Acta Mathematica Sinica, English Series

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Robust solutions of uncertain complex-valued quadratically constrained programs

Da Chuan Xu, Zheng Hai Huang

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1279-1290

In this paper, we discuss complex convex quadratically constrained optimization with uncertain data. Using S-Lemma, we show that the robust counterpart of complex convex quadratically constrained optimization with ellipsoidal or intersection-of-two-ellipsoids uncertainty set leads to a complex semidefinite program. By exploring the approximate S-Lemma, we give a complex semidefinite program which...

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Binormal operator and *-Aluthge transformation

Chang Sen Yang, Yan Feng Ding

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1369-1378

Let T = U|T| be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation $$ \tilde T^{( * )} = |T^ * |^{\tfrac{1} {2}} U|T^ * |^{\tfrac{1} {2}} $$ is called the Aluthge transformation and $$ \tilde T_n $$ means the n-th Aluthge transformation. Similarly, the transformation $$ \tilde T^{( * )} = |T^ * |^{\tfrac{1} {2}} U|T^ * |^{\tfrac{1} {2}} $$ ...

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Intersections of homogeneous Cantor sets with their translations

Mei Feng Dai

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1313-1318

We pursue the study on homogeneous Cantor sets with their translations. We get the fractal structure of intersection I(t), and find that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from shifting numbers with the coding of t. Concretely, a very brief calculation formula of the measure with the coding of t is given.

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Existence of positive solutions for a singular p-Laplacian differential equation

Xia Li, Zheng An Yao, Wen Shu Zhou

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1331-1344

In this paper, we are concerned with the existence of positive solutions for a singular p-Laplacian differential equation $$ (\varphi _p (u'))' + \frac{\beta } {r}\varphi _p (u') - \gamma \frac{{|u'|^p }} {u} + g(r) = 0,0 < r < 1, $$ subject to the Dirichlet boundary conditions: u(0) = u(1) = 0, where ϕ _p(s) = |s|^p−2 s,p > 2,...

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Bifurcations of travelling wave solutions for a two-component camassa-holm equation

Ji Bin Li, Yi Shen Li

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1319-1330

By using the method of dynamical systems to the two-component generalization of the Camassa-Holm equation, the existence of solitary wave solutions, kink and anti-kink wave solutions, and uncountably infinite many breaking wave solutions, smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the...

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The viability property of controlled jump diffusion processes

Shi Ge Peng, Xue Hong Zhu

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1351-1368

In this paper, we first give a comparison theorem of viscosity solution to some nonlinear second order integrodifferential equation. And then using the comparison theorem, we obtain a necessary and sufficient condition for the viability property of some controlled jump diffusion processes which can keep the solution within a constraint K.

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The number of rooted essential maps on surfaces

Wen Zhong Liu, Yan Pei Liu

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1303-1312

In this paper, general functional equations of rooted essential maps on surfaces (orientable and nonorientable) are deduced and their formal solutions are presented. Further, three explicit formulae for counting essential maps on S ₂, $$ \widetilde{N_3 } $$ and $$ \widetilde{N_4 } $$ are given. In the same time, some known results can be derived.

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On sums of a prime and four prime squares in short intervals

Guang Shi Lü, Xian Meng Meng

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1291-1302

In this paper, we prove that each sufficiently large integer N ≢ 1 (mod 3) can be written as $$ N = p + p_1^2 + p_2^2 + p_3^2 + p_4^2 , $$ , with $$ \left| {p - \frac{N} {5}} \right| \leqslant U,\left| {p_j - \sqrt {\frac{N} {5}} } \right| \leqslant U,j = 1,2,3,4, $$ where U = N ^{9/20 + ɛ} and p, p _j are primes.

article

Weighted estimates for multilinear commutators of Marcinkiewicz integral

Pu Zhang

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1387-1400

Let µ be the n-dimensional Marcinkiewicz integral and µ_b→ the multilinear commutator of µ. In this paper, the following weighted inequalities are proved for ω ∈ A _∞ and 0 < p < ∞, $$ ||\mu (f)||_{L^p (\omega )} \leqslant C||Mf||_{L^p (\omega )} and||\mu _{\vec b} (f)||_{L^p (\omega )} \leqslant C||M_{L(logL)^{{1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-\nulldelimiterspace} r}} }...

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Infinitely many solutions for a class of fourth order elliptic equations in R ^N

Min Bo Yang, Zi Fei Shen

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1269-1278

With the aids of variational method and concentration-compactness principle, infinitely many solutions are obtained for a class of fourth order elliptic equations with singular potential $$ \Delta ^2 u = \mu \frac{{|u|^{2^{**} (s) - 2} u}} {{|x|^s }} + \lambda k(x)|u|^{r - 2} u,u \in H^{2,2} (R^N ). $$

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A generalization of Hardy spaces H ^p by using atoms

Eiichi Nakai

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1243-1268

Let X = (X, d, µ) be a space of homogeneous type in the sense of Coifman and Weiss. The purpose of this paper is to generalize the definition of Hardy space H ^p(X) and prove that the generalized Hardy spaces have the same property as H ^p(X). Our definition includes a kind of Hardy-Orlicz spaces and a kind of Hardy spaces with variable exponent. The results are new even for the ℝⁿ case. Let (...

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On the index of finite determinacy of vector fields with resonances

Vincent Naudot, Jia Zhong Yang

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1401-1408

In this paper we study normal forms for a class of germs of 1-resonant vector fields on ℝⁿ with mutually different eigenvalues which may admit extraneous resonance relations. We give an estimation on the index of finite determinacy from above as well as the essentially simplified polynomial normal forms for such vector fields. In the case that a vector field has a zero eigenvalue, the result leads...

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Lipschitz algebras and peripherally-multiplicative maps

A. Jiménez-Vargas, Moisés Villegas-Vallecillos

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1233-1242

Let X be a compact metric space and let Lip(X) be the Banach algebra of all scalarvalued Lipschitz functions on X, endowed with a natural norm. For each f ∈ Lip(X), σ _π(f) denotes the peripheral spectrum of f. We state that any map Φ from Lip(X) onto Lip(Y) which preserves multiplicatively the peripheral spectrum: $$ \sigma _\pi (\Phi (f)\Phi (g)) = \sigma _\pi...

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A new contraction principle in menger spaces

Binayak S. Choudhury, Krishnapada Das

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1379-1386

In the present work we introduce a new type of contraction mapping by using a specific function and obtain certain fixed point results in Menger spaces. The work is in line with the research for generalizing the Banach’s contraction principle. We extend the notion of altering distance function to Menger Spaces and obtain fixed point results.

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Trees with non-regular fractal boundary

Jing Hu Yu, Yi Ming Ding

Acta Mathematica Sinica, English Series > 2008 > 24 > 8 > 1345-1350

For any given 0 < α < β < ∞, we construct a tree such that under tree metric, the Hausdorff dimension of the corresponding boundary is α, but both the Packing dimension and the boxing dimension are β. Applying the connection between tree and iterated functions system, nonregular fractal sets on real line are constructed. Moreover, the method adopted in our paper is different from those which...

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Acta Mathematica Sinica, English Series

Robust solutions of uncertain complex-valued quadratically constrained programs

Binormal operator and *-Aluthge transformation

Intersections of homogeneous Cantor sets with their translations

Existence of positive solutions for a singular p-Laplacian differential equation

Bifurcations of travelling wave solutions for a two-component camassa-holm equation

The viability property of controlled jump diffusion processes

The number of rooted essential maps on surfaces

On sums of a prime and four prime squares in short intervals

Weighted estimates for multilinear commutators of Marcinkiewicz integral

Infinitely many solutions for a class of fourth order elliptic equations in R ^N

A generalization of Hardy spaces H ^p by using atoms

On the index of finite determinacy of vector fields with resonances

Lipschitz algebras and peripherally-multiplicative maps

A new contraction principle in menger spaces

Trees with non-regular fractal boundary

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