DEGREE OF APPROXIMATION BY SUPERPOSITIONS OF A SIGMOIDAL FUNCTION

In this paper we study the degree of approximation by superpositions of a sigmoidal function.We mainly consider the univariate case.If f is a continuous function,we prove that for any bounded sigmoidal function σ,d_(n,σ)(f)≤‖σ‖ω(f,1/(n+1)).For the Heaviside function H(x),we prove that d_(n,H)(f)≤ω(f,1/(2(n+1))). If f is a continuous funnction of bounded variation,we prove that d_(n,σ)(f)≤‖σ‖/(n+1)V(f)and d_(n,H)(f)≤ 1/(2(n+1))V(f).For he Heaviside function,the coefficient 1 and the approximation orders are the best possible.We compare these results with the classical Jackson and Bernstein theorems,and make some conjec- tures for further study.

Generation of Superpositions of Coherent Interaction States Along a Straight Line via Raman Interaction

We present an alternative scheme for preparing the superpositions of coherent states along a straight line of a cavity field using degenerate atom-cavity field Raman interaction. In the scheme, a collection of A-type three-level atoms is orderly sent through the cavity to interact with the cavity field adjusted by a microwave source connected to it, followed by state-selective measurements. In this way, we can prepare the superpositions of several coherent states along a straight line with arbitrary weighting factors for the cavity field. In the scheme, the coherence of the atom-cavity system may be maintained and the second microwave field is unnecessary, which is prior to the previous scheme.

Possible Linear Superpositions for Special Solutions of the Fifth-Order KdV-Type Equations

Using Lou and Ni's deformation and mapping idea in nonlinear equations to a set of fifth order KdV type equations, it is found that some types of solitary wave solutions and periodic solutions with special velocities can be linearly superposed to new exact solutions.

Quantum State Engineering by Superpositions of Coherent States along a Straight Line in Cavity Quantum Electrodynamics

A scheme is proposed for generating the superpositions of several coherent states in a cavity field with dispersive cavity quantum electrodynamics(QED).In the scheme,a sequence of atoms interact dispersively with the cavity field,connected with a microwave source,and is manipulated by classical fields,followed by state-selective measurements.In this way,the cavity field is collapsed onto a superposition of several coherent states along a straight line with controllable coefficients.This scheme provides the possibility for quantum state engineering via coherent-state superpositions in cavity QED for the first time.

Teleportation of Unknown Superpositions of Collective Atomic Coherent States

We propose a scheme to teleport an unknown superposition of two atomic coherent states with different phases. Our scheme is based on resonant and dispersive atom-field interaction. Our scheme provides a possibility of teleporting macroscopic superposition states of many atoms first time.

Generation of superpositions of coherent states for an atomic sample in cavity QED

This paper proposes a scheme for generation of superpositions of coherent states of the effective bosonic mode in a collection of atoms. In the scheme an atomic sample interacts with a slightly detuned cavity mode and a resonant strong classical field. Under ceitain conditions the atomic system evolves from a coherent state to a superposition of coherent states.

Generation of Superpositions of Two-Mode Coherent States for the Motion of Two Trapped Ions

We propose a method for the generation of superpositions of two-mode coherent states for the center-of-mass and relative vibrational modes of two trapped ions. In the scheme the ions are driven by a single travelling-wave laser field tuned to the carrier. Then a measurement of the internal states collapses the vibrational modes to the entangled coherent state.

Dense coding scheme using superpositions of Bell-states and its NMR implementation

Dense coding using superpositions of Bell-states is proposed. The generalized Grover's algorithm is used to prepare the initial entangled states, and the reverse process of the quantum algorithm is used to determine the entangled state in the decoding measurement. Compared with the previous schemes, the superpositions of two Bell-states are exploited. Our scheme is demonstrated using a nuclear magnetic resonance (NMR)quantum computer. The corresponding manipulations are obtained. Experimental results show a good agreement between theory and experiment. We also generalize the scheme to transmit eight messages by introducing an additional two-state system.

Approximation to Continuous Functions of Several Variables and Nonlinear Functions, Operators by Superpositions of Some Functions of One Variable

At the Second International Conference of Mathematics, Hilbert raised 23 problems, which promoted the development of mathematics in the 20th century. Among them, the 13th problem can be described as follows: Can the solution of any algebraic equation be expressed by superpositions of some functions of one variable? Hilbert conjectured that the solution of the equation x^7+ax^3 +bx^2+cx+1=0, viewed as a function of a, b, c,

Hidden Vacuum Rabi Oscillations:Dynamical Quantum Superpositions of On/Off Interaction between a Single Quantum Dot and a Microcavity

We show that it is possible to realize quantum superpositions of switched-on and-off strong light-matter interaction in a single quantum dot-semiconductor microcavity system.Such superpositions enable the observation of counterintuitive quantum conditional dynamics effects.Situations are possible where cavity photons as well as the emitter luminescence display exponential decay but their joint detection probability exhibits vacuum Rabi oscillations.Remarkably,these quantum correlations are also present in the nonequilibrium steady state spectra of such coherently driven dissipative quantum systems.

A Wave Superposition-Finite Element Method for Calculating the Radiated Noise Generated by Volumetric Targets in Shallow Water

A combined method of wave superposition and finite element is proposed to solve the radiation noise of targets in shallow sea.Taking the sound propagation of spherical sound source in shallow sea as an example,the radiation sound field of the spherical sound source is equivalent to the linear superposition of the radiation sound field of several internal point sound sources,and then the radiated noise induced by spherical sound source can be predicted quickly.The accuracy and efficiency of the method are verified by comparing with the numerical results of finite element method,and the rapid prediction of underwater radiated noise of cylindrical shell is carried out based on the method.The results show that compared with the finite element method,the relative error of the calculation results under different simulation conditions does not exceed 0.1%,and the calculation time is about 1/10 of the finite element method,so this method can be used to solve the radiated noise of shallow underwater targets.

Enhanced soft magnetic properties of SiO_(2)-coated FeSiCr magnetic powder cores by particle size effect

It has been known that metal FeSiCr powders with large average particle sizes have been typically employed to prepare magnetic powder cores(SMCs),with few studies reported on the influence of magnetic properties for original powders with various average particle sizes less than 10m.In this work,SiO_(2)-coated FeSiCr SMCs with different small particle sizes were synthesized using the sol-gel process.The contribution of SiO_(2)coating amount and voids to the soft magnetic properties was elaborated.The mechanism was revealed such that smaller particle sizes with less voids could be beneficial for reducing core loss in the SMCs.By optimizing the core structure,permeability and magnetic loss of 26 and 262 kW/cm^(3)at 100 kHz and 50 mT were achieved at a particle size of 4.8m and ethyl orthosilicate addition of 0.1 mL/g.The best DC stacking performance,reaching 87%,was observed at an ethyl orthosilicate addition rate of 0.25 mL/g under 100 Oe.Compared to other soft magnetic composites(SMCs),the FeSiCr/SiO_(2)SMCs exhibit significantly reduced magnetic loss.It further reduces the magnetic loss of the powder core,providing a new strategy for applications of SMCs at high frequencies.

First-Arrival Picking Method for Active Source Data with Ocean Bottom Seismometers Based on Spatial Waveform Variation Characteristics

The precision and reliability of first-arrival picking are crucial for determining the accuracy of geological structure inversion using active source ocean bottom seismometer(OBS)refraction data.Traditional methods for first-arrival picking based on sample points are characterized by theoretical errors,especially in low-sampling-frequency OBS data because the travel time of seismic waves is not an integer multiple of the sampling interval.In this paper,a first-arrival picking method that utilizes the spatial waveform variation characteristics of active source OBS data is presented.First,the distribution law of theoretical error is examined;adjacent traces exhibit variation characteristics in their waveforms.Second,a label cross-correlation superposition method for extracting highfrequency signals is presented to enhance the first-arrival picking precision.Results from synthetic and field data verify that the proposed approach is robust,successfully overcomes the limitations of low sampling frequency,and achieves precise outcomes that are comparable with those of high-sampling-frequency data.

Design and Implementation of a High-Sensitivity Magnetic Sensing System Based on GMI Effect

A high-sensitivity magnetic sensing system based on giant magneto-impedance(GMI)effect is designed and fabricated.The system comprises a GMI sensor equipped with a gradient probe and an signal acquisition and processing module.A segmented superposition algorithm is used to increase target signal and reduce the random noise.The results show that under unshielded,room temperature conditions,the system achieves successful detection of weak magnetic fields down to 2 pT with a notable sensitivity of 1.84×10^(8)V/T(G=1000).By applying 17 overlays,the segmented superposition algorithm increases the power proportion of the target signal at 31 Hz from6.89%to 45.91%,surpassing the power proportion of the 2 Hz low-frequency interference signal.Simultaneously,it reduces the power proportion of the 20 Hz random noise.The segmented superposition process effectively cancels out certain random noise elements,leading to a reduction in their respective power proportions.This high-sensitivity magnetic sensing system features a simple structure,and is easy to operate,making it highly valuable for both practical applications and broader dissemination.

Improving the Quality of Public Space in Old Residential Areas Based on Superposition and Intervention:A Case Study of Baihuazhou District of Nanchang City

With the development of cities,China has entered an era of stock planning.The functions of old residential areas in urban development can no longer meet the needs of residents in the new era for a better life,and new standards have been put forward for the quality of public spaces for living.This paper analyzes and sorts out the characteristics and problems of the public space in the old residential areas of Baihuazhou district.Combining superposition and intervention techniques,the spatial,historical,cultural and other characteristic elements of the residential areas are preserved to the greatest extent.The public space in the old residential areas of Baihuazhou District is updated and reconstructed in order to achieve the goal of improving the quality of its public space.

Malus’ Law Derived from Deterministic Particle Behavior

A polarized beam of energy is usually interpreted as a set of particles, all having the same polarization state. Difference in behavior between the one and the other particle is then explained by a number of counter-intuitive quantum mechanical concepts like probability distribution, superposition, entanglement and quantized spin. Alternatively, I propose that a polarized beam is composed of a set of particles with a cosine distribution of polarization angles within a polarization area. I show that Malus’ law for the intensity of a beam of polarized light can be derived in a straightforward manner from this distribution. I then show that none of the above-mentioned counter-intuitive concepts are necessary to explain particle behavior and that the ontology of particles, passing through a polarizer, can be easily and intuitively understood. I conclude by formulating some questions for follow-up research.

On the Proof of the Contradiction of Set Theory

The article is devoted to proving the inconsistency of set theory arising from the existence of strange trees. All steps of the proof rely on common informal set-theoretic reasoning, but they take into account the prohibitions that were introduced into axiomatic set theories in order to overcome the difficulties encountered by the naive Cantor set theory. Therefore, in fact, the article is about proving the inconsistency of existing axiomatic set theories, in particular, the ZFC theory.

Completion of the Proof of the Contradiction of Set Theory

The article is devoted to completing the proof of the inconsistency of set theory. In this article and in the two preceding ones, all steps of the proof are based on generally accepted informal set-theoretic reasoning, but consider the prohibitions that were included in axiomatic set theories in order to overcome the difficulties encountered by the naive Cantor set theory. Therefore, in fact, the articles are about proving the inconsistency of existing axiomatic set theories, in particular, the ZFC theory.

Spatial, Temporal and Coherent Superposition of Quantum States, a Reinterpretation of the Quantum Superposition Principle

Spatial, temporal and coherent superposition of quantum states is considered. A consistent interpretation of the simultaneous superposition of stationary quantum states within material wave packets is proposed.

Forming mechanism of ultrasonic vibration assisted compression

To study the mechanism of ultrasonic vibration assisted forming,the static and vibration assisted compression tests of aluminum 1050 were carried out via a 25 kHz high-frequency ultrasonic vibration device.It is found that vibration reduces the flow resistance and improves the surface topography.The force reduction level is proportional to the ultrasonic vibration amplitude.By using numerical simulation of static and vibration assisted compression tests,the deformation characteristics of material were investigated.Throughout the vibration,the friction between the materials and tools reduces.The stress superposition and friction effects are found to be two major reasons for reducing the force.However,the force reduction because of stress superposition and friction effects is still less than the actual force reduction from the tests,which suggests that softening effect may be one of the other reasons to reduce the force.