What is Existence: Definition and 569 Discussions

Simple question: Let f be a real-valued function defined an open interval U. If f is twice differentiable at u, then f' is continuous at u right? Does that mean that there exists a neighborhood around u where f' exists for all points in this neighborhood?

While it is widely accepted that there is no existence of the aether, what does this say about Dirac's argument which contradicts mainstream belief? This comes about 50 years after it was suggested that there was no aether. Is this a special case? I was told that aether theory is not excluded...

Homework Statement show the initial value problem x(dy/dx)=4y, y(0)=1 has no solution. does this contradict the existence theorem. please explain The Attempt at a Solution it is easy to find out a general solution is y=C*e^(4x), C is a constant. and for any x the right part of the...

Homework Statement Prove that there exists two infinite sequences <an> and <bn> of positive integers such that the following conditions hold simultaneously: i) 1<a1<a2<a3...; ii) an<bn<(an)^2 for all n>=1 iii)(an) - 1 divides (bn) - 1 for all n>=1 iv)(an)^2 -1 divides (bn)^2 - 1 for all...

Is it true that 0.1% of the mass in an uncontrolled nuclear chain reaction gets converted into energy via E=Mc^2? And if this is so, then does the mass technically go to infinity in accordance with the Lorentz transformation before turning into energy?

Homework Statement The theorem for a unique solution to a DE says: Let R be a rectangular region in the xy plane that contains the point (xo,yo). If f(x,y), which = dy/dx and the partial derivative of f(x,y) are continuous on R, then a unique solution exists in that region. Question...

Hi, I don't know if this has been discussed (or is trivial or even silly). I was wondering sometimes there is reliance on indirect proofs in mathematics. I was wondering can it be proven in any case where an indirect proof exists that a direct proof does not?

Homework Statement a) Verify that both y1(t)= 1-t and y2(t)= (-t^2)/4 are solutions of the initial value problem y-prime = (-t + (t^2 + 4y)^(1/2)) / 2 , for y(2) = -1 Where are these solutions valid? b) Explain why the existence of two solutions of the given problem does not...

I had a brief conversation with a professor of mine and he presented, in short an argument against the existence of black holes. I'm sure you've heard it or a variation of it. It goes something like this: An advisor and his student are near a black hole. They are both wearing a watch, the...

Homework Statement Let \mathcal{V} \subset \mathbb{R}^n be open and f: \mathcal{V} \rightarrow \mathbb{R}^n be continous. Assume that f has partial derivates which are continous. Then the autonomous differential equation \frac{dx}{dt}(t) = f(x(t)) on the region D = \mathbb{R} \times...

Homework Statement Let F(x) = (x-a)^2(x-b)^2 + x. Show that the output \frac{a+b}{2} exists for some value x. Homework Equations Quadratic formula. x^2 \geq 0.The Attempt at a Solution Hmm I've tried setting the two equal but that doesn't look nice (if I multiply everything out). It's easy to...

Homework Statement Given the IVP problem \rm{(1+x^2)^{-1} \frac{dx}{dt} = \frac{\alpha \cdot 2 \pi}{4} \cdot cos(t)} \iff \rm{(1+x^2)^{-1} dx = \frac{\alpha \cdot 2 \pi}{4} \cdot cos(t)} dt Find the maximal solution with the initial conditio condition x(0) = 0. Then alpha is...

Hey my smart *** English teacher is always asking us to try and prove our existence to him... When someone tries to say "Well you can see me can't you?" He'll reply "Have your eyes never deceived you before? Is that lake in the middle of the desert that always moves away from you when you get...

Forgive me if this is a stupid question, I'll be trying to come up to speed over the next few weeks.. I am trying to recall a model of the universe that stuck in my head from a few years ago. -Does space-time's existence rely on the presence of mass? -When people say that space-time...

Assume GL(n,q) is the general linear group of nxn matrices with entries in the finite field with q elements. Define a Singer Cycle to be an element of GL(n,q) of order (q^n)-1. How can we show that such an element always exists? That is, for all n and q. Thanx in advance.

From this site: http://www.angelfire.com/md2/timewarp/sartre.html we read: What is Being? What is Nothingness? How are they related? For Sartre, Being is objective, it is what is. Being is in-itself. Existence, on the other hand, has a subjective quality in relation to human reality...

So I'm studying infinite limits in my calculus text (seemed close enough to good old arithematic to put in general math, though), and the following rule is mentioned: Given two functions f(x) and g(x) defined for all real numbers, when given the quotient f(x)/g(x) where f(c) is not 0 and g(c)...

Homework Statement given this ODE with initial conditions y(1)=0 \[ (x + y^2 )dx - 2xydy = 0 \] Homework Equations solving this ODE gives us \[y = \sqrt {x\ln (x)} \] as we can see this equation is true only for x>=1 in order to use the theorem on existence and uniqueness we isulate...

I remember before reading bits and pieces about how if we have a function of two variables, say f = f(x,y), then it must be true that d/dx(df/dy) = d/dy(df/dx), where the "d"'s are partials. Can anyone guide me to what this theorem is called or to its implications? Also, does it work in...

Hi all. I'm currently working on a problem that has led me to an integral equation of the form: u(t)=\int_0^t K(t,\tau)f(\tau)\, \mathrm{d}\tau \qquad t\in (0,T) or simply u=Kf. I've managed to prove the following: K :L^2(0,T)\rightarrow L^2 (0,T) K is compact. u\in L^2(0,T)...

ultra sensitive microphone or technique for listening to inner ear A friend has very loud tinnitus. He says it is constantly louder than anything else he can hear. Nothing can be heard by an independant observer such as myself, but it is possible that a sound with a small power level is so...

Hi! Thanks for reading! :) Homework Statement Y(x) is the solution of the next DFQ problem: y' = [(y-1)*sin(xy)]/(1+x^2+y^2), y(0) = 1/2. I need to prove that for all x (in Y(x)'s definition zone), 0<Y(x)<1. Homework Equations I just know that this excercise is under the title of "The...

Fix b >1, \ y >0 , and prove that there is a unique real x such that b^{x} = y . Here is the outline: (a) For any positive integer n , b^{n}-1 \geq n(b-1) . Why do we do this? (b) So b-1 > n(b^{1/n}-1) . (c) If t>1 and n > (b-1)/(t-1) then b^{1/n} < t . etc.. Is...

1. What are the limits in population density achieved by each of the species when grown independently and when combined with another species? This is looking at Gause's struggle to existence on paramecium species (bursaria, caudatum, and aurelia). I believe that the answer to this question is...

This is a bit philosophical. What does it mean to say that a mathematical object exists? To add some concrete thoughts, I recently read the following: "The empty set has the property that for all objects x, the statement 'x is in the empty set' is false." But this statement reeks of all...

Homework Statement Suppose f is a real function on (0,1] and f is Riemann-integrable on [c,1] for every c>0. Define \int_0^1 f(x) dx = \lim_{c\to 0} \int_c^1 f(x) dx if this limit exists and is finite. Construct a function f such that the above limit exists, although it fails to exist with...

Let A be a constant. Let f(t) be an integrable function in any interval. Let h(t) be defined on [0, oo[ such that h(0) = 0 and for any other "t", h(t) = (1 - cos(At)) / t It is not difficult to see that h is integrable on [0, b] for any positive "b", so fh is also integrable in...

As I understand it, according to the Copenhagen interpretation of QM, nothing can be said to exist until it is observed. I have also read that it is impossible to observe virtual particles in an experiment. How is it then that virtual particles can be said to exist?

Its my understanding that Bayes' Theorem was presented by a clergyman who used it to answer whether God exists. Does anyone on this forum knows about the conclusion he reached through his research.

Homework Statement y y\prime = 3 y(2) = 0 Homework Equations Solve and find two different solutions. The Attempt at a Solution F = \frac{3}{y} \frac{\partial F}{\partial y} = \frac{-3}{y^2} Where do I go from here?

Homework Statement Hello. My following problem is partially about the maths concept involved but is largely to do with what the question is actually asking? It's from an online quiz and a printscreen of it has been provided as an attachment. Homework Equations See attachments for the...

I ask this because it seems to me there is a striking paradox to a God's existence claim. I do think that what is and what isn't are conceptions that depend on our perception of the universe. So how, then, if we take down this road of thought, can we speak of existence outside the physical...

Homework Statement Prove that for all x there exists and x if it is an element of the positive integers, then there is an integer y and an integer z. Homework Equations The Attempt at a Solution I know that the contrapositive would be "If there is not an element of the positive...

So, I was recently think about something. Don't ask me how I thought of this, but positrons and anti-protons were on my mind all day and I thought of something. I thought about inertia. I thought about it and the more and more I did I figured that there must be something within matter that makes...

Hello! Can anyone help me with the following question about black holes? Let us consider a massive star which at the end of its evolution collapses into a black hole (say a Schwarzschild black hole, for the sake of simplicity). An observer far away, in its coordinate time, will never see the...

How do you prove the existence but not necessarily the value of a limit?

I am familiar with the existence and uniqueness of solutions to the system \dot{x} = f(x) requiring f(x) to be Lipschitz continuous, but I am wondering what the conditions are for the system \dot{q}(x) = f(x) . It seems like I could make the same argument for there existing a...

[SOLVED] Existence of solution to integral equation Homework Statement There's k:[0,1]²-->R square integrable and the operator T from L²([0,1],R) to L²([0,1],R) defined by T(u)(x)=\int_{0}^{1}k(x,y)u(y)dy (a) Show that T is linear and continuous. (b) If ||k||_2 < 1, show that for any f in...

[SOLVED] Evidence for the existence of neutrinos. [solved]

Homework Statement I was talking with my maths lecturer about how he knew certain special differential equations such as y'' = y has only y = e^x + e^-x as a possible solution. I understand the superposition principle but not why only y = e^x satisfies the DE. Why can't there be some other...

Because of the existence of sterile neutrino, how and why it gives effect on the big bang nucleosynthesis more than active-active neutrino oscillation. Its a question on behalf of my friend. thanks for any help

Homework Statement It is a theorem in my book that if f and g are two Lebesgue integrable complex valued functions on R, then the integral \int_{-\infty}^{+\infty}|f(x-y)g(y)|dy is finite for almost all x in R. Why not all? f is integrable, hence bounded, say, by M. Therefor, whatever x, we...

I have a question. This question, you should know, is based on something that I actually haven't heard in a classroom or anywhere legitimate, so if the assumption that I have that I'm asking my question about is false, just let me know :) Alright, the thing I'm assuming is true is that...

After mulling over special relativity for a while, I have come to some conclusions that seem correct, but I don't know if I'm on the right track. I'd like to know what some of you think. Please keep in mind that I only recently started learning about relativity, so I'm somewhat of a layperson...

I'm reading a proof where there's a conclusion: "Since zW\subset W, there is an eigenvector v\neq 0 of z in W, zv=\lambda v." There W is a subspace of some vector space V, and z is a matrix, in fact a member of some solvable Lie algebra \mathfrak{g}\subset\mathfrak{gl}(V). (Could be irrelevant...

Let w be a 1-form on smooth manifold M. Then is there a vector field X such that locally w(X)=f where f:M-->R continuous? How can I prove it? Thanks.

I have 2 maps f and h such f :\, (\mathcal{X}, \mathbb{E}) \rightarrow (\mathcal{Y}, \mathbb{K}) h :\, (\mathcal{X}, \mathbb{E}) \rightarrow (\mathcal{Z}, \mathbb{G}) where \mathbb{K} and \mathbb{G} are \sigma-algebras on the spaces Y and Z respectively, and \mathbb{E} =...

Could you please explain the theory intuitively and provide a proof to it. I understand how to apply it but i want to understand the logic behind it.

I just don't understand the idea behind it. I hate it when they throw these theories at us without proofs or elaborate explanations and just ask us to accept and applym mthem. Anyone care to enlighten me?

Homework Statement Suppose f(t,x) is a continuous vector valued function on \mathbb{R} \times \mathbb{R}^n. If f is locally Lipschitz with respect to x with the property that \|f(t,x)\| \le C \|x\| for some positive constant C > 0, then prove that the maximum interval of the existence of the...