Variational methods for degenerate Kirchhoff equations

For a degenerate autonomous Kirchhoff equation which is set on $\mathbb{R}^N$ and involves the Berestycki-Lions type nonlinearity, we cope with the cases $N=2,3$ and $N\geq5$ by using mountain pass and symmetric mountain pass approaches and by using Clark theorem respectively.Comment: 28 pages, no figure

Multiple solutions for a Kirchhoff-type equation with general nonlinearity

This paper is devoted to the study of the following autonomous Kirchhoff-type equation $$-M\left(\int_{\mathbb{R}^N}|\nabla{u}|^2\right)\Delta{u}= f(u),~~~~u\in H^1(\mathbb{R}^N),$$ where $M$ is a continuous non-degenerate function and $N\geq2$. Under suitable additional conditions on $M$ and general Berestycki-Lions type assumptions on the nonlinearity $f$, we establish several existence results of multiple solutions by variational methods, which are also naturally interpreted from a non-variational point of view.Comment: 18 pages, no figures. Minor modifications. Accepted for publication by Advances in Nonlinear Analysis and available online now as ahead of prin

An autonomous Kirchhoff-type equation with general nonlinearity in RN\mathbb{R}^N

We consider the following autonomous Kirchhoff-type equation \begin{equation*} -\left(a+b\int_{\mathbb{R}^N}|\nabla{u}|^2\right)\Delta u= f(u),~~~~u\in H^1(\mathbb{R}^N), \end{equation*} where $a\geq0,b>0$ are constants and $N\geq1$. Under general Berestycki-Lions type assumptions on the nonlinearity $f$, we establish the existence results of a ground state and multiple radial solutions for $N\geq2$, and obtain a nontrivial solution and its uniqueness, up to a translation and up to a sign, for $N=1$. The proofs are mainly based on a rescaling argument, which is specific for the autonomous case, and a new description of the critical values in association with the level sets argument.Comment: 20 pages. Major changes and added reference

Biharmonic maps in two dimensions

Biharmonic maps between surfaces are studied in this paper. We compute the bitension field of a map between surfaces with conformal metrics in complex coordinates. As applications, we show that a linear map from Euclidean plane into $(\mathbb{R}^2, \sigma^2dwd\bar w)$ is always biharmonic if the conformal factor $\sigma$ is bi-analytic; we construct a family of such $\sigma$, and we give a classification of linear biharmonic maps between $2$-spheres. We also study biharmonic maps between surfaces with warped product metrics. This includes a classification of linear biharmonic maps between hyperbolic planes and some constructions of many proper biharmonic maps into a circular cone or a helicoid.Comment: 20 page

Nonlinear scalar field equations with general nonlinearity

Consider the nonlinear scalar field equation \begin{equation} \label{a1} -\Delta{u}= f(u)\quad\text{in}~\mathbb{R}^N,\qquad u\in H^1(\mathbb{R}^N), \end{equation} where $N\geq3$ and $f$ satisfies the general Berestycki-Lions conditions. We are interested in the existence of positive ground states, of nonradial solutions and in the multiplicity of radial and nonradial solutions. Very recently Mederski [30] made a major advance in that direction through the development, in an abstract setting, of a new critical point theory for constrained functionals. In this paper we propose an alternative, more elementary approach, which permits to recover Mederski's results on the scalar field equation. The keys to our approach are an extension to the symmetric mountain pass setting of the monotonicity trick, and a new decomposition result for bounded Palais-Smale sequences

Inequalities between the Chern numbers of a singular fiber in a family of algebraic curves

In a family of curves, the Chern numbers of a singular fiber are the local contributions to the Chern numbers of the total space. We will give some inequalities between the Chern numbers of a singular fiber as well as their lower and upper bounds. We introduce the dual fiber of a singular fiber, and prove a duality theorem. As an application, we will classify singular fibers with large or small Chern numbers.Comment: 23 page

Certification of Boson Sampling Devices with Coarse-Grained Measurements

A boson sampling device could efficiently sample from the output probability distribution of noninteracting bosons undergoing many-body interference. This problem is not only classically intractable, but its solution is also believed to be classically unverifiable. Hence, a major difficulty in experiment is to ensure a boson sampling device performs correctly. We present an experimental friendly scheme to extract useful and robust information from the quantum boson samplers based on coarse-grained measurements. The procedure can be applied to certify the equivalence of boson sampling devices while ruling out alternative fraudulent devices. We perform numerical simulations to demonstrate the feasibility of the method and consider the effects of realistic noise. Our approach is expected to be generally applicable to other many-body certification tasks beyond the boson sampling problem.Comment: 8 pages including Supplemental Materials, 7 figures, 3 table

The compensation for the edge focusing of chicane bump magnets by harmonic injection in CSNS/RCS

In the Rapid Cycling Synchrotron(RCS) of China Spallation Neutron Source(CSNS), transverse painting injection is employed to suppress the space-charge effects. The beta-beating caused by edge focusing of the injection bump magnets leads to tune shift and shrinkage of the acceptance, which may result in additional beam loss. In RCS, the main quadrupoles are excited by White resonant power supplies, and the exciting current cannot be arbitrarily programed. Generally, this kind of perturbation could be corrected by trim-quadrupole, however, we don't have trim-quadrupole in CSNS/RCS. A new method based on the harmonic injection is firstly introduced to compensate the beta-beating caused by edge focusing of the chicane bump magnets at RCS. In this paper, the principle and feasibility of compensation scheme were presented. The simulation study was done on the application of the new method to the CSNS/RCS, and the results show the validity and effectiveness of the method.Comment: 6 pages, 13 figure

MIMO Relaying Broadcast Channels with Linear Precoding and Quantized Channel State Information Feedback

Multi-antenna relaying has emerged as a promising technology to enhance the system performance in cellular networks. However, when precoding techniques are utilized to obtain multi-antenna gains, the system generally requires channel state information (CSI) at the transmitters. We consider a linear precoding scheme in a MIMO relaying broadcast channel with quantized CSI feedback from both two-hop links. With this scheme, each remote user feeds back its quantized CSI to the relay, and the relay sends back the quantized precoding information to the base station (BS). An upper bound on the rate loss due to quantized channel knowledge is first characterized. Then, in order to maintain the rate loss within a predetermined gap for growing SNRs, a strategy of scaling quantization quality of both two-hop links is proposed. It is revealed that the numbers of feedback bits of both links should scale linearly with the transmit power at the relay, while only the bit number of feedback from the relay to the BS needs to grow with the increasing transmit power at the BS. Numerical results are provided to verify the proposed strategy for feedback quality control.Comment: 13pages appeared in IEEE Transactions on Signal Processin

Canonical Class Inequality for Fibred Spaces

We establish the canonical class inequality for families of higher dimensional projective manifolds. As an application, we get a new inequality between the Chern numbers of 3-folds with smooth families of minimal surfaces of general type over a curve, $c_1^3<18c_3$.Comment: 9 page