Function Definition and 1000 Threads

Homework Statement On a certain island, there is a population of snakes, foxes, hawks and mice. Their populations at time t are given by s(t),  f (t), h(t), and m(t) respectively. The populations grow at rates given by the differential equations s'=(8/3)s - f - (1/3)h - (1/6)m f'=(2/3)s + f -...

Homework Statement Let ##f:X\rightarrow Y## with X = Y = ##\mathbb{R}^2## an euclidean topology. ## f(x_1,x_2) =( x^2_1+x_2*sin(x_1),x^3_2-sin(e^{x_1+x_2} ) )## Is f continuous? Homework Equations f is continuous if for every open set U in Y, its pre-image ##f^{-1}(U)## is open in X. or if...

I don't even know what a conjecture is y=-1/2[cos(x+pi)+cos(x-pi)]

f(x)=sinx+cosx getting really frustrated with my math teacher. gives us forumlas for things but then barely shows us how to use them if at all and then throws problems at that we have to make sense of ourself. why can't math teachers teach? anyway, the question is express f(x)=sinx+cosx in...

Hi everyone! Sorry for the bad english! So, just a quick doubt... Does things collapse from a wave of probability into a quantum field or is the wave in the quantum field the probabilistic wave itself? An example to make it clearer: Suppose we have an atom, it enters an atom interferometer, it...

Hello, guys. I'm currently working on a physics problem that requires me to evaluate the inverse Laplace of the function in the attached file. When b = 0, "y" vanishes, and all one has to do is to look up the Laplace table for the inverse. However, non-zero b has been giving me a headache. I...

Homework Statement Allow f:ℤ→ℤ be defined by, for all n∈ℤ f(n) = {n-1 if n is even, n+5 if n is odd Prove that ran(f) = ℤ Homework EquationsThe Attempt at a Solution I am unsure of how exactly to prove this due to the fact now I am working with a piecewise function. Here is what I have so...

Hi everyone, I'm working through the boundary conditions and I could not figure out what to do with the last boundary condition (when z=L) I know that the values for K are: How so? 1. Homework Statement A hollow right angle cylinder of radius a and length l. The sides and bottom are...

H(t) = t^3-6t^2+5t+30 this is a yo yo 30 inches above ground at t =0, at 4 secs it is 18 inches above ground. Please tell me how these figures are derived; t^3,6t^2, 5t; I realize the 30 is initial position. I am 81 but very curious. Thank you.

Homework Statement If ##f\left(x\right) = x^2 + 2x + 2##, find two functions ##g## for which ##\left(f \circ g\right)\left(x\right) = x^2 - 4x + 5##. Homework Equations If ##f\left(x\right) = x^2 + 2x + 2##, then ##\left(f \circ g\right)\left(x\right) = g\left(x\right)^2 + 2g\left(x\right) +...

Homework Statement Find the useful denial of a injective function and a surjective function. Homework EquationsThe Attempt at a Solution I know a one to one function is (∀x1,x2 ∈ X)(x1≠x2 ⇒ f(x1) ≠ f(x2)). So would the useful denial be (∃x1,x2 ∈ X)(x1 ≠ x2 ∧ f(x1) = f(x2))? I know a onto...

Homework Statement 1. A weight hangs from a spring. If a force is applied to the weight at t = 0 seconds, it will start moving up and down. The following equation gives the distance d, in centimetres, of the weight from its equilibrium point: d=4(sin5t-4cos6t) At what times during the...

Homework Statement [/B] A random variable x has a probability function ##G(t)##. Show that the probability that ##x## takes an even value is ## \frac 1 2 ( 1+G(-1))##Homework EquationsThe Attempt at a Solution [/B] ##G(t)= \sum_{k=0}^\infty p_k t^k ##... ## 1=P(X=even)+ P(X=odd)##...1 ##G(-1)=...

Homework Statement The density of a rod in function of space is given as ##\rho (x)=\frac{c}{x^2}## 1. What kind of density is this? 2. What is the dimension of ##c##? 3. What is the mass of the rod in the intervals - [1 m, 2 m], - (1 m, 2 m), - (0 m, 1 m), - [0 m, 1 m]? 4. Can a plate with...

1. Homework Statement Consider a potential field $$V(r)=\begin{cases}\infty, &x\in(-\infty,0]\\\frac{\hslash^2}{m}\Omega\delta(x-a), &x\in(0,\infty)\end{cases}$$ The eigenfunction of the wave function in this field suffices...

Consider a potential cavity $$V(r)=\begin{cases}\infty, &x\in(-\infty,0]\\\frac{\hslash^2}{m}\Omega\delta(x-a), &x\in(0,\infty)\end{cases}$$ The eigenfunction of the wave function in this field suffices $$-\frac{\hslash^2}{2m}\frac{d^2\psi}{dx^2}+\frac{\hslash^2}{m}\Omega\delta(x-a)\psi=E\psi$$...

wikipedia says: "The exponential function, g: R → R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) = −1, showing that g is not onto (surjective). However, if the codomain is restricted to the positive real numbers R+, then g becomes bijective; its inverse is the...

Homework Statement We've been given a set of hints to solve the problem below and I'm stuck on one of them Let f:[a,b]->R , prove, using the hints below, that if f is continuous and if f(a) < 0 < f(b), then there exists a c ∈ (a,b) such that f(c) = 0 Hint let set S = {x∈[a,b]:f(x)≤0} let c =...

Homework Statement A solid body begins to rotate around a fixed axis with angular acceleration ##\beta=\beta_0\cosφ##, where ##\beta_0## is a constant vector, ##φ##, is the angle of rotation of the body from initial position. Determine the angular velocity of this body as a function of the...

Hi all, I am slightly confused with regard to some ideas related to the GCE and CE. Assistance is greatly appreciated. Since the GCE's partition function is different from that of the CE's, are all state variables that are derived from the their respective partition functions still equal in...

Homework Statement If possible, calculate the following limit: \lim_{(x,y)\rightarrow (0,0)} {\frac{2x^2 + 3y^2}{5xy}} Homework Equations N/A The Attempt at a Solution [/B] I tried using both parametric and polar equations to find the limit, but neither worked. Setting either x or y...

Why can't G and its derivative be continuous in the relation below? $$p(x)\dfrac{dG}{dx} \Big|_{t-\epsilon}^{t+\epsilon} +\int_{t-\epsilon}^{t+\epsilon} q(x) \;G(x,t) dx = 1$$

In several places, for example https://xxx.lanl.gov/pdf/chao-dyn/9406003v1, it is claimed that the Riemann zeta function is a fractal under the assumption of a positive result for the Riemann Hypothesis, because (1) the Voronin Universality Theorem, and (2) if the RH is true, then the zeta...

How do you prove 1.85 is valid for all closed surface containing the origin? (i.e. the line integral = 4pi for any closed surface including the origin)

E(X) of probability density function f(x) is \int x f(x) dx E(X2) of probability density function f(x) is \int x^2 f(x) dx Can I generalize it to E(g(x)) of probability density function f(x) = \int g(x). f(x) dx ? I tried to find E(5 + 10X) from pdf. I did two ways: 1. I found E(X) then using...

Homework Statement Find a function where the domain is integers, codomain is real numbers, and image isn't equal to codomain. Homework EquationsThe Attempt at a Solution I know that it means that when I plug in an integer I will obtain a real number, but how do I make it so that the image is...

Hi. If I have a function for example f( x ) = x2 + x then to obtain f( -x) I just put (-x) in place of the x in f(x) so I get f( -x) = x2 - x Am I right so far ? So f(x) and f(-x) look like different functions but if you put a negative number in f(-x) it flips the -x back to +x so are f(x) and...

Homework Statement Obtain Maclaurin Series for: f(x) = sin(x2)/x Homework Equations f(x) = ∑f(n)(c) (x-c)n / n! (for Maclaurin c = 0) The Attempt at a Solution I know that sin(x2) = x2 - (x2*3/3! +... from the final answer I see, that this is just multiplied to 1/x. This bothers me...

For a standard second-order system, its transfer function is G(s) = ω2/(s2 + 2ζω + ω2) where ω is the natural frequency and ζ is the damping ratio. But for a non-standard second-order transfer function, G(s) = (2s + 1)/(s2 + 2s + 5), what are its natural frequency and damping ratio? Thank you!

Homework Statement We have two semi-infinite coplanar planes defined by z=0, one corresponding to x<0 set at potential zero, and one corresponding to x> set to potential ##V_0##. a) Find the Green function for the potential in this region b) Find the potential ##\Phi(r)## for all points in...

hi(Smile) I need some start help finding the two constants a and b: Do i start like this?: $$\lim_{{x}\to{6-}} (20) = \lim_{{x}\to{6+}} (8x+a)$$

I've been playing around with this function of two variables: xy (eax-eay)/(xeax-yeay ) where a is a positive parameter. Now, when x and y are very large and close to each other in size, this seems to easily lead to catastrophic cancellation and computational problems. One thing that seems...

Say we have ##P_k(z)## a family of entire functions, and they depend analytically on ##k## in ##\Delta##. Assume ##P_k(z)## is nonzero on ##S^1## for all ##k##. How do I see that for each ##t \ge 0##, we have that$$\sum_{|z| < 1, P_k(z) = 0} z^t$$is an analytic function of ##k##? Here, the zeros...

Homework Statement -2^x = y Homework EquationsThe Attempt at a Solution When I plug this function in my graphing calculator, it appears to be 2^x reflected across the x axis. This doesn't make sense to me. For example, for x values of 1 and 2, the value of y is not on the same half of the...

Homework Statement $$f(x+\frac{1}{x}) = x^2+\frac{1}{x^2}\\ f(1) = ?$$ Homework EquationsThe Attempt at a Solution I thought maybe I can find a ##x## that ##x+\frac{1}{x} =1## that I can substitute, but that ##x## is a imaginary number so I am not sure what should I do next.

Homework Statement Find the following: $$ \int_0^\inf \frac{x^{a+1}e^{-x/\delta}}{\delta^{a+1}\Gamma(a+1)} dx; \, a > 1 , \delta >0 , 0 \leq x \leq \inf$$ Homework Equations - The Attempt at a Solution The numerator in the integral is constant, so it can be taken outside the integral. I then...

Homework Statement [/B] I am trying to determien the characteristic function of the function: $$ f(x)= ae^{-ax}$$ $$\therefore E(e^{itx}) =\int_0^\infty e^{itx}ae^{-ax} dx = a \cdot \frac{e}{it-a} |_0 ^ \infty $$ But I am not sure how to evaluate the integral. Wolfram alpha suggests this...

Homework Statement I have simulated Langevin equation (numerically in Matlab) for some specific conditions, so I have obtained the solution ##X(t)##. But now, with the solution I have obtained, I have to calculate ## <X(t)|x_0>, <X^2(t)|x_0>-(<X(t)|x_o>)^2 ## and the conditional correlation...

Homework Statement Find the inverse function of ##f(x) =x^4+2x^2, x>0## Homework Equations ##f(f^{-1}(x)) = x## The Attempt at a Solution My only progress so far is ##x^4+2x^2 = x^2(x^2+2)## Then I am stuck. Since my progress is close to nothing so I don’t expect a complete...

Let's say we have a Dirac field ##\Psi## and a scalar field ##\varphi## and we want to compute this correlation function $$<0|T \Psi _\alpha (x) \Psi _\beta (y) \varphi (z_1) \varphi (z_2)|0>$$ $$= \frac {1}{i} \frac{\delta}{\delta \overline{\eta}_\alpha(x)} i \frac{\delta}{\delta \eta_\beta(y)}...

I need a little help with understanding a differential relationship between functions. If g and f are vector fields and f(g(x,y),q(x,y))=∇2g(x,y) How could you, if possible, express ∂f/∂g explicitly? Please help a bit confused.

Homework Statement Determine function $$f(x) = \log(\sqrt{x^2+1}+x)$$ Is odd or even. Homework Equations ##\log(a+b) = \log(a) + \log(1+b/a)## The Attempt at a Solution First I thought it is a even function without considering the x at the end, which of course isn't the actual case. Then I...

I have the next function: f(x)= -2(x+1)/(x-1)^2 x is from [0,1)U(1,3] I need to find the image of the function which is (-infinity, -2].I tried with the derivative but when I solve f'(x)=0 I obtain x=-3 which is not from x interval.I don't know how to continue.

Given a scalar function, we consider the following transformation: $$\delta f(x) = f'(x') - f(x) $$ Given a coordinate transformation $$x' = g(x)$$ But since ##f(x)## is a scalar isn't it true that ##f'(x') = f(x) ## Then the variation is always zero? What am I missing?

Homework Statement Find the first and second derivatives of ##\displaystyle f(x)=\frac {1} {x^2+6}## Homework EquationsThe Attempt at a Solution [/B] ##\displaystyle f(x)=\frac {1} {x^2+6}## ##\displaystyle f(x)=(x^2+6)^{-1}## ##\displaystyle f'(x)=-1(2x)(x^2+6)^{-2}## ##\displaystyle...

Hello! As the topic suggests I´m interested which functions space square waves span? Lets say we define them as https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8953debf86627276f45bf8822140ff2bbaee56 . Do they span the same space as the sines and cosines in Fourier analysis? :/ Thanks!

Homework Statement ##3f(x)+2f(\frac{1}{x}) = x##, solve ##f(x)## Homework Equations Not sure.Maybe the ones of inverse functions. The Attempt at a Solution The only thing that I came up so far is that the function’s highest order term is ##x## because if there are higher orders,it will show...

Homework Statement Differentiate ##F(x)=4^{3x}+e^{2x}## Homework EquationsThe Attempt at a Solution I have an exam coming up and need some help with this problem. ##F(x)=4^{3x}+e^{2x}## ##f(x)=4^{3x}## ##G(x)=e^{2x}## First I need to find f'(x) and g'(x), which I thought I did correctly...

##E = \frac{1}{2}(kx^2+m \dot{x}^2)## ## \frac{2E - kx^2}{m}=\dot{x}^2## ##\frac{dx}{dt} = \sqrt{\frac{2E - kx^2}{m}}## or ## dt = \sqrt{\frac{m}{2E - kx^2}}dx ## ⇒##= \frac{1}{\sqrt{\frac{2E - kx^2}{m}}}dx## My Question please help me. 1. I know ##T = 2\pi\sqrt{\frac{m}{k}} .## but i don't...

Can the wave function for entangled particles be solved perturbatively? Are there virtual processes involved with this? Thanks again.