Bounded Definition and 514 Threads

Let $u_t=u_xx,\,t>0,\,x\in\mathbb R$ and $u(x,0)=xe^{-|x|}.$ Show that $|u(x,t)|\le \dfrac K{\sqrt t}$ for all $t>0$ and $x\in\mathbb R$ where $K$ is a constant. So I apply Fourier transform, then $\mathcal F(u_t)=\mathcal F(u_xx)$ then $\dfrac{{\partial \mathcal F(u)(w,t)}}{{\partial t}} = -...

Consider the set $S=\left\{ z\in \mathbb{C}:\text{Re}(z)>0,\text{ }\arg (z)\in \left( -\dfrac{\pi }{4},\dfrac{\pi }{4} \right) \right\},$ and a function $f\in H(S)\cap C(\overline S)$ so that for each $z\in\partial S$ is $|f(z)|\le1$ and for all $z=x+yi\in S$ is $|f(z)|\le e^{\sqrt x}.$ Prove...

Homework Statement If {S_n} is a sequence whose values lie inside an interval [a,b], prove {S_n/n} is convergent. We don't know Cauchy sequence yet. All we know is the definition of a bounded sequence, and convergence and divergence of sequences. Along with comparison tests and Squeeze...

Homework Statement Let a_n = 1 + 1/(1*2) + 1/(2*3) + ... + 1/(n*[n+1]). Prove {a_n} is bounded above. Homework Equations 1/(2*3) = 1/2 - 1/3 The Attempt at a Solution I accidentally left my notebook at school and I have no idea how to do this without my class notes. The book...

I just ran into this problem and have no idea how to solve it. Basically I'm trying to prove that all orders of derivative of the given function is bounded by the function on the right. I'm pretty sure the inequality is true, but I really have no clue on how to prove it. I thought about using...

Homework Statement Let f:[-1,1] \times \mathbb{R} \to\mathbb{R} be a function. If f is defined by: (i) f(x,y) = 3\exp(x-y^2) then is the derivative with respect to y bounded? If f is defined by: (ii) f(x,y) = 7\exp(y^2-x) then is the derivative with respect to y bounded...

Homework Statement Show that if X is a bounded random variable, then E(X) exists.Homework Equations The Attempt at a Solution I am having trouble of finding out where to begin this proof.This is what I got so far: Suppose X is bounded. Then there exists two numbers a and b such that P(X > b)...

If I have a spread of electrical charges contained inside a Gaussian surface, and if I cause those electrical charges to move at relativistic speeds, the electric fields of those charges should be subject to relativistic contraction. What happens then to electric flux that cuts through that...

Hi Suppose (X,d) is a bounded metric space. Can we extend (X,d) into (X',d') such that (X',d') is compact and d and d' agree on X? ( The reason for asking the question: To prove a theorem in Euclidean space, I found it convenient to first extend the bounded set in question to a compact one (...

Question: Find the exact value of the are of the region bounded by: x^3, the x-axis and x=1 and x=4 Answer is 3.75 I tried finding the anti derivative so 1/4(x)^4, and therefore I've got 1/4(4)^4 - 1/4, which isn't the correct answer

Homework Statement Give an example of a bounded subset of Q which has no least upper bound in Q. Explain why your answer has this property. Homework Equations The Attempt at a Solution [1/8, 1/4, 3/8, 1/2, 5/8, 3/4...infinity] is this correct?

I came across this proof and have a question about the bolded portion: Consider the following objection to the bolded: In order for \mathcal{G} to be an open cover of K its sets must contain all of the points of K. The sets of \mathcal{G} are B_r(p) for some fixed p, and so as r gets...

Hi, If I have an additive function which is f(x+y)=f(x)+f(y), the question is how can we prove that if this function has a limit at each real number then there is a number a greater than zero and M greater than zero such that |f(x)|\leq M, for all x\in[-a,a],

Homework Statement Find area of regions bounded by x^2 + y^2 = 9, y = 2x, x-axis in the first quadrant The attempt at a solution So, i drew the graph of y against x in my copybook, and circle with origin (0,0), radius = 3 units. The line y = 2x cuts through the circle. Transforming to...

Homework Statement Evaluate. ∫∫D x2 dAxy, bounded by 5x2 + 4xy + y2 = 1 Homework Equations ∫∫D H(x,y) dAxy = ∫∫D H(u,v)\frac{\delta(x,y)}{\delta(u,v)}dAuv The Attempt at a Solution So I understand I'm supposed to find a change of variables to transform the ellipse into a circle...

I'm not sure if I am confusing myself or not, but a friend and I were trying to figure this out. Basically, I know that if a sequence is bounded, we are guaranteed at least one convergent subsequences. However, is it possible for a bounded sequence to have infinitely many of such subsequences?

Hi! I want to learn a course of "general relativity". For this, I've realized that I have to master the differential geometry. So, I've chosen Lee's book called " introduction to smooth manifolds". In the appendix of the book, some required knowledege of integrations on an euclidean space...

First, is one light wave (or perhaps half wave) possible that stretches across the universe, such that each end of the wave (or half wave) is on opposite sides of the event horizon of the universe, which is the distance light has traveled since the beginning of the universe. Second, is this...

I have a point in 3D specified by its coordinates (x0, y0, z0) I have a line in 3D specified and bounded by its end points (x1, y1, z1) and (x2, y2, z2) How do I calculate the minimum distance between the point and the line, keeping in mind that it may not be the perpendicular distance...

Homework Statement Find a sequence of differentiable functions $f_n\colon [a,b]\rightarrow\mathbb(R)$ s.t.: --there exists $M>0$ with $|f_n'(x)|\leq M$ for all $n\in\mathbb{N}$ and $x\in[a,b]$; --for all $n\in\mathbb{N}$, $|f_n(a)|\leq M$; --$(g_n)$ is a convergent subsequence with...

I'm working on a problem for my analysis class. Here it is: Let f be differentiable on an open subset S of R. Suppose there exists M > 0 such that for all x in S, |f'(x)| ≤ M, i.e. the derivative is bounded. Show that f is uniformly continuous on S. I'm not too sure that this question is...

Homework Statement Find the absolute minimum and maximum of F(x,y,z) = x2 - 2x - y2 + z2 on the ellipsoid G(x,y,z) = x2 + 4y2 + z2 = 4 Homework Equations The Attempt at a Solution I was thinking of trying to solve this by using Lagrange multipliers. So, finding the gradients: Fx = 2x - 2 = Gx...

Homework Statement A bounded monotone sequence converges. Proof for bounded monotone increasing sequence and decreasing sequence. Does both them converges?Homework Equations So, I used the least upper bound and great lower bound to prove increasing sequence and decreasing sequence...

Hey, I'm trying to prove that uv=>0 is bounded so I can state that an entire function is constant when f = u + iv, when f is entire. I have worked out the rest but I'm struggling to prove that its bounded, Can you say u=>0, v=>0 then u + v => 0, and that bounded from below?

Say f is a non-negative, integrable function over a measurable set E. Suppose \int_{E_k} f\; dm \leq \epsilon for each positive integer k, where E_k = E \cap [-k,k] Then, why is it true that \int_E f\; dm \leq \epsilon \quad ? I know that \bigcup_k E_k = E and intuitively it seems...

Hello all. I am having a very serious problem. The question states: Find the value(s) of δ such that the solution of the initial-value problem y'' − 4y = sin x; where y(0) = δ and y'(0) = 0 is bounded. I have no problem "solving"...

Homework Statement Is the point P(0,2) in the region bounded below by y=x I'm not quite sure i understand this question. the 'bounded below' part mainly. Is it asking if the point is below the x axis? The wording is confusing me Homework Equations The Attempt at a Solution I...

Homework Statement This is an example taken from the textbook lesson and there's one part I don't understand: Find the area of the region bounded above by the spiral r = pi/(3θ) and below by the polar axis, between r = 1 and r = 2. SOLUTION: Double integral of r(dθ)(dr) with boundaries...

Homework Statement Want to prove that [0,1] in R is compact. Let \bigcup_{\alpha\in A} I_{\alpha} be an open cover of [0,1]. By open sets in R. Let E={t\in[0,1] s.t. [0,t] is covered by a finite number of the open cover sets I_{\alpha}}. Prove that E\neq\emptyset. The Attempt at a...

Homework Statement Let f and g be real-valued functions defined on A ⊆ R and let c ∈ R be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that limx→c g(x) = 0. Prove that limx→c f(x)g(x) = 0. Homework Equations The Attempt at a Solution This isn't a very hard...

Im having a little trouble about how to go about defining this signal. It has a sqrt(-1) in it raised to a power so this is where i get confused. No doubt my poor algebra skills may be holding me back from understanding this problem. The signal is x(k)=j^-k u(k) I need to determine: A...

Homework Statement Suppose that the function f|(a,b)→ℝ is uniformly continuous. Prove that f|(a,b)→ℝ is bounded. Homework Equations A function f|D→ℝ is uniformly continuous provided that whenever {un} and {vn} are sequences in D such that lim (n→∞) [un-vn] = 0, then lim (n→∞) [f(un) -...

Can someone give me an example of a bounded function f defined on a closed interval [a,b] such that f does not attain its sup (or inf) on this interval? Obviously, f cannot be continuous, but for whatever reason (stupidity? lack of imagination?) I can't think of an example of a discontinuous...

Hi, just a quick question. Let f be real function s.t. the limit of f as x approaches a equals L. Is f bounded? i.e. is it sufficient to assume a function is bounded if it has a limit. Thanks to all who may reply.

Hello everybody, A few years ago i tried to join a mathematics department and in the relevant exams i came up against the following problem. I apologise beforehand if the statement of the problem is a little bit ambiguous because i do not remember it exactly. However, I am sure you will get...

Homework Statement Let A be a positive definite n\times n real matrix, b\in\mathbb{R}^n, and consider the quadratic polynomial Q(x)=\frac{1}{2}\langle{x, Ax\rangle}-\langle{b, x\rangle}. Show that Q is bounded below. 2. The attempt at a solution I have to come up with a constant m so that...

Find the area of region bounded by y = 4 -x^2 ; y = 2 - x; x = - 2 and x = 3 I've calculated the area to be 11/3 however the answer given is 49/6. Is the answer correct? I've recalculated this twice and there seems to be nothing wrong with my working...

I have been consulting different sources of analysis notes. My confusion comes from these two definitions \begin{defn} Let S be a non-empty subset of $\mathbb{R}$. \begin{enumerate} \item $S$ is Bounded above $ \Longleftrightarrow\exists\,M > 0$ s.t. $\forall\, x\in S$, $x\leq M$...

Homework Statement Hello everyone, I was wondering if someone could check my work on this problem as I'm not sure it's right. I'm in Calc II right now and we are doing finding volume bounded by curves rotated around an axis. So, here is the problem, y=cos(pix)+1, y=4x^(2)-9, x=0; about...

Suppose we're in a general normed space, and we're considering a sequence \{x_n\} which is bounded in norm: \|x_n\| \leq M for some M > 0. Do we know that \{x_n\} has a convergent subsequence? Why or why not? I know this is true in \mathbb R^n, but is it true in an arbitrary normed space? In...

Homework Statement Theorem 1.4: Show that every Cauchy sequence is bounded. Homework Equations Theorem 1.2: If a_n is a convergent sequence, then a_n is bounded. Theorem 1.3: a_n is a Cauchy sequence \iff a_n is a convergent sequence. The Attempt at a Solution By Theorem 1.3, a...

Can a function f: (a,b) in R be bounded and diffferentiablle, but have an unbounded derivative. I believe it can, but can not think of any examples where this is true. Anyone have any ideas?

Hi, All: Let f R-->R be differentiable. If |f'(x)|<M< oo, then f is uniformly continuous, e.g., by the MVTheorem. Is this conditions necessary too, i.e., if f:R-->R is differentiable and uniformly continuous, does it follow that |f'(x)|<M<oo ? Thanks.

Am I right in thinking that the statement "f:[a,b]-->R is of bounded variation" is equivalent to the statements "f:[a,b]-->R has bounded range" and "f;[a,b]-->R is a bounded function".

Homework Statement Find the volume bounded by the paraboloid z= 2x2+y2 and the cylinder z=4-y2. Diagram is included that shows the shapes overlaying one another, with coordinates at intersections. (Will be given if necessary) Homework Equations double integral? function1-function2...

Hello, I'm working through an analysis textbook on my own, and came across a true/false question I was hoping someone could help me with. The question is: If A is a bounded set, then s = sup A is a limit point of A. I think that the statement is false, as I came up with what I think is...

Homework Statement Prove that every convergent sequence is bounded. Homework Equations Definition of \lim_{n \to +\infty} a_n = L \forall \epsilon > 0, \exists k \in \mathbb{R} \; s.t \; \forall n \in \mathbb{N}, n \geq k, \; |a_n - L| < \epsilon Definition of a bounded sequence: A...

In a book I'm reading, it defines a bounded bilinear mapping \omega: X\times Y\rightarrow W, where X,Y and W are all normed linear spaces as \left\| \omega(\xi,\eta)\right\| \leq b \left\| \xi \right\| \left\| \eta \right\| So it uses \left\| \xi \right\| \left\| \eta \right\| as a norm on...

Homework Statement Find the volume bounded by the following surfaces: z = 2(x^{2}+y^{2}) z = 18 y = \frac{1}{\sqrt{x}} y = -\frac{1}{\sqrt{x}} x\geq0 The Attempt at a Solution I have no Idea how to attempt it! I mean, I will, somehow. But want to know a straight-forward way. Would...

Why a chaotic system always bounded? What factor control the boundedness?