Approximate Diagonal Integral Representations and Eigenmeasures for Lipschitz Operators on Banach Spaces

A new stochastic approach for the approximation of (nonlinear) Lipschitz operators in normed spaces by their eigenvectors is shown. Different ways of providing integral representations for these approximations are proposed, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex. We use the recently introduced notion of eigenmeasure and focus attention on procedures for extending a function for which the eigenvectors are known, to the whole space. We provide information on natural error bounds, thus giving some tools to measure to what extent the map can be considered diagonal with few errors. In particular, we show an approximate spectral theorem for Lipschitz operators that verify certain convexity properties.

Ground states for critical fractional Schr\”{o}dinger-Poisson systems with vanishing potentials

This paper deals with a class of fractional Schr\”{o}dinger-Poisson system \[\begin{cases}\displaystyle (-\Delta )^{s}u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=a (x)f(u), &x \in \R^{3}\\ (-\Delta )^{s}\phi=K(x)|u|^{2^*_s-1}, &x \in \R^{3}\end{cases} \]with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where $s \in (\frac{3}{4},1)$, $ 2^*_s = \frac{6}{3-2s}$ is the fractional critical exponent. The problem is set on the whole space and compactness issues have to be tackled. By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on $V, K, a$ and $f$.

In a holistic perspective, time is absolute and relativity a direct consequence of the conservation of total energy

We are taught to think that the description of relativistic phenomena requires distorted time and distance. The message of this essay is that, in a holistic perspective, time and distance are universal coordinate quantities, and relativity is a direct consequence of the conservation of energy. Instead of the kinematics/metrics-based approach of the theory of relativity, the dynamic universe (DU) approach starts from the dynamics of space as a whole and expresses relativity in terms of locally available energy instead of locally distorted time and distance. In such an approach, e.g., the frequency of atomic clocks at different states of motion and gravitation is obtained from the quantum mechanical solution of the characteristic frequencies, and the unique status of the velocity of light becomes understood via its linkage to the rest of space. In the kinematic/metrics-based theory of relativity, we postulate the principle of relativity, Lorentz covariance, the equivalence principle, the constancy of the speed of light, and the rest energy of mass objects. The conservation of momentum and energy is honored in local frames of reference, and time and distance are parameters in frame-to-frame observations. In the dynamics-based DU, the whole space is studied as a closed energy system and the energy in local structures is derived conserving the overall energy balance. Any local state of motion and gravitation in space is related, through a system of nested energy frames, to the state of rest in hypothetical homogeneous space, which serves as the universal frame of reference. Relativity of observations appears as a direct consequence of the overall energy balance and the linkage of local to the whole—with time and distance as universal coordinate quantities. DU postulates spherically closed space and zero-energy balance of motion and gravitation. DU does not need the relativity principle or any other postulates of the theory of relativity. Primarily, the theory of relativity is an empirically driven mathematical description of observations, with postulates formulated to support the mathematics. DU relies on mathematics built on the conservation of an overall zero-energy balance as the primary law of nature, which makes DU more like a metaphysically driven theory. Both approaches produce precise predictions. The choice is philosophical—nature is not dependent on the way we describe it.

Prediction of Conductive Anomalies Ahead of the Tunnel by the 3D-Resitivity Forward Modeling in the Whole Space

Water inrush in tunneling poses serious harm to safe construction, causing economic losses and casualties. The prediction of water hazards before tunnel excavations becomes an urgent task for governments or enterprises to ensure security. The three-dimensional (3D) direct current (DC) resistivity method is widely used in the forward-probing of tunnels because of its low cost and highly sensitive response to water-bearing structures. However, the different sizes of the tunnel will distort the distribution of the potential field, which causes an inaccurate prediction of water-bearing structures in front of the tunnels. Some studies have pointed out that the tunnel effect must be considered in the quantitative interpretation of the data. However, there is rarely a predicted model considering the tunnel effect to be reported in geophysical literature. We developed a predicted model algorithm by considering the tunnel effect for forward-probing in tunnels. The algorithm is proven to be feasible using a slab analytic model. By simulating a large number of models with different tunnel sizes, we propose an equation, which considers the tunnel effect and can predict the water-bearing structures ahead of the tunnel face. The Monte Carlo method is used to evaluate the quality of the predicted model by simulating and comparing 10,000 random models. The results show that the proposed method is accurate to forecast the water-rich structures with small errors.

Subspace Detours Meet Gromov–Wasserstein

In the context of optimal transport (OT) methods, the subspace detour approach was recently proposed by Muzellec and Cuturi. It consists of first finding an optimal plan between the measures projected on a wisely chosen subspace and then completing it in a nearly optimal transport plan on the whole space. The contribution of this paper is to extend this category of methods to the Gromov–Wasserstein problem, which is a particular type of OT distance involving the specific geometry of each distribution. After deriving the associated formalism and properties, we give an experimental illustration on a shape matching problem. We also discuss a specific cost for which we can show connections with the Knothe–Rosenblatt rearrangement.

Prephase-Based Equivalent Amplitude Tailoring for Low Sidelobe Levels of 1-Bit Phase-Only Control Metasurface under Plane Wave Incidence

<p>A prephase synthesis method is proposed for sidelobe level (SLL) suppression of a 1-bit phase-only control metasurface under plane wave incidence. The array factor of the metasurface with N×N unit cells shows that controlling the number of prephases with varying values over the reflective surface realizes equivalent amplitude tailoring. Different from optimizing the prephase distribution, selection of the numbers of 0 and π/2 prephases in specific N regions is used to suppress the SLLs. Therefore, the parameters in the optimization can be dramatically reduced from N² to N. The prephase distribution is then designed based on the optimized number of prephases and a symmetric matrix for SLL suppression in the whole space. The SLLs are further suppressed by optimizing some of the unit cell states based on similar equivalent amplitude tailoring. Simulation and measurement of a set of 1-bit reflective metasurfaces with 20×20 unit cells verify that the phase-only control metasurface realizes SLL suppression to -13 dB for multiple beam directions from -30 to 30 degrees with a 10-degree step under normal plane wave incidence.</p>

Prephase-Based Equivalent Amplitude Tailoring for Low Sidelobe Levels of 1-Bit Phase-Only Control Metasurface under Plane Wave Incidence

<p>A prephase synthesis method is proposed for sidelobe level (SLL) suppression of a 1-bit phase-only control metasurface under plane wave incidence. The array factor of the metasurface with N×N unit cells shows that controlling the number of prephases with varying values over the reflective surface realizes equivalent amplitude tailoring. Different from optimizing the prephase distribution, selection of the numbers of 0 and π/2 prephases in specific N regions is used to suppress the SLLs. Therefore, the parameters in the optimization can be dramatically reduced from N² to N. The prephase distribution is then designed based on the optimized number of prephases and a symmetric matrix for SLL suppression in the whole space. The SLLs are further suppressed by optimizing some of the unit cell states based on similar equivalent amplitude tailoring. Simulation and measurement of a set of 1-bit reflective metasurfaces with 20×20 unit cells verify that the phase-only control metasurface realizes SLL suppression to -13 dB for multiple beam directions from -30 to 30 degrees with a 10-degree step under normal plane wave incidence.</p>

Decay of weak solutions to Vlasov equation coupled with a shear thickening fluid

We show that the energy norm of weak solutions to Vlasov equation coupled with a shear thickening fluid on the whole space has a decay rate the energy norm $E(t) \leq {C}/{(1+t)^{\alpha }}, \forall t \geq 0$ for $\alpha \in (0,3/2)$ .

Existence of groundstates for Choquard type equations with Hardy–Littlewood–Sobolev critical exponent

In this paper, we consider a class of Choquard equations with Hardy–Littlewood–Sobolev lower or upper critical exponent in the whole space $\mathbb{R}^{N}$ R N . We combine an argument of L. Jeanjean and H. Tanaka (see (Proc. Am. Math. Soc. 131:2399–2408, 2003) with a concentration–compactness argument, and then we obtain the existence of ground state solutions, which extends and complements the earlier results.

Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective

In the mean field integrate-and-fire model, the dynamics of a typical neuron within a large network is modeled as a diffusion-jump stochastic process whose jump takes place once the voltage reaches a threshold. In this work, the main goal is to establish the convergence relationship between the regularized process and the original one where in the regularized process, the jump mechanism is replaced by a Poisson dynamic, and jump intensity within the classically forbidden domain goes to infinity as the regularization parameter vanishes. On the macroscopic level, the Fokker-Planck equation for the process with random discharges (i.e. Poisson jumps) are defined on the whole space, while the equation for the limit process is on the half space. However, with the iteration scheme, the difficulty due to the domain differences has been greatly mitigated and the convergence for the stochastic process and the firing rates can be established. Moreover, we find a polynomial-order convergence for the distribution by a re-normalization argument in probability theory. Finally, by numerical experiments, we quantitatively explore the rate and the asymptotic behavior of the convergence for both linear and nonlinear models.