\underset{x\rightarrow b}{lim}f(x)$

Define $f:\Bbb R^2\to \Bbb R^2$ by $f(x,y)=(x+2y+y^2+|xy|,2x+y+x^2+|xy|)$ Prove that $f$ is differentiable at $(0,0)$. In order to prove that $f$ is differentiable at $(0,0)$ we must have $\lim _{|h|\to 0}\dfrac{|f(h)-f(0)|}{|h|}= 0$ But here $\lim_{h^2+k^2\to 0}\dfrac{h+2k+k^2+|hk|}{h^2+k^2}$...

When I need to sketch: $y=a^x-a^{2x}, a>0$, do I need to graph two functions? One when $0<a<1$ and the other when $a>1$? I didn't get any difference in the ascending descending intervals. Because $y'=lna(a^x-2a^{2x})$ so y'=0 when $x=log_a0.5$ and $y''(log_a0.5)<0$ So it has a max point. Than...

I'm wondering about invariant subspaces. If we have an endomorphism $f$ in $\mathbb{R^3}$ such as its matrix in canonical basis is \begin{pmatrix} 1 & 1 & 0 \\ -1 & 2 & 1 \\ 1 & 0 & 1 \\ \end{pmatrix} and $V=Vect{(1,1,1)}$, $Z=Vect((1,0,0),(0,1,-1))$ Is $V$ invariant by $f$ ? And how...

I was wondering about the following fact: $\forall x (x = f(y)) \implies \forall y (y = f(x))$ I think this is not true, but I can't find a counterexample but not I can prove it is wrong. Any ideas?

Suppose we have an $(n\times n)$-matrix $A(t)$ and an $n$-vector $b(t)$ that depend continously on $t$. Let $u\colon\mathbb{R}\to\mathbb{R}_{\geq 0}$ some function. In a proof, I found the following inequality: $$ 2\lVert A(t)\rVert u + 2\lVert b(t)\rVert\sqrt{u}\leq(2\lVert A(t)\rVert+\lVert b(...

A shop sells both hot and cold drinks. Hot drink sales occur at the instants of a Poisson process with expectation 30 drinks per hour.Cold drink sales occur at the instants of a Poisson process with expectation 20 drinks per hour. 60% of customers purchasing a drink are female, 40% of customers p...

Does every polynomial $a_0+ a_1x+ ... + x^n = 0$ where $a_i \in \mathbb{C}$ have a solution?

So i have a sequence defined by $a_1 =1$ and $a_{n+1} = a_n + \frac{1}{a_n}$ and i would like to know $\lim_{n\rightarrow \infty} a_n$. I have said that the sequence $a_n$ is unbounded and thus the limit does not exist. But I don't know how to rigorously prove that the sequence is unbounded and i...

I just want check if I got this right X~N(150, 60^2) N = 500 We take a sample n = 100 from the sample we take the sample mean x̄ What is P(x̄ < 120) ? When I tried doing this I got insanely large Z scores, did I do something wrong?

To establish, for which values of real parameter $\alpha$, the integral $$\int_0^{1/2}\frac{1}{x^\alpha \log x}dx$$ exists finite. For me, this problem is very difficult. Any suggestions please?

I have a positive operator A on the Hilbert space $\mathcal{H}$. I must prove that $\|A\|=\sup_{x \ne 0}\frac{(Ax,x)}{(x,x)}$. I am only able to get one inequality: Assume x is nonzero: $\frac{(Ax,x)}{(x,x)}\le \|Ax\|\|x\|/\|x\|^2\le \|A\|$. So $\|A\|\ge\sup_{x \ne 0}\frac{(Ax,x)}{(x,x)}$. An...

Let matrix $A=\frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -1 \\ 1 & 1 \\ \end{bmatrix} $. Check if $A$ is a rotation matrix in $\mathbb{R^2}$ by angle $\theta=\frac{\pi}{4}$. Entries of a matrix $A$ in trigonometric form are $A= \begin{bmatrix} \cos(-...

Consider the following series: $$\sum_{n=1}^\infty \frac{\cos(nx(n-1) - \cos(nx+(n+1)}{2(n+x^2)}$$ Now, for every $n\ > 1$, we have cancellations as the series is telescoping. Having said that, would it be right to claim that the series equlas: $$\frac{1}{2x^2} + \frac{1}{2(1+x^2)}$$

Find the sum function of the following power series $\sum\limits _{n=1}^{\infty}\frac{x^{2n+1}}{n}$ I actually felt like I had the right idea on this one, my solution is: Moving one $x$ out of the sum and then differentiating we get $$\sum\limits _{n=1}^{\infty}\frac{x^{2n+1}}{n}=x\sum\lim...

Let $$c_n=\sum_{k=1}^{2n-1}\frac{1}{k^a}\left(\frac{1}{(2n-k)^a}-\frac{1}{(2n+1-k)^a}\right)$$ Can we show that $\sum_{n=1}^\infty c_n$ converge?

Help please, If I have a: $X\sim U[0,a]$ s.t. $a>0$ $Y|X=x\sim U[0,x]$ then: $$f_Y (y)= \int_{-\infty}^{\infty} f_{X,Y}(x,y)dx = \int_{0}^{a} f_{Y|X}(y)\cdot f_X(x)dx=\int_{0}^{a} \frac{1}{ax} dx = \frac{1}{a}(ln(a)-ln(0))$$ How do I get past this? Thanks for any help!

Consider the matrix exponential map $H \mapsto e^{i H t}$ acting on the Gaussian unitary ensemble (GUE) of Hermitian matrices. I would expect that for large $t$, the resulting measure on the unitary group approaches Haar measure -- is that right? Is there a simple heuristic argument showing tha...

I need a textbook (or set of online lecture notes) on calculus of variation which focuses on the following topics "Variation of a functional, Euler-Lagrange equation, Necessary and sufficient conditions for extrema , Variational methods for boundary value problems in ordinary and partial diff...

Let $A = \left( {\begin{array}{*{20}{c}} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array}} \right)$. What is numerical range of $A$?

A=| 3 sqrt(2) sqrt(6)| |sqrt(2) 1 sqrt(3)| | sqrt(6) sqrt(3) 4 | Hi im suppose to find Eigenvalues and eigen vectors for this Problem but when i find the det(A-xI) i get a complicated polynomial with imaginary numbers as roots. and then when i try to solve for the e...

This following is excerpted from Category Theory by S. Awodey. "Theorem 1.6. Every category C with a set of arrows is isomorphic to one in which the objects are sets and the arrows are functions." Remark 1.7. This shows us what is wrong with the naive notion of a “concrete” category of sets and ...

Let $a_0 < a_1 < a_2 < \cdots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n \geqslant 1$ such that $$ a_n < \frac {a_0 + a_1 + \cdots + a_n} {n} \leqslant a_{n + 1}. \qquad \qquad \qquad \qquad (1) $$

If $G$ is a nonzero abelian group show that $$Hom_{\mathbb Z}(G,\frac{\mathbb Q}{\mathbb Z}) \neq 0$$

If $R$ is a ring. What is the relation between $R^{-1}$ and the set of units of $R$? Are they the same?

I am solving a problem on calculus of variation in which $F(x,y,y')$ is given as $F(x,y,y')=e^yy'^2$ After solving Euler equation I got this $2y'' +2y'-y'^2=0$. I don't know how to proceed further. Please guide me. Thanks in advance.

question in calculus: f(x) = [x], show the derivative at point x=2 (by definition of limits) Thanx (:

All definitions of a Winding Number I came across (or all equivalents), were only for a closed path. Could a Winding Number be defined for a path which is not closed? Will this definition be of any use?

Let $a,b,c,d$ be integers with $1\leq a,b,c,d\leq10$, and $x$ be real such that $ax^3-bx^2-cx-d=0$. How to find the maximum of $x$ ? Thanks in advance.

I want to prove the following theorem. Theorem: If lim(x,y)→(a,b) f(x,y)=L and limx→a f(x,y)and limy→b f(x,y)exist then limx→a limy→b f(x,y)= limy→blimx→a f(x,y) =L. This is how I tried this. Assuming lim(x,y)→(a,b) f(x,y)=L and limx→a f(x,y)and limy→b f(x,y)exist, ∀ϵ>0,∃δ1>0;0<[(x-a)^2+(y...

This question of mine is, if not completely then at least partially, motivated by the proposed proof of the $abc$ conjecture which was given by Mochizuki. Now suppose that I want to use consequences of the $abc$ conjecture in order to prove some other results and that I also did not read the pro...

The quantum group $U_q(sl_3)$ is generated by $E_1, E_2, F_1, F_2, K_1, K_2, K_1^{-1}, K_2^{-1}$ subject to some relations. I read some papers and there is a notation $K_{\lambda}$, where $\lambda$ is a weight. I think that we have $\lambda = a \alpha_1 + b \alpha_2$ for some numbers $a,b$. For e...

I'm reading a paper on Truncated distributions, where some asymptotic tests are considered to determine if a data sample is truncated or not. I'm confused about the following line: An asymptotic test based on $L(\hat{\alpha})$ (this is a known statistic) rejects $H_0$ on asymptotic level $...

Let $(E,r)$ be a metric space, $D_E[0,\infty)$ be the Skorohod space on $[0,\infty)$ takes value in $E$. Consider the function $$(D_E[0,\infty)\times[0,\infty),\mathcal B(D_E[0,\infty)\times \mathcal B([0,\infty))\to (E,\mathcal B(E))$$ $$(x,t)\mapsto x_t$$ I am wondering whether this map is meas...

$$ \frac{1}{x} <E $$ E always positive. how can I get to expression like this: $ x > something $ in 1 stage? if it's not possible in 1 stage (although this is what my teacher did) another answers is also good :)

Is part inside loop at a vertex considered as region in graph theory? Edit

I want to prove that $\sum_{n=1}^\infty n^{ln(x)} $ is convergent for x in $(0, \frac {1}{e})$ interval and divergent for $ x \geq \frac {1}{e} $. I am lost on how to prove it. Could someone please show me or give me a hint? I assume I would need to use one of the criterias for determining conv...

I have to use the definition of uniform continuity to disprove that $cos(\frac{pi}{x})$ is uniformly continuous, but I don't know how to do that.

Consider the non-negative matrix $B$ with dominant eigenvalue $\mu$. Let $s\ge\mu$. Let $R$ be the resolvent of $B$, i.e. $R=(sI - B)^{-1}$. What conditions must $s$ and $B$ satisfy such that all the column sums of $R$ are equal?

I'm trying to find a formula that will allow me to calculate the sum total of a progression (not sure if that's the word) in a spreadsheet. 1 + 0.79 + 0.79*0.79 + 0.79*0.79*0.79 + ... I can simplify the parts between the plus signs and graph them with 0.79^x But how do I calculate the sum of t...

Let $p$ be a positive integer. For each nonnegative integer $k$, write $[k]$ for the set $\{0,1,2,\ldots,k\}$. Also, we define $[-1]:=\emptyset$. We say that an integer $k\geq -1$ is $p$-splittable if there is a partition of $[k]$ into $p$ subsets $A_1$, $A_2$, $\ldots$, $A_p$ such that $\s...

I am reading through "Matrix Computations" by Gene H. Golub and Charles F. Van Loan and have come across a proof on the properties of Tridiagonal Decomposition that seems to gloss over parts I do not understand. The Theorem states that: If $Q^TAQ=T$ is the tridiagonal decomposition of symmetr...

Real numbers $a_1, a_2, \cdots, a_n$ are given. For each $i$ ($1 \leqslant i \leqslant n$) define $$ d_i = \max \{a_j : 1 \leqslant j \leqslant i \} - \min \{a_j : i \leqslant j \leqslant n \} $$ and let $$ d = \max \{d_i : 1 \leqslant i \leqslant n\}. $$ (a) Prove that, for any real numbers $x_...

1 2 3 4 ...10..n n 2n 3n ...100..n^2 Blockquote what is the sum of this table? i think that it maybe solvable with series. i'll be grateful if anyone could help me.

I'm trying to find the extrema of f(x,y)= cos(x²-y²) constrained to x²+y²=1. Using Lagrange Multipliers I get this far: -x(sin(2x²-1)=λx -y(sin(-2y²+1)=λy But I don't know how to proceed after this. Could anyone explain to me what I have to do next please?

Recently I came across the Riemmann representation of the Zeta function as follows: Zeta(s) = (2^s)*(pi^s-1)*sin(pi*s/2)*Gamma(1-s)*Zeta(1-s) Now, I went ahead to calculate the term Gamma(1-s) for s=2, knowing beforehand the values of the otehr terms, namely, Zeta(2) = pi^2/6 (Bassel problem)...

x' = λ - ρx - βxz; y' = βxz - δy; z' = py - cz; x0=43100; y0 = 0, z0 = 0.0033, λ = 388, ρ = 0.009 δ = 0.18, p = 50000, c = 23, β=3.61e-8 my code is as follows: f = @(t,x) [388-0.009*x(1)-0.0000000361*x(1)*x(3); 0.0000000361*x(1)*x(3) - 0.18*x(2); 50000*x(3) - 23*x(3)]; tspan = 0:7:84; [t,xa] = o...

How many solutions are there to the equation x1 + x2 + x3 + x4 + x5 + x6 = 25 where each xi is a non-negative integer and 3 ≤ x1 ≤ 10

Working through a textbook on algorithms (CLRS intro to algorithms) and just wanted to see if someone could help me understand one of the exercises at the end of a chapter. Problem: Is n^(2+1) = O(n^2)? The book uses = loosely to convey the "set of" notation, so it's more like "is n^(2+1) con...

for $u$ scalaire function defined on $\mathbb{R}^n$ we have this equality : $\nabla(|\nabla u|^{p-2})=(p-2)|u|^{p-4}|\nabla u|^{p-4}\nabla u\nabla\nabla u$ my question is : what doese it mean $\nabla\nabla u$ because i know only the signification of the gradient of a function , but here will be ...

I want to prove that $\sum_{n=1}^\infty cos(n^2\pi)\cdot(\sqrt{n+11}-\sqrt{n+2}) $ is convergent (or divergent). I am lost on how to prove it. Could someone please show me or give me a hint? I assume I would need to use one of the criterias for determining convergence? I tried to multiplicate ...

Consider the 2-dimensional system of non-linear ODEs, semplified instance of a predator-prey population model $\dot x=\alpha x (1-x)-xy$ $\dot y = y(x-y)-\beta y$ with $\alpha = 1 ,\beta = 1/2$ Why does $x(0) > 0$ and $y(0) > 0$ imply that $x(t) > 0$ and $y(t) > 0$ for all times (populations...

I have this question i need to solve - let p be a prime number. sort groups of order $p^2$ Any help or hint that will help me begin solving this question will help.

I'll start by stating the problem text: "A string is streched along the x-axis from $x=0$ to $x= \pi/2$ and is made to vibrate by a force proportional to - $f(x)sin( \omega t)$. The amplitude, $y(x)$ , of small vibrations is then given as the solution to: $$ y'' + y = f(x) $$ Where $y(0) = y(\pi...

By making the substitution $x=\pi -t$ show that; $$\int_{0}^{\pi} xf(\sin x)dx = \frac{1}{2}\pi \int_{0}^{\pi} f(\sin x)dx$$ where $f(\sin x) $ is a given function of $\sin x$. I get $dx = -dt$ and that $\sin (\pi -t) = \sin t$ which gives me; $$\int_{0}^{\pi} xf(\sin x)dx =\int_{0}^{\pi} tf(\s...

A little earlier I posted a question Isomorphism between ring of polynomial functions on unithyperbola and Laurent polynomials which was marked as duplicate by somebody who didn't want to help me further. He said that my question is a duplicate of Localization in a ring However I don't see wh...

At a lecture about factor rings the lecturer gave us the following example: $$\mathbb{C} = \mathbb{R}[t]/(t^2 + 1)\mathbb{R}[t]$$ He said that it was quite obvious and skipped the explanation. But it is not obvious for me at all and I have some questions: How could $(t^2 + 1)\mathbb{R}[t]$ ...

In this example here: I am having trouble understanding the steps in applying the product rule, like what is set as u, du, v, du? Please help clarify.

So the algebraic multiplicity is largest power k such that (x-lambda)^(k) which is a root of the characteristic polynomial. Geometric multiplicity is the dimension of an eigenspace for each lambda. Eigenspace= Ker(T-lambda.I) I set a basis for this eigenspace to be {v_1,...,v_m) I extended the...

It is given a relation R={(1,1); (2,2); (3,3); (4,4); (5,5); (3,5); (4,3); (4,2); (4,5); (2,5)} for a set A={1, 2, 3, 4, 5}. I found this relation is reflexive, anti-symetric and transitive, i.e an order relation. If order relations exists, there must be unique "factor" that is valuable for all...

would please someone help me out with that. I print Screen for everything I have done so far. The assumption is missing .

Let $a$ and $b$ be positive integers. Show that if $4ab - 1$ divides $(4a^2 - 1)^2$, then $a = b$.

I have seen elsewhere that: $y=\sin x/ x$ has a horizontal asymptote of $x=0$, as it approaches that line as x tends to +/- infinity. Now, why does it not have an asymptote at $y=0$ or $x=1$, as the curve tends towards but never touches these lines? (Which satisfies the definition given by wol...

I have tried using rsolve in Maple to obtain a recursion formula from an ordinary differential equation with summations. I get Is there some reason for Maple not calculating the sums? It seems rsolve is not used at all; however, it seems to work if I don't use sums in $x(t)$ and $u(t)$. I a...

$e^{3n}\: f \: g^{2n}\: h$ n ∈ ℕ∪{0} For Example: fh eeefggh eeeeeefggggh I am stuck at the $e^{3n}$ and $g^{2n}$ part. This is what I have so far: S→EfGh E→eE∣ϵ G→gG∣ϵ Please correct me if I am wrong. Thanks

How many cuts are needed to divide a circle into $N$ pieces (not necessarily equal pieces)? Is there any formula for this?

**Let enter image description here be the augmented matrix for a linear system. For what values of a and b does the system have: a) A unique solution; b) Infinitely many solutions; c) No solutions.

Consider X = ℝ with the lower limit topology, and Y = ℝ with the upper limit topology. The set [0,1) × [0,1) is an open set in the product topology on X × Y.

Consider $A:\mathcal{D}(A)\subset L^{2}[0,1]\to L^{2}[0,1]$ given by $$A:=-\frac{d^{2}}{dx^{2}},\qquad\mathcal{D}(A):=C^{2}_{0}(0,1)$$ Now, I assume that the closure of $A$ is its extension defined on $C^{2}_{0}[0,1]$. Then am I right in thinking that this is this densely defined if $\overline{C...

What's the difference between $R^{4}$ and $R^{1,3}$? I know that the first one has metric Kronecker delta $\delta_{ij}$. Does the second one have Minkowski metric $g_{\mu \nu}$?

It is given a relation R={(1,1); (2,2); (3,3); (4,4); (5,5); (3,5); (4,3); (4,2); (4,5); (2,5)} for a set A={1,2,3,4,5}. I found this relation is reflexive, anti-symetric and transitive, i.e. an order realation. If order relation exists, there must be unique factor that is valuable for all orde...

this question in given in discreet mathematics by kenneth rosen. Each inhabitant of a remote village always tells the truth or always lies. A villager will only give a "yes " or a "No" response to a question a tourist asks. Suppose you are a tourist visiting this area and come to a fork in the r...

If $X$ is a normed vector space and $M$ is a proper closed subspace, I want to show that for any $\epsilon>0$ there exists an $x\in X$ such that $\|x\|=1$ and $\|x+M\|\geq 1-\epsilon$. Is there anything wrong with arguing as follows: Suppose by contradiction there exists some $\epsilon >0$ such ...

I have the following problem, and I don't know what I'm missing: Let $\Omega = D(0,1)\setminus\{0\}$. I want to find all $f \in C^2(\Omega)$, such that $$\Delta_{\Bbb R^2} f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = r \cos\varphi = g(r,\varphi) \qquad (x,y...

I get the Extended Euclidean Algorithm but I don't understand what this question is asking me: In this question we consider arithmetic modulo 60 ("clock arithmetic" on the clock with numbers {0,1,2,3,...,58,59}). Which of the following statements apply? a. 50 X 50 = 40 modulo 60 b. 17 + 50 =...

Homework Question We have been given the homework question that is in the picture. I am not sure if we understand correctly what is asked of us. Would be grateful for any help.

Is this series convergentor divergent. I first thought it was convergent due to pointwise limits existing at every value of x but now im not sure wether or ot I am right. My series is fn(x)=(x^n)/(1+x^n) with x ∈ [0,1]. How do I decide wether this series is convergent or divergent.

Can someone please explain the questionmarked extrapolation in the following image?

Let $S$ and $T$ be sets, and let ${\phi}:S\, {\rightarrow}\,T $ be a mapping. Let $*$ be an operation on $S$. Let $x$ and $y$ be any elements of $S$ and let $a$ be any of the set's cancellable elements (under $*$). Then, from the above, we have that ${\forall}x,y,a\,{\in}\,S:x\,*\,a=y\,*\,a{\im...

I solved a) and b). But I cant seem to get a grip of what characterizes the functionals for which we want continuity in the general case. Hints please!

I am encountering dual bases for the first time in the context of algebraic number theory, mainly in proofs regarding the existence of an integral basis for $\mathcal{O}_K$ and its ideals. I am wondering about motivation for defining a dual basis and intuitively what it is. So far I have been wor...

Let a and b be numbers in the set S = {0, 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10} such that b is the inverse of a (mod 11) and a and b are not equal. How many such subsets {a, b} of S are there?

Let there be a natural number $a,$ that can be expressed as $a = x^p - y^p$ where $x, y$ and $p$ are natural numbers, each two of them being pairwise co-prime and $p$ is an odd prime. Then can $x$ and $y$ be found out uniquely in terms of $a$ and $p$? If yes, how?

The sequence goes: 1,$\frac{2}{3}$,$\frac{7}{9}$,$\frac{20}{27}$,$\frac{61}{81}$ I tried using the common difference method of analysis and found the second row follows the rule: multiply by $\frac{-1}{3}$, Third row onwards seems to follow the rule: multiply by $\frac{1}{3}$. I can't figure ou...

I have $X$ and $Y$ which are independent random variables following the normal distribution. How should I prove that a random variable ($Y-2X$, $X+3Y$) has bivariate normal distribution? And how should I find the values of parameters for it? Thanks in advance!