CALCULUS ON FRACTALS BASED UPON LOCAL FIELDS─In memory of Founding Editor Professor M.T.Cheng with great respect and deep sorrow

The main contents in this note are: 1. introduction; 2. locally compact groups and local fields; 3. calculus on fractals based upon local fields; 4. fractional calculus and fractals; 5. fractal function spaces and PDE on fractals.

ON RIESZ TYPE KERNELS OVER LOCAL FIELDS

Based on a representation lemma. Riesz type kernels on the local field K and on the integer ring O in K are coitstructed. Furthermore, we discuss approximation theorems for the Lipschitz class Lip(L ;α) ana the Lp boundedness of such operators motivated by the open problem: Does σηfa,s→f for f ∈L1(O) (see M. H. Taible-son [6] and [5])?

Some New Inequalities for Wavelet Frames on Local Fields

Wavelet frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of science and engineering. Finding general and verifiable conditions which imply that the wavelet systems are wavelet frames is among the core problems in time-frequency analysis. In this article, we establish some new inequalities for wavelet frames on local fields of positive characteristic by means of the Fourier transform. As an application, an improved version of the Li-Jiang inequality for wavelet frames on local fields is obtained.

Construction Theory of Function on Local Fields

We establish the construction theory of function based upon a local field Kp as underlying space. By virture of the concept of pseudo-differential operator, we introduce ＂fractal calculus＂ （or, p-type calculus, or, Gibbs-Butzer calculus）. Then, show the Jackson direct approximation theorems, Bermstein inverse approximation theorems and the equivalent approximation theorems for compact group D（C Kp） and locally compact group Kp^＋-（= Kp）, so that the foundation of construction theory of function on local fields is established. Moreover, the Jackson type, Bernstein type, and equivalent approximation theorems on the HOlder-type space C^σ（Kp）, σ 〉0, are proved; then the equivalent approximation theorem on Sobolev-type space Wr（Kp）, σ≥0, 1≤r 〈∞, is shown.

LOCAL HARDY SPACES ON HEISENBERG GROUP OVER LOCAL FIELDS

Let G be a Heisenberg group over local field K. In the paper we give some local maximal functions which define the Hardy spaces h(P)(G) equivalently, obtain the atomic decomposition of local Hardy spaces. We also give some bounded convolution operators h(P)(G), from which we can characterize h(P)(G) by sqare functions.

CHAINS OF FUNCTION SPACES OVER EUCLIDIAN SPACES AND LOCAL FIELDS

The Lipschitz class Lip(K, α) on a local field K is defined in [10], and an equivalent relationship between the Ho¨lder type space Cα(K)[9] and Lip(K,α) is given. In this note, we give a 'chain of function spaces' over Euclidian space by defining higher order continuous modulus in R, and point out that there is no need of higher order continuous modulus for describing the chain of function spaces over local fields.

ON HAUSDORFF DIMENSION OF CERTAIN RIESZ PRODUCT IN LOCAL FIELDS

In this paper, we consider the Riesz product dμ =^∞∏j=1（1＋ajRexbjλj（x））dx in local fields, and we obtain the upper and lower bound of its Hausdorff dimension.

Hierarchical Management Scheme by Local Fields

In the present paper we propose a new hierarchical management scheme within suitable extensions of 2-adic numbers, in which the subfields are treated as classes, and the communications are realized by computation on elliptic curves. The security of our system is much stronger than that of the traditional public key cryptographic systems.

Strategies for discovery and optimization of thermoelectric materials： Role of real objects and local fields

Thermoelectric materials provide a renewable and eco-friendly solution to mitigate energy shortages and to reduce environmental pollution via direct heat-to-electricity conversion. Discovery of the novel thermoelectric materials and optimization of the state-of-the-art material systems lie at the core of the thermoelectric society, the basic concept behind these being comprehension and manipulation of the physical principles and transport properties regarding thermoelectric materials. In this mini-review, certain examples for designing high-performance bulk thermoelectric materials are presented from the perspectives of both real objects and local fields. The highlights of this topic involve the Rashba effect, Peierls distortion, local magnetic field, and local stress field, which cover several aspects in the field of thermoelectric research. We conclude with an overview of future developments in thermoelectricity.

On Some Computational Problems in Local Fields

Lattices in Euclidean spaces are important research objects in geometric number theory,and they have important applications in many areas,such as cryptology.The shortest vector problem(SVP)and the closest vector problem(CVP)are two famous computational problems about lattices.In this paper,we consider p-adic lattices in local fields,and define the p-adic analogues of SVP and CVP in local fields.The authors find that,in contrast with lattices in Euclidean spaces,the situation is different and interesting.The SVP in Euclidean spaces corresponds to the Longest Vector Problem(LVP)in local fields.The authors develop relevant algorithms,indicating that these problems are computable.

Distributional Dimension of Fractal Sets in Local Fields

The distributional dimension of fractal sets in R^n has been systematically studied by Triebel by virtue of the theory of function spaces. In this paper, we first discuss some important properties about the B-type spaces and the F-type spaces on local fields, then we give the definition of the distributional dimension dimD in local fields and study the relations between distributional dimension and Hausdorff dimension. Moreover, the analysis expression of the Hausdorff dimension is given. Lastly, we define the Fourier dimension in local fields, and obtain the relations among all the three dimensions. Keywords local field, B-type space, F-type space, distributional dimension, Hausdorff dimension Fourier dimension

Lipschitz classes on local fields

The Lipschitz class Lipαon a local field K is defined in this note,and the equivalent relationship between the Lipschitz class Lipαand the Holder type space C~α(K)is proved.Then,those important characteristics on the Euclidean space R^n and the local field K are compared,so that one may interpret the essential differences between the analyses on R^n and K.Finally,the Cantor type fractal functionθ(x)is showed in the Lipschitz class Lip(m,K),m＜(ln 2/ln 3).

Basic Results of Gabor Frame on Local Fields

Basic facts for Gabor frame {Eu（m）bTu（n）ag}m, n∈P on local field are investigated. Accurately, that the canonical dual of frame {Eu（m）bTu（n）ag}m,n∈P also has the Gabor structure is showed; that the product ab decides whether it is possible for {Eu（m）bTu（n）ag}m,n∈P to be a frame for L^2（K） is discussed; some necessary conditions and two sufficient conditions of Gabor frame for L^2 （K） are established. An example is finally given.

Generalized Multiresolution Structures in Reducing Subspaces of Local Fields

In this article,we introduce the notion of general multiresolution structure(GMS)in the reducing subspace over local fields.We show that the GMS is admitted by an arbitrary reducing subspace and characterize all those GMSs which admit a pyramids decomposition.Towards the culmination,we obtain a frame-like expansion for signals in reducing subspaces in terms of GMS over local fields.

A Note on Additive Characters of Locally Compact Fields

In this note, we give an elementary and constructive proof for that the additive character group of a locally compact field is isomorphic to itself as an additive topological group.

Cortico-striatal gamma oscillations are modulated by dopamine D3 receptors in dyskinetic rats

Long-term levodopa administration can lead to the development of levodopa-induced dyskinesia.Gamma oscillations are a widely recognized hallmark of abnormal neural electrical activity in levodopa-induced dyskinesia.Currently,studies have reported increased oscillation power in cases of levodopa-induced dyskinesia.However,little is known about how the other electrophysiological parameters of gamma oscillations are altered in levodopa-induced dyskinesia.Furthermore,the role of the dopamine D3 receptor,which is implicated in levodopa-induced dyskinesia,in movement disorder-related changes in neural oscillations is unclear.We found that the cortico-striatal functional connectivity of beta oscillations was enhanced in a model of Parkinson’s disease.Furthermore,levodopa application enhanced cortical gamma oscillations in cortico-striatal projections and cortical gamma aperiodic components,as well as bidirectional primary motor cortex(M1)↔dorsolateral striatum gamma flow.Administration of PD128907(a selective dopamine D3 receptor agonist)induced dyskinesia and excessive gamma oscillations with a bidirectional M1↔dorsolateral striatum flow.However,administration of PG01037(a selective dopamine D3 receptor antagonist)attenuated dyskinesia,suppressed gamma oscillations and cortical gamma aperiodic components,and decreased gamma causality in the M1→dorsolateral striatum direction.These findings suggest that the dopamine D3 receptor plays a role in dyskinesia-related oscillatory activity,and that it has potential as a therapeutic target for levodopa-induced dyskinesia.

Hippocampal ischemia causes deficits in local field potential and synaptic plasticity

The long-term enhancement in glutamate receptor mediated excitatory responses has been observed in stroke model. This pathological form of plasticity, termed post-ischemic long-term potentiation （i-LTP）, points to functional reorganization after stroke. Little is known, however, about whether and how this i-LTP would affect subsequent induction of synaptic plasticity. Here, we first directly confirmed that i-LTP was induced in the endothelin-l-induced ischemia model as in other in vitro models. We also demonstrated increased expression of NR2B, CaMKII and p-CaMKII, which are reminiscent of i-LTP. We further induced LTP of field excitatory post- synaptic potentials （fEPSPs） on CA1 hippocampal neurons in peri-infarct regions of the endothelin-l-induced mini-stroke model. We found that LTP of fEPSPs, induced by high-frequency stimulation, displayed a progressive impairment at 12 and 24 hours after ischemia. Moreover, using in vivo multi-channel recording, we found that the local field potential, which represents electrical property of cell ensembles in more restricted regions, was also dam- pened at these two time points. These results suggest that i-LTP elevates the induction threshold of subsequent synap- tic plasticity. Our data helps to deepen the knowledge of meta-synaptic regulation of plasticity after focal ischemia.

Novel Biological Based Method for Robot Navigation and Localization

The capability and reliability are crucial characteristics of mobile robots while navigating in complex environments. These robots are expected to perform many useful tasks which can improve the quality of life greatly. Robot localization and decisionmaking are the most important cognitive processes during navigation. However, most of these algorithms are not efficient and are challenging tasks while robots navigate through complex environments. In this paper,we propose a biologically inspired method for robot decision-making, based on rat’s brain signals. Rodents accurately and rapidly navigate in complex spaces by localizing themselves in reference to the surrounding environmental landmarks. Firstly, we analyzed the rats’ strategies while navigating in the complex Y-maze, and recorded local field potentials(LFPs), simultaneously.The recorded LFPs were processed and different features were extracted which were used as the input in the artificial neural network(ANN) to predict the rat’s decision-making in each junction. The ANN performance was tested in a real robot and good performance is achieved. The implementation of our method on a real robot, demonstrates its abilities to imitate the rat’s decision-making and integrate the internal states with external sensors, in order to perform reliable navigation in complex maze.

Effect of Lorentz local field correction on propagation of ultrashort laser pulse in one-dimensional para-nitroaniline (PNA) molecules

This paper investigates the effect of Lorentz local field correction （LFC） on the propagation of ultrashort laser pulses in a para-nitroaniline molecular medium under resonant and nonresonant conditions by solving numerically the full-wave Maxwell-Bloch equations beyond slowly-varying envelope approximation and rotating-wave approximation. The effect of the LFC is considerably obvious when pulses with large areas propagate in the dense molecular medium. In the case of resonance, the group velocity of the sub-pulses split from the incident pulse along propagation is severely decreased by the LFC, especially for the latest sub-pulse. However, in the case of nonresonance, the influence of the LFC on the temporal evolution of the pulse is less obvious and lacks homogeneity with an increase in incident pulse area, propagation distance and molecular density.

Simultaneous low extinction and high local field enhancement in Ag nanocubes

We theoretically investigate surface plasmon resonance properties in Au and Ag cubic nanoparticles and find a novel plasmonic mode that exhibits simultaneous low extinction and high local field enhancement properties. We analyse this mode from different aspects by looking at the distribution patterns of local field intensity, energy flux, absorption and charge density. We find that in the mode the polarized charge is highly densified in a very limited volume around the corner of the nanocube and results in very strong local field enhancement. Perturbations of the incident energy flux and light absorption are also strongly localized in this small volume of the corner region, leading to both low absorption and low scattering cross section. As a result, the extinction is low for the mode. Metal nanoparticles involving such peculiar modes may be useful for constructing nonlinear compound materials with low linear absorption and high nonlinearity.