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In the past, research on a single low dimensional activation function in networks has led to internal covariate shift and gradient deviation problems. A relatively small research area is how to use function combinations to provide property completion for a single activation function application. We propose a network adversarial method to address the aforementioned challenges. This is the first method to use different activation functions in a network. Based on the existing activation functions in the current network, an adversarial function with opposite derivative image properties is constructed, and the two are alternately used as activation functions for different network layers. For complex situations, we propose a method of high-dimensional function graph decomposition(HD-FGD), which divides it into different parts and then passes through a linear layer. After integrating the inverse of the partial derivatives of each decomposed term, we obtain its adversarial function by referring to the computational rules of the decomposition process. The use of network adversarial methods or the use of HD-FGD alone can effectively replace the traditional MLP+activation function mode. Through the above methods, we have achieved a substantial improvement over standard activation functions regarding both training efficiency and predictive accuracy. The article addresses the adversarial issues associated with several prevalent activation functions, presenting alternatives that can be seamlessly integrated into existing models without any adverse effects. We will release the code as open source after the conference review process is completed.

In this paper, we show that for $\alpha\in(1/2,5/4)$, there exists a force $f$ and two distinct Leray-Hopf flows $u_1,u_2$ solving the forced fractional Navier-Stokes equation starting from rest. This shows that the J.L. Lions exponent is sharp in the class of Leray-Hopf solutions for the forced fractional Navier-Stokes equation., Comment: 14 pages; to appear in Bull. Lond. Math. Soc

In this paper, we study the $2$D cubic nonlinear Schr\"odinger equation (NLS) with the partial harmonic potential. First, we prove the local well-posedness in Bourgain spaces by establishing a key bilinear estimate associated with the partial harmonic oscillator. Then, we give the polynomial bound of the Sobolev norms for the solutions using the method of the Planchon, Tzvetkov, and Visciglia., Comment: 19pages

We prove a Mikhlin--H\"ormander multiplier theorem for the partial harmonic oscillator $H_{\textup{par}}=-\pa_\rho^2-\Delta_x+|x|^2$ for $(\rho, x)\in\R\times\R^d$ by using the Littlewood--Paley $g$ and $g^\ast$ functions and the associated heat kernel estimate. The multiplier we have investigated is defined on $\mathbb R \times \mathbb N$., Comment: 14 pages, no figure. All comments are welcome

In this paper, our goal is to establish the Sobolev space associated to the partial harmonic oscillator. Based on its heat kernel estimate, we firstly give the definition of the fractional powers of the partial harmonic oscillator $$\AH=-\partial_{\rho}^2-\Delta_x+|x|^2,$$ and show that its negative powers are well defined on $L^p(\mathbb R^{d+1})$ for $p\in [1,\infty]$. We then define associated Riesz transforms and show that they are bounded on classical Sobolev spaces by the calculus of symbols. Secondly, by a factorization of the operator $\AH$, we define two families of Sobolev spaces with positive integer indices, and show the equivalence between them by the boundedness of Riesz transforms. Moreover, the adapted symbolic calculus also implies the boundedness of Riesz type transforms on the Sobolev spaces associated to the partial harmonic oscillator $\AH$. Lastly, as applications of our results, we obtain the revised Hardy-Littlewood-Sobolev inequality, the Gagliardo-Nirenberg-Sobolev inequality, and Hardy's inequality in the potential space $L_{\AH}^{\alpha, p}$., Comment: 23 pages. All comments are welcome

In this paper, we investigate the $W^{s,p}$-boundedness for stationary wave operators of the Schr\"odinger operator with inverse-square potential $$\mathcal L_a=-\Delta+\tfrac{a}{|x|^2}, \quad a\geq -\tfrac{(d-2)^2}{4},$$ in dimension $d\geq 2$. We construct the stationary wave operators in terms of integrals of Bessel functions and spherical harmonics, and prove that they are $W^{s,p}$-bounded for certain $p$ and $s$ which depend on $a$. As corollaries, we solve some open problems associated with the operator $\mathcal L_a$, which include the dispersive estimates and the local smoothing estimates in dimension $d\geq 2$. We also generalize some known results such as the uniform Sobolev inequalities, the equivalence of Sobolev norms and the Mikhlin multiplier theorem, to a larger range of indices. These results are important in the description of linear and nonlinear dynamics for dispersive equations with inverse-square potential., Comment: 57 pages, 1 figure, 1 table, Improved previous result from $L^p$-boundedness to $W^{s, p}$-boundedness

In this paper, we prove a limiting absorption principle for high-order Schr\"odinger operators with a large class of potentials which generalize some results by A. Ionescu and W. Schlag. Our main idea is to handle the boundary operators by the restriction theorem of Fourier transform. Two key tools we use in this paper are the Stein--Tomas theorem in Lorentz spaces and a sharp trace lemma given by S. Agmon and L. H\"ormander, Comment: 15 pages. All comments are welcome

In this paper, we investigate that the H\"older regularity of solutions to the time fractional Schr\"odinger equation of order $1<\alpha<2$, which interpolates between the Schr\"odinger and wave equations. This is inspired by Hirata and Miao's work which studied the fractional diffusion-wave equation. First, we give the asymptotic behavior for the oscillatory distributional kernels and their Bessel potentials by using Fourier analytic techniques. Then, the space regularity is derived by employing some results on singular Fourier multipliers. Using the asymptotic behavior for the above kernels, we prove the time regularity. Finally, we use mismatch estimates to prove the pointwise convergence to the initial data in H\"older spaces. In addition, we also prove H\"older regularity result for the Schr\"odinger equation., Comment: 25 pages

In this paper, we consider the Cauchy problem for the fractional Schr\"odinger equation $i D_t^\alpha u + (-\Delta)^{\frac{\beta}{2}} u =0$ with $0<\alpha<1$, $\beta>0$. We establish the dispersive estimates for the solutions. In particular, we prove that the decay rates are sharp., Comment: 11 pages

The semilinear space-time fractional Schr\"odinger equation is considered. First, we give the explicit form for the fundamental solutions by using the Fox $H$-functions in order to to establish some $L^s$ decay estimates. After that, we give some space-time estimates for the mild solutions from which the local well-posedness is derived on some proper Banach space., Comment: 23pages