Ngô Quốc Anh

Tháng Mười Hai 2, 2009

MuPAD

Chuyên mục: Linh Tinh — Ngô Quốc Anh @ 22:21

MuPAD was a Computer algebra system (CAS). Originally developed by the MuPAD research group at the University of Paderborn, Germany, it was developed by the company SciFace Software GmbH & Co. KG in cooperation with the MuPAD research group and partners from some other universities since 1997.

I found MuPAD a very cool tool for plotting graph of functions. I will show you in details some examples.

1. We start with the following function

f(x,y)=-(x^2+y^2)(\cos (3y) + \sin (7y)).

To plot its graph, I use the following

f := plot::Function3d(exp(-x^2-y^2)*(cos(3*y)+sin(7*x)), x=-2..2, y=-2..2, Submesh=[1,1], FillColorFunction=((x, y, z) -> [(z+2)/4, 0.5, (2-z)/4])): plot(f)

and this is what we get

exp(-x^2-y^2)*(cos(3*y)+sin(7*x))

To obtain the following picture

you need

f := plot::Function3d(exp(-x^2-y^2)*(cos(3*y)+sin(7*x)),
x=-2..2, y=-2..2,
Submesh=[1,1]):
graphLines:=plot::modify(f, Filled = FALSE, XLinesVisible = FALSE, YLinesVisible = FALSE,
ZContours = [Automatic, 15], LineColorFunction=((x, y, z) -> [(z+2)/4, 0.5, (2-z)/4])):
linesProjection := plot::Transform3d(
[0, 0, -1.5], // shift vector
[1, 0, 0, // transformation matrix
0, 1, 0,
0, 0, 0],
graphLines):
plot(f,linesProjection)

2. With the following function

{\rm Re}(\cos (x+iy).

To plot its graph, I use the following

f := plot::Function3d(Re(cos(x+I*y)), x=-2..2, y=-2..2, Submesh=[1,1], FillColorFunction=((x, y, z) -> [(z+2)/4, 0.5, (2-z)/4])): plot(f)

and this is what we getRe(cos(x+I*y)3. If our function has singularities, for example, Gamma function f(x,y)=|\Gamma (x+iy)|, we have

Re(cos(x+I*y)f := plot::Function3d(abs(gamma(x+I*y)), x=-4.5..4.5, y=-3..3, Submesh=[3,3], XSubmesh = 1, YSubmesh = 2): plot(f)

4. If you intend to plot sereval graphs, the following is a good example.

sin(x^2 + y^2), cos(x^2 - y^2)

Here we used functions \sin(x^2 + y^2), \cos(x^2 - y^2). The code I used is

plotfunc3d(sin(x^2 + y^2), cos(x^2 - y^2), x = 0..1, y = 0..2)

5. The Mobius band can be drawn by using the following

x := cos(alpha) * (1 + r * cos(alpha/2)):
y := sin(alpha) * (1 + r * cos(alpha/2)):
z := r * sin(alpha/2):
plot(plot::Surface([x,y,z], r = -0.5 .. 0.5, alpha = -PI .. PI,
Mesh = [35, 31], LineColor = RGB::Black.[0.2]),
Axes = None, Scaling = Constrained)

6. If you need a transparent picture, probably you need to add a factor like abs(sin(z)) in the FillColorFunction. Following is an example

K := (x,y,z)->[abs(sin(x)), abs(sin(y)), abs(sin(z)), abs(sin(z))]:
F1 := plot::Function3d(
1.7*x*exp(-1/2*(x^2+y^2)),
x=-4..4, y=-4..4,
FillColorFunction=K):
F2 := plot::Function3d(-1, x=-4..4, y=-4..4):
plot(F1, F2)

The original one is

7. I will end this topic by showing some more pictures obtained by zooming a little bit closed.

sin(x

sin(x

sin(x

sin(x

plotfunc3d(sin(x - -PI)*sin(y - -PI), Submesh = [4, 4], x = -PI .. PI, y = -PI .. PI)

sin(x

sin(x

In conclusion, if you want to plot a general surface in 3D, you need its parametrization in order to draw. For details, I prefer the reader to this paper.

Tháng Mười 30, 2009

A characteristic of essentially bounded functions

In this topic, we prove the following statement

Statement: Let (X,\mathcal B, m) be a probability space. Let h \in L^2(m). Then h is essentially bounded iff h \cdot f \in L^2(m) for all f \in L^2(m).

Proof. If h is bounded, then by using the Holder inequality one has

\displaystyle\int_X {{{\left| {h \cdot f} \right|}^2}dm}\leq \underbrace {\sqrt {\int_X {{{\left| h \right|}^2}dm} } }_{ \leqslant c}\sqrt {\int_X {{{\left| f \right|}^2}dm} }<+\infty

for all f \in L^2(m). Conversely, we suppose h is such that h \cdot f \in L^2(m) whenever f \in L^2(m). Let

\displaystyle X_n = \{ x \in X : n-1 \leq |h(x)| < n\}, \quad \forall n>0.

Then \{X_n\}_1^\infty partitions X. Let

\displaystyle f\left( x \right) =\sum\limits_{n = 1}^\infty{\frac{1}{{n\sqrt {m\left( {{X_n}} \right)} }}{\chi_{{X_n}}}\left( x \right)} ,

where it is understood that the n-term is omiited if m(X_n)=0. Then

\displaystyle\int_X {{{\left| f \right|}^2}dm}=\int_X {{{\left({\sum\limits_{n = 1}^\infty {\frac{1}{{n\sqrt {m\left( {{X_n}}\right)} }}{\chi _{{X_n}}}\left( x \right)} } \right)}^2}dm}\leq \sum\limits_{n = 1}^\infty{\frac{1}{{{n^2}}}}<\infty

which implies f \in L^2(m). Since

\displaystyle\int_X {{{\left| {hf} \right|}^2}dm}=\sum\limits_{n \in F} {\int_{{X_n}} {{{\left| {hf} \right|}^2}dm}}\geq\sum\limits_{n \in F}{{{\left( {\frac{{n - 1}}{n}}\right)}^2}}

where F = \left\{ {n:m\left( {{X_n}} \right) \ne 0} \right\}. Sincc h \cdot f \in L^2(m) we have that F is finite and therefore h is essentially bounded.

Tháng Mười 28, 2009

The weak and weak* topologies: A few words

Chuyên mục: Giải tích 7 (MA4247), Linh Tinh, Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 2:48

The weak and weak* topologies are the weakest in which certain linear functionals are continuous.

We start with a normed linear space X. The dual space of X, denoted by X', is the collection of all continuous linear functionals, i.e., the set of all mapping \ell : X \to \mathbb R satisfying

\ell(ax)=a \ell (x), \ell(x+y)=\ell(x)+\ell(y)

and

\displaystyle\lim_{n \to \infty} \ell(x_n) = \ell(x) when \displaystyle\lim_{n \to \infty} \|x_n - x\|=0.

Definition 1. In X, the strong topology is the norm topology, i.e., we can talk about an open set of X, for example U in the following sense: U \subset X is said to be open if and only if for each x_0 \in U, there exists \varepsilon>0 such that \{ x \in X: \|x-x_0\|<\varepsilon\} \subset U.

Claim 1. Bounded linear functionals are continuous in the strong topology.

Proof. We first recall that a linear functional \ell is said to be bounded if there is a positive number c such that |\ell (x)| \leq c\|x\| for all x \in X.

Now we assume \ell is continuous but not bounded; then for any choice of c=n, one has \ell(x_n) > n \|x_n\|. Clearly, x_n can be replaced by any multiple of x_n; if we normalize x_n so that

\displaystyle \|x_n\|=\frac{1}{\sqrt{n}}

then x_n \to 0 but \ell (x_n) \to \infty. This shows the lack of boundedness implies the lack of continuity.

Now we assume \ell is bounded. For arbitray x_n and x, one gets

|\ell(x_n)-\ell (x)| = |\ell (x_n-x)| \leq c\|x_n-x\|;

this shows that boundedness implies continuity.

Definition 2. In X, the weak topology is the weakest topology in which all bounded linear functionals are continuous.

The open sets in the weak topology are unions of finite intersections of sets of the form

\{ x : a< \ell(x) < b\}.

Clearly, in an infinite-dimensional space the intersection of a finite number of sets of the above form is unbounded. This shows that every set that is open in the weak topology is unbounded. In particular, the balls

\{ x : \|x\|<R\}

opens in the strong topology, are not open is the weak topology.

Definition 3. In X' the dual space of X, the weak* topology is the crudest topology in which all linear functionals

x: X' \to \mathbb R, x(\ell) := \ell(x)

are continuous.

If X' is nonreflexive, the weak* topology is genuinely coarser than the weak topology, as will be clear from the following theorem due to Alaoglu

Theorem (Alaoglu). The closed unit ball in X' is compact in the weak* topology.

We end this topic by the following theorem

Theorem. The closed unit ball in X is compact in the weak topology if and only if X is reflexive.

Tháng Chín 18, 2009

Moving plane method via a small example inspired by Gidas, Ni, and Nirenberg

Chuyên mục: Linh Tinh, Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 1:41

In this section we will use the moving plane method to discuss the symmetry of solutions. The following result was first proved by Gidas, Ni and Nirenberg.

Theorem. Suppose u \in C(\bar B_1) \cap C^2(B_1) is a positive solution of

-\Delta u = f(u) in B_1 and u=0 on \partial B_1

where f is locally Lipschitz in \mathbb R. Then u is radialy symmetric in B_1 and \frac{\partial u}{\partial r}(x)<0 for x \ne 0.

The original proof requires that solutions be C^2 up to the boundary. Here we give a method which does not depend on the smoothness of domains nor the smoothness of solutions up to the boundary.

Statement. Suppose that ­ is a bounded domain which is convex in x_1 direction and symmetric with respect to the plane \{x_1 = 0\}. Suppose u \in C^2(\Omega)\cap C(\bar\Omega) is a positive solution of

-\Delta u = f(u) in \Omega and u=0 on \partial \Omega

where f is locally Lipschitz in \mathbb R. Then u is radialy symmetric in x_1 and D_{x_1}u(x)<0 for x \ne 0.

Idea of proof.  Write x=(x_1, y)\in\Omega for y \in \mathbb R^{n-1}. We will prove that

u(x_1,y)<u(x_1^\star,y) for any x_1>0 and x_1^\star<x_1 with x_1^\star+x_1>0.

Then by letting x_1^\star \to -x_1, we get u(x_1,y)\leq u(-x_1,y) for any x_1. Then by changing the direction x_1 \to -x_1 we get the symmetry.

Proof. Let a=\sup x_1 for (x_1, y)\in\Omega. For 0<\lambda<a, define

\Sigma_\lambda=\{x \in \Omega: x_1>\lambda\}

T_\lambda = \{x_1=\lambda\}

\Sigma'_\lambda= the reflection of \Sigma_\lambda with the respect to T_\lambda

x_\lambda=(2\lambda-x_1,x_2,...,x_n) for x=(x_1,x_2,...,x_n).

In \Sigma_\lambda we define

w_\lambda(x)=u(x)-u(x_\lambda) for x\in \Sigma_\lambda.

Then we have by Mean Value Theorem

\Delta w_\lambda+c(x,\lambda)w_\lambda=0 in \Sigma_\lambda

w_\lambda \leq 0 and w_\lambda \not \equiv 0 on \partial \Sigma_\lambda.

where c(x,\lambda) is a bounded function in \Sigma_\lambda.

We need to show w_\lambda<0 in \Sigma_\lambda for any \lambda \in (0,a). This implies in particular that w_\lambda assumes along \partial \Sigma_\lambda \cap \Omega its maximum in \Sigma_\lambda. By Hopf Lemma we have for any such \lambda \in (0,a)

D_{x_1}w_\lambda\bigg|_{x_1=\lambda} = 2D_{x_1}u\bigg|_{x_1=\lambda}<0.

For any \lambda close to a, we have w_\lambda<0 by the maximum principle for narrow domain. Let (\lambda_0, a) be the largest interval of values of \lambda such that w_\lambda<0 in \Sigma_\lambda. We want to show that \lambda_0=0. If \lambda_0>0, by continuity, w_\lambda \leq 0 in \Sigma_{\lambda_0} and w_{\lambda_0} \not \equiv 0 on \partial \Sigma_{\lambda_0}. Then the Strong Maximum Principle implies w_{\lambda_0}<0 in \Sigma_{\lambda_0}. We will show that for any small \varepsilon>0

w_{\lambda_0-\varepsilon} <0 in \Sigma_{\lambda_0-\varepsilon}.

Fix \delta>0 (to be determined). Let K be a closed subset in \Sigma_{\lambda_0} such that |\Sigma_{\lambda_0} - K| <\frac{\delta}{2}. The fact that w_{\lambda_0}<0 in \Sigma_{\lambda_0} implies

w_{\lambda_0}(x)\leq -\eta <0 for any x \in K.

By continuity we have

w_{\lambda_0-\varepsilon}<0 in K.

For \varepsilon>0 small, |\Sigma_{\lambda_0-\varepsilon}-K|<\delta. We choose \delta in such a way that we may apply the Maximum Principle for domain with small volume to w_{\lambda_0-\varepsilon} in \Sigma_{\lambda_0-\varepsilon}-K. Hence we get

w_{\lambda_0-\varepsilon} \leq 0 in \Sigma_{\lambda_0-\varepsilon}-K

and then

w_{\lambda_0-\varepsilon}(x) <0 in \Sigma_{\lambda_0-\varepsilon} -K.

Therefore we obtain for any small \varepsilon>0

w_{\lambda_0-\varepsilon}(x) <0 in \Sigma_{\lambda_0-\varepsilon}.

This contradicts the choice of \lambda_0.

latex

Tháng Chín 13, 2009

An algebraic identity for 9th level

Chuyên mục: Các Bài Tập Nhỏ, Linh Tinh — Ngô Quốc Anh @ 21:16

In this topic I will show you how to prove

\sqrt[3]{2} + \sqrt[3]{{20}} - \sqrt[3]{{25}} = 3\sqrt {\sqrt[3]{5} - \sqrt[3]{4}}

In order to prove that fact, we just do as following: by using

(a+b+c)^2=a^2+b^2+c^2+2(ab+ac+bc)

we obtain

{\Big( {\sqrt[3]{2} + \underbrace {\sqrt[3]{{20}}}_{\sqrt[3]{{{2^2}}}\sqrt[3]{5}} - \underbrace {\sqrt[3]{{25}}}_{\sqrt[3]{{{5^2}}}}} \Big)^2}=\sqrt[3]{{{2^2}}} + \sqrt[3]{{{2^4}}}\sqrt[3]{{{5^2}}} + \sqrt[3]{{{5^4}}} + 2\left( {\underbrace {\sqrt[3]{2}\sqrt[3]{{{2^2}}}}_2\sqrt[3]{5} - \sqrt[3]{2}\sqrt[3]{{{5^2}}} - \sqrt[3]{{{2^2}}}\underbrace {\sqrt[3]{5}\sqrt[3]{{{5^2}}}}_5} \right).

Therefore

{\left( {\sqrt[3]{2} + \sqrt[3]{{20}} - \sqrt[3]{{25}}} \right)^2} = \sqrt[3]{{{2^2}}} + 2\sqrt[3]{2}\sqrt[3]{{{5^2}}} + 5\sqrt[3]{5} + 2\left( {2\sqrt[3]{5} - \sqrt[3]{2}\sqrt[3]{{{5^2}}} - 5\sqrt[3]{{{2^2}}}} \right).

Clearly

\sqrt[3]{{{2^2}}} + 2\sqrt[3]{2}\sqrt[3]{{{5^2}}} + 5\sqrt[3]{5} + 2\left( {2\sqrt[3]{5} - \sqrt[3]{2}\sqrt[3]{{{5^2}}} - 5\sqrt[3]{{{2^2}}}} \right) = 9\left( {\sqrt[3]{5} - \sqrt[3]{{{2^2}}}} \right).

Thus

{\left( {\sqrt[3]{2} + \sqrt[3]{{20}} - \sqrt[3]{{25}}} \right)^2} = 9\left( {\sqrt[3]{5} - \sqrt[3]{4}} \right)

which yields

\left| {\sqrt[3]{2} + \sqrt[3]{{20}} - \sqrt[3]{{25}}} \right| = 3\sqrt {\sqrt[3]{5} - \sqrt[3]{4}} .

Finally, by using the fact that (a+b)^3=a^3+3a^2b+3ab^2+b^3 we get

{\left( {\sqrt[3]{2} + \sqrt[3]{{20}}} \right)^3} - 25 = \underbrace {\left( {22 + 3\sqrt[3]{{{2^2}}}\sqrt[3]{{20}} + 3\sqrt[3]{2}\sqrt[3]{{{{20}^2}}}} \right) - 25}_{3\left( {\sqrt[3]{{{2^2}}}\sqrt[3]{{20}} + \sqrt[3]{2}\sqrt[3]{{{{20}^2}}} - 1} \right)} > 0

which implies

{\sqrt[3]{2} + \sqrt[3]{{20}} - \sqrt[3]{{25}}}>0.

In other words,

\sqrt[3]{2} + \sqrt[3]{{20}} - \sqrt[3]{{25}} = 3\sqrt {\sqrt[3]{5} - \sqrt[3]{4}}.

QE in Department of Mathematics, National University of Singapore, August 2009

I have just passed QE held in August 2009 for my first attendance, I hereby show you the analysis paper

Question 1 [10 marks]. Suppose f and g are both measurable functions on the interval (0,1) such that for all t \in \mathbb R^1

| \{ x \in(0,1):f(x)\geq t\}| = |\{ x\in (0,1):g(x)\geq t\}|

Assume that f and g both are monotone decreasing and continuous from left. Can you conclude that f(x)=g(x) for all x \in (0,1)? Give the reason to support your answer.

Question 2 [10 marks]. Compute the volume of the region bounded by

{\left( {{a_{11}}x + {a_{12}}y + {a_{13}}z} \right)^2} + {\left( {{a_{21}}x + {a_{22}}y + {a_{23}}z} \right)^2} + {\left( {{a_{31}}x + {a_{32}}y + {a_{33}}z} \right)^2} = 1

where the determinant of the 3 \times 3 matrix (a_{ij}) is NOT equal to zero.

Question 3 [10 marks]. Let D be a measureable set in \mathbb R^n with finite measure. Suppose \phi(x,t) is a real valued continuous function on D \times \mathbb R^1 such that for almost every x \in D, \phi(x,t) is a continuous function of t and for every real number t, \phi(x,t) is measurable function of x. If \{f_n\} is a sequence of measurable functions on D that converges to f in measure, show that \{\phi(x,f_n(x))\} converges to \phi(x,f(x)) in measure.

Question 4 [10 marks]. Find the function

I\left( y \right) =\displaystyle\int\limits_0^\infty {{e^{ - a{x^2}}}\cos \left( {yx} \right)dx}

if a>0 is a constant. Justify your answer.

Question 5 [10 marks]. Compute the intergal

\displaystyle\int\limits_0^\pi {\frac{{x\sin x}}{{1 + {a^2} - 2a\cos x}}dx}

where a>0 is a constant.

Question 6 [10 marks]. Supposet f(z) is a holomorphic function on the complex plane \mathbb C. If f locally keeps the area invariant, what will the function f be?

Question 7 [10 marks]. Is there an analytic function f on \Delta (unit disk in the complex plane with center 0) such that |f(z)|<1 for |z|<1 with f(0)=\frac{1}{2} and f'(0)=\frac{3}{4}? If so, find such an f. Is it unique?

Question 8 [10 marks]. Let m<n be two positive integers and \Omega and G be open subsets in \mathbb R^n and \mathbb R^m respectively. Does there exist a map f :\Omega \to G such that f and the inverse of f are both C^1? Justify your answer.

Question 9 [10 marks]. Is there a square integrable function f on [0,\pi] such that both inequalities

\displaystyle\int\limits_0^\pi{{{\left( {f\left( x \right)-\sin x} \right)}^2}dx}\leq\frac{4}{9}

and

\displaystyle\int\limits_0^\pi{{{\left( {f\left( x \right)-\cos x} \right)}^2}dx}\leq\frac{1}{9}

hold? Justify your answer.

Question 10 [10 marks]. Let \alpha_k for k=1,2,...,n be n real numbers such that 0<\alpha_k<\pi for any k. Define

\alpha=\displaystyle\frac{1}{n}\sum\limits_{k=1}^{n}{\alpha_k}.

Show that

\displaystyle{\left( {\prod\limits_{k = 1}^n {\frac{{\sin {\alpha _k}}}{{{\alpha _k}}}} } \right)^{\frac{1}{n}}} \leq \frac{{\sin \alpha }}{\alpha }.

Tháng tám 29, 2009

On a polynomials of degree n, having all its zeros in the unit dis

I found a very useful inequality involving a polynomials of degree n, having all its zeros in the unit disk from the following paper, doi:10.1016/j.jmaa.2009.07.049, published in J. Math. Anal. Appl.

Statement. If P(z) is a polynomial of degree n, having all its zeros in the disk |z| \leq 1, then

\left| {zP'(z)} \right| \geq \displaystyle\frac{n} {2}\left| {P(z)} \right|

for |z|=1.

Proof. Since all the zeros of P(z) lie in $latex|z| \leq 1$. Hence if z_1, z_2,...,z_n are the zeros of P(z), then |z_j| \leq 1 for all j =1,2,...,n. Clearly,

\displaystyle\Re \frac{{{e^{i\theta }}P'\left( {{e^{i\theta }}} \right)}}{{P\left( {{e^{i\theta }}} \right)}} = \sum\limits_{j = 1}^n {\Re \frac{{{e^{i\theta }}}}{{{e^{i\theta }} - {z_j}}}} ,

for every point e^{i\theta}, 0 \leq \theta < 2 \pi which is not a zero of P(z). Note that

\displaystyle\sum\limits_{j = 1}^n {\Re \frac{e^{i\theta }}{e^{i\theta } - z_j}}\geq\sum\limits_{j = 1}^n{\frac{1}{2}}=\frac{n}{2}.

This implies

\displaystyle\left| {\frac{{{e^{i\theta }}P'\left( {{e^{i\theta }}} \right)}}{{P\left( {{e^{i\theta }}} \right)}}} \right| \geq \Re \frac{{{e^{i\theta }}P'\left( {{e^{i\theta }}} \right)}}{{P\left( {{e^{i\theta }}} \right)}} \geq \frac{n}{2},

for every point e^{i \pi}, 0 \leq \theta < 2\pi. Hence

\left| {zP'(z)} \right| \geq \displaystyle \frac{n}{2}\left| {P(z)} \right|

for |z|=1 and this completes the proof.

Tháng tám 18, 2009

An other limit supremum of sin function

In the topic we showed that for any irrational \alpha the limit \mathop {\lim }\limits_{n \to \infty } \sin \left( {n\alpha \pi } \right) does not exist. In this topic, we consider the following limit

\mathop {\overline {\lim } }\limits_{n \to \infty } \sin \left( {nx} \right) .

To be precise, we prove that

\mathop {\overline {\lim } }\limits_{n \to \infty } \sin \left( {nx} \right) = 1

for almost every x \in [0,2\pi].

Solution. Let

A = \left\{ {x \in \left( {0,2\pi } \right): \frac{x} {\pi } \notin \mathbb{Q}} \right\}.

Then A is a measurable set of measure 2\pi. Moreover, for any x \in A,

\mathop {\overline {\lim } }\limits_{n \to \infty } \sin \left( {nx} \right) = 1.

Indeed for any x \in A, since

\left\{ {k\frac{x} {\pi } - 2l: l \in \mathbb{Z}} \right\}

is dense subgroup of \mathbb R there are sequences \{k_n\} and \{l_n\} of \mathbb Z such that

\mathop {\lim }\limits_{n \to \infty } \left( {{k_n}\frac{x} {\pi } - {l_n}} \right) = \frac{1} {2}.

Since

\frac{1} {2} \notin \left\{ {k\frac{x} {\pi } - 2l: k,l \in \mathbb{Z}} \right\}

\{k_n\} admits a subsequence \{k'_n\} either increasing to +\infty or decreasing to -\infty. If \mathop {\lim }\limits_{n \to \infty } {{k'}_n} =  + \infty then

\mathop {\lim }\limits_{n \to \infty } \sin \left( {{{k'}_n}x} \right) = \mathop {\lim }\limits_{n \to \infty } \sin \left( {...

Otherwise \mathop {\lim }\limits_{n \to \infty } \left( { - 3{{k'}_n}} \right) =  + \infty and

\mathop {\lim }\limits_{n \to \infty } \sin \left( { - 3{{k'}_n}x} \right) = \mathop {\lim }\limits_{n \to \infty } \sin \lef...

Tháng tám 17, 2009

The use of equivalence in measure

In mathematics, and specifically in measure theory, equivalence is a notion of two measures being “the same”. Two measures are equivalent if they have the same null sets.

Definition. Let (X, \Sigma) be a measurable space, and let \mu, \nu : \Sigma \to [0, +\infty] be two measures. Then \mu is said to be equivalent to \nu if

\mu (A) = 0 \iff \nu (A) = 0

for measurable sets A in \Sigma, i.e. the two measures have precisely the same null sets. Equivalence is often denoted \displaystyle{\mu \sim \nu} or \mu \approx \nu.

In terms of absolute continuity of measures, two measures are equivalent if and only if each is absolutely continuous with respect to the other:

\mu \sim \nu \iff \mu \ll \nu \ll \mu.

Equivalence of measures is an equivalence relation on the set of all measures \Sigma \to [0, +\infty].

Examples.

  1. Gaussian measure and Lebesgue measure on the real line are equivalent to one another.
  2. Lebesgue measure and Dirac measure on the real line are inequivalent.

Application. Let \mu be a finite measure on \mathbb R, and define

f\left( x \right) = \int\limits_{ - \infty }^{ + \infty } {\frac{{\ln \left| {x - t} \right|}} {{\sqrt {\left| {x - t} \right...

Show that f(x) is finite a.e. with respect to the Lebesgue measure on \mathbb R.

Proof. Let

g\left( x \right) = \left\{ \begin{gathered}   \frac{{\ln \left| x \right|}} {{\sqrt {\left| x \right|} }},x \ne 0, \hfill \\...

then g \in L^1(\mathbb R, d\nu) where

d\nu \left( x \right) = \frac{{dx}} {{1 + {x^2}}}

and

f\left( x \right) = \int_{ - \infty }^{ + \infty } {g\left( {x - t} \right)d\mu \left( t \right)} ,x \in \mathbb{R}.

Clearly

\int_{ - \infty }^{ + \infty } {\left| {f\left( x \right)} \right|d\mu \left( x \right)}  \leqslant \int_{ - \infty }^{ + \in...

and by Fubini’s Theorem

\int_{ - \infty }^{ + \infty } {\left( {\int_{ - \infty }^{ + \infty } {\left| {g\left( {x - t} \right)} \right|d\mu \left( t...

then

\int_{ - \infty }^{ + \infty } {\left| {f\left( x \right)} \right|d\mu \left( x \right)}  \leqslant \int_{ - \infty }^{ + \in...

Since

\begin{gathered}   \int\limits_{ - \infty }^{ + \infty } {\left( {\int\limits_{ - \infty }^{ + \infty } {\left| {g\left( {x -...

Thus the following function

x \mapsto \int\limits_{ - \infty }^{ + \infty } {\left| {g\left( {x - t} \right)} \right|d\mu \left( t \right)}

finite a.e. with respect to the measure \nu. The conclusion follows from that the measure \nu and the Lebesgue measure are equivalent.

Source: http://en.wikipedia.org/wiki/Equivalence_(measure_theory)

Tháng tám 16, 2009

On the stability of the Runge-Kutta 4 (RK4)

In the literature, the so-called RK4 is given as following: we first define the following coefficients

\displaystyle \begin{gathered}{K_{1}}= f\left({{t_{i}},{y_{i}}}\right),\hfill\\ {K_{2}}= f\left({{t_{i}}+\frac{1}{2}\Delta t,{y_{i}}+\frac{1}{2}\Delta t{K_{1}}}\right),\hfill\\ {K_{3}}= f\left({{t_{i}}+\frac{1}{2}\Delta t,{y_{i}}+\frac{1}{2}\Delta t{K_{2}}}\right),\hfill\\ {K_{4}}= f\left({{t_{i}}+\Delta t,{y_{i}}+\Delta t{K_{3}}}\right),\hfill\\ \end{gathered}

then

\displaystyle {y_{i + 1}} = {y_i} + \frac{{\Delta t}} {6}\left( {{K_1} + 2{K_2} + 2{K_3} + {K_4}} \right).

This is the most important iterative method for the approximation of solutions of ordinary differential equations

\displaystyle y' = f(t, y), \quad y(t_0) = y_0.

This technique was developed around 1900 by the German mathematicians C. Runge and M.W. Kutta. In order to study its stability, we use the model problem

\displaystyle y' = \lambda y, \quad \Re \lambda < 0.

In other words, we replace f(t,y) by \lambda y. Then the stability condition for time step \Delta comes from the following condition

\displaystyle \left| \frac{y_{i+1}}{y_i} \right| \leq 1.

Applying the above discussion to RK4 method, we see that

\displaystyle\begin{gathered}{K_{4}}=\lambda\left({{y_{i}}+\Delta t{K_{3}}}\right),\hfill\\ {K_{3}}=\lambda\left({{y_{i}}+\frac{1}{2}\Delta t{K_{2}}}\right),\hfill\\ {K_{2}}=\lambda\left({{y_{i}}+\frac{1}{2}\Delta t{K_{1}}}\right),\hfill\\ {K_{1}}=\lambda{y_{i}},\hfill\\ \end{gathered}

which implies

\displaystyle\begin{gathered}{K_{4}}=\lambda\left({{y_{i}}+\Delta t\lambda\left({{y_{i}}+\frac{1}{2}\Delta t\lambda\left({{y_{i}}+\frac{1}{2}\Delta t{y_{i}}}\right)}\right)}\right) =\lambda{y_{i}}\left({1+\Delta t\lambda\left({1+\frac{1}{2}\Delta t\lambda\left({1+\frac{1}{2}\Delta t}\right)}\right)}\right),\hfill\\ {K_{3}}=\lambda\left({{y_{i}}+\frac{1}{2}\Delta t\lambda\left({{y_{i}}+\frac{1}{2}\Delta t{y_{i}}}\right)}\right) =\lambda{y_{i}}\left({1+\frac{1}{2}\Delta t\lambda\left({1+\frac{1}{2}\Delta t}\right)}\right),\hfill\\ {K_{2}}=\lambda\left({{y_{i}}+\frac{1}{2}\Delta t\lambda{y_{i}}}\right) =\lambda{y_{i}}\left({1+\frac{1}{2}\Delta t\lambda }\right),\hfill\\ {K_{1}}=\lambda{y_{i}},\hfill\\ \end{gathered}

which yields

\displaystyle\begin{gathered}{K_{4}}=\lambda\left({{y_{i}}+\Delta t\lambda\left({{y_{i}}+\frac{1}{2}\Delta t\lambda\left({{y_{i}}+\frac{1}{2}\Delta t{y_{i}}}\right)}\right)}\right) =\lambda{y_{i}}\left({1+\Delta t\lambda\left({1+\frac{1}{2}\Delta t\lambda\left({1+\frac{1}{2}\Delta t}\right)}\right)}\right),\hfill\\ {K_{3}}=\lambda\left({{y_{i}}+\frac{1}{2}\Delta t\lambda\left({{y_{i}}+\frac{1}{2}\Delta t{y_{i}}}\right)}\right) =\lambda{y_{i}}\left({1+\frac{1}{2}\Delta t\lambda\left({1+\frac{1}{2}\Delta t}\right)}\right),\hfill\\ {K_{2}}=\lambda\left({{y_{i}}+\frac{1}{2}\Delta t\lambda{y_{i}}}\right) =\lambda{y_{i}}\left({1+\frac{1}{2}\Delta t\lambda }\right),\hfill\\ {K_{1}}=\lambda{y_{i}},\hfill\\ \end{gathered}

Thus

\displaystyle\frac{{{y_{i+1}}}}{{{y_{i}}}}= 1+\frac{{\Delta t}}{6}\left[{\lambda+2\lambda\left({1+\frac{1}{2}\Delta t\lambda }\right)+2\lambda\left({1+\frac{1}{2}\Delta t\lambda\left({1+\frac{1}{2}\Delta t}\right)}\right)+\lambda\left[{1+\Delta t\lambda\left({1+\frac{1}{2}\Delta t\lambda\left({1+\frac{1}{2}\Delta t}\right)}\right)}\right]}\right]

which is nothing but

\displaystyle\frac{{{y_{i+1}}}}{{{y_{i}}}}= 1+\Delta t\lambda+\frac{{{{\left({\Delta t\lambda }\right)}^{2}}}}{2}+\frac{{{{\left({\Delta t\lambda }\right)}^{3}}}}{6}+\frac{{{{\left({\Delta t\lambda }\right)}^{4}}}}{{24}}.

Therefore, the stability condition is given as follows

\displaystyle \left| {1 + z + \frac{{{z^2}}} {2} + \frac{{{z^3}}} {6} + \frac{{{z^4}}} {{24}}} \right| \leq 1, \quad \Re z < 0.

Bài viết cũ hơn »

Blog at WordPress.com.