Existence Definition and 543 Threads

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.

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...

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...

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 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...

In http://arxiv.org/abs/1010.1939, Eq 26 & 27, Rovelli used 2 limits to define the current spin foam models. But he doesn't know if those limits exist. In http://arxiv.org/abs/1010.5437, Rovelli and Smerlak further elaborated properties of the limits, assuming their existence. Frank Hellmann...

I have attatched my Paradox for the existence of 4,5 and 7 using Brocard's problem . I don't know where i have gone wrong as 4,5,7 exist, surely.

Let A = [a b; c d] a 2x2 matrix with complex entries. Suppose that A is row-reduced and also that a+b+c+d =0 . Prove that there are exactly three such matrices... so i realize that there are seven possible 2x2 matrices that are row-reduced. [1 0; 0 1], [0 1; 1 0], [0 0; 1 0], [0 0;0 1]...

We've done a little bit on existence/uniqueness of solutions, and there's one thing that's a little confusing to me. We have a theorem which (paraphrased) says that if you have a linear ODE with an initial value problem, then a solution exists on the largest open interval containing t0 on which...

I have been thinking out of the box and I have come to the conclusion that something cannot exist without consuming something else. I know this sounds really wacky (for want of a better phrase :)) which is why I posted this in the Quantum forum, lol. We know plants consume nutrients and use...

How do you prove that if \textrm{card}(X)\leq\textrm{card}(Y) is not true, then \textrm{card}(X)\geq\textrm{card}(Y) must be true? In other words, if we know that no injection X\to Y exists, how do we prove that an injection Y\to X must exist? This is not the same thing as what...

Homework Statement Consider the IVP compromising the ODE. dy/dx = sin(y) subject to the initial condition y(X) = Y Without solving the problem, decide if this initial value problem is guaranteed to have a unique solution. If it does, determine whether the existence of that solution is...

why do minority carriers exist? in a p-type material, why don't the minority carriers recombine (and get annihlated) with excess holes?

Hi all, I was reading the book by Herbert Federer on Geometric Measure Theory and it seems he proves the existence of the Tensor Product quite differently from the rest. However it is not clear to me how to prove the existence of the linear map "g" in his construction. He defines F as the...

Is pure quantum gravity known to exist? I had thought it exists in 3D, but Strominger writes http://arxiv.org/abs/0906.1313 "Determining Z for pure 3D quantum Einstein gravity - if it exists - is an important open problem" Eg. Does the Turaev-Viro model not describe 3D QG?

Homework Statement for the differential equation t^2y''-2ty'+2y=0 with the general solutions y=C(t) + D(t^2) where C and D are constants. given the inital solution y(0)=1 and y'(0)=1 there are no solutions that exist. Why does this not contradict the Existence and Uniqueness Theorem...

I know this probably sounds weird, but I have a research problem that requires "random" analog circuits. Basically what this means is that I create Spice netlists by randomly adding linear and/or nonlinear components of random types with random node and parameter values. This works fine and I...

In certain philosophy discussions the concept of a square circle sometimes comes up as an example of something that can be proven not to exist. It occurred to me that the impossibility of its existence depends on: 1. the definitions one uses for square and circle; and 2. the geometry in...