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The zeroth-order general Randić index of graph $ G = (V_G, E_G) $, denoted by $ ^0R_{\alpha}(G) $, is the sum of items $ (d_{v})^{\alpha} $ over all vertices $ v\in V_G $, where $ \alpha $ is a pertinently chosen real number. In this paper, we obtain the sharp upper and lower bounds on $ ^0R_{\alpha} $ of trees with a given domination number $ \gamma $, for $ \alpha\in(-\infty, 0)\cup(1, \infty) $ and $ \alpha\in(0, 1) $, respectively. The corresponding extremal graphs of these bounds are also characterized.

In this paper, we investigate a nonlinear inverse problem of identifying a time-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation by using the measured data at a bound...

In this paper, a stationary one-dimensional Poisson-Nernst-Planck model with permanent charge is studied under the assumption that n - 1 positively charged ion species have the same valence and the permanent charge is small. By expanding the singular solutions of Poisson-Nernst-Planck model with respect to small permanent charge, the explicit formulae for the zeroth order approximation and the first order approximation of individual flux can be obtained. Based on these explicit formulae, the effects of small permanent charges on individual flux are investigated.

Define the zeroth-order Randić index as R0(G)=∑x∈V(G)1dG(x),{R^0}\left( G \right) = \sum\nolimits_{x \in V\left( G \right)} {{1 \over {\sqrt {{d_G}} \left( x \right)}},} where dG(x) denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs, respectively, to be super edge-connected in terms of the zeroth-order Randić index.

This report presents a method with high spatial and temporal accuracy for estimating solutions of Navier-Stokes equations at high Reynolds number. It employs Crank-Nicolson time discretization along with the zeroth-order ap-proximate deconvolution model of turbulence to regularize the flow prob-lem; solves a deviation of the Navier Stokes equation instead. Both theoreti-cal and computational findings of this report illustrate that the model pro-duces a high order of accuracy and stability. Furthermore, measurements of the drag and lift coefficients of a benchmark problem verify the potential of the model in this kind of computations.

The combined relativistic energy approach and relativistic many-body perturbation theory with the zeroth order Dirac-Kohn-Sham one-particle approximation are used for estimating the energies and the E1 radiative transitions amplitudes (oscillator strengths) for the low-excited states of the francium. The comparison with available theoretical and experimental (compilated) data is performed. The important point is linked with an accurate accounting for the complex exchange-correlation (polarization) effect contributions and using the optimized one-quasiparticle representation in the relativistic many-body perturbation theory zeroth order that significantly provides a physically reasonable agreement between theory and precise experiment.

The stiffness matrix in a vector finite-element method has a nullspace whose size grows linearly with the matrix size. Finding it numerically can be computationally expensive. However, the nullspace is important in many applications from solving the low-frequency breakdown problem to developing fast solvers. In this article, an analytical method to generate the nullspace of the stiffness matrix is developed, not only for the zeroth-order but also for any higher order vector basis function in arbitrary unstructured meshes. Using the mesh information, we are able to analytically construct the nullspace without any difficulty, thus avoiding solving an eigenvalue problem for finding the nullspace. The proposed analytical method has been applied to a variety of 2-D and 3-D irregular meshes for various orders of vector basis functions. Comparisons with the reference nullspace obtained from a brute-force eigenvalue solution have validated the proposed analytical method. The proposed work has also been applied to efficiently solve the low-frequency breakdown problem in a full-wave solver and perform fast layout parasitics extraction.

This letter considers the iterative numerical optimization of time-varying cost functions where no gradient information is available at each iteration. In this case, the proposed algorithm estimates a directional derivative by finite differences. The main contributions are the derivation of error bounds for such algorithms and proposal of optimal algorithm parameter values, e.g., step-sizes, for strongly convex cost functions. The algorithm is applied to tackle a source localization problem using a sensing agent where the source actively evades the agent. Numerical examples are provided to illustrate the theoretical results.

Let G = (V,E), V = {v1, v2,..., vn}, be a simple connected graph of order n, size m with vertex degree sequence ∆ = d1 ≥ d2 ≥ ··· ≥ dn = d > 0, di = d(vi). Denote by G a complement of G. If vertices vi and v j are adjacent in G, we write i ~ j, otherwise we write i j. The general zeroth-order Randic coindex of ' G is defined as 0Ra(G) = ∑i j (d a-1 i + d a-1 j ) = ∑ n i=1 (n-1-di)d a-1 i , where a is an arbitrary real number. Similarly, general zerothorder Randic coindex of ' G is defined as 0Ra(G) = ∑ n i=1 di(n-1-di) a-1 . New lower bounds for 0Ra(G) and 0Ra(G) are obtained. A case when G has a tree structure is also covered.

Tunable microwave devices have the benefits of added functionality, smaller form factor, lower cost, and lightweight, and are in great demand for future communications and radar applications as they can extend the operation over a wide dynamic range. Current tunable technologies include several schemes such as ferrites, semiconductors, microelectromechanical systems (MEMS), and ferroelectric thin films. While each technology has its own pros and cons, ferroelectric thin film-based technology has proved itself as the potential candidate for tunable devices due to its simple processes, low power consumption, high power handling, small size, and fast tuning. A tunable Composite Right Left-Handed Zeroth Order Resonator (CRLH ZOR) is introduced in this chapter and it relies mainly on the latest advancement in the ferroelectric materials. It is common that for achieving optimum performance for the resonant structure, this involves the incorporation of an additional tuning by either mechanical means (i.e. with tuning screws) or other coupling mechanisms. The integration between electronic tuning and High-Temperature Superconducting (HTS) components yields a high system performance without degradation of efficiency. This leads not only low-loss microwave components that could be fine-tuned for maximum efficiency but will provide a tunable device over a broadband frequency spectrum as well. The dielectric properties of the ferroelectric thin film, and the thickness of the ferroelectric film, play a fundamental role in the frequency or phase tunability and the overall insertion loss of the circuit. The key advantages of using ferroelectric are the potential for significant size-reduction of the microwave components and systems and the cabibility for integration with microelectronic circuits due to the utilization of thin and thick ferroelectric film technology. In this chapter, ZOR is discussed and the conceptual operation is introduced. The ZOR is designed and simulated by the full-wave analysis software. The response is studied using electromagnetic characteristics with the applied electric field, ferroelectric thickness, and the operating temperature.

The unoccupied electronic structures of three closed-shell, highly popular organoiron complexes ([Fe(CO)5], [(η5-C5H5)Fe(CO)(μ-CO)]2, and [(η5-C5H5)2Fe]; 0, I, and II, respectively) have been investigated both experimentally and theoretically by combining original gas-phase X-ray absorption spectroscopy (XAS) outcomes recorded at the C and O K-edge with results of scalar relativistic time-dependent density functional calculations carried out within the zeroth order regular approximation. Experimental evidence herein discussed complement the Fe L2,3-edges XAS ones we recently recorded, modeled, and assigned for the same complexes (Carlotto et al. Inorg. Chem. 2019, 58, 5844). The first-principle simulation of the C and O K-edge features allowed us to univocally identify the electronic states associated to the ligand-to-metal charge transfer (LMCT) transitions both in I and in II. At variance to that, LMCT transitions with sizable oscillator strengths do not play any role in determining neither the C nor the O K-edge spectral pattern of 0. The higher π-acceptor capability of the CO ligand, regardless of its terminal or bridging coordination, with respect to [(η5-C5H5)]- is herein ultimately confirmed.

We present here a relativistic density functional theory investigation of the penta- and heptavalent states of gold and roentgenium, employing the ZORA (zeroth order regular approximation to the Dirac equation) Hamiltonian, including spin-orbit coupling at the two-component level, and large all-electron relativistic Slater-type quadruple-ζ quadruple polarization (ZORA-STO-QZ4P) basis sets. Unsurprisingly, our calculations confirm the stability of the experimentally known complexes AuF

In this letter, a triple-band zeroth-order-resonance (ZOR) epsilon-negative antenna using two series pseudo-open termination (P-OT) unit cells is proposed, for the case where shunt ZOR $({\boldsymbol{f}_{\boldsymbol{sh}}})$ is higher than series ZOR $({\boldsymbol{f}_{\boldsymbol{se}}})$ . Epsilon-negative transmission line (ENG TL) resonates a shunt ZOR, while the two series P-OT unit cells resonate two series ZORs ( ${\boldsymbol{f}_{\boldsymbol{se}}}$ and ${\boldsymbol{f}_{\boldsymbol{se1}}}$ ). Both the P-OT unit cells create different capacitances with the ENG TL to resonate ${\boldsymbol{f}_{\boldsymbol{se}}}$ and ${\boldsymbol{f}_{\boldsymbol{se}1}}$ . An electric field distribution vector is used to verify the resonances for zeroth order, and it is found that all the three resonances follow zeroth-order characteristics. The P-OT unit cells can be placed anywhere around the ENG TL to get proper matching. The measured resonances are 5.95 GHz $({\boldsymbol{f}_{\boldsymbol{se}1}})$ , 6.26 GHz $({\boldsymbol{f}_{\boldsymbol{se}}})$ , and 6.91 GHz $({\boldsymbol{f}_{\boldsymbol{sh}}})$ covering the satellite transmission frequencies of the C-band. The measured peak gains are 1.14, 1.39, and 2.19 dBi for $({\boldsymbol{f}_{\boldsymbol{se}1}})$ , ${\boldsymbol{f}_{\boldsymbol{se}}}$ , and ${\boldsymbol{f}_{\boldsymbol{sh}}}$ , respectively. ${\boldsymbol{f}_{\boldsymbol{se}1}}$ and ${\boldsymbol{f}_{\boldsymbol{se}}}$ have peak gain at the left and the right side of the $\varphi = {\text{0}^{\circ}}$ plane, respectively, while ${\boldsymbol{f}_{\boldsymbol{sh}}}$ has peak gains at both the right and the left sides of the $\varphi = {\text{0}^{\circ}}$ plane.

In this paper, a triple-band and dual-band zeroth order resonance (ZOR) epsilon negative antennas are proposed using two pseudo-open termination unit cells for the case where the shunt ZOR ( f sh ) is higher than the series ZOR ( f s e 1 ). The epsilon negative transmission line (ENG TL) resonates f sh while the series ZORs ( f s e 1 and f s e 2 ) are excited by the series capacitance of the two pseudo-open termination unit cells. If the series capacitances of the two pseudo-open termination unit cells are different then f s e 1 and f s e 2 are excited, else only f s e 1 is excited. f s e 1 and f s e 2 along with f sh results in triple-band ZOR antenna, while a single f s e 1 with f sh results in dual-band ZOR antenna. The pseudo-open termination unit cells can be placed anywhere around the ENG TL to get proper matching. All the resonances for both the antennas follow zeroth order characteristics which are verified using electric field vector. The measured peak gains of the triple-band ZOR antenna for f s e 2 , f s e 1 and f sh are 1.14 dBi, 1.39 dBi, and 2.19 dBi, respectively, with dipole mode for f sh . For dual-band ZOR antenna, the measured peak gains for f s e 1 is 1.4 dBi, and f sh is 2.05 dBi, with dipole mode for f s e 1 and f sh .

A parallel nonlinear energy sink (NES) is proposed and analyzed. The parallel NES is composed of a vibro-impact (VI) NES and a cubic NES. The dynamical equation is given, and the essential analytical investigation is carried out to deal with the cubic nonlinearity and impact nonlinearity. Multiple time-scale expansion is introduced, and the zeroth order is derived to give a rough outline of the system. The underlying Hamilton dynamic equation is given, and then the optimal stiffness is expressed. The clearance is regarded as a critical factor for the VI. Based on the periodical impact treatment by analytical investigation, the relationships of the cubic stiffness, the clearance, and the zeroth-order attenuation amplitude of the linear primary oscillator (LPO) are obtained. A cubic NES under the optimal condition is compared with the parallel NES. Harmonic signals, harmonic signals with noises, and the excitation generated by a second-order filter are considered as the potential excitation forces on the system. The targeted energy transfer (TET) in the designed parallel NES is shown to be more efficient.

A mosaicking method for echelles is proposed. The “zero-order” and diffraction-order interference fringes of the echelle can be imaged simultaneously on the detector using a mirror–echelle structure designed. The five-dimensional mosaicking errors are divided into two categories for elimination using the “zero order” and the diffraction order. During the experiments, adjustment devices for the mosaicking errors are installed on two echelles for these two categories to avoid mechanical coupling. As theoretical analysis, we follow the experimental steps to obtain a mosaicked echelle.

Let G=(V,E), V={v1, v2,..., vn}, be a simple connected graph with n vertices, m edges and vertex degree sequence ? = d1?d2 ?...? dn = ? > 0, di = d(vi). General zeroth-order Randic index of G is defined as 0R?(G) = ?ni =1 d?i , where ? is an arbitrary real number. In this paper we establish relationships between 0R?(G) and 0R?-1(G) and obtain new bounds for 0R?(G). Also, we determine relationship between 0R?(G), 0R?(G) and 0R2?-?(G), where ? and ? are arbitrary real numbers. By the appropriate choice of parameters ? and ?, a number of old/new inequalities for different vertex-degree-based topological indices are obtained.

Let G = (V,E ), V = [v \,v2, . . . , vn}, be a simple connected graph with n vertices, m edges and vertex degree sequence d1 > d2 > ■■■ > dn > 0, di = d(vi ). General zeroth-order Randic index of G is defined as °Ra (G) = Ση=ι d" > and general sum-connectivity index as Xa(G) = (di + d j)α, where a is an arbitrary real number. In this paper we establish a relationship between 0Rα+β (G), ^ α -β ^ ) and °Ra (G), as well as χ α+β (G), χ α -β ^ ) and Xa (G), where α and β are arbitrary real numbers. By the appropriate choice of parameters α and β, a number of new/old inequalities that reveal relationships between various vertex and edge degree-based topological indices are obtained.

In milling, two different self-excited vibrations have been reported; regenerative and mode coupling chatter. The regenerative chatter mechanism has been extensively studied and validated with tests whereas mode coupling chatter mechanism was reported a long time ago for threading operations. The presented mode coupling chatter models were based only on current vibrations of the system but not on the delayed vibrations. With the increase in the research carried out on robotic milling operations, the low frequency severe self-excited vibrations at high spindle speeds were claimed to be mode coupling chatter by many researchers. However, the justification of mode coupling chatter mechanism is absent from predicted stability boundaries and a strong evidence distinguishing it from regenerative chatter mechanism is not present. Additionally, mode coupling chatter models applied to milling was based on threading operations and hence they were not capturing the characteristics of intermittent milling process. Therefore, this paper focuses on the diagnosis of mode coupling chatter in robotic milling. Mode coupling chatter principles are applied to milling process considering its intermittent characteristics. The zeroth order approximation for mode coupling chatter mechanism is adapted for milling and extended to multi frequency approximation. Mode coupling chatter stability boundaries are calculated, explored with tests and compared with regenerative chatter stability boundaries. Results show that mode coupling chatter stability boundaries are very low and vaguely dependent on the spindle speed. This contradicts the stability observed in the experimental tests. Hence, it is concluded that mode coupling chatter in milling is not possible, because the assumption of the chip thickness depending only on current vibrations does not apply to milling operations. The novelty of the presented paper is the theoretical and experimental justification that mode coupling chatter is not possible in milling operations.

Resonant planetary systems contain at least one planet pair with orbital periods librating at a near-integer ratio (2/1, 3/2, 4/3, etc.) and are a natural outcome of standard planetary formation theories. Systems with multiple adjacent resonant pairs are known as resonant chains and can exhibit three-body resonances -- characterized by a critical three-body angle. Here we study three-body angles as a diagnostic of resonant chains through tidally-damped N-body integrations. For each combination of the 2:1, 3:2, 4:3, and 5:4 mean motion resonances (the most common resonances in the known resonant chains), we characterize the three-body angle equilibria for several mass schemes, migration timescales, and initial separations. We find that under our formulation of the three-body angle, which does not reduce coefficients, 180 deg is the preferred libration center, and libration centers shifted away from 180 deg are associated with non-adjacent resonances. We then relate these angles to observables, by applying our general results to two transiting systems: Kepler-60 and Kepler-223. For these systems, we compare N-body models of the three-body angle to the zeroth order in e approximation accessible via transit phases, used in previous publications. In both cases, we find the three-body angle during the Kepler observing window is not necessarily indicative of the long-term oscillations and stress the role of dynamical models in investigating three-body angles. We anticipate our results will provide a useful diagnostic in the analysis of resonant chains., Comment: Accepted for publication in Astronomical Journal

Surface roughness parameters are an important factor affecting surface wear resistance, but the relevance between the wear resistance and the surface roughness parameters has not been well studied. This paper based on the finite element simulation technology, through the grey incidence analysis (GIA) method to quantitatively study the relevance between the wear amount of per unit sliding distance (&Delta, Vs) and the surface texture roughness parameters under dry friction conditions of the different surface topography. A zeroth order six-variables grey model, GM(0,6), for prediction the wear characteristic parameter &Delta, Vs was established, and the experiment results verified that the prediction model was accurate and reasonable.

We consider the zeroth order model of the family of approximate deconvolution models of Stolz and Adams. We propose and analyze fully discrete schemes using discontinuous finite elements. Optimal error estimates are derived. The dependence of these estimates with respect to the Reynolds number Re is O ( Re e Re ) , which is an improvement with respect to the classical continuous finite element method where the dependence is O ( Re e Re 3 ) , Layton [1] .

In this research, the polarization and bandwidth of a zeroth‐order resonators (ZORs) loaded microstrip antenna are improved by employing a grid polarization filter cover and a metallic cavity. The polarization improvement is owing to three operations, (a) the filter cover allows co‐polarization wave passing through only, (b) the ZORs partly rotate the reflected cross‐polarizations (X‐polarizations) wave to be co‐polarization, (c) the metallic cavity suppresses the fringe fields of the cover. The bandwidth enhancement is owing to the polarization cover that reflects the X‐polarization, which forms a resonant cavity. The improvements of the simulated cross‐polarizations range from 17.7 to 35.3 dB in the band. The measured bandwidth is increased from 940 MHz (17.22%) of the referenced antenna to 1390 MHz (24.5%) of the proposed antenna. The measured results verify the simulations.

We introduce a multimodel approach to solve coupled cluster equations, employing a quasi Newton algorithm for the ground state and an Olsen algorithm for the excited states. In these algorithms, both of which can be viewed as Newton algorithms, the Jacobian matrix of a lower level coupled cluster model is used in Newton equations associated with the target model. Improvements in convergence then implies savings for sufficiently large molecular systems, since the computational cost of macroiterations scales more steeply with system size than the cost of microiterations. The multimodel approach is suitable when there is a lower level Jacobian matrix that is much more accurate than the zeroth order approximation. Applying the approach to the CC3 equations, using the CCSD approximation of the Jacobian, we show that the time spent to determine the ground and valence excited states can be significantly reduced. We also find improved convergence for core excited states, indicating that similar savings will be obtained with an explicit implementation of the core-valence separated CCSD Jacobian transformation., 22 pages and 3 figures

We introduce the problem of computing a parsing where each phrase is of length at most $m$ and which minimizes the zeroth order entropy of parsing. Based on the recent theoretical results we devise a heuristic for this problem. The solution has straightforward application in succinct text representations and gives practical improvements. Moreover the proposed heuristic yields structure whose size can be bounded both by $|S|H_{m-1}(S)$ and by $|S|/m(H_0(S) + \cdots + H_{m-1})$, where $H_{k}(S)$ is the $k$-th order empirical entropy of $S$. We also consider a similar problem in which the first-order entropy is minimized.

Despite the great achievements of the modern deep neural networks (DNNs), the vulnerability/robustness of state-of-the-art DNNs raises security concerns in many application domains requiring high reliability. Various adversarial attacks are proposed to sabotage the learning performance of DNN models. Among those, the black-box adversarial attack methods have received special attentions owing to their practicality and simplicity. Black-box attacks usually prefer less queries in order to maintain stealthy and low costs. However, most of the current black-box attack methods adopt the first-order gradient descent method, which may come with certain deficiencies such as relatively slow convergence and high sensitivity to hyper-parameter settings. In this paper, we propose a zeroth-order natural gradient descent (ZO-NGD) method to design the adversarial attacks, which incorporates the zeroth-order gradient estimation technique catering to the black-box attack scenario and the second-order natural gradient descent to achieve higher query efficiency. The empirical evaluations on image classification datasets demonstrate that ZO-NGD can obtain significantly lower model query complexities compared with state-of-the-art attack methods., accepted by AAAI 2020

A generally applicable approach for the calculation of relativistic one-electron properties with two-component wave functions is presented. The formalism is explicitly evaluated for the example of quasi-relativistic wavefunctions obtained within the zeroth order regular approximation (ZORA). The wide applicability of the scheme is demonstrated for the calculation of parity ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetry violating properties, which are important for searches of physics beyond the standard model of particle physics. The quality of the ZORA results is shown exemplarily for the molecules RaF and TlF by comparison to data from four-component calculations as far as available. Finally, the applicability of RaF in experiments that search for $\mathcal{P,T}$-violation not only in the electronic but also in quark sector is demonstrated., 11 pages, 3 tables

In this paper, a compact electrically small wideband coplanar waveguide (CPW) -fed zeroth order resonating (ZOR) antenna is proposed. The presented antenna is composed of unit cell of the composite right/left handed (CRLH) artificial structure. In order to enhance the bandwidth, the CPW structure was used to reduce the quality factor while occupying a small area. The simulation results show that the proposed antenna achieves a 10 dB bandwidth of 157% and the peak gain of 3.33dBi with a compact size of 0.02910×0.04810 at the operating frequency. Compared with other ZOR antennas, it has a smaller size and wider bandwidth.

For a graph G without isolated vertices, the inverse degree of a graph G is defined as I D ( G ) = &sum, u &isin, V ( G ) d ( u ) - 1 where d ( u ) is the number of vertices adjacent to the vertex u in G. By replacing - 1 by any non-zero real number we obtain zeroth-order general Randić index, i.e., 0 R &gamma, ( G ) = &sum, V ( G ) d ( u ) &gamma, where &gamma, &isin, R - { 0 } . Xu et. al. investigated some lower and upper bounds on I D for a connected graph G in terms of connectivity, chromatic number, number of cut edges, and clique number. In this paper, we extend their results and investigate if the same results hold for &gamma, &lt, 0 . The corresponding extremal graphs have also been identified.

We prove that the information-theoretic upper bound on the minimax regret for zeroth-order adversarial bandit convex optimisation is at most $O(d^{2.5} \sqrt{n} \log(n))$, where $d$ is the dimension and $n$ is the number of interactions. This improves on $O(d^{9.5} \sqrt{n} \log(n)^{7.5}$ by Bubeck et al. (2017). The proof is based on identifying an improved exploratory distribution for convex functions., Comment: To appear in Mathematical Statistics and Learning. 22 pages, 6 figures

This work presents an implementation of the original orbital-free Hohenberg-Kohn density functional theory in a form that is able to predict chemical bonding in molecules. The method is completely parameter-free and does not require analytical functional approximations. Instead, the proposed method is based on the idea that atoms are meaningful pieces of a molecule and thus, a promolecule, build from frozen spherical atomic entities, serves as a suitable model for the latter. This idea is imposed on the physical equations, originating from density functional theory converted into a bifunctional formalism. The viewpoint proposed in this study offers a new strategic way of subsequent approximation levels in orbital-free density functional theory. In this work the zeroth order approximation is shown to predict chemical bonding in molecules, providing a concept of the chemical bond without involving orbitals.

In this paper, we prove a Szego type limit theorem on $$\ell ^2(\mathbb {Z}^d)$$ . We take the self-adjoint operator $$H=-\Delta +V$$ on $$\ell ^2(\mathbb {Z}^d)$$ , where $$\displaystyle (\Delta u)(\mathbf{n}) = \sum \nolimits _{|\mathbf{n}-\mathbf{k}|=1} (u(\mathbf{k}) - u(\mathbf{n}))$$ and the operator V is the multiplication by a positive sequence $$\{V(\mathbf{n}), \mathbf{n}\in \mathbb {Z}^d\}$$ with $$V(\mathbf{n}) \rightarrow \infty $$ as $$|\mathbf{n}| \rightarrow \infty . $$ We take the orthogonal projection $$\pi _{\lambda }$$ onto the subspace, in $$\ell ^2(\mathbb {Z}^d)$$ , spanned by eigenfunctions of H with eigenvalues $$\le \lambda $$ . Let B be a zeroth order self-adjoint pseudo-difference operator associated with symbol $$b \in S_{1,0, \infty }(\mathbb {T}^d\times \mathbb {Z}^d)$$ . We then show for “nice functions” f, that $$\begin{aligned} \lim _{\lambda \rightarrow \infty } \frac{Tr(f(\pi _\lambda B\pi _\lambda ))}{Tr(\pi _\lambda )} = \lim _{\lambda \rightarrow \infty } \frac{1}{(2\pi )^d} \frac{\sum _{V(\mathbf{n}) \le \lambda } \int _{\mathbb {T}^d} f(b(\mathbf{x},\mathbf{n})) ~ d\mathbf{x}}{\sum _{V(\mathbf{n})\le \lambda } 1}. \end{aligned}$$

EgeUn###, This paper is devoted to the inverse problem of identifying an unknown spacewise-dependent zeroth-order coefficient p(x) in the 1D diffusion equation u(t) = (k(x)u(x))(x) - p(x)u from boundary Dirichlet measured output f(t) := u(0, t), t is an element of [0, T]. Compactness and Lipschitz continuity of the input-output operator Phi[p] := u(x, t; p)vertical bar x=0(+), Phi[ . ] : p subset of H-1(0, l) bar right arrow L-2(0, T) are proved. Then an existence of a quasi-solution of the inverse problem is obtained. We prove Frechet differentiability of the Tikhonov functional and derive an explicit gradient formula for the Frechet gradient through the solutions of the direct and corresponding adjoint problems solutions. This allows to use gradient-type algorithms for the numerical solution of the considered inverse problem.

Define the zeroth-order Randic index $$R^0(G)=\sum _{x\in V(G)}\frac{1}{\sqrt{d_G(x)}}$$, where $$d_G(x)$$ denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super-edge-connected in terms of the zeroth-order Randic index, respectively.

Let G be a connected graph with order n, minimum degree δ = δ(G) and edge-connectivity λ = λ(G). A graph G is maximally edge-connected if λ = δ, and super edge-connected if every minimum edgecut consists of edges incident with a vertex of minimum degree. Define the zeroth-order general Randic index $$R_\alpha ^0\left( G \right) = \sum\limits_{x \in V\left( G \right)} {d_G^\alpha \left( x \right)} $$ , where dG(x) denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super edge-connected in terms of the zeroth-order general Randic index for −1 ≤ α < 0, respectively.

In this paper, a calibration method is proposed which eliminates the zeroth order effect in lateral shearing interferometry. An analytical expression of the calibration error function is deduced, and the relationship between the phase-restoration error and calibration error is established. The analytical results show that the phase-restoration error introduced by the calibration error is proportional to the phase shifting error and zeroth order effect. The calibration method is verified using simulations and experiments. The simulation results show that the phase-restoration error is approximately proportional to the phase shift error and zeroth order effect, when the phase shifting error is less than 2° and the zeroth order effect is less than 0.2. The experimental result shows that compared with the conventional method with 9-frame interferograms, the calibration method with 5-frame interferograms achieves nearly the same restoration accuracy.

This paper deals with the asymptotic derivation of thin and thick nonlocal plate models at different orders from three-dimensional stress gradient elasticity, through the power series expansions of the displacements in the thickness ratio of the plate. Three nonlocal asymptotic approaches are considered: a partial nonlocality following the thickness of the plate, a partial nonlocality following the two directions of the plates and a full nonlocality (following all the directions). The three asymptotic approaches lead at the zeroth order to a nonlocal Kirchhoff–Love plate model, but differ in the expression of the length scale. The nonlocal asymptotic models coincide at this order with the stress gradient Kirchhoff–Love plate model, only when the nonlocality is following the two directions of the plate and expressed through a nabla operator. This asymptotic model also yields the nonlocal truncated Uflyand–Mindlin plate model at the second order. However, the two other asymptotic models lead to equations that differ from the current existing nonlocal engineering models (stress gradient engineering plate models). The natural frequencies for an all-edges simply supported plate are obtained for each model. It shows that the models provide similar results for low orders of frequencies or small thickness ratio or nonlocal lengths. Moreover, only the asymptotic model with a partial nonlocality following the two directions of the plates is consistent with a stress gradient plate model, whatever the geometry of the plate.