Value sharing problems for differential and difference polynomials of meromorphic functions in a non-Archimedean field

In this paper we discuss the uniqueness problem for differential and difference polynomials of the form (f ^nm (z)f ^nd (qz + c))^(k) for meromorphic functions in a non-Archimedean field.     

On functional equations for meromorphic functions and applications

Donovan’s conjecture states that there exist only finitely many Morita equivalence classes of p-blocks with a given defect. This conjecture was shown by Radha Kessar to be equivalent to two other conjectures, one of which is that the basic algebras of p-blocks with a given defect can all be defined over a single finite field. We shall show that this latter conjecture is equivalent to the seemingly stronger statement that all p-blocks with a given defect can be defined over a single finite field.     

Sharp inequalities of singular values of smooth kernels

New inequalities of singular values of the integral operators with smoothL ² kernels are obtained and shown by examples to be sharp if the kernels satisfy also certain boundary conditions. These results are based on an idea of Gohberg-Krein by which the singular values of the integral operators are interrelated to the eigenvalues of some two point boundary value problems.Dedicate to Professor Ky Fan on the occasion of his 85th birthday     

Flocking Dynamics of the Inertial Spin Model with a Multiplicative Communication Weight

In this paper, we study a flocking dynamics of the deterministic inertial spin (IS) model. The IS model was introduced for the collective dynamics of active particles with an internal angular momentum, or spin. When the generalized moment of inertia becomes negligible compared to spin dissipation (overdamped limit) and mutual communication weight is a function of a relative distance between interacting particles, the deterministic inertial spin model formally reduces to the Cucker–Smale (CS) model with constant speed constraint whose emergent dynamics has been extensively studied in the previous literature. We present several sufficient frameworks leading to the asymptotic mono-cluster flocking, in which spins and relative velocities tend to zero asymptotically. We also provide several numerical simulations for the decoupled and coupled inertial spin models to see the effect of the C–S velocity flocking and compare them with our analytical results.

     

Spectral mapping theorems for exponentially bounded c-semigroups in Banach spaces

LetZ be a generator of an exponentially boundedC-semigroup {S _t } _t≥0 in a Banach space and letT _t =C ⁻¹ S _t . We show that the spectral mapping theorems such as exp(tσ(Z)) ⊂ σ(T _t ) and exp(tσ _p (Z)) ⊂ tσ _p (T _t ) ⊂ exp(tσ _p (Z)) ⋃ {0} for everyt≥0 hold. The present studies were supported by the Basic Science Research Institute Program, Ministry of Education, 1987.     

有机溶液中圆锥泡声致发光

在U形管声致发光装置的基础上建立了一套新型的声致发光装置——直管圆锥泡声致发光装置. 利用此装置以有机溶液为液体介质得到了超强的发光脉冲并测量得到了其发光光谱. 结果表明发光光谱为一从紫外光至可见光波长范围的连续谱,上面叠加有C₂的d³Π_g→d³Π_u跃迁形成的五个序列谱带,分别对应于Δv=-2,Δv=-1,Δv=0,Δv< 关键词： 声致发光 光谱 斯旺带 振动温度     

New estimates for the Laplacian,the div–curl,and related Hodge systems

We establish new estimates for the Laplacian, the div–curl system, and more general Hodge systems in arbitrary dimension, with an application to minimizers of the Ginzburg–Landau energy. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Kato's inequality when Δu is a measure

We extend the classical version of Kato's inequality in order to allow functions u∈L¹_loc such that Δu is a Radon measure. This inequality has been recently applied by Brezis, Marcus, and Ponce to study the existence of solutions of the nonlinear equation ?Δu+g(u)=μ, where μ is a measure and $\text{g}\text{:}\text{R}\text{→}\text{R}$ is a nondecreasing continuous function. To cite this article: H. Brezis, A.C. Ponce, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Gagliardo–Nirenberg inequalities and non-inequalities: The full story

We investigate the validity of the Gagliardo–Nirenberg type inequality

(1)${6f6}_{W^{s,p}(\mathrm{\Omega })}?{6f6}_{W^{s_1,p_1}(\mathrm{\Omega })}^{\theta }{6f6}_{W^{s_2,p_2}(\mathrm{\Omega })}^{1?\theta },$

with $\mathrm{\Omega }?{\mathbb{R}}^N$. Here, $0\le s_1\le s\le s_2$ are non negative numbers (not necessarily integers), $1\le p_1,p,p_2\le \infty$, and we assume the standard relations

$s=\theta s_1+(1?\theta )s_2,1/p=\theta /p_1+(1?\theta )/p_2\text{ for some }\theta \in (0,1).$

By the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when $s_1,s_2,s$ are integers. It turns out that (1) holds for “most” of values of $s_1,\dots ,p_2$, but not for all of them. We present an explicit condition on $s_1,s_2,p_1,p_2$ which allows to decide whether (1) holds or fails.