Positive Definition and 987 Threads

I have a question relating to solving for both a and b in the following question: Find positive integers a and b such that: $\displaystyle \left(\sqrt[3]{a}+\sqrt[3]{b}-1 \right)^2=49+20\sqrt[3]{6}$ This one appears to be tough because it doesn't seem right to expand the left hand side and...

What happens to the Reimann tensor at the event horizon of a black hole? Do some of the 24 components become zero or infinite? What happens to parallel transport of a vector on the surface of an event horizon that is different than on a surface outside the event horizon? I'm newly educated...

eigenvalues of a compact positive definite operator! Let A be a compact positive definite operator on Hilbert space H. Let ψ1,...ψn be an orthonormal set in H. How to show that <Aψ1,ψ1>+...+<Aψn,ψn> ≤ λ1(A)+...+λn(A), where λ1≥λ2≥λ3≥... be the eigenvalues of A in decreasing order. Can...

Hi all, I am working with some beta functions. I want to show that the following is positive and bounded between 0 and 1. Is it possible to show this? $$ \frac{ \frac{B( a + b , \frac{2}{ c} )}{B(a, \frac{2}{c}) } - \big\{\frac{B( a + b , \frac{1}{ c} )}{B(a, \frac{1}{c}) }\big\}^{2} }{...

...plate of a different capacitor, why won't the positive end of a battery (lets say AA) pass current to the negative plate of a different AA battery. Does it have something to do with the chemical reaction that happens inside the battery?

Homework Statement Here is the problem given. The answer is E. Homework Equations No equations are given. Assumed use of standard equations for electric field and force involving electric field. The Attempt at a Solution I tried using a few equations (E = F/q0), (F = mv^2)...

My question relates to constructional geometry & matrices aren't to be involved in the solution because stated Math level is up to O Levels... The figure below shows shear with y=3 as invariant line & shear-factor of 3 My question is if you are provided the original polygon & asked to do shear...

on page 261 of this paper by J. Vermeer (http://www.math.technion.ac.il/iic/e..._pp258-283.pdf ) he writes The following assertions are equivalent. a) A is similar to a Hermitian matrix b) A is similar to a Hermitian matrix via a Hermitian, positive definite matrix c) A is similar to A*...

Hi! So, when we're calculating the potential energy of a mass in space we make it negative because we assign infinity as the reference point. Now, to find the potential energy of something I understand that we find the work that is done to it to move it from the reference point to another...

If I parameterized a surface by the vector function r(a,b) I would then proceed to find the normal to the surface by crossing ra and rb. But how would I determine which normal raXrb or rbXra has the positive orientation? Im assuming you would first need to know if the concave side of the...

Hi, If A is some nonsquare matrix that is possible rank-deficient, then am I right to understand that (A^T)(A) is a positive semidefinite matrix? Does there exist an inverse (A^T A)^-1? Thanks for any help

The linear Diophantine equations: ax+by=c, a,b,c is natural numbers. If c is a multiple of gcd(a,b), there is infinite integer solutions, and I know how to find x,y. However, I wonder how to find positive integer solution x,y only.

Homework Statement Hello everyone. My problem is as follows: In a spontaneous process where two bodies at different temperatures T_{1} and T_{2}, where T_{1}>T_{2}, are put together until they reach thermal equilibrium. The number of atoms or molecules of the first is N_{1} and N_{2} for the...

Homework Statement How come A person picks a 1.00-kg box of macaroni from off the shelf and lowers it 0.77 m into a shopping cart. The work done on the macaroni by the person is negative work however when person picks up a 1.00-kg box of macaroni from off the shelf and lowers it 0.77 m into a...

Homework Statement d=1352846320 The question asks for fxy. The rate of change of (the rate of change of f in the x direction) in the y-direction. We know fx is negative because as f moves in the x-direction, f decreases. But how do I know the rate of change of fx in the...

This post in influenced by 3 new threads in our cosmology forum. Recent observational data favors positive curvature of our Universe. The question I have, however, is why positive curvature implies spatially finite Universe? Yes, it might look quite obvious if we embed curved space into higher...

Homework Statement Show that if a 6x6 matrix A has a negative determinant, then A has at least one positive eigenvalue. Hint: Sketch the graph for the characteristic polynomial of A. Homework Equations Characteristic polynomial: (-\lambda)^n + (\text{tr}A)(-\lambda)^{n-1} + ...

Homework Statement Are there any positive integers n, a and b such that 96n+88=a^2+b^2 Homework Equations The Attempt at a Solution It resembles the Pythagorean theorem but I'm not sure how that would help me solve it. I factored the LHS 2^3((2^2)(3)n+11)=a^2+b^2 How do I...

Hello all! *please explain the terms 'positive definite function' and 'semi-definite function'.* CONTEXT: I am reading a book on the stability analysis of non-linear models. In the chapter for 'neighborhood stability analysis',I came across the "Lyapunov function V". V has the following...

A theorem from number theory states that, if a and b are nonzero integers, then there exists a smallest positive linear combination of a and b. This is my proof: Let S be a set such that S = {w\inNatural numbers : w=am+bn} , where a and b are positive integers, m and n are any integers...

Homework Statement Consider a Hilbert space with a (not necessarily orthogonal) basis \{f_i\} Show that G=\sum_i |f_i\rangle\langle f_i| has strictly positive eigenvalues. Homework Equations The Attempt at a Solution I know that G=\sum_i |f_i\rangle\langle f_i| is hermitian...

Let n be a positive integer and a be a positive divisor of n. Is there any general formula to find the number of positive divisors b of n such that (a,b)=1 ?.

Homework Statement Plot the waveforms for capacitor voltage VC, output voltage Vo, and diode voltage Vd given that Vs is a 20 Vpp triangle wave with period T. Use CVD model with diode VON = 0.7 V. Homework Equations KVLs? The Attempt at a Solution From my basic...

1. Three consecutive positive integers are such that the sum of the squares of the first two and the product of the other two is 46. Find the numbers. Variables: x. Three numbers: (x), (x + 1), (x + 2) 2. (I think, although I'm not sure.) x2 + (x + 1)2 + (x + 1)(x + 2) = 46 3. x2...

I am wondering if there is a systematic way to fix the phase of complex eigenvectors. For example e^{i \theta}(1,\omega,\omega^2) where e^{i \theta} is an arbitrary phase and \omega and \omega^2 are the cube roots of unity, is an eigenvector of the cyclic matrix \left(\begin{matrix}0&...

Homework Statement For what positive integers n does 15|2^{2n}-1 Homework Equations We know 2^{2n}\equiv1mod15 I was thinking this might be helpful but not sure x^{2} ≡ −1 (mod p) is solvable if and only if p ≡ 1 (mod 4) The Attempt at a Solution I think that the answer is...

I know this is a basic question, but why are positive frequency modes so important when expanding a field operator. Furthermore, what do they represent?

Good Morning, The equations under investigation: Electrons: J = enμE + eD(dn/dx) Holes: J = epμE - eD(dp/dx) n or p = concentration of electrons or holes respectively D = diffusion constant μ = mobility The question in mind is as follows: If holes are...

Homework Statement How much work is done by the electric field in moving a particle from (a,a,0) to (a,a,a) in a region where the electric field is: E = zye_x + yxey + xyezHomework Equations F=qE W = integral F dot dl V(2)-V(1)= - integral E dot dl The Attempt at a Solution I know how to do...

Question: Show that the derivative of f(x) = (x-a)m (x-b)n vanishes at some point between a and b if m and n are positive integers. My attempt: f(x) = (x-a)m (x-b)n f '(x) = m(x-a)m-1 (x-b)n + n(x-a)m (x-b)n-1 f '(x) = [(x-a)n-1 (x-b)n-1 ] [(m)(x-b) +(n)(x-a)] And this is as far as I got.

Homework Statement Why is weight positive if gravity is negative? Homework Equations W = mg The Attempt at a Solution So say there is some weight of an object, and the object's mass is multiplied by the acceleration due to gravity pointing downwards, so the weight becomes negative...

Homework Statement Two small beads having positive charges 3q and q are fixed at opposite ends of a horizontal insulating rod, extending from the origin to the point x=d. A third small charged bead is free to slide on the rod. At what point is the third bead in equilibrium? Explain whether it...

$$ P(r,\theta) = \frac{1}{\pi}\left(\frac{1}{2} + \sum_{n = 1}^{\infty} r^n\cos\theta\right) = \frac{1}{2\pi}\frac{1 - r^2}{1 - 2r\cos\theta + r^2} $$Prove that $P(r,\theta) > 0$ for all $r$ and $\theta$ where $0\leq r < 1$ and $-\pi\leq\theta\leq\pi$. How can I start this?

Prove that: d/dx x^-n = -nx^-n-1 Use the factorization of a difference of nth powers given in this section (not using quotient rule) My attempt gets me from the definition of the derivative to (1/n^(n-1)) n times... I need the negative. I get nx^(-n-1) instead of -nx^(n-1).

Homework Statement if M is dxd positive semi-definite matrix, then: M = L'(or L transpose) * L where L is a matrix of dimensions rxd How can I get L given M and the number of rows, r? for example, if M is 800x800 p.s.d, I want L such that it is a 30x800 matrix and so M (800x800) = L'...

Homework Statement Lifting a 20lb bucket of sand a distance of 100 ft at a rate of 2ft/s. I would assume: [W=F*d] = [W = MA*d] = [W = M(v/t)*d] and at a constant velocity: rate of v/t=0 so [W = M(0)*d] = [W = 0*d] = [W = 0] However, a video online shows that you'd take 20lb*100 ft...

1.Homework Statement BACKGROUND: A positive helium ion He+ (mass 6.7 x 10-27 kg, charge 2e) is released from rest from the surface of a +1000 V electrode, as shown in the diagram below. It crosses (in vacuum) the gap between the electrodes, passes through a small hole in a 0 V electrode...

For a positive integer $n$, let $$f_n(\theta)=\tan \frac{\theta}{2}(1+\sec \theta)(1+\sec 2\theta)(1+\sec 4 \theta)\cdots (1+\sec2^n \theta)$$ Find the value of (i) $f_2 \left(\dfrac{\pi}{16} \right)$ (ii) $f_3 \left(\dfrac{\pi}{32} \right)$ (iii) $f_4 \left(\dfrac{\pi}{64} \right)$ (iv)...

for the given circuit and waveform , i am not getting how output for ωt < π is Vs + E1 ? given Vs is input and equal to Vm Sinωt

Hi guys, So, these two formulas are making me very confused: UE=kQq/r ΔUE=-WE Ok, here is the problem: Let's imagine that we have two positive charges. One of them is static, the other one can move. If we place the positive charge close to the source charge, the movable...

Homework Statement (i) Show that if A is symmetric positive semi-definite, then there exists a symmetric matrix B such that A=B^2. (ii) Let A be symmetric positive definite. Find a matrix B such that A=B^2.Homework Equations The Attempt at a Solution For part 1, I used: B = Q\sqrt{\Lambda}...

Exploiting quasistatic approximation, if one wishes to calculate self-inductance of any loop, he is led to the following double line integral: \oint\oint\frac{d\vec{l_{1}}\cdot d\vec{l_{2}}}{r}, where r is the distance between the length elements \vec{dl_{1}} and \vec{dl_{2}}. Is this...

Hi experts, I still do not understand how positive charge is distributed inside a conductor. In case of extra electrons is it quite evident, as they are free to move and repel each other, so they go as far away as they can. But what exactly happens if electrons are removed from a conductor...

I am trying to understand the IR spectra of liquid. I can get the autocorrelation function of atoms' velocity, <v_{i}(0)v_{i}(t)> make a Fourier Transformation, the vibrational density of state (VDOS) can be obtained. Does the VDOS always be positive? Or it can also take negative value...

Can someone just confirm if I'm doing these two problems correctly ... 1. How much energy is needed to place 4 positive charges, each of magnitude +5mC, at the vertices of a square of side 2.5cm 2. Choose one way of assembling the charges and calculate the potential at each empty vertex as...

I just want to make sure I'm understanding negative work correctly. An example problem where there is a bullet with an initial velocity of x m/s and a final velocity of 0 m/s. It is stopped by a wood block and the question asks to determine the amount of work done on the bullet by the block. If...

Hi, Acceleration is defined as the change in velocity over the time interval. Is acceleration ever negative, or is it always positive? If I were to throw a rock up in the air, is it always accelerating at 10 m/s22 even up until it stops and starts falling back down? Thanks,

Homework Statement Assume that x is a positive multiple of 5 and is greater than 5. If 2x + 1 < 100, how many values for x are possible? Homework Equations The Attempt at a Solution How I solved the problem First manipulated inequality: 2x+1<100 => 2x < 99 => x <...

What is the fundamental premise of "positive thinking" The only quantifiable effect of positive or negative thinking that I can think of would be how each mindset influences your actual behavior. But do positive thinkers believe that it also somehow produces effects via some unknown mechanism...

Homework Statement I need to understand how signed multiplication is done in 2's to 10's complement.Homework Equations The complement D' of a number D with m digits in base r is rm - D. So, the complement of 010 in binary (2 in dec) is 1000 - 010 = 110 (-2, but 6 if it were unsigned). We add a...