LpL^p-boundedness of wave operators for fourth order Schr\"odinger operators with zero resonances on R3\mathbb{R}^3

Let $H = \Delta^2 + V$ be the fourth-order Schr\"odinger operator on $\mathbb{R}^3$ with a real-valued fast-decaying potential $V$. If zero is neither a resonance nor an eigenvalue of $H$, it has recently been shown by Goldberg and Green \cite{GoGr21} that the wave operators $W_\pm(H, \Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for $1 < p < \infty$. Additionally, it has been proved by the authors in \cite{MWY23} that these wave operators are unbounded at the endpoints, i.e., for $p=1$ and $p=\infty$. In this paper, our primary focus is to further establish the $L^p$-bounds of $W_\pm(H, \Delta^2)$ that exhibit all types of singularities at the zero energy threshold. We first prove that $W_\pm(H, \Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for $1 < p < \infty$ in the first kind resonance case. We then proceed to establish for the second kind resonance case that they are bounded on $L^p(\mathbb{R}^3)$ for $1 < p < 3$, but not for $3 \le p < \infty$. In the third kind resonance case, we show that $W_\pm(H, \Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for $1<p<3$. Remarkably, we can also prove that $W_\pm(H, \Delta^2)$ remain bounded on $L^p(\mathbb{R}^3)$ for $3\le p<\infty$ if in addition $H$ has a zero eigenvalue, but no $p$-wave zero resonance and all zero eigenfunctions are orthogonal to $x_ix_jx_kV$ in $L^2(\mathbb{R}^3)$ for all $i,j,k=1,2,3$ with $x=(x_1,x_2,x_3)$ and that, without such additional conditions, they are unbounded on $L^p(\mathbb{R}^3)$ for $3\le p<\infty$. These results precisely delineate all $L^p$-bounds of the wave operators $W_\pm(H, \Delta^2)$ in $\mathbb{R}^3$ with the exception of the endpoints $p=1$ and $p=\infty$. As a result, $L^p$-$L^q$ decay estimates are derived for solutions of fourth-order Schr\"odinger equations and beam equations with zero resonance singularities.Comment: 70 pages! Any comments are welcome

Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schr\"odinger operators in dimension three

This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,\Delta^2)$ associated with fourth-order Schr\"odinger operators $H=\Delta^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim \langle x\rangle^{-\mu}$ for some $\mu>0$. A recent work by Goldberg and Green \cite{GoGr21} has demonstrated that wave operators $W_\pm(H,\Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1<p<\infty$ under the condition that $\mu>9$, and zero is a regular point of $H$. In this paper, we aim to further establish endpoint estimates for $W_\pm(H,\Delta^2)$ in two significant ways. First, we provide counterexamples that illustrate the unboundedness of $W_\pm(H,\Delta^2)$ on the endpoint spaces $L^1(\mathbb{R}^3)$ and $L^\infty(\mathbb{R}^3)$, even for non-zero compactly supported potentials $V$. Second, we establish weak (1,1) estimates for the wave operators $W_\pm(H,\Delta^2)$ and their dual operators $W_\pm(H,\Delta^2)^*$ in the case where zero is a regular point and $\mu>11$. These estimates depend critically on the singular integral theory of Calder\'on-Zygmund on a homogeneous space $(X,d\omega)$ with a doubling measure $d\omega$.Comment: 29 pages. Any comments are welcome

The Jacobian Consistency of a One-Parametric Class of Smoothing Functions for SOCCP

Second-order cone (SOC) complementarity functions and their smoothing functions have been much studied in the solution of second-order cone complementarity problems (SOCCP). In this paper, we study the directional derivative and B-subdifferential of the one-parametric class of SOC complementarity functions, propose its smoothing function, and derive the computable formula for the Jacobian of the smoothing function. Based on these results, we prove the Jacobian consistency of the one-parametric class of smoothing functions, which will play an important role for achieving the rapid convergence of smoothing methods. Moreover, we estimate the distance between the subgradient of the one-parametric class of the SOC complementarity functions and the gradient of its smoothing function, which will help to adjust a parameter appropriately in smoothing methods

Emergent Bio-Functional Similarities in a Cortical-Spike-Train-Decoding Spiking Neural Network Facilitate Predictions of Neural Computation

Despite its better bio-plausibility, goal-driven spiking neural network (SNN) has not achieved applicable performance for classifying biological spike trains, and showed little bio-functional similarities compared to traditional artificial neural networks. In this study, we proposed the motorSRNN, a recurrent SNN topologically inspired by the neural motor circuit of primates. By employing the motorSRNN in decoding spike trains from the primary motor cortex of monkeys, we achieved a good balance between classification accuracy and energy consumption. The motorSRNN communicated with the input by capturing and cultivating more cosine-tuning, an essential property of neurons in the motor cortex, and maintained its stability during training. Such training-induced cultivation and persistency of cosine-tuning was also observed in our monkeys. Moreover, the motorSRNN produced additional bio-functional similarities at the single-neuron, population, and circuit levels, demonstrating biological authenticity. Thereby, ablation studies on motorSRNN have suggested long-term stable feedback synapses contribute to the training-induced cultivation in the motor cortex. Besides these novel findings and predictions, we offer a new framework for building authentic models of neural computation

Fabrication and Thermoelectric Properties of Graphene/ Bi

Graphene/Bi2Te3 thermoelectric materials were prepared by spark plasma sintering (SPS) using hydrothermal synthesis of the powders as starting materials. The X-ray diffraction (XRD) and field emission scanning electron microscope (FE-SEM) were used to investigate the phase composition and microstructure of the as-prepared materials. Electrical resistivity, Seebeck coefficient, and thermal conductivity measurement were applied to analyze the thermoelectric properties. The effect of graphene on the performance of the thermoelectric materials was studied. The results showed that the maximum dimensionless figure of merit of the graphene/Bi2Te3 composite with 0.2 vol.% graphene was obtained at testing temperature 475 K, 31% higher than the pure Bi2Te3

Electric-field Control of Magnetism with Emergent Topological Hall Effect in SrRuO3 through Proton Evolution

Ionic substitution forms an essential pathway to manipulate the carrier density and crystalline symmetry of materials via ion-lattice-electron coupling, leading to a rich spectrum of electronic states in strongly correlated systems. Using the ferromagnetic metal SrRuO3 as a model system, we demonstrate an efficient and reversible control of both carrier density and crystalline symmetry through the ionic liquid gating induced protonation. The insertion of protons electron-dopes SrRuO3, leading to an exotic ferromagnetic to paramagnetic phase transition along with the increase of proton concentration. Intriguingly, we observe an emergent topological Hall effect at the boundary of the phase transition as the consequence of the newly-established Dzyaloshinskii-Moriya interaction owing to the breaking of inversion symmetry in protonated SrRuO3 with the proton compositional film-depth gradient. We envision that electric-field controlled protonation opens a novel strategy to design material functionalities

Reversible manipulation of the magnetic state in SrRuO3 through electric-field controlled proton evolution

Ionic substitution forms an essential pathway to manipulate the structural phase, carrier density and crystalline symmetry of materials via ion-electron-lattice coupling, leading to a rich spectrum of electronic states in strongly correlated systems. Using the ferromagnetic metal SrRuO3 as a model system, we demonstrate an efficient and reversible control of both structural and electronic phase transformations through the electric-field controlled proton evolution with ionic liquid gating. The insertion of protons results in a large structural expansion and increased carrier density, leading to an exotic ferromagnetic to paramagnetic phase transition. Importantly, we reveal a novel protonated compound of HSrRuO3 with paramagnetic metallic as ground state. We observe a topological Hall effect at the boundary of the phase transition due to the proton concentration gradient across the film-depth. We envision that electric-field controlled protonation opens up a pathway to explore novel electronic states and material functionalities in protonated material systems

Optimasi Portofolio Resiko Menggunakan Model Markowitz MVO Dikaitkan dengan Keterbatasan Manusia dalam Memprediksi Masa Depan dalam Perspektif Al-Qur`an

Risk portfolio on modern finance has become increasingly technical, requiring the use of sophisticated mathematical tools in both research and practice. Since companies cannot insure themselves completely against risk, as human incompetence in predicting the future precisely that written in Al-Quran surah Luqman verse 34, they have to manage it to yield an optimal portfolio. The objective here is to minimize the variance among all portfolios, or alternatively, to maximize expected return among all portfolios that has at least a certain expected return. Furthermore, this study focuses on optimizing risk portfolio so called Markowitz MVO (Mean-Variance Optimization). Some theoretical frameworks for analysis are arithmetic mean, geometric mean, variance, covariance, linear programming, and quadratic programming. Moreover, finding a minimum variance portfolio produces a convex quadratic programming, that is minimizing the objective function ðð¥with constraintsð ð 𥠥 ðandð´ð¥ = ð. The outcome of this research is the solution of optimal risk portofolio in some investments that could be finished smoothly using MATLAB R2007b software together with its graphic analysis

Search for heavy resonances decaying to two Higgs bosons in final states containing four b quarks

A search is presented for narrow heavy resonances X decaying into pairs of Higgs bosons (H) in proton-proton collisions collected by the CMS experiment at the LHC at root s = 8 TeV. The data correspond to an integrated luminosity of 19.7 fb(-1). The search considers HH resonances with masses between 1 and 3 TeV, having final states of two b quark pairs. Each Higgs boson is produced with large momentum, and the hadronization products of the pair of b quarks can usually be reconstructed as single large jets. The background from multijet and t (t) over bar events is significantly reduced by applying requirements related to the flavor of the jet, its mass, and its substructure. The signal would be identified as a peak on top of the dijet invariant mass spectrum of the remaining background events. No evidence is observed for such a signal. Upper limits obtained at 95 confidence level for the product of the production cross section and branching fraction sigma(gg -> X) B(X -> HH -> b (b) over barb (b) over bar) range from 10 to 1.5 fb for the mass of X from 1.15 to 2.0 TeV, significantly extending previous searches. For a warped extra dimension theory with amass scale Lambda(R) = 1 TeV, the data exclude radion scalar masses between 1.15 and 1.55 TeV