Positive Definition and 964 Threads

Hello. Homework Statement Let x \in \mathbb R^n and t \in \mathbb R. Prove the following equivalence: \left \| x \right \|_2 \leq t \ \ \Leftrightarrow \ \ \begin{pmatrix} t \cdot I_n & x \\ x^T & t \end{pmatrix} \text{is positive semidefinite } Homework Equations \left...

I accelerate a mass upward by 2G (2×force of gravity) over some change in height Δh_{1}, then I apply only \frac{1}{2}G over some other change in height Δh_{2}. If, over Δh_{2}, the mass still moving upward (but accelerating downwards), am I doing positive work over Δh_{2} even though the mass...

Homework Statement If $$f(x)>0$$ is continuous for all $$x\ge0$$ and the improper integral $$\int_0^{\infty}f(x) dx$$ exists, then $$\lim_{x\rightarrow\infty}f(x)=0.$$ 2. Relevant I think this assertion is false. A counterexample can be constructed along the following lines of...

the number of ordered pairs of positive integers $x,$y such that $x^2 +3y$ and $y^2 +3x$ are both perfect squares my solution...

Homework Statement Prove that \lim_{x\to0}\frac{e^\frac{-1}{x^2}}{x^n}=0 for any positive integer n. Homework Equations The Attempt at a Solution I've tried using a combination of induction and l'hopital's rule to no avail. Perhaps I am over complicating it? All help is...

Hi all. I have a quick question In this wikilink en.wikipedia.org/wiki/Positive-definite_matrix in the first example I don't get how they get 2x1^2 -2x1x2+2x2^2-2x2x3+2x3^2 in the third line. Can anyone bother to explain? Thanks a lot

Homework Statement The exact question is in question one part ii of http://www.xtremepapers.com/papers/CIE/Cambridge%20International%20A%20and%20AS%20Level/Mathematics%20(9709)/9709_s10_qp_11.pdfHomework Equations The Attempt at a Solution For the first one I just pluged in a value for x, and...

Hi! Homework Statement 1. Substituting an ansatz \Psi(x)= u(p) e^{(-i/h) xp} into the Dirac equation and using \{\gamma^i,\gamma^j\} = 2 g^{ij}, show that the Dirac equation has both positive-energy and negative-energy solutions. Which are the allowed values of energy? 2. Starting...

Hi this is my first subject in this forum and i hope you can help 1.we have been given an assignment to analyze this non-ideal op amp with both positive and negative feedback and to be honest i don't even know where to start i know i should post an attempt of trying to solve but honestly...

Homework Statement if n is a positive integer than √(4n-2) is irrational. Homework Equations The Attempt at a Solution √(4n-2) Assume is rational then by definition of rationality √(4n-2)=p/q for some integers p,q where q≠0 so √(2(2n-1))=p/q by factoring out the...

Who first assigned "negative" to electrons and "positive" to protons? a sort of history question... the fact that the actual flow of charge (which are electrons in metal conductors) is in the opposite direction of the "positive" current flow in circuits has always been an annoyance to me...

Suppose I have a matrix that looks like this [,1] [,2] [1,] 2.415212e-09 9.748863e-10 [2,] -2.415212e-09 5.029136e-10 How do I make it positive definite? I am not looking for specific numerical value answer, but a general approach to this problem. I have heard...

if electricity is flow of electrons and also positive and negative charges attract then why electricity needs a conducting medium. electrons has mass and it should just travel through vaccum then why in vacuum electricity cann't travel please reply!

Homework Statement Here is the question with the answer: http://dl.dropbox.com/u/64325990/phys153Q/21.5.PNG The Attempt at a Solution I initially thought it would be A but it isn't and does anyone know why?

help me understand a line in an "ATA is positive, semi-definite" proof I am looking at a proof for why ATA is positive semi-definite when A is nxn and it has this line. vTAATv = ATv • ATv ≥ 0. I understand what vTAATv means and the purpose of proving that it's nonnegative, etc... My...

Homework Statement d) When is the particle moving in the positive direction? f(t) = cos(πt/4), t ≤ 10 Homework Equations f '(t) = -(π/4)sin(π(t)/4) The Attempt at a Solution 0 < -(π/4)sin(π(t)/4) (π(t)/4)>πn t>4n 0<=n<=2 t>4 t>8 The answer says 8>t>4 im probably...

Hello, Second term of a geometric series is 48 and the fourth term is 3... Show that one possible value for the common ratio, r, of the series is -1/4 and state the other value. ar=48, ar^3= 3... so ar^3/ar=3/48 which simplifies to r^2 = 1/16, therefore r = 1/4 Can anyone explain where...

Homework Statement find the least positive integer n for which 5^{n} \equiv 1 (mod17) or 5^{n} \equiv -1 (mod 17) Homework Equations The Attempt at a Solution I really don't understand and method to doing these problems as I can't use a calculator and I can only work out...

The original problem is as follows: IF E,F are measurable subset of R and m(E),m(F)>0 then the set E+F contains interval. After several hours of thought, I finally arrived at conclusion that If I can show that m((E+c) \bigcap F) is nonzero for some c in R, then done. But such a...

I need any help with the next question I posted in math.stackexchange, thanks. http://math.stackexchange.com/questions/119105/symplectic-positive-definite-matrix

Given $20$ pairwise distinct positive integers each less than $70$. Prove that among their pairwise differences there are at least four equal numbers.

Hello there, I have no idea how to solve the differential equation y''(x) + A sin(y(x)) - B = 0 , where A and B are positive real numbers. I do also have initial conditions: y(0) = 0 and y'(0) = 0. I would be grateful for any help.

Red is for + and Black is for -. If I place red rod into positive side of battery and place black rod into negative side of battery, it displays +9V. If my hand is touching the black rod, and the red rod is touching the ground, then it displays a negative value, I would like to know...

Let ABC be a triangle. Prove that $sin^2\frac{A}{2}+sin^2\frac{B}{2}+sin^2\frac{C}{2}+2sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}=1$. Conversely, prove that if x, y and z are positive real numbers such that $x^2y^2+z^2+2xyz=1$, then there is a triangle ABC such that $x=sin\frac{A}{2}...

Given a symmetric matrix A=\left(\begin{array}{ccccc} \sum a_{1s} & & & & \\ & \ddots & & a_{ij} \\ & & \ddots & & \\ &a_{ij} & & \ddots & \\ & & & & \sum w_{as} \end{array}\right) \in\mathbb{R}^{n\times n}, with strictly positive entries a_{ij}, and with the...

Homework Statement I have a bode plot with a positive phase. Homework Equations this is a MATLAB code for the thing % bode phase plot w = logspace(1, 4, 100); G = 100*(1+0.017i*w)./(1i.*w.*(1+0.05i.*w).*(1+0.0025i.*w).*(1+0.001i.*w)); fi = atand(imag(G)./real(G)); semilogx(w, fi); % x-axis...

Probably the stupidest question I have ever asked, but is it possible to prove that the multiplication of two negatives yields a positive? Go easy on me I've asked better questions :D BiP

I don't really understand the concept of what makes work positive or negative for example if 3 charges are brought together from infinite what is the work done on them? I am extremely confused about what make work positive and negative.

Homework Statement This is for a physics class but at this point in the problem it's basic math... which for some reason I can just not figure out. I need to find at what time x(t) = 0 and I keep getting the negative time... I want the first positive time. Homework Equations x(t) =...

hey friend can anybody give answer? why (-) * (-) = (+)

Hi Let \( c>0 \) be a real number. Then I am trying to prove that \( \forall\; n\in\mathbb{N}\; (c^{1/n} >0) \). I let \(n\) be arbitrary and then tried to use method of contradiction. But ran into difficulties. Is there another approach ?

someome please help me with this problem: "Any real numbers x and y with 0 < x < y, there exist positive integers p and q such that the irrational number s =( p√2)/q is in the interval (x; y)."

2 unit negative chargees and a positive charge +q are along the same line. what should be the magnitude of charge q and in which position must it be placed so that the three charges remain in equilibrium...

I'm curious, is the square of any function always positive? It seems obvious that it's always positive because if you have a function (F), an input (x) and an output (y) then you have y = F(x) And if you square the function then (F(x))2 = y2 which means that every value in the range is...

Homework Statement Prove: (n)^(1/n) < 1 + sqrt(2/n) for all positive n.Homework Equations The Attempt at a Solution Using induction, base case is easy enough to prove, however proving it holds for n+1 is where I am hitting a wall. So the problem is reduced to proving: (n+1)^(1/(n+1)) <...

Homework Statement a)Determine at least three limit points for the set {sin(n): n a positive integer} b)How many limit points does the set {sin(n): n a positive integer} have? The Attempt at a Solution For a it seems that it wouldn't have a limit point because sin(n) would not converge to...

one-point compactification of space of matrices with non-negative trace Hi I'm a physicist and my question is a bit text-bookey but it is also part of the proof that the universe had a beginning...so could I ask anyway...You got q which is a continuous function of a 3 by 3 matrix where if any...

Homework Statement What proportion of 2x2 symmetric matrices with entries belonging to [0, 1] have a positive determinant? Homework Equations A^{T} = A If A = [[a, b], [c, d]] Then det(A) = ad - bc. But A is symmetric, so c = b. So det(A) = ad - b^2 So, in order for A to have a...

Homework Statement Problem 2.6.3. in "Foundations of modern analysis", by Avner Friedman. Let f be a measurable function. Prove that f is integrable if and only if f+ and f- are integrable, or if and only if |f| is integrable. Homework Equations Friedman defines "integrable" like this: An...

Homework Statement Suppose that p and q are prime numbers and that n = pq. Use the principle of inclusion-exclusion to find the number of positive integers not exceeding n that are relatively prime to n. Homework Equations Inclusion-Exclusion The Attempt at a Solution The...

Prove that for positive real numbers a,b (a+1/b+1)^(b+1) is greater than or equal to (a/b)^(b). The case in which a<b is easy to prove, but after trying to represent the inequality with an integral, I'm a bit stumped. Any ideas?

Homework Statement Consider the function f(a)= 1 ∫ [g(x)-(anxn+an-1xn-1+...+a0)]2 dx 0 where a=(a0,a1,...an) and g is some known function defined on [0,1]. From this, we can show that Thus, the Hessian of f at a = [2/(j+k+1)] j=0,1,2,...n; k=0,1,2,...,n. Fact: This Hessian...

Hello people, Im working on a project and this problem came up: I have a symmetric matrix whose elements are complex variables, and i know that this matrix is positive semi-definite. I have to derive a criterion for the matrix's elements, so that if it's satisfied by them then the matrix...

Homework Statement For which positive real numbers a does the series Ʃo→∞ a^log(n) converge. Here logarithms are to the base e Homework Equations Im afraid I'm not sure where to start, I'm not sure which topics would be applicable to this question. If someone could point...

What would happen if I were to set up two extreamly powerful magnets one above each other. Say the strength of each magnet was about 30 tesla. If both the magnets were positive and repelled each other with extreme force, if protons were in the middle of this repelling field, could those protons...

This is copied from Paul's online math notes There is one final topic that we need to touch on before leaving this section. As we noted back in the section on radicals even though √9=3 there are in fact two numbers that we can square to get 9. We can square both 3 and -3. The same will...

Homework Statement Let L be a simple compact Lie group, and \Delta_+ is the set of positive roots. I have previously shown that if \alpha\in\Delta_+ and \alpha_i is a simple root, then s_i\alpha\in \Delta_+ where s_i is the Weyl reflection associated with \alpha_i. Now, let \delta =...

What would happen hypothetically if there was a positive charged particle the same size as an electron (that wasn't antimatter), that carried current in the opposite direction in a wire to a normal conventional current in another wire? According to the Lorentz force they would attract so again...

I've been trying to use Taylor's theorem with h = (y-x)/2 to show that a twice differentiable function for which the second derivative is positive is midpoint convex (ie, f( (1/2)*(x+y) ) \leq (1/2) * (f(x)+f(y)) ). (It's not a homework problem.) The problem I end up with this is that I'm not...

Often people asks how to obtain a positive definite matrix. I would like to make a list of all possible ways to generate positive definite matrices (I consider only square real matrices here). Please help me to complete it. Here M is any matrix, P any positive definite matrix and D any...