Ngô Quốc Anh

Tháng Năm 28, 2008

Lebesgue’s density theorem

Chuyên mục: Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 3:25

In mathematics, Lebesgue’s density theorem states that for any Lebesgue measurable set A, the “density” of A is 1 at almost every point in A. Intuitively, this means that the “edge” of A, the set of points in A whose “neighborhood” is partially in A and partially outside of A, is negligible.

Let μ be the Lebesgue measure on the Euclidean space Rn and A be a Lebesgue measurable subset of Rn. Define the approximate density of A in a ε-neighborhood of a point x in Rn as

d_\varepsilon(x)=\frac{\mu(A\cap B_\varepsilon(x))}{\mu(B_\varepsilon(x))}

where Bε denotes the closed ball of radius ε centered at x.

Lebesgue’s density theorem asserts that for almost every point of A the density

d(x)=\lim_{\varepsilon\to 0} d_{\varepsilon}(x)

exists and is equal to 1.

In other words, for every measurable set A the density of A is 0 or 1 almost everywhere in Rn. However, it is a curious fact that if μ(A) > 0 and μ(Rn\A) > 0, then there are always points of Rn where the density is neither 0 nor 1.

For example, given a square in the plane, the density at every point inside the square is 1, on the edges is 1/2, and at the corners is 1/4. The set of points in the plane at which the density is neither 0 nor 1 is non-empty (the square boundary), but it is negligible.

Tháng Năm 23, 2008

Chuyên mục: Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 0:47

Wenjun Liu and Quoc Anh Ngo
An Ostrowski-Gruss Type Inequality on Time Scales

In this paper we derive a new inequality of Ostrowski-Gruss type on time scales and thus unify corresponding continuous and discrete versions. We also apply our result to the quantum calculus case.

~~> http://eureka.vu.edu.au/~rgmia/v11n2/OSTROWSKI-GRUSS.pdf

KQ11 – On an Integral Inequality

Chuyên mục: Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 0:45

S.S. Dragomir and Q. A. Ngo
On an Integral Inequality

Let [a,b]\to [0,\infty)$ be continuous function and [a,b]\to [0,\infty)$ be non-decreasing, differentiable on $ (a,b)$. In this short paper, we prove that the following inequality

$\displaystyle \int^b_x f^\beta(t){\mathrm{d\mspace{-2mu}}}t\ge\int^b_x g^\beta(t){\mathrm{d\mspace{-2mu}}}t $

holds for $ 0 < \alpha < \beta$ and for all $ x \in [a,b]$ provided

$\displaystyle \int^b_x f^\alpha(t){\mathrm{d\mspace{-2mu}}}t\ge\int^b_x g^\alpha(t){\mathrm{d\mspace{-2mu}}}t $

holds for all $ x \in [a,b]$.

~~> http://eureka.vu.edu.au/~rgmia/v11n1/Kq25_en.pdf

Tháng Năm 18, 2008

Đề thi Cao học ĐHKHTN Hà Nội 2008

Chuyên mục: Đề Thi — Ngô Quốc Anh @ 21:57

Đợt 1
Đại số: 2007_ds_1.pdf
Giải tích: 2008_gt_1.pdf

Đợt 2
Đại số: 2008_ds_2.pdf
Giải tích: 2008_gt_2.pdf

Tháng Năm 14, 2008

KQ7 – A generalization of Ostrowski inequality on time scales for k points

Chuyên mục: Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 16:37
http://img404.imageshack.us/img404/7636/clipboard02ap3.jpg
arXiv:0804.3230 [ps, pdf, other]

Title: A generalization of Ostrowski inequality on time scales for k points
Authors: Wenjun Liu, Quoc Anh Ngo
Comments: 10 pages
Subjects: Functional Analysis (math.FA); General Mathematics (math.GM)

In this paper we first generalize the Ostrowski inequality on time scales for k points and then unify corresponding continuous and discrete versions. We also point out some particular Ostrowski type inequalities on time scales as special cases.

To appear in
About this Journal
http://img61.imageshack.us/img61/2083/clipboard02uo7.jpg
~~> http://dx.doi.org/10.1016/j.amc.2008.05.124

Tháng Năm 10, 2008

An integral inequality

Chuyên mục: Các Bài Tập Nhỏ, Giải Tích 2 — Ngô Quốc Anh @ 3:48

Let [0,1]\rightarrow \mathbb{R}^{ + } be a continuous function and let be a natural number. Prove that

\int\limits_{0}^{1}f^{n}(x^{n})dx\geq \frac {1}{n}\left(\frac {n +1}{n }\right)^{n - 1}\left(\int\limits_{0}^{1}f(x)dx\right)^{n}.

Solution. Recall Hôlder inequality

, , \frac {1}{a} + \frac {1}{b} = 1,

|\int_{K}h.g|\leq(\int_{K}|h|^{a})^{1/a}(\int_{K}|g|^{b})^{1/b}.

Then

\int_{0}^{1}f(x)dx = \int_{0}^{1}y^{\frac {n - 1}{n^2}}(f(y)y^{\frac {1 - n}{n^2}})dy\leq (\int_{0}^{1}y^{\frac {1}{n}}dy)^{\frac {n - 1}{n}}(\int_{0}^{1}f^{n}(y)y^{\frac {1}{n} - 1}dy)^{\frac {1}{n}}

taking the n-power and in the second integral at RHS , do change variable will give the inequality.

Tháng Năm 5, 2008

KQ10 – New Proof on some sharp double integral Inequalities of the Hermite-Hadamard Type

Chuyên mục: Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 1:15
http://img404.imageshack.us/img404/7636/clipboard02ap3.jpg

arXiv:0805.0433 [ps, pdf, other]

Vu Nhat Huy, Wenjun Liu, Quoc Anh Ngo

(Submitted on 4 May 2008)
Title: New Proof on some sharp double integral Inequalities of the Hermite-Hadamard Type

Abstract: In this paper, we derive a new proof on some sharp double integral inequalities of the Hermite-Hadamard type. Our approach is mainly based on well-known Taylor’s theorem with the integral remainder.

Comments: 5 pages, 0 figure
Subjects: Functional Analysis (math.FA)
MSC classes: 26D15
Cite as: arXiv:0805.0433v1 [math.FA]

Tháng Năm 2, 2008

Strong Maximum Principle Revisited

Chuyên mục: Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 3:24

Suppose satisfies . If we knew that u \in C^2(\Omega) \cap C^1(\overline\Omega), then we would deduce from the strong maximum principle that there exists an s.t.

u(x) \ge \epsilon \, \mathrm{dist}(x,\partial\Omega) \quad \text{ for any } x \in \Omega.

But assume we only know that . Is the above inequality still true?

Of course not, if we want to use and then unfortunately, the function

u(x) = \exp\left(-\frac 1{1-|x|}\right)

doesn’t work because the inequality is not satisfied:

-u''(x) = u(x) (1-|x|)^{-4} (1-2|x|).

Nevertheless, if , then the function

u(x) = (1-|x|)^s - (1-|x|)^r, \quad \frac 12 < s < 1 < r,

doesn’t satisfy the property that “ for some constant ” while

-\Delta u = \left( s(1-s) + r(r-1)(1-|x|)^{r-s} \right) (1-|x|)^{s-2} + 2\gamma(r-s)\delta \ge 0.

Here is the Dirac measure.

Tháng Năm 1, 2008

KQ9 – Existence results for quasilinear elliptic boundary value problems via topological methods

Chuyên mục: Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 16:46
http://img404.imageshack.us/img404/7636/clipboard02ap3.jpg
arXiv:0805.0075 [ps, pdf, other]

Title: Existence results for quasilinear elliptic boundary value problems via topological methods
Authors: Quoc Anh Ngo
Comments: 4 pages
Subjects: Analysis of PDEs (math.AP)

In this paper, existence and localization results of $C^1$-solutions to elliptic Dirichlet boundary value problems are established. The approach is based on the nonlinear alternative of Leray-Schauder.

Two times derivable real function

Chuyên mục: Giải Tích 1 — Ngô Quốc Anh @ 16:41

Two times derivable real function
RMO 2008, 11th Grade, Problem 3

Let be a function, two times derivable on for which there exist such that\frac { f(b)-f(a) }{b-a} \neq f'(c) , for all . Prove that .

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