energy bound
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AbstractWe use the behavior of the $$L_{2}$$ L 2 norm of the solutions of linear hyperbolic equations with discontinuous coefficient matrices as a surrogate to infer stability of discontinuous Galerkin spectral element methods (DGSEM). Although the $$L_{2}$$ L 2 norm is not bounded in terms of the initial data for homogeneous and dissipative boundary conditions for such systems, the $$L_{2}$$ L 2 norm is easier to work with than a norm that discounts growth due to the discontinuities. We show that the DGSEM with an upwind numerical flux that satisfies the Rankine–Hugoniot (or conservation) condition has the same energy bound as the partial differential equation does in the $$L_{2}$$ L 2 norm, plus an added dissipation that depends on how much the approximate solution fails to satisfy the Rankine–Hugoniot jump.

AbstractNonequilibrium information thermodynamics determines the minimum energy dissipation to reliably erase memory under time-symmetric control protocols. We demonstrate that its bounds are tight and so show that the costs overwhelm those implied by Landauer’s energy bound on information erasure. Moreover, in the limit of perfect computation, the costs diverge. The conclusion is that time-asymmetric protocols should be developed for efficient, accurate thermodynamic computing. And, that Landauer’s Stack—the full suite of theoretically-predicted thermodynamic costs—is ready for experimental test and calibration.

Abstract Given a Hermitian line bundle $$L\rightarrow M$$ L → M over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as $$\epsilon \rightarrow 0$$ ϵ → 0 , of couples $$(u_\epsilon ,\nabla _\epsilon )$$ ( u ϵ , ∇ ϵ ) critical for the rescalings $$\begin{aligned} E_\epsilon (u,\nabla )=\int _M\Big (|\nabla u|^2+\epsilon ^2|F_\nabla |^2+\frac{1}{4\epsilon ^2}(1-|u|^2)^2\Big ) \end{aligned}$$ E ϵ ( u , ∇ ) = ∫ M ( | ∇ u | 2 + ϵ 2 | F ∇ | 2 + 1 4 ϵ 2 ( 1 - | u | 2 ) 2 ) of the self-dual Yang–Mills–Higgs energy, where u is a section of L and $$\nabla $$ ∇ is a Hermitian connection on L with curvature $$F_{\nabla }$$ F ∇ . Under the natural assumption $$\limsup _{\epsilon \rightarrow 0}E_\epsilon (u_\epsilon ,\nabla _\epsilon )<\infty $$ lim sup ϵ → 0 E ϵ ( u ϵ , ∇ ϵ ) < ∞ , we show that the energy measures converge subsequentially to (the weight measure $$\mu $$ μ of) a stationary integral $$(n-2)$$ ( n - 2 ) -varifold. Also, we show that the $$(n-2)$$ ( n - 2 ) -currents dual to the curvature forms converge subsequentially to $$2\pi \Gamma $$ 2 π Γ , for an integral $$(n-2)$$ ( n - 2 ) -cycle $$\Gamma $$ Γ with $$|\Gamma |\le \mu $$ | Γ | ≤ μ . Finally, we provide a variational construction of nontrivial critical points $$(u_\epsilon ,\nabla _\epsilon )$$ ( u ϵ , ∇ ϵ ) on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren’s existence result for (nontrivial) stationary integral $$(n-2)$$ ( n - 2 ) -varifolds in an arbitrary closed Riemannian manifold.

Abstract In this paper, we analyze static traversable wormholes via Noether symmetry technique in modified Gauss–Bonnet $$f(\mathcal {G})$$f(G) theory of gravity (where $$\mathcal {G}$$G represents Gauss–Bonnet term). We assume isotropic matter configuration and spherically symmetric metric. We construct three $$f(\mathcal {G})$$f(G) models, i.e, linear, quadratic and exponential forms and examine the consistency of these models. The traversable nature of wormhole solutions is discussed via null energy bound of the effective stress–energy tensor while physical behavior is studied through standard energy bounds of isotropic fluid. We also discuss the stability of these wormholes inside the wormhole throat and conclude the presence of traversable and physically stable wormholes for quadratic as well as exponential $$f(\mathcal {G})$$f(G) models.

Abstract We prove a nontrivial energy bound for a finite set of affine transformations over a general field and discuss a number of implications. These include new bounds on growth in the affine group, a quantitative version of a theorem by Elekes about rich lines in grids. We also give a positive answer to a question of Yufei Zhao that for a plane point set $P$ for which no line contains a positive proportion of points from $P$, there may be at most one line, meeting the set of lines defined by $P$ in at most a constant multiple of $|P|$ points.

Majorana zero modes (MZMs) that obey the non-Abelian statistics have been intensively investigated for potential applications in topological quantum computing. The prevailing signals in tunneling experiments “fingerprinting” the existence of MZMs are the zero-energy bound states (ZEBSs). However, nearly all of the previously reported ZEBSs showing signatures of the MZMs are observed in difficult-to-fabricate heterostructures at very low temperatures and additionally require applied magnetic field. Here, by using in situ scanning tunneling spectroscopy, we detect the ZEBSs upon the interstitial Fe adatoms deposited on two different high-temperature superconducting one-unit-cell iron chalcogenides on SrTiO3(001). The spectroscopic results resemble the phenomenological characteristics of the MZMs inside the vortex cores of topological superconductors. Our experimental findings may extend the MZM explorations in connate topological superconductors toward an applicable temperature regime and down to the two-dimensional (2D) limit.