Homoclinic points for convex billiards

In this paper we investigate some generic properties of a billiard system on a convex table. We show that generically, every hyperbolic periodic point admits some homoclinic orbit.Comment: 12 pages, 1 figur

LpL^p Christoffel-Minkowski problem: the case 1<p<k+11< p<k+1

We consider a fully nonlinear partial differential equation associated to the intermediate $L^p$ Christoffel-Minkowski problem in the case $1<p<k+1$. We establish the existence of convex body with prescribed $k$-th even $p$-area measure on $\mathbb S^n$, under an appropriate assumption on the prescribed function. We construct examples to indicate certain geometric condition on the prescribed function is needed for the existence of smooth strictly convex body. We also obtain $C^{1,1}$ regularity estimates for admissible solutions of the equation when $p\ge \frac{k+1}2$.Comment: 23 page

Homoclinic intersections for geodesic flows on convex spheres

In this paper we study some generic properties of the geodesic flows on a convex sphere. We prove that, $C^r$ generically ($2\le r\le\infty$), every hyperbolic closed geodesic admits some transversal homoclinic orbits

Low Complexity Hybrid Precoding and Channel Estimation Based on Hierarchical Multi-Beam Search for Millimeter-Wave MIMO Systems

In millimeter-wave (mmWave) MIMO systems, while a hybrid digital/analog precoding structure offers the potential to increase the achievable rate, it also faces the challenge of the need of a low-complexity design. In specific, the hybrid precoding may require matrix operations with a scale of antenna size, which is generally large in mmWave communication. Moreover, the channel estimation is also rather time consuming due to the large number of antennas at both Tx/Rx sides. In this paper, a low-complexity hybrid precoding and channel estimation approach is proposed. In the channel estimation phase, a hierarchical multi-beam search scheme is proposed to fast acquire $N_{\rm{S}}$ (the number of streams) multipath components (MPCs)/clusters with the highest powers. In the hybrid precoding phase, the analog and digital precodings are decoupled. The analog precoding is designed to steer along the $N_{\rm{S}}$ acquired MPCs/clusters at both Tx/Rx sides, shaping an equivalent $N_{\rm{S}}\times N_{\rm{S}}$ baseband channel, while the digital precoding performs operations in the baseband with the reduced-scale channel. Performance evaluations show that, compared with a state-of-the-art scheme, while achieving a close or even better performance when the number of radio-frequency (RF) chains or streams is small, both the computational complexity of the hybrid precoding and the time complexity of the channel estimation are greatly reduced.Comment: 11 pages, 9 figures. This paper reports a Low Complexity Hybrid Precoding and Channel Estimation method for mmWave communication

Codebook Design for Millimeter-Wave Channel Estimation with Hybrid Precoding Structure

In this paper, we study hierarchical codebook design for channel estimation in millimeter-wave (mmWave) communications with a hybrid precoding structure. Due to the limited saturation power of mmWave power amplifier (PA), we take the per-antenna power constraint (PAPC) into consideration. We first propose a metric, i.e., generalized detection probability (GDP), to evaluate the quality of \emph{an arbitrary codeword}. This metric not only enables an optimization approach for mmWave codebook design, but also can be used to compare the performance of two different codewords/codebooks. To the best of our knowledge, GDP is the first metric particularly for mmWave codebook design for channel estimation. We then propose an approach to design a hierarchical codebook exploiting BeaM Widening with Multi-RF-chain Sub-array technique (BMW-MS). To obtain crucial parameters of BMW-MS, we provide two solutions, namely a low-complexity search (LCS) solution to optimize the GDP metric and a closed-form (CF) solution to pursue a flat beam pattern. Performance comparisons show that BMW-MS/LCS and BMW-MS/CF achieve very close performances, and they outperform the existing alternatives under the PAPC.Comment: 14 pages, 10 figures. Hierarchical codebook design for mmWave channel estimation with a hybrid precoding structure. Submitted to TW

Enabling UAV Cellular with Millimeter-Wave Communication: Potentials and Approaches

To support high data rate urgent or ad hoc communications, we consider mmWave UAV cellular networks and the associated challenges and solutions. To enable fast beamforming training and tracking, we first investigate a hierarchical structure of beamforming codebooks and design of hierarchical codebooks with different beam widths via the sub-array techniques. We next examine the Doppler effect as a result of UAV movement and find that the Doppler effect may not be catastrophic when high gain directional transmission is used. We further explore the use of millimeter wave spatial division multiple access and demonstrate its clear advantage in improving the cellular network capacity. We also explore different ways of dealing with signal blockage and point out that possible adaptive UAV cruising algorithms would be necessary to counteract signal blockage. Finally, we identify a close relationship between UAV positioning and directional millimeter wave user discovery, where update of the former may directly impact the latter and vice versa.Comment: This paper explores the potentials and approaches to exploit mmWave communication to establish a UAV cellular. It is to appear in IEEE Communications Magazin

Full-Duplex Millimeter-Wave Communication

The potential of doubling the spectrum efficiency of full-duplex (FD) transmission motivates us to investigate FD millimeter-wave (FD-mmWave) communication. To realize FD transmission in the mmWave band, we first introduce possible antenna configurations for FD-mmWave transmission. It is shown that, different from the cases in micro-wave band FD communications, the configuration with separate Tx/Rx antenna arrays appears more flexible in self-interference (SI) suppression while it may increase some cost and area versus that with the same array. We then model the mmWave SI channel with separate Tx/Rx arrays, where a near-field propagation model is adopted for the line-of-sight (LOS) path, and it is found that the established LOS-SI channel with separate Tx/Rx arrays also shows spatial sparsity. Based on the SI channel, we further explore approaches to mitigate SI by signal processing, and we focus on a new cancellation approach in FD-mmWave communication, i.e., beamforming cancellation. Centered on the constant-amplitude (CA) constraint of the beamforming vectors, we propose several candidate solutions. Lastly, we consider an FD-mmWave multi-user scenario, and show that even if there are no FD users in an FD-mmWave cellular system, the FD benefit can still be exploited in the FD base station. Candidate solutions are also discussed to mitigate both SI and multi-user interference (MUI) simultaneously.Comment: This paper explores the combination of full duplex communication and millimeter wave communication. (To appear in IEEE Wireless Communications

Hierarchical Codebook Design for Beamforming Training in Millimeter-Wave Communication

In millimeter-wave communication, large antenna arrays are required to achieve high power gain by steering towards each other with narrow beams, which poses the problem to efficiently search the best beam direction in the angle domain at both Tx and Rx sides. As the exhaustive search is time consuming, hierarchical search has been widely accepted to reduce the complexity, and its performance is highly dependent on the codebook design. In this paper, we propose two basic criteria for the hierarchical codebook design, and devise an efficient hierarchical codebook by jointly exploiting sub-array and deactivation (turning-off) antenna processing techniques, where closed-form expressions are provided to generate the codebook. Performance evaluations are conducted under different system and channel models. Results show superiority of the proposed codebook over the existing alternatives.Comment: 13 pages, 11 figures. To appear in IEEE Trans. Wireless Commn. This paper proposes the BMW-SS approach to design a fully-hierarchical codebook for mmWave communication

Three-flavor Nambu--Jona-Lasinio model at finite isospin chemical potential

QCD at finite isospin chemical potential $\mu_{\text I}$ possesses a positively definite fermion determinant and the lattice simulation can be successfully performed. While the two-flavor effective models may be sufficient to describe the phenomenon of pion condensation, it is interesting to study the roles of the strangeness degree of freedom and the U$_{\rm A}(1)$ anomaly. In this paper, we present a systematic study of the three-flavor Nambu--Jona-Lasinio model with a Kobayashi-Maskawa-'t Hooft (KMT) term that mimics the U$_{\rm A}(1)$ anomaly at finite isospin chemical potential. In the mean-field approximation, the model predicts a phase transition from the vacuum to the pion superfluid phase, which takes place at $\mu_{\rm I}$ equal to the pion mass $m_\pi$. Due to the U$_{\rm A}(1)$ anomaly, the strangeness degree of freedom couples to the light quark degrees of freedom and the strange quark effective mass depends on the pion condensate. However, the strange quark condensate and the strange quark effective mass change slightly in the pion superfluid phase, which verifies the validity of the two-flavor models. The effective four-fermion interaction of the Kobayashi-Maskawa-'t Hooft term in the presence of the pion condensation is constructed. Due to the U$_{\rm A}(1)$ anomaly, the pion condensation generally induces scalar-pseudoscalar interaction. The Bethe-Salpeter equation for the mesonic excitations is established and the meson mass spectra are obtained at finite isospin chemical potential and temperature. Finally, the general expression for the topological susceptibility $\chi$ at finite isospin chemical potential $\mu_{\rm I}$ is derived. In contrast to the finite temperature effect which suppresses $\chi$, the isospin density effect leads to an enhancement of $\chi$.Comment: Version punlished in PR

Topological Susceptibility in Three-Flavor Quark Meson Model at Finite Temperature

We study $U_A(1)$ symmetry and its relation to chiral symmetry at finite temperature through the application of functional renormalization group to the $SU(3)$ quark meson model. Very different from the mass gap and mixing angel between $\eta$ and $\eta'$ mesons which are defined at mean field level and behavior like the chiral condensates, the topological susceptibility includes a fluctuations induced part which becomes dominant at high temperature. As a result, the $U_A(1)$ symmetry is still considerably broken in the chiral symmetry restoration phase.Comment: 9 pages, 5 figure