The fundamental gap of a kind of two dimensional sub-elliptic operator

This paper is concerned at the minimization fundamental gap problem for a class of two-dimensional degenerate sub-elliptic operators. We establish existence results for weak solutions, Sobolev embedding theorem and spectral theory of sub-elliptic operators. We provide the existence and characterization theorems for extremizing potentials $V(x)$ when $V(x)$ is subject to $L^\infty$ norm constraint

Extremal properties of the first eigenvalue and the fundamental gap of a sub-elliptic operator

We consider the problems of extreming the first eigenvalue and the fundamental gap of a sub-elliptic operator with Dirichlet boundary condition, when the potential $V$ is subjected to a $p$-norm constraint. The existence results for weak solutions, compact embedding theorem and spectral theory for sub-elliptic equation are given. Moreover, we provide the specific characteristics of the corresponding optimal potential function

Some controllability results of a class of N-dimensional parabolic equations with internal single-point degeneracy

This paper investigates the controllability of a class of $N$-dimensional degenerate parabolic equations with interior single-point degeneracy. We employ the Galerkin method to prove the existence of solutions for the equations. The analysis is then divided into two cases based on whether the degenerate point $x=0$ lies within the control region $\omega_0$ or not. For each case, we establish specific Carleman estimates. As a result, we achieve null controllability in the first case $0\in\omega_0$ and unique continuation and approximate controllability in the second case $0\notin\omega_0$

Efficient Semi-Supervised Federated Learning for Heterogeneous Participants

Federated Learning (FL) has emerged to allow multiple clients to collaboratively train machine learning models on their private data. However, training and deploying large-scale models on resource-constrained clients is challenging. Fortunately, Split Federated Learning (SFL) offers a feasible solution by alleviating the computation and/or communication burden on clients. However, existing SFL works often assume sufficient labeled data on clients, which is usually impractical. Besides, data non-IIDness across clients poses another challenge to ensure efficient model training. To our best knowledge, the above two issues have not been simultaneously addressed in SFL. Herein, we propose a novel Semi-SFL system, which incorporates clustering regularization to perform SFL under the more practical scenario with unlabeled and non-IID client data. Moreover, our theoretical and experimental investigations into model convergence reveal that the inconsistent training processes on labeled and unlabeled data have an influence on the effectiveness of clustering regularization. To this end, we develop a control algorithm for dynamically adjusting the global updating frequency, so as to mitigate the training inconsistency and improve training performance. Extensive experiments on benchmark models and datasets show that our system provides a 3.0x speed-up in training time and reduces the communication cost by about 70.3% while reaching the target accuracy, and achieves up to 5.1% improvement in accuracy under non-IID scenarios compared to the state-of-the-art baselines.Comment: 16 pages, 12 figures, conferenc

Nonlinear vibrations of beams with spring and damping delayed feedback control

The primary, subharmonic, and superharmonic resonances of an Euler–Bernoulli beam subjected to harmonic excitations are studied with damping and spring delayed-feedback controllers. By method of multiple scales, the non-linear governing partial differential equation is transformed into linear differential equations directly. Effects of the feedback gains and time-delays on the steady state responses are investigated. The velocity and displacement delayed-feedback controllers are employed to suppress the primary and superharmonic resonances of the forced nonlinear oscillator. The stable vibration regions of the feedback gains and time-delays are worked out based on stablility conditions of the resonances. It is found that proper selection of feedback gains and time-delays can enhance the control performance of beam’s nonlinear vibration. Position of the bifurcation point can be changed or the bifurcation can be eliminated

In situ three-dimensional reconstruction of mouse heart sympathetic innervation by two-photon excitation fluorescence imaging

Background Sympathetic nerve wiring in the mammalian heart has remained largely unexplored. Resolving the wiring diagram of the cardiac sympathetic network would help establish the structural underpinnings of neurocardiac coupling. New Method We used two-photon excitation fluorescence microscopy, combined with a computer-assisted 3-D tracking algorithm, to map the local sympathetic circuits in living hearts from adult transgenic mice expressing enhanced green fluorescent protein (EGFP) in peripheral adrenergic neurons. Results Quantitative co-localization analyses confirmed that the intramyocardial EGFP distribution recapitulated the anatomy of the sympathetic arbor. In the left ventricular subepicardium of the uninjured heart, the sympathetic network was composed of multiple subarbors, exhibiting variable branching and looping topology. Axonal branches did not overlap with each other within their respective parental subarbor nor with neurites of annexed subarbors. The sympathetic network in the border zone of a 2-week-old myocardial infarction was characterized by substantive rewiring, which included spatially heterogeneous loss and gain of sympathetic fibers and formation of multiple, predominately nested, axon loops of widely variable circumference and geometry. Comparison with Existing Methods In contrast to mechanical tissue sectioning methods that may involve deformation of tissue and uncertainty in registration across sections, our approach preserves continuity of structure, which allows tracing of neurites over distances, and thus enables derivation of the three-dimensional and topological morphology of cardiac sympathetic nerves. Conclusions Our assay should be of general utility to unravel the mechanisms governing sympathetic axon spacing during development and disease