What is Existence: Definition and 570 Discussions

I was playing trying to work through a proof in Apostol's Calculus and can't quite understand a step noted. This is from chapter 3, theorem 1.35. Every nonnegative real number has a unique nonnegative square root. The part where you are establishing the set S as nonempty so you can use LUB it...

Existence of strictly convex functions with same "ordering" as convex one Consider any real-valued convex function c : R^n \rightarrow R. I am interested in whether there exists some strictly convex function d, that satisfies d(x) > d(y) if c(x) > c(y). That is, given a convex function, can we...

Can anyone point me to some publications or archives which feature developments in solving the N-S existence and smoothness problem? Basically, I'd like to read up about how far people have gone towards solving the problem, e.g. a new method to analyze the equations. Also, what fields of...

I've read that mesons consist of a quark and an antiquark. So, here's my question. Why don't the quark and the anti-quark annihilate with each other (like they ususally do)? For example, the pi(0) meson consists of the up and the anti-up quark and the eta meson consists of the down and the...

How will the LHC prove the existence of these two particles? How is it possible to prove they exist, and what means will they use to find out?

Homework Statement Is there a theorem that states the following? Let P= \{ P_{1}, . . . , P_{n} \} be the set of n distinct points in \mathbb{R}^{n-1} and P'= \{ P'_{1}, . . . , P'_{n} \} also a set of points in \mathbb{R}^{n-1}. If for all i,j |P_{i} - P_{j}|=|P'_{i} - P'_{j}| then there...

I know this topic has been addressed on this forum ad nauseam and please redirect my post if it is posted incorrectly. The basis for my view is that the scientific method will not be able to distinguish between different views which make the same observational predictions. I do understand...

Let's say there is a chamber with 1000 O2 molecules. According to wikipedia, a perfect vacuum is an arbitrary space in which there is an absence of matter. Of course, the space that consists of the entire volume inside the chamber isn't a vacuum because of those 1000 molecules, but then isn't it...

Be {xn} a sequence that satisfies the condition 0 ≤ x_{m+n} ≤ x_{m} + x_{n}. Prove that lim_{n ->∞} xn/n exists. I'm kind of lost in this.

Can somebody help me out here? Consider y'(t) = y(t)[a(t) - b(t)y(t)] where a,b : (-infinite, +infinite) → (0, +infinity) and there exists M>0 such that: (1/M) ≤ a(t), b(t) ≤ M, for all t in the reals Claim: There exists a unique positive solution Phi(t) defined for all t in the reals in...

Homework Statement Prove the following statement is true or not: the statement: Let (X,d) be a non-empty metric space and A is a non-empty subset of X. Then if A' is not empty, then A' is infinte. Homework Equations Definition of limit point and its negation. The Attempt at a...

In mathematics, people construct and define things. And along with these, I found something very confusing: Statement 1: if a thing exists, then the thing is defined. I think this statement is true in mathematics. But this kind of statement I can't translate to first order calculus...

Homework Statement Show that there exists one root int (0,2) of the following function: f(x)=(1-x^2)^2-√((1-x^2)*(1-1/2*x^2)) Homework Equations The Attempt at a Solution I first found: f(0)=0 and f(2)=7.268 But, i don't know what to do now. I'm not sure if it has...

Homework Statement In the following case: x'(t) = log (3t (x(t) - 2)) does the theorem of existence and unicity guarantee a unique solution for the initial value problem x(3) = 5, justify your answer?Homework Equations x'(t) = log (3t (x(t) - 2))The Attempt at a Solution Ok what I would do is...

dy/dx=(sinx)/y Initial condition is y(pi/2)=1 The solution to the IVP is y=(1-2cosx)^.5 That I know is correct, but they're saying the interval of existence is when pi/3<x<5*pi/3. Is that wrong? I think it should include the π/3 and 5π/3.

Ok so ill give an example, x'(t) = log(3t(x(t)-2)) is differential equation where t0 = 3 and x0 = 5 The initial value problem is x(t0) = x0. So what i'de do is plug into initial value problem to get x(3) = 5, so on a graph this plot would be at (5,3)? Then plop conditions into differential...

I have begun to read about the hyperreals, and am wondering whether the natural extensions of real-valued functions to hyperreal-valued functions is simply a definition of the hyperreals, or can it be proved? Or is it accepted as an axiom? For example, if f(x) = sin(x), then is the existence...

Homework Statement lim(x,y)->(1,0) of ln((1+y^2)/(x^2+xy)) Homework Equations The Attempt at a Solution Used two paths, x=1 y=0 both gave my lim=0 so I tried x=rsin y=rcos, in attempt to use ε-δ to prove it. got to ln((1+r^2sin^2)/(r^2cos(cos+sin))) not sure where to go...

Hello. I just came to realize that in mathematics, a problem is defined as a set of strings. But then, for example, if I state a problem as "Find an addition of 1 and 2," how are strings (e.g. find) connected? Is any form of a string that connects these strings that is a string?

Homework Statement x/√(x2+y12)-(l-x)/√((l-x)2+y22)=0 How do I prove that the above equation has a solution for x in ℝ and that the solution is unique? (y1, y2, and l are constants.) Homework Equations x√((l-x)2+y22)-(l-x)√(x2+y12)=0 x√((l-x)2+y22)+x√(x2+y12)=l√(x2+y12)...

Dear MHB members, Suppose that $p,f$ are locally essentially bounded Lebesgue measurable functions and consider the differential equation $x'(t)=p(t)x(t)+f(t)$ almost for all $t\geq t_{0}$, and $x(t_{0})=x_{0}$. By a solution of this equation, we mean a function $x$, which is absolutely...

I'm having trouble understanding what uniqueness is/means. When given a slope/direction field I don't know what I should be looking for if asked to determine if a given initial condition has a unique solution. Example: \textit{y' = }\frac{(x - 1)}{y} With this equation I can see that as long...

Hello, So I am confused on whether the statement that "a function f exists at all points in an open subset U of (say) R" , indicates that it is well-defined on all the points in that subset i.e will the function have a real value on all the points in the subset? Also, can the derivative of...

Homework Statement Let f(t) be a signal whose time period is T. 1. if f(t+T/2)=f(t) proof that the Fourier series representation will contain no odd harmonics 2. if f(t+T/2)=-f(t) proof that the Fourier series representation will contain no even harmonics Homework Equations The...

What is a proof that every Riemann surface has a non-constant meromorphic function? Is it even true? I was wondering this because if it is true then for compact Riemann surfaces without boundary one can use the meromorphic function to produce a meromorphic 1 form whose degree - the sum of the...

Happy new year from France. I am reading books on elementary particle and i see that their gauge bosons may be neutral or have opposite charge. They live in semisimple Lie algebras. So I searched in math books how to prove that in a semisimple Lie algebra if α is a root so is -α. I found...

Hi everyone, I'm not quite sure how to proceed to show existence (and perhaps uniqueness) of the following system of (first order) differential equations: \dot{x}=f(t_1,x,y,z) \dot{y}=g(t_2,x,y,z) \dot{z}=h(t_3,x,y,z) where \dot{x}=\frac{\partial x}{\partial t_1}...

Hi everyone, I'm not quite sure how to proceed to show existence (and perhaps uniqueness) of the following system of (first order) differential equations: \dot{x}=f(t_1,x,y,z) \dot{y}=g(t_2,x,y,z) \dot{z}=h(t_3,x,y,z) where \dot{x}=\frac{\partial x}{\partial t_1}, \dot{y}=\frac{\partial...

Is it possible for a particle to exist according to one reference frame and simultaneously not exist according to another? If energy is relative, can a collision between two particles have enough energy to produce new particles according to its own reference frame but not have said amount of...

Homework Statement Let f(t) = t if 0<t<3 et if t>3 a. Is f(t) piece-wise continuous? b. Is f(t) of exponential order α? Either prove it by producing an M, T and α that satisfies the definition, or show that no such constants exist. c. Does the Laplace transform of f(t) exist...

The trade off in uncertainty in momentum vs position in the HUP leaves me confused. In the context where the momentum is measured to increasingly greater precision and consequently position becomes less refined, is any logical inference made about the existence of the object i.e. is it...

If p is prime and (a, p) = 1, show that x^2 \cong a (mod p) has solutions if a^{\frac{p-1}{2}} \cong 1 (mod p) and does not have solutions if a^{\frac{p-1}{2}} \cong -1 (mod p) . So because of Euler's theorem i know that a^{p-1} \cong 1 (mod p) and from the hypothesis...

Perrin got his nobel prize because of his experiment on brownian motion, which is thought to have proven the existence of atoms i cannot understand that if you see an array of dots from STM, i believe you demonstrate the existence of atoms but i cannot see why perrin's experiment or...

I recently watched another program with Stephen Hawking called "Did God Create The Universe?" However this topic isn't religion related but my questions popped up due to this. I am having problem understanding how the universe could pop up from nothing when time did not exist. They said this...

Homework Statement Given M \in N, show that there exists an X \in N such that for all n \geq X , n^2+n+1 \succ M Homework Equations The Attempt at a Solution Since both M and X are natural numbers and I am just trying to prove the existence of a certain natural number X, I...

I asked this question before but I totally misunderstood what it was asking. Basically, I need to find that there exists a sequence {a_k} such that it converges to x for some x in R. Since the real numbers are equivalence classes of convergent Cauchy sequences the result seems fairly obvious...

Why is the existence of The Big Bang "agreed" upon? From all my research and studies mathmatical evidence shows the existence of Black holes. From some of the most fundamental physics we have our Conservation of Energy and Black holes are at the moment considered points where it is possible...

I'm trying to figure out how to show that the limit of a function of two or more variables does not exist. I know that to do this we must show that the limit from 2 different pathways is not equal to the same thing, but, I want to know how to figure out what pathways to check, assuming you...

Do you find this argument by this author that SR implies "at least one continuum other than our own spacetime" flawed or reasonable? According to the special theory of relativity, observers stationary relative to one another will measure the time in the rest frame of an entity moving relative...

Homework Statement Show that for every a* = (a1, 1/a1), there exists another point of the form (a, 1/a) in a ball (i.e. circle, since we're in R2) of radius r, centered at a*, for any r > 0. The Attempt at a Solution This is actually only a part of the whole problem, but I just can't put it...

Hi I know it's easy to prove that if a vectorfield is the gradien of a potential, \vec F = \nabla V, then \nabla \times F = 0. But how about the converse relation? Can I prove that if \nabla \times F = 0, then there exist a salar potential such that \vec F = \nabla V? I get as far as...

Homework Statement Let X={1/n: n\inN} (where N is the set of natural numbers) i) Does inf(X) exist? ii) What is inf(X)? Homework Equations The Attempt at a Solution I think I should try to prove inf(X) exists by considering it a Lower Limit, but I don't know how to go about...

Hi, I saw a proof/argument done today that I think was wrong: It is finding the limit as a->oo of the integral from 0 to b<oo: Int_(0..b) Sqr[x(1 +cos(ax))]dx , where Sqr is the square root Now, the argument given was that one could find a bound for the oscillation of...

Homework Statement Solve the Cauchy problem: (t2 + 1)y' + etsin(t) y = sin(t) t2 y(0) = 0 Homework Equations y'(t,y) + p(t)y = g(t,y) Integrating factor e(integral of p(t)) The Attempt at a Solution I tried finding an integrating factor, but it came out ugly. I couldn't solve the...

What is dark energy? What is it composed of?

Hello, I have a PDE: 3*u_x + 2*u_y = 0, and I am interested in determining initial values such that there is a unique solution, there are multiple solutions, and there are no solutions at all. What theorem(s)/techniques would be of use to me for something like this? Regards, Dan

Homework Statement Prove that if x in R and epsilon > 0 are arbitrary, then there exist r in Q such that |x - r | < epsilonHomework Equations The Attempt at a Solution I'm stumped on this one. I tried using the reverse triangle inequality, but I seemingly hit dead ends with it.

Hi, Just wanted to ask a question regarding existence and uniqueness of solutions to SDEs. Say you have shown existence and uniqueness of a solution to an SDE that the process [tex ] X_{t} [/tex ]a particular process follows (by showing drift and diffusion coefficients are Lipschitz). If you...

Hi everybody, I was wondering this: "What is the probability, given all the information (including scientific evidence and accepted theories), of having this existence (I'm not talking about life and consciousness) just right how it is?" I have no idea about any kind of research or study...

Hi everybody, I was wondering this: "What is the probability, given all the information (including scientific evidence and accepted theories), of having this existence (I'm not talking about life and conciousness) just right how it is?" I have no idea about any kind of research or study...