Global existence of critical nonlinear wave equation with time dependent variable coefficients

In this paper, we establish global existence of smooth solutions for the Cauchy problem of the critical nonlinear wave equation with time dependent variable coefficients in three space dimensions {equation}\partial_{tt}\phi-\partial_{x_i}\big(g^{ij}(t,x)\partial_{x_j}\phi\big)+\phi^5=0, mathbb{R}_t \times \mathbb{R}_x^3,{equation} where $\big(g_{ij}(t,x)\big)$ is a regular function valued in the spacetime of $3\times3$ positive definite matrix and $\big(g^{ij}(t,x)\big)$ its inverse matrix. Here and in the sequence, a repeated sum on an index in lower and upper position is never indicated. In the constant coefficients case, the result of global existence is due to Grillakis \cite{Grillakis1}; and in the time-independent variable coefficients case, the result of global existence and regularity is due to Ibrahim and Majdoub \cite{Ibrahim}. The key point of our proofs is to show that the energy cannot concentrate at any point. For that purpose, following Christodoulou and Klainerman \cite{Chris}, we use a null frame associated to an optical function to construct a geometric multiplier similar to the well-known Morawetz multiplier. Then we use comparison theorem originated from Riemannian Geometry to estimate the error terms. Finally, using Strichartz inequality due to \cite{Smith} as Ibrahim and Majdoub \cite{Ibrahim}, we obtain global existence

Global Existence of the Critical Semilinear Wave Equations with Variable Coefficients Outside Obstacles

In this paper, we consider exterior problem of the critical semilinear wave equation in three space dimensions with variable coefficients and prove global existence of smooth solutions. Similar to the constant coefficients case, we show that the energy cannot concentrate at any point $(t,x)\in(0,\infty)\times\Omega$. For that purpose, following Ibrahim and Majdoub \cite{Ibrahim}, we use a geometric multiplier close to the well-known Morawetz multiplier used in the constant coefficients case. Then we use comparison theorem from Riemannian Geometry to estimate the error terms. Finally, using Strichartz inequality as in Smith and Sogge \cite{Sogge}, we get the global existence

The zero inertia limit of Ericksen-Leslie's model for liquid crystals

In this paper we study the zero inertia limit that is from the hyperbolic to parabolic Ericksen-Leslie's liquid crystal flow. By introducing an initial layer and constructing an energy norm and energy dissipation functional depending on the solutions of the limiting system, we derive a global in time uniform energy bound to the remainder system under the small size of the initial data.Comment: 56 pages; all coments wellcome

Learning Disentangled Representations for Timber and Pitch in Music Audio

Timbre and pitch are the two main perceptual properties of musical sounds. Depending on the target applications, we sometimes prefer to focus on one of them, while reducing the effect of the other. Researchers have managed to hand-craft such timbre-invariant or pitch-invariant features using domain knowledge and signal processing techniques, but it remains difficult to disentangle them in the resulting feature representations. Drawing upon state-of-the-art techniques in representation learning, we propose in this paper two deep convolutional neural network models for learning disentangled representation of musical timbre and pitch. Both models use encoders/decoders and adversarial training to learn music representations, but the second model additionally uses skip connections to deal with the pitch information. As music is an art of time, the two models are supervised by frame-level instrument and pitch labels using a new dataset collected from MuseScore. We compare the result of the two disentangling models with a new evaluation protocol called "timbre crossover", which leads to interesting applications in audio-domain music editing. Via various objective evaluations, we show that the second model can better change the instrumentation of a multi-instrument music piece without much affecting the pitch structure. By disentangling timbre and pitch, we envision that the model can contribute to generating more realistic music audio as well

Multitask learning for frame-level instrument recognition

For many music analysis problems, we need to know the presence of instruments for each time frame in a multi-instrument musical piece. However, such a frame-level instrument recognition task remains difficult, mainly due to the lack of labeled datasets. To address this issue, we present in this paper a large-scale dataset that contains synthetic polyphonic music with frame-level pitch and instrument labels. Moreover, we propose a simple yet novel network architecture to jointly predict the pitch and instrument for each frame. With this multitask learning method, the pitch information can be leveraged to predict the instruments, and also the other way around. And, by using the so-called pianoroll representation of music as the main target output of the model, our model also predicts the instruments that play each individual note event. We validate the effectiveness of the proposed method for framelevel instrument recognition by comparing it with its singletask ablated versions and three state-of-the-art methods. We also demonstrate the result of the proposed method for multipitch streaming with real-world music. For reproducibility, we will share the code to crawl the data and to implement the proposed model at: https://github.com/biboamy/ instrument-streaming.Comment: This is a pre-print version of an ICASSP 2019 pape

Entropy dynamics of a dephasing model in a squeezed thermal bath

We study the entropy dynamics of a dephasing model, where a two-level system (TLS) is coupled with a squeezed thermal bath via non-demolition interaction. This model is exactly solvable, and the time dependent states of both the TLS and its bath can be obtained exactly. Based on these states, we calculate the entropy dynamics of both the TLS and the bath, and find that the dephasing rate of the system relies on the squeezing phase of the bath. In zero temperature and high temperature limits, we prove that both the system and bath entropy increases monotonically. Moreover, we find that the dephasing rate of the system relies on the squeezing phase of the bath, and this phase dependence cannot be precisely derived from the Born-Markovian approximation which is widely adopted in open quantum systems.Comment: 9.3 pages, 2 figure

On well-posedness of Ericksen-Leslie's parabolic-hyperbolic liquid crystal model in compressible flow

We study the Ericksen-Leslie's parabolic-hyperbolic liquid crystal model in compressible flow. Inspired by our study for incompressible case \cite{Jiang-Luo-arXiv-2017} and some techniques from compressible Navier-Stokes equations, we prove the local-in-time existence of the classical solution to the system with finite initial energy, under some constraints on the Leslie coefficients which ensure the basic energy law is dissipative. Furthermore, with an additional assumption on the coefficients which provides a damping effect, and the smallness of the initial energy, the global classical solution can be established.Comment: arXiv admin note: text overlap with arXiv:1709.0637

Hit Song Prediction for Pop Music by Siamese CNN with Ranking Loss

A model for hit song prediction can be used in the pop music industry to identify emerging trends and potential artists or songs before they are marketed to the public. While most previous work formulates hit song prediction as a regression or classification problem, we present in this paper a convolutional neural network (CNN) model that treats it as a ranking problem. Specifically, we use a commercial dataset with daily play-counts to train a multi-objective Siamese CNN model with Euclidean loss and pairwise ranking loss to learn from audio the relative ranking relations among songs. Besides, we devise a number of pair sampling methods according to some empirical observation of the data. Our experiment shows that the proposed model with a sampling method called A/B sampling leads to much higher accuracy in hit song prediction than the baseline regression model. Moreover, we can further improve the accuracy by using a neural attention mechanism to extract the highlights of songs and by using a separate CNN model to offer high-level features of songs

Coupled Self-Organized Hydrodynamics and Navier-Stokes models: local well-posedness and the limit from the Self-Organized Kinetic-fluid models

A coupled system of self-organized hydrodynamics and Navier-Stokes equations (SOH-NS), which models self-propelled particles in a viscous fluid, was recently derived by Degond et al. \cite{DMVY-2017-arXiv}, starting from a micro-macro particle system of Vicsek-Navier-Stokes model, through an intermediate step of a self-organized kinetic-kinetic model by multiple coarse-graining processes. We first transfer SOH-NS into a non-singular system by stereographic projection, then prove the local in time well-posedness of classical solutions by energy method. Furthermore, employing the Hilbert expansion approach, we justify the hydrodynamic limit from the self-organized kinetic-fluid model to macroscopic dynamics. This provides the first analytically rigorous justification of the modeling and asymptotic analysis in \cite{DMVY-2017-arXiv}.Comment: 42 pages. arXiv admin note: text overlap with arXiv:1706.05666 by other author

Cultural evolution and personalization

In social sciences, there is currently no consensus on the mechanism for cultural evolution. The evolution of first names of newborn babies offers a remarkable example for the researches in the field. Here we perform statistical analyses on over 100 years of data in the United States. We focus in particular on how the frequency-rank distribution and inequality of baby names change over time. We propose a stochastic model where name choice is determined by personalized preference and social influence. Remarkably, variations on the strength of personalized preference can account satisfactorily for the observed empirical features. Therefore, we claim that personalization drives cultural evolution, at least in the example of baby names