Could you please help to prove that $col(AB) \subseteq col(B)$ and $col(AB) \subseteq col(A)$ given $Ax=b$ and $Bx=c$ and $ABx=d$ or in other words given the column spaces $col(A)$ and $col(B)$ and $col(AB)$?

I just have a fast question. How do you solve the following? Intuition? 0.8log(9-0.2x)+0.2log(2+0.8x)=0.6log(10-0.2x)+0.4log(3+0.8x) Thanks

Let $*$ be an operation such that $(xy)^* = y^*x^*$, e.g. if $xy$ are 2-by-2 matrices and $*$ is "take the inverse". Is there a name for such a property ?

prove thath $K(G) \leq K(G-e)$. $K(G)$ is domination number of $G$ $ K(G-e)$ is domination number of $G$ $e$ is an edge of $G$

I first expanded the equation into $\frac{1}{(z-i)(z+i)}$, then I apply the theorem that $$\underset{z=i}{Res}\frac{1}{(z-i)(z+i)}=\lim_{z\rightarrow i}(z-i)\frac{1}{(z-i)(z+i)}=\frac{1}{2i} $$ but the correct answer should be $\frac{-i}{2}$, which step is wrong?

I have the standard definition of an odd-function from wikipedia: Again, let f(x) be a real-valued function of a real variable. Then f is odd if the following equation holds for all x and -x in the domain of $-f(x) = f(-x)$ Can anyone help me how to do this? Do I have to show, that it...

Is it possible to find two complex numbers $w,z$ and a complex exponents $\alpha$ such that the principal values of $z^\alpha w^\alpha$ and $(zw)^\alpha)$ are different?

How to solve the equation of sqrt(x)+sqrt(y)=sqrt (N) (where N=2205) in integers? How in general to solve the similar equations?

I have the linear problem as it follows. I have 3 different types of devices. Type A, Type B, Type C. At any given moment, there is exactly one type of each device installed. So one device A, one device B and one device C. Each type of device has its own probability to get damaged, so I want to...

I need help calculate : $\underset{n\rightarrow\infty}{lim}\left(\cfrac{1}{n^{2}}+\cfrac{2}{n^{2}}+...+\cfrac{n}{n^{2}}\right)$

I have a nonlinear equation, that I would like to resolve. So I would like to use a PI controler for that. Is it possible to make my equation as a feedback and to use a PI controler? Thank's.

let (X,d) be a metric space. which of the following is possible? (A) X has exactly 3 dense subsets (B) X has exactly 4 dense subsets (C)X has exactly 5 dense subsets (D) X has exactly 6 dense subsets

how do you solve the following? My intuition tells me to take exp and then raise it to five to get a polynomial. But, it stops there. Any ideas? 0.8ln(9-0.2x)+0.2ln(2+0.8x)=0.6ln(10-0.2x)+0.4ln(3+0.8x)

I have a conceptual problem with the relation between two integrable functions that are equal a.e.. Here there is a possible setting, to make things more concrete. Let $(X, \Sigma, \mu)$ a measure space, and let $\phi \in [0,1]^X$ be a measurable (and clearly integrable) function, such that for ...

How can I integrate $$\int \sqrt{1-sin(x)} \ dx$$ Can't seem to figure it out. Thank you.

Integrate sin^3xcos^4x dx by substitute u=cos x Can anyone help me cuz i am new to this integration. A little show on working step much appreciate

Is $(-2)^{\sqrt{2}}$ a complex number and why? Is there some reason if this is a complex number why we can't define it to be an integer?

For a project about Systems Theory we need to study the proof of the Routh-Hurwitz test by Meinsma, 1995. The theorem says the following. A nonconstant polynomial $p(s)=p_0s^n+p_1s^{n-1}+\ldots+p_n$ with $p_i\in\mathbb{R}$ and $p_0$ not equal to zero is stable if and only if $p_1$ is nonzero, $p_...

I try to understand the Max-Flow-Min-Cut Theorem, but somewhat fail to apply it to the Project Selection Problem. My difficulty can be illustrated by the example in this wikipedia article: https://en.wikipedia.org/wiki/Max-flow_min-cut_theorem#Project_selection_problem Graph Three projects $(...

Hej out there. I have a assignment, and two of the questions are bugging me. 1. show that in the equation x^n + y^n =z^n have a (none trivial) whole number solution, so x^n + y^n =z^n ( mod p) for a prim number p, but not the other way around 2. show that Fermats last theorem, can be shown by x...

Using Peano Axioms I have formalized the following: x is the square of an odd prime number For some odd prime number x' , x is its square IF x is some odd prime number, THEN x is the square of x' IF x' is some odd number AND x' is some prime, THEN x is the square of x' [x' is some odd number AN...

Let $f:[a,b]\rightarrow\mathbb{R}$ be a continuous function. Show that there exists $c\in[0,1]$ such that $\int_0^1 x^2 \! f(x) \, \mathrm{d}x = 1/3 f(c)$ I am stuck with this one. $f$ is said to be continuous on [a,b] so how can we conclude about [0,1]? Also can you please give me a clue abou...

I am using these as references: How to find the dimensions of a rectangle if its area is to be a maximum? Does the symmetry of a parabola in finding the maximum area of a rectangle under said parabola matter? (related) Problem: Find the maximum area of a rectangle inside a parabola whose e...

I cannot figure this out help. This deals with Mathmatical induction problem. I've tried factoring but it doesn't work out.

I cannot find a number that has two representations in this ring, please help

A graph with n nodes "shortest" is the sum of the edges weight of the shortest path tree to the root.minimal is the sum of the edges weight in MST. How Can I find the upper bound of shortest/minimal??

Let the manifold $S$ in $R^n$ be defined by $g(x)=0$. If $p$ is a point not on $S$, and $q$ is the point of $S$ which is closest to $p$, show that the line from $p$ to $q$ is perpendicular to $S$ at $q$, Hint: Minimize $f(x)=|x-p|^2$ First off I don't understand why that is the function to minim...

For an even $n \in \mathbb{N}$, the sign of $\tau \in S_n$ is 1 if and only if the number of disjoint cycles in $\tau$ is even? And it is odd for odd $n$?

The diameter of the planet Earth is about 8000 miles. Imagine someone has tied a rope around the Earth ar the equator. You come along and as a practical joke put yardsticks all the way around the Earth and prop up the rope all the way around the equator. Estimate IN FEET to within 5% how much rop...

Let $f$ be an entire function such that $if(z)= \bar f(z)$ for all $\Bbb C$. Show that $f$ is the constant function. I didn't understand it but I took some notes which seem like gibberish to me now. Maybe someone could clarify: $\bullet$ I had drawn a line on a graph that has an arg of $\pi/4$...

I'm running into a lot of questions on linear algebra lately where, instead of using constant numbers in matrices and vectors (these are questions which I have no problems in answering), they use a mix of constants and a variable a. I'm finding these difficult to answer. For example: 1) Determin...

I have (n+1)! / (n+2)! which simplifies to 1/(n+2) But I dont understand how this works Could someone explain the theory of factorials divide like this ?

For an i.i.d. sample of random variables Xi distributed according to a normal distribution, known variance. I found a sufficient statistic—the sample mean. How do I check if other statistic like (X1+2X2)/3 or ∑Xi/(n+1) are also sufficient. Thank you.

I am see my notes about curves on complex spaces and I do not understand why it is so... I need help. I have this example: My notes said that I can defined $\theta: [a,b] \rightarrow \mathbb{R} /$ $\theta(t)=arg(\gamma(t))$ when $a\leq t \leq t_1$ $\theta(t)=arg(\gamma(t))+2\pi$ when $t_1< t

I do not know how to deal with this problem: show that a real polynomial in two variables which does not change sign has only a finite number of roots

Three friend walk into a bar, sit down, and when waiter comes to them, they ask him how much is the beer. The waiter tells them that the beer is $10$ dollars. They order three beers and each of them gives him $10$ dollars. A few moments later the boss of the bar arrives and asks waiter how much ...

Is that true? $x^y \equiv x^z \pmod p$ if and only if $y \equiv z \pmod p$ If yes, then how to prove it?

This may sound stupid but, is this correct?: Is 100 a hundred percent "better" than 50? (Better as higher or something like that) If it is correct: How "better" is 50 than 30? and How much "worse" is 30 than 50? Don't kill me, please

Hey need help to calculate : $\underset{n\rightarrow\infty}{lim}\cfrac{n^{2}}{\left(2+\frac{1}{n}\right)^{n}}$

Basically, I have written codes for composite trapezoidal, and simpson numberical integration rules. I want to now to do graphs for the error (Number of rectangles vs error) of each of these methods. Is there a built-in function that do this in matlab? or do we have to write the code for it?

How to check if transformation $F:\Bbb{P}^2\to \Bbb{P}^5$, given as follows $F([x,y,z])=([x^2,y^2,z^2,xy,yz,zx])$, is smooth, immersion or embedding?

The questions is: The width of a right triangle is twice its height. Find all angles in the triangle. How do I proceede? :)

Lets say there are 200 tickets in a lottery system. Six tickets are drawn one by one, without replacement. These six tickets win a prize. Lets say you bought 1 ticket, what is the probability that you win a prize? The obvious answer would just be 6/200 = 3%, but isn't it more than that since the...

Consider $Y = \frac{1-\sin(2X)}{1-\sin(X)}$, where $X$ is a random variable. How to find the distribution of Y?

I have looked at the following source here (part (b)) for this proof. But I have a hard time seeing that it is completely correct. If we take $r = min(r_1, r_2)$ as in the reference how can we be sure that $B_r(x) \subset A_1 \cap A_2$ What if the portion of $A_1$ or $A_2$ that we needed for t...

Where m and n are positive integers. Prove or give a counter example.

From a sample of 1751 army hospitals, estimate the mean expenses for a full time equivalent employee for all US army hospitals using a 90% confidence interval given x = 6563 and s = 2484. Work: 1.645(2484/41.845) 1.645(59.362) 97.65 6563+- 97.65 6465.35---6660.65 ANswer I think I did that o...

Prove that for every real number x with $x\geqq -3$ there exists a real number y such that $ \frac{y(y-6)}{3}=x $ $\forall x\in \Bbb R \wedge x\geqq-3 \ \ \exists y \in \Bbb R : \frac{y(y-6)}{3}=x \\ $ To prove this directly I must solve the equation for y and then plug the result into the le...

How do I show that my series "the sum from n=1 to infinity of 1/n^3" is convergent using the integral test ?

I'm reading Ken Shoemake's explanation of quaternions in David Eberly's book Game Physics. In it, he defines the rotation matrix for a quaternion $q = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} + w\mathbf{1}$ to be the product of two matrices: $\begin{align} \mathbf{Q\overleftarrow{Q}}^T &= \begin...

I asked this question recently (Basis of differential one-form confusion), thought I understood the answer, but now realise I don't. 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 $f$, b...

The question is as follows: Given vector field $$V = \left(\frac{1-y}{x^2 + (y-1)^2}, \frac{x}{x^2+(y-1)^2}\right)$$ Evaluate $$\int_{l_1}V \bullet dr\text{, }\int_{l_2}V \bullet dr$$ Where $l_1$ and $l_2$ are given as $$l_1: x^2+(y-1)^2=1\text{, } l_2: x^2+(y-4)^2=1$$ According to the ...

I want to prove this statement. Q(sqrt(2), sqrt(3), ..... , sqrt(n) ) = Q(sqrt(2) + sqrt(3) + .... + sqrt(n) ) for any n >1 It looks very hard problem. How can i approach this one? Thanks!

Can anybody please provide me proof of Cantor set is uncountable which is independent of ternary expansion. Thanks in advance.

The modified Bessel differential equation is always presented as $$r^2 \frac{\partial^2 f(r)}{\partial r^2} + r\frac{\partial f(r)}{\partial r} - (r^2 + n^2)f(r) = 0$$ with solutions $$f(r) = AI_n(r) + BK_n(r)$$ But if I had $$r^2 \frac{\partial^2 f(r)}{\partial r^2} + r\frac{\partial f(r)}{...

I have been assigned to do the question I've attached. I have managed to do a,b, and c. Now I have 2 questions: (I'll use normal brackets for inner product brackets) Firstly, in part (a), I used that lim as n-->∞ of (An(x)-A(x)|y) = ∞ of: (An(x)-A(x)|y), and I don't know how to prove this contin...

Can someone explain what this formula is doing? =EXP(x)/(1+EXP(x))*100 If you are not familiar with EXP, EXP calculates the e^x. Thanks

This is from Spivak Calculus on Manifolds, section 5.3 I have done part a, but I am stuck on part (b) and have been for a day now: let $p = (p_1,p_2,p_3)$ then $w(p)(v_p,w_p) = \dfrac{p_1 dy \wedge dz (v_p,w_p) + p_2 dz \wedge dx (v_p,w_p) + p_3dx \wedge dy (v_p,w_p)}{|p|^{3/2}}$ but I am uns...

I posted early a bit a same question but i wanted to know if i'am doing alright another one. Let X be the number of customers buying a book in a bookstore e-shop. Assume X has a Poisson distribution with a mean of 1 books bought every 10 minutes. (a) What is the probability that no one will buy ...

I have fitted a polynomial surrogate model $y = f(x)$ to several sets of data and need to find the inverse of the function, i.e. $x = f^{-1}(y)$. The data is very well fit by polynomials - errors <1E-15. And all data is in the range [0,2]. However some of the polynomial surrogate models are quit...

p is prime and a have a multipl. inverse in Z/pZ. prove: If a^(p-1)/2 ≡1 mod p then the congruence x^2≡a(mod p) has a solution x.

Problem: Assume we have the next recursive sequence: $x_n=\sqrt[3]{6+x_{n-1}}$, $x_1 = \sqrt[3]{6}$ Prove that exists such constant C (C $\neq$ 0) that: $\lim_{n\to+\infty}\frac{12^n(2-x_n)}{C}=1$ Can anyone help me with it?

I know this solution has already been shown but each time I have seen examples of this proof I get lost at the last step. \begin{align} \cos^2 x + \sin^2 x &= \sum_{r = 0}^{\infty} \frac{(-1)^r}{(2r)!}\biggl(\sum_{k = 0}^r \binom{2r}{2k}(-1)^{2k}\biggr)x^{2r} + \sum_{r = 0}^{\infty} \frac{(-1)^r...

Suppose that we have two vector fields $V,W \in TM$ with $M$ a differentiable manifold. We define the Lie Bracket as $[V,W](f)=V(W(f))-W(V(f))$ and it's easy to show that this bracket is indeed a vector field. I feel like I'm missing something but then why is not possible to define the Lie Brac...

I need to understand these proofs, but I haven't managed to find any material on these questions (especially the latter two). Any help proving these and explaining them would be awesome. Prove that P is in NP, Prove that polynomial-time reduction from NP-hard language L1 to L2 implies L2 is al...

Here is the link to the question and the answer: https://gyazo.com/d98918d0e0eaabe88606c0314aff0aca Here is the link to what I have done: https://gyazo.com/20ddc3822c0c7bca888cd3334dd78281 What happens to 3PI? Where does it go?

if $limit_{n \to \infty} a_{1} + a_{2} + a_{3} + ... +a_{n}= L$ So, $limit_{n \to \infty} \frac{1}{a_{1}}+ \frac{1}{a_{2}} + \frac{1}{a_{3}} + ... + \frac{1}{a_{n}}= \frac{1}{L} $ ?

Company is planning to pay a dividend of 5\$ per share (dividend for previous year). Investor that wants to buy a shares of this company assumes that dividend will be stable (Thus will not change in future). What is the maximum price that investor can pay for one share, assuming that he will rece...

I am finding the divisor of $f = (x_1/x_0) − 1 $on X, where X = $V ( x_1^2 + x_2 ^2 − x_0^ 2 ) ⊂ \mathbb P^ 2 $. Characteristic is not 2. I am totally new to divisor. So the plan in my mind is first find the prime divisor, and find ord(f) on every prime divisor. But I am having trouble in the fi...

Use a triple integral to find the volume of the solid bounded below by z = x 2+y 2 and bounded above by $$z = 8 − 2sqrt(x^2 + y^2) $$ How would you set up this integral?

Determine the number of correct digits in the number $x$ given its relative error $E_r$ (a): ~ $x=0.4785, E_r=0.2\times 10^{-2}$ (b) :~ $x=386.4, E_r=0.3$ (c): ~ $x=86.34, E_r=0.3$ For the problem (b), we have $E_r=0.3<0.5=\frac{1}{2}<\frac{1}{2} \frac{10}{4}=\frac{1}{2}\frac{1}{(3+1)\times ...

Consider the iteration matrix for the general splitting method $M=I-N^{-1}A$ where $N$ is any invertible matrix. Show that if $\lambda =1$ is an eigenvalue of $M$. then $A$ cannot be invertible. I tried: $$Mx = \lambda x$$ $$(I-N^{-1}A)x = x$$ $$I-N^{-1}A = I$$ $$0 = N^{-1}A$$ But i'm not sur...

A matching $M$ in a undirected graph $G(V,E)$ is a subset of the edges of $E$ such that no two edges in $M$ are incident to a common vertex. A perfect matching ${M}'$ is one in which every vertex is matched. Example Image So I want to express the set ${M}'$ with a mathematical-logic notation. ...

My work is having a raffle tomorrow for a grand prize. 200 tickets have been sold, and there is only 1 Grand prize to win. I have bought six tickets. What are the odds I will win the prize?

1) Since power series $\sum \limits_{n=0}^{\infty}a_nx^n$ converges at point $x=1$ then radius of convergence $R\geqslant 1$. Then $\sum \limits_{n=0}^{\infty}a_nx^n$ converges absolutely for $x<1$ since power series converges absolutely at any interior point of $(-R,R)$. 2) Hence $\sum \limits...

If I have a set A consisting of all the functions f that rotate a plane by some number of degrees a E R, then what is the identity? Since the identity is supposed to be unique, should it be f that rotates the plane by 360 or f that rotates the plane by 0?

Here's the problem: Let r be the remainder when (a−1)n + (a+1)n is divided by a2. For example, if a = 7 and n = 3, then r = 42: 63 + 83 = 728 ≡ 42 mod 49. And as n varies, so too will r, but for a = 7 it turns out that rmax = 42. For 3 ≤ a ≤ 1000, find ∑ rmax. Source What I...

Does such intersection exists? im thinking about $An=(3+1/n;4+1/n)$ since $\bigcap An = [3,4] $ so its closed and bounded then its compact. Can someone please say whether its correct or not?

I am really having difficulty with this problem. I understand part a, as I did d/dx[x^2y'+(x^2'-2x)y]=0 and then took the integral of both sides to get lny==x+2lnx+C. What I don't understand is how the integrating factor makes an equation exact and how to prove that.

I'm given the integral $$\int \limits_0 ^{+ \infty} \frac{e ^ {-\cos t} \cdot \sin (t ^ \beta)}{t^\alpha} dt \qquad a,b \in \mathbb{R}$$ and I need to test the absolute convergence. I split it in two parts, namely $$\int \limits_0 ^{1} \frac{e ^ {-\cos t} \cdot \sin (t ^ \beta)}{t^\alpha} dt + \...

I need some help with this problem, Suppose that R1 and R2 are reflexive relations on a set A. Show that R1 ⊕ R2 is irreflexive?

Consider the polynomial (−3x + 4y)^8. What is the coefficient of x^3y^5 in this polynomial? I need help clarifying this. When I create a Pascals Triangle I get the answer to be 56. Yet when my professor posted the answer to this problem, the answer was -8*7*27*1024.. How does that make any sense...

Let $g\in \mathbb{Q}[X]$ be irreducible with $\text{deg}\;g=3$. We assume that only one of the roots of $g$ is in $\mathbb{R}$. So, $\alpha \in \mathbb{R}$. Let $L\subset\mathbb{C}$ be the splitting field of $g$ over $\mathbb{Q}$. Show that for every homomorphism $\tau: \mathbb{Q}(\alpha)\right...

This are my steps: $0$ is certainly in $ℝ^n$ If $at \in ℝ$, then $atv=t(av)=ℝ(av)=a(ℝv)$ If $t+s \in ℝ$, then $(t+s)v=tv+sv=ℝ(tv+sv)=(ℝt+ℝs)v$. Since I felt my steps are partly erroneous, could anyone fix it please?

Let A = $\begin{bmatrix}1 & 0\\1 & 0\end{bmatrix}$ Find two matrices B and C with AB = AC, and B does not equal C. I always have trouble with problems like this. Here, I know $\begin{bmatrix}1\\0\end{bmatrix}$ and $\begin{bmatrix}1\\1\end{bmatrix}$ works, but I've always done it with trial and ...

How do I prove that the determinant is the only function that has the properties of the determinant (multilinearity,being alternating, $det(I)=1$). In my notes we started to prove it like: suppose there is another such function(say g)... But from then on I can't make sense of it.

How many terms of the Maclaurin series of $f (x) = \ln(1 + x)$ are needed to compute ln(1.2) with an error of at most 0.0001?

The proof looks easy, but how can you prove using official terms?

Let $x$ be nonnegative real number. The function $f_n$ is defined by: $$ \begin{align} f_n : &\mathbb{R}_{+}\mapsto\mathbb{R}\\ &x\phantom{R}\mapsto 3x^ne^{-x^2}-1. \end{align} $$ Let $v_n$ be a real number for all $n>1$ such that : $$ v_n >1,\;\text{and}\;f_n(v_n)=0. $$ Calculate the limit...

I know that $\{1,x,x^2 \}$ is a linear independent family on $[-1,1]$, because $ax^2+bx+c$ has at most 2 roots, therefore if it equals $0$ for all $x$ of $[-1,1]$, it implies that $a=b=c=0$. However, I am wondering if the following proof is right or not: if $ax^2+bx+c=0$ on $[-1,1]$, it is true ...

The region in R 3 bounded by the planes y = 1, y = −x, z = −x and, the coordinate planes, x = 0 and z = 0 $$ \int_{0}^{1} \int_{0}^{-y}\int_{0}^{-x} \;dz \;dx\;dy $$ I feel like I am a bit off with the bounds here? Can someone please confirm?

I am trying to use FEM to approximate the solution to the following BVP: $-\frac{d}{dx}[a(x)u'(x)]+b(x)u(x)=f(x)$, on [0,1] where $u(0)=0$ and $u(1)=1$. I am using the Galerkin method with hat functions: ${\varphi _j}$$\left( x \right)$ = $\left\{ {\begin{array}{*{20}{c}}{\frac{{x - {x_{j - 1}}...

Let $f_i \colon [0,1]\rightarrow \mathbb{R}$ be a sequence of functions which converge to $f_\infty$ pointwise. How can I prove that $\lim_{i\rightarrow \infty} \int f_i d\lambda= \int f_\infty d\lambda$, when $||f_i||_2\leq 1$? My attemp is to use Egorov and then try to use the bound on the $l...

Integrate 1/(x^3-25x) Between limits of 4,3. Think it involves the 'Difference in squares' substitution. I ended up with an answer of 0.3242 which is wrong as it is 0.023. Any help and pointers would be greatly appreciated. Matt

for $\int x^2sin(x) \mathrm{d}x$ with x = 0.2 and 2.87 I am not exactly sure how to approach this problem.

I finished my engineering career (Electronic Engineering), but i feel that my mathematical knowledge is deficient in important areas (like Fourier series, Laplace, differential equations.. etc). I'm willing to learn all over again mathematics, but i don't know where to start. Actually did try to...

I'm going through all of the proofs in The Incredible Proof Machine and need a hint for one of the proofs. (The Incredible Proof Machine is an online graphical proof tool.) Given: $(\forall x.P(x)) \to A $ Prove: $\exists x.P(x) \to A $ It seems like a trivial proof and here's my hand-waving at...

The function is quite simple and takes into account all integers > 2. It seems to spike the highest when the number is a prime, also numbers that are not primes are visibly under the prime line (red). Would this be worth looking into further? Thank you! Pattern

Let X be a set. All unions are disjoint unions. Suppose $f:X \rightarrow (\emptyset) \cup X \cup X^2$ is a bijection. Is there a injection $g: X \cup X^3 \rightarrow X^2 \cup X^4$? My first idea is to notice $g$ is equal to $h: X \times (\emptyset) \cup X^2 \rightarrow X^2 \times (\emptyset) \...

Prove that there are $\epsilon >0$ and $M>0$ s.t. $|h|<\epsilon \implies |f(y)-f(x)|<M.|y-x|.$ A hint is enough, thank you.

I'm trying hard to figure out how (x-a)^2 + (y-b)^2 = r^2 can be written as y = b + sqrt(r^2 - (x-a)^2) it says that you’ll want to have y as a function of x I don't now how to make math symbols like squaring on this website like x^y

Is this a doubly asymptotic triangle in the poincare half plane or a singly asymptotic triangle?

How do I derive the identities for the nk cases from the n=k case?(Proving Newton's identities)

F1 = ∀x∃y R(x,y) F2 = ∀x (∀yR(x,y) -> ∃z S(x,z)) F3 = ∀x∀y∀z (R(x,y) -> (S(x,z)-> S(y,z))) G = ∀x∃yS(x,y) Using resolution show that F1,F2,F3 |= G. So far I have done the following a)Prenex form of the formula b)Skolem form of the formula 1= Prenex ∃x∀y R(x,y) 1= Skolem ∀y R(f(y...

In Andersons prove of Tuttes Theorem using induction, they choose a maximal $X \in V(G)$ with $$q(G-X) = |X|$$ where $q(G-X)$ denotes the number of odd components in $G-X$. I don't understand the following statement from the prove: The subgraph $G-X$ has no odd components, because if $S$ is a...

I know the answer is [4^(2x+1)][x^(4x)][(1+ln(2x))] but I do not understand how it is attained.

An arc of a curve $y=\frac{|x|}{1+x^2}$ is rotating around $O_{x^{-}}$ axis. Find the volume of a shape after rotation. Local maximum of a function $y$ are $M_1(1,\frac{1}{2}),M_2(-1,\frac{1}{2})$. I don't understand how to find the volume of a shape. I know the limits of integration (integrati...

Let $A=cR$ where $c\ne 0$ is a column in $ℝ^m$ and $r \ne 0$ is a row in $ℝ^n$. Prove col $A=$ span{$c$} and row $A=$ span{$r$}. Could you give me an approach?

I am having troubles from the last line. It says if form $\omega$ is exact then $\omega = dr$ I understand how they are using stoke's theorem however I don't understand why $$\int_{\partial S^1} r = 0$$ - could someone explain this please?

I have no troubles to solve an inequality using this command: solve(df(x) > 0, x) However, I need to solve it within a specified range (-1..1). Is that possible?