Gradient Learning under Tilted Empirical Risk Minimization

Gradient Learning (GL), aiming to estimate the gradient of target function, has attracted much attention in variable selection problems due to its mild structure requirements and wide applicability. Despite rapid progress, the majority of the existing GL works are based on the empirical risk minimization (ERM) principle, which may face the degraded performance under complex data environment, e.g., non-Gaussian noise. To alleviate this sensitiveness, we propose a new GL model with the help of the tilted ERM criterion, and establish its theoretical support from the function approximation viewpoint. Specifically, the operator approximation technique plays the crucial role in our analysis. To solve the proposed learning objective, a gradient descent method is proposed, and the convergence analysis is provided. Finally, simulated experimental results validate the effectiveness of our approach when the input variables are correlated.     

Weyl families of transformed boundary pairs

Let $(\mathfrak{L},\mathrm{\Gamma })$ be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space $\mathfrak{H}$. Let $M_{\mathrm{\Gamma }}$ be the Weyl family corresponding to $(\mathfrak{L},\mathrm{\Gamma })$. We cope with two main topics. First, since $M_{\mathrm{\Gamma }}$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation $M_{\mathrm{\Gamma }}(z)$, for some $z\in \mathbb{C}\setminus \mathbb{R}$, becomes a nontrivial task. Regarding $M_{\mathrm{\Gamma }}(z)$ as the (Shmul'yan) transform of $zI$ induced by Γ, we give conditions for the equality in $\overline{M_{\mathrm{\Gamma }}(z)}\subseteq \overline{M_{\overline{\mathrm{\Gamma }}}(z)}$ to hold and we compute the adjoint $M_{\overline{\mathrm{\Gamma }}}{(z)}^{\ast }$. As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for $T^{+}$ is nonempty. Based on the criterion for the closeness of $M_{\mathrm{\Gamma }}(z)$, we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family $M_{\mathrm{\Gamma }}$ corresponding to a boundary relation Γ for $T^{+}$ is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$ with its Weyl family $M_{{\mathrm{\Gamma }}^{\prime }}$. The transformation scheme is either ${\mathrm{\Gamma }}^{\prime }=\mathrm{\Gamma }{V}^{-1}$ or ${\mathrm{\Gamma }}^{\prime }=V\mathrm{\Gamma }$ with suitable linear relations V. Results in this direction include but are not limited to: a 1-1 correspondence between $(\mathfrak{L},\mathrm{\Gamma })$ and $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$; the formula for $M_{{\mathrm{\Gamma }}^{\prime }}-M_{\mathrm{\Gamma }}$, for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple $(\mathfrak{L},{\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1)$ with $ker\mathrm{\Gamma }=T$ and $T_0=T_0^{\ast }$ (second scheme, Hilbert space case).     

三重兼并费米子拓扑半金属MoP高压超导相的低压截获

碳化钨结构的三重兼并费米子拓扑半金属MoP高压下发生超导转变，然而，高压超导相能否在低压甚至常压截获尚不清楚.本文对两个密封于金刚石对顶砧中的MoP高压超导相在室温条件下进行快速卸压，通过测量低温电阻对温度的依赖关系，验证能否在低压甚至常压截获高压超导相.实验结果表明：初始压力61.5 GPa的样品#1,超导电性可截获的最低压力为44.2 GPa;初始压力95.0 GPa的样品#2,超导电性可截获的最低压力为3.5 GPa.     

The local Borg–Marchenko uniqueness theorem of matrix-valued Dirac-type operators for coefficients locally smooth at the right endpoint

We provide an expression of the Weyl–Titchmarsh matrix associated with a self-adjoint matrix-valued Dirac-type operator defined on [0, b) for . As a concrete application of it, we establish asymptotics of the difference $(M_{+}-{\stackrel{\sim }{M}}_{+})(z)$ of two Weyl–Titchmarsh matrices $M_{+},{\stackrel{\sim }{M}}_{+}$ corresponding to two Dirac-type operators $L=Jd/dx-B(x)$, $\stackrel{\sim }{L}=Jd/dx-\stackrel{\sim }{B}(x)$ with $B=\stackrel{\sim }{B}$ a.e. on [0, a], in $L_2{(0,b)}^{2m}$ with , when the $B(x),\stackrel{\sim }{B}(x)$ are sufficiently smooth in a right neighborhood of the point a and their right derivatives at a coincide up to a certain order. In addition to this, we also provide new proofs of the local Borg–Marchenko theorem and asymptotic high-energy expansion of Weyl–Titchmarsh matrices associated with Dirac-type operators.     

On the volume functional of compact manifolds with harmonic Weyl tensor

The main aim of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold M with boundary $\partial M$ and under the harmonic Weyl tensor condition. In particular, we prove that a critical metric with a harmonic Weyl tensor on a simply connected compact manifold with the boundary isometric to a standard sphere ${\mathbb{S}}^{n-1}$ must be isometric to a geodesic ball in a simply connected space form ${\mathbb{R}}^n$, ${\mathbb{H}}^n$, and ${\mathbb{S}}^n$. To this end, we first conclude the classification of such critical metrics under the Bach-flat assumption and then we prove that both geometric conditions are equivalent in this situation.     

Floquet Transmission in Weyl/Multi-Weyl and Nodal-Line Semimetals through a Time-Periodic Potential Well

A quantum pumping protocol through which the quasiparticles of Weyl/multi-Weyl and nodal-line semimetals are subjected to a time-periodic rectangular potential well is considered. The presence of an oscillating potential of frequency ω creates equispaced Floquet side-bands with spacing $\hslash \omega$. As a result, a Fano resonance is observed when the difference in the Fermi energy (i.e., the energy of the incident quasiparticle), and the energy of one of the (quasi)bound state levels of the well, coincides with the energy of an integer number of photons (each carrying energy quantum $\hslash \omega$). Using the Floquet theory and the scattering matrix approach in the zero-temperature non-adiabatic pumping limit, characteristic Fano resonance patterns are found in the transmission coefficients. The inflection points in the pumped shot noise spectra also serve as a proxy for the corresponding Fano resonances. Therefore, the pumped shot noise is also numerically evaluated. Finally, the existence of the Fano resonance points is correlated to the (quasi)bound states of the well, by explicitly calculating the bound states of the static well (which are a subset of the bound states of the driven system). Since semimetals with anisotropic dispersions are considered, all the features observed depend on the orientation of the potential well.