Dielectric capacitors reveal great potential in the application of high power and/or pulsed power electronic devices owing to their ultrafast charge-discharge rate and ultrahigh power density. Among various dielectric capacitors, the environment-friendly lead-free dielectric ceramics have drawn extensive investigations in recent years. Nevertheless, the relatively small recoverable energy storage density (W rec ) is still an obstacle for their application. Herein, the (0.55−x)BiFeO 3 -0.45SrTiO 3 -xBaTiO 3 ternary ceramics with 0.1 wt% MnO 2 were prepared by the solid-state reaction, and achieved enhanced relaxor behavior as well as breakdown strength E b . As a result, the x = 0.12 ceramic exhibited superior comprehensive energy storage performance of large E b (50.4 kV/mm), ultrahigh W rec (7.3 J/cm 3 ), high efficiency η (86.3%), relatively fast charge-discharge speed (t 0.9 = 6.1 μs) and outstanding reliability under different frequency, fatigue, and temperature, indicating that the BiFeO 3 -based relaxor ferroelectric ceramics are prospective alternatives for electrostatic energy storage.

We construct a group (an HNN extension of a free group) with polynomial isoperimetric function, linear isodiametric function and non-simply connected asymptotic cones.

The standard Universal Approximation Theorem for operator neural networks (NNs) holds for arbitrary width and bounded depth. Here, we prove that operator NNs of bounded width and arbitrary depth are universal approximators for continuous nonlinear operators. In our main result, we prove that for non-polynomial activation functions that are continuously differentiable at a point with a nonzero derivative, one can construct an operator NN of width five, whose inputs are real numbers with finite decimal representations, that is arbitrarily close to any given continuous nonlinear operator. We derive an analogous result for non-affine polynomial activation functions. We also show that depth has theoretical advantages by constructing operator ReLU NNs of depth 2k 3 + 8 and constant width that cannot be well-approximated by any operator ReLU NN of depth k, unless its width is exponential in k.

We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define  a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent if $X_{G_1}^{H} = X_{G_2}^{H}$, and use this idea to study uniqueness results for $H$-chromatic symmetric functions, with a particular emphasis on the case $H$ is a complete bipartite graph. We also show that several of the classical bases of the space of symmetric functions, i.e. the monomial symmetric functions, power sum symmetric functions, and elementary symmetric functions, can be realized as $H$-chromatic symmetric functions. Moreover, we show that if $G$ and $H$ are particular types of multipartite complete graphs we can derive a set of $H$-chromatic symmetric functions that are a basis for $\Lambda^n$. We end with some conjectures and open problems.

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