Function Definition and 1000 Threads

1.To shift the graph of a function : Vertical Shifts : ## y=f(x) +h## where the graph shifts ##k## units up if ##k## is positive and downwards when ##k## is negative. Horizontal Shifts : ##y=f(x+h)## where the graph shifts to the left by ##h## units when positive and to the right when ##h## is...

If I'm given a function ##f(x)##, say it has continuos first derivative, then I expand it as ##f(x + \Delta x) = f(x) + (df / dx) \Delta x##. If instead, I'm given ##f^{-1}(x)## how do I go about expanding it? Will this be just ##f^{-1}(x + \Delta x) = f^{-1}(x) + (df^{-1} / dx) \Delta x##?

Hi, I want to program an GARCH model for exchange rates. To do this, I calculated the residuals. Next, I did the following (in python) def main(): vP0 = (0.1, 0.05, 0.92) a = minimize(garch_loglike, vP0, eps, bounds = ((0.0001, None), (0.0001, None), (0.0001, None))...

Is there anyone here that know and understand the OVGF method who can help me? I have some doubts about it, and there is almost nothing about it in literature.

From ##\vec r''(t)## we integrate to get $$\vec r'(t) = \left(-\sin(t)+C_1\right)\hat i + \left(6\cos(2t)+C_2\right)\hat j - \left(9.8t+C_3\right)\hat k$$ Solving for the C constants using ##\vec r'(0) = 1\hat i + 6\hat j + 0\hat k##, ##\vec r'(0) = <C_1, C_2, C_3>## ##=<1, 6, 0>## So we now...

I know that due to causality g(t-t')=0 for t<t' and I also know that for t>t', we should get g(t-t')=\frac{sin(\omega_0(t-t'))}{\omega_0} But I can't seem to get that to work out. Using the Cauchy integral formula above, I take one pole at -w_0 and get \frac{ie^{i\omega_0(t-t')}}{2\omega_0} and...

For 1) I found two ways but I get difference results. The first way is I use P(A|B) = P(A and B)/P(B). I get P(X<1|Y<1)=(∫_0^1▒∫_0^1▒〖3/4 (2-x-y)dydx〗)/(∫_0^1▒∫_0^1▒〖3/4 (2-x-y)dydx〗+∫_1^2▒∫_0^(2-x)▒〖3/4 (2-x-y)dydx〗)=6/7 The 2nd method is I use is f(x│y)=f(x,y)/(f_X (x)...

Let $m_n$ be the smallest value of the function: $$f_n(x)=\sum_{k=0}^{2n}x^k.$$ Show, that $m_n\to\frac{1}{2}$ as $n \to \infty$. Source: Nordic Math. Contest

Hi, I am curious about: $$x^m \delta^{(n)}(x) = (-1)^m \frac {n!} {(n-m)!} \delta^{(n-m)}(x) , m \leq n $$ I understand the case where m=n and m>n but not this. Just testing the left hand side with m=3 and n=4 and integrating by parts multiple times, I get -6. With the same values, the...

I am working with a polynomial and wish to integrate over one of it's branch surfaces with high precision. The function is: ## -z^2 + z^3 + w (-4 z + 3 z^2) + w^3 (-2 + 8 z + 4 z^2 - 4 z^3) + w^2 (-z^3 - 9 z^4) + w^4 (6 - 8 z^2 + 7 z^3 + 8 z^4)=0## So I first solve the associated...

As you can see, I've tried using KCL at node A to find the 2nd order ODE that describes this circuit in terms of the capacitor voltage. The problem I run into, however, is that I can't find anything to put the node voltage at A in terms of. I've tried (not shown here) doing mesh current as well...

Form solid state physics, we know that the volume of k-space per allowed k-value is ##\triangle{\mathbf{k}}=\dfrac{8\pi^3}{V}## ##\sum_{\mathbf{k}}F(\mathbf{k})=\dfrac{V}{(2\pi)^3}\sum_{\mathbf{k}}F(\mathbf{k})\triangle{\mathbf{k}}##...

OK, so I'm trying to work out a few ideas regarding locality. I've studied at the undergrad level in the past (including quantum), but with professors that slaved away at proving math constructs and never bothered to indulge in clarifying the context of any concepts, so I'm pretty weak here...

The graph of the Planck blackbody function has an interesting feature:## \\ ## ## \rho_o=\frac{\int\limits_{0}^{\lambda_{max}} L_{BB}(\lambda,T) \, d \lambda}{\int\limits_{0}^{+\infty} L_{BB}(\lambda, T) \, d \lambda} \approx .2500 ##, where ## \lambda_{max} ##, in an exact derivation of...

I am strugglin with this step in my assignment. I am dealing with a centrifuge with a known moment of inertia. I should write the expression for a torque of the motor and express it as a function of angular velocity. Can you help me please?

Would you please explain what an implicit function in general is? Why ##y^2+x^2=c## is assumed as implicit even though it can be expressed in terms of ##y##? ##y^2=c-x^2## and then ##y=\sqrt |x|## Thank you.

Hi all. I'm trying to find a formula that will calculate the probability distribution of a stock price after X days, using the assumption that the price change follows a normal distribution. In the spreadsheet, you can see the simulation I've made of the probability distribution of the price of...

Hi everyone. I'm currently trying to create a function/expression based on several variables. I've so far figured out the rules that the variables should follow but I'm struggling to put them together into a formula. I'm hoping that someone here might tell me if this is even possible, and give...

Hey, I've got this problem that I've been trying to crack for a while. I can't find any info for multi-variable expected values in my textbook, and I couldn't find a lot of stuff that made sense to me online. Here's the problem. Find $E(C)$ Find $Var(C)$ I tried to get the limits from the...

Hi everyone, I was thinking about the complex part of the dielectric function. To my understanding there's good physical explanation of it. is a superimposed description of dispersion phenomena occurring at multiple frequencies. Say I only have the real part such as the one below, and would...

Hi everyone, I understand that the grand-canonical partition function is given by $$Z = \sum_i e^{-\beta(E_i - \mu N_i)}$$ Is there any interpretation to the quantity ##E_i - \mu N_i## here? In the canonical ensemble this would simply be energy of the ##i##th state, so I suppose this would be...

Is there such a thing as an antiderivative of a multivariable function? I haven't put too much thought into this yet but I wanted to ask anyways. Sticking for now just to two variables, I was observing that double integrals are always definite integrals, whereas in the single-variable case, we...

How can I calculate ∂/∂t(∫01 f(x,t,H(x-t)*a)dt), where a is a constant, H(x) is the Heaviside step function, and f is I know it must have something to do with distributions and the derivative of the Heaviside function which is ∂/∂t(H(t))=δ(x)... but I don't understand how can I work with the...

Homework Statement Assuming a Salpeter IMF with upper and lower mass limits of 0.1 and 20 M⊙ respectively, calculate: (i) the mass point at which half the mass formed in a stellar cluster lies in more massive systems and half in less massive systems. ii) the mass point at which half the...

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...