Presence Of Critical Points Research Articles

An efficient and accurate numerical method for the Bessel transform with an irregular oscillator and its error analysis

In this paper we present and analyze efficient numerical schemes for computing the Bessel transforms with an irregular oscillator. Especially in the presence of critical points, e.g., endpoints, zeros and stationary points for the general oscillator g(t), we derive a series of new quadrature formulae for such transforms and carry out rigorous analysis for the proposed numerical methods. The error analysis demonstrates that this methods exhibit high asymptotic order, and the accuracy improves drastically with either increasing the frequency ω or adding more nodes. Compared with the existing modified Filon-type method, the established approaches show higher precision and order of the error estimate at the same computational cost. The extensive numerical examples are provided to verify the theoretical results and illustrate the efficiency and accuracy of the proposed method.

ENO and WENO schemes using arc-length based smoothness measurement

A novel procedure is given for choosing smoothest stencil to construct less oscillatory ENO schemes. The procedure is further used to define the smoothness parameter in the non-linear weights of new WENO schemes. The main significant features of these new ENO and WENO schemes are that they are less oscillatory and achieve their relevant order of accuracy in the presence of critical points in the exact solution. It is shown theoretically as well as computationally in L1 and L∞ norm. Moreover, computational results are given to show less oscillatory behaviour of the new WENO scheme compared to WENO5-JS and WENO5-Z schemes.

Asymptotic Analysis and Numerical Methods for Oscillatory Infinite Generalized Bessel Transforms with an Irregular Oscillator

In this work, we perform a complete asymptotic analysis and the construction of affordable quadrature rules for a class of oscillatory infinite Bessel transform with a general oscillator. Especially in the presence of critical points, e.g., endpoints, zeros and stationary points, we first derive a series of useful asymptotic expansions in inverse powers of the frequency parameter $$\omega $$. The resulting asymptotic expansions clarify the large $$\omega $$ behavior of the transform and provide powerful tools for designing quadrature rules and conducting error analysis. As a consequence, efficient and affordable new modified Filon-type methods for computing the transform numerically are proposed. Particularly, we carry out the rigorous error analysis and obtain asymptotic error estimates in inverse powers of $$\omega $$. Numerical examples can confirm our analysis. The accuracy can be improved greatly by either adding more derivatives interpolation at endpoints or adding more interior nodes. Moreover, only using a small number of nodes and multiplicities, we can obtain the required accuracy level. For fixed number of nodes and multiplicities, the higher accuracy can be achieved with the larger values of $$\omega $$.

Converging photospheric vortex flows close to the polarity inversion line of a fully emerged active region

Abstract. We report on the occurrence of vortexes in flow fields obtained from the evolution of the line-of-sight component of the photospheric magnetic field in a region around the polarity inversion line (PIL) of a fully emerged active region. Based on a local linear approximation for the flow field, we identify the presence of critical points and classify them according to the eigenvalues of the Jacobian matrix of the linear transformation. Converging vortexes are associated with the presence of a particular kind of critical point, known as the attracting focus. We identified 12 converging vortexes in the analyzed period and detected the occurrence of other types of critical points, which indicate the complexity of the flow field around the PIL. The detected vortexes show a clockwise preferred sense of rotation with approximately 67 % of the cases. A geometrical analysis of the velocity structures produced an average value of D‾=1.63±0.05 for the fractal dimension, which is very close to the one obtained for isotropic homogeneous turbulence (D=5/3). This suggests that the flow around the PIL is turbulent in nature.

An improved non-linear weights for seventh-order weighted essentially non-oscillatory scheme

In this article, the construction and implementation of a seventh order weighted essentially non-oscillatory scheme is reported for solving hyperbolic conservation laws. Local smoothness indicators are constructed based on L1-norm, where a higher order interpolation polynomial is used with each derivative being approximated to the fourth order of accuracy with respect to the evaluation point. The global smoothness indicator so constructed ensures the scheme achieves the desired order of accuracy. The scheme is reviewed in the presence of critical points and verified the numerical accuracy, convergence with the help of linear scalar test cases. Further, the scheme is implemented to non-linear scalar and system of equations in one and two dimensions. As the formulation is based on method of lines, to move forward in time linear strong-stability-preserving Runge–Kutta scheme (lSSPRK) for the linear problems and the fourth order nonlinear version of five stage strong stability preserving Runge–Kutta scheme (SSPRK(5, 4)) for nonlinear problems is used.

New Stability Estimates for the Inverse Medium Problem with Internal Data

A major problem in solving multi-waves inverse problems is the presence of critical points where the collected data completely vanishes. The set of these critical points depend on the choice of the boundary conditions, and can be directly determined from the data itself. To our knowledge, in the most existing stability results, the boundary conditions are assumed to be close to a set of CGO solutions where the critical points can be avoided. We establish in the present work new weighted stability estimates for an electro-acoustic inverse problem without assumptions on the presence of critical points. These results show that the Lipschitz stability far from the critical points deteriorates near these points to a logarithmic stability.

A continuation algorithm for planar implicit curves with singularities

Continuation algorithms usually behave badly near to critical points of implicitly defined curves in R2, i.e., points at which at least one of the partial derivatives vanishes. Critical points include turning points, self-intersections, and isolated points. Another problem with this family of algorithms is their inability to render curves with multiple components because that requires finding first a seed point on each of them. This paper details an algorithm that resolves these two major problems in an elegant manner. In fact, it allows us not only to march along a curve even in the presence of critical points, but also to detect and render curves with multiple components using the theory of critical points.

Adsorption properties of CdS-CdTe system semiconductors

The adsorption of carbon(II) oxide and ammonia on nanofilms of solid solutions and binary compounds of the CdS-CdTe system is studied by means of piezoquartz microweighing, FTIR IR, and measuring electroconductivity. Allowing for the conditions and composition of semiconductor systems, we determine the mechanisms and principles of adsorption processes by analyzing the α p = f(T), α T = f(p), and α T = f(t) experimental dependences; IR spectra; the thermodynamic and kinetic characteristics of adsorption; the acid-base, electrophysical, and other characteristics of adsorbents; the electron nature of adsorbate molecules; and the obtained acid-base characteristics: the composition and adsorption characteristics and composition state diagrams. Previous statements on the nature and retention of local active centers responsible for adsorption and catalytic processes upon changes in their habitus and composition (as components of systems of the AIIIBV-AIIBVI and AIIBVI-AIIBVI types) on the surface of diamond-like semiconductors are confirmed. Specific features of the behavior of (CdS) x (CdTe)1 − x solid solutions are identified in addition to general features with binary compounds (CdS, CdTe), as is demonstrated by the presence of critical points on acid-base characteristics-composition and adsorption characteristics-composition diagrams. On the basis of these diagrams, the most active adsorbents (with respect to CO and NH3) used in designing highly sensitive and selective sensors are identified.

Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes

In the reconstruction step of (2r-1) order weighted essentially non-oscillatory conservative finite difference schemes (WENO) for solving hyperbolic conservation laws, nonlinear weights αk and ωk, such as the WENO-JS weights by Jiang et al. and the WENO-Z weights by Borges et al., are designed to recover the formal (2r-1) order (optimal order) of the upwinded central finite difference scheme when the solution is sufficiently smooth. The smoothness of the solution is determined by the lower order local smoothness indicators βk in each substencil. These nonlinear weight formulations share two important free parameters in common: the power p, which controls the amount of numerical dissipation, and the sensitivity ε, which is added to βk to avoid a division by zero in the denominator of αk. However, ε also plays a role affecting the order of accuracy of WENO schemes, especially in the presence of critical points. It was recently shown that, for any design order (2r-1), ε should be of Ω(Δx2) (Ω(Δxm) means that ε⩾CΔxm for some C independent of Δx, as Δx→0) for the WENO-JS scheme to achieve the optimal order, regardless of critical points. In this paper, we derive an alternative proof of the sufficient condition using special properties of βk. Moreover, it is unknown if the WENO-Z scheme should obey the same condition on ε. Here, using same special properties of βk, we prove that in fact the optimal order of the WENO-Z scheme can be guaranteed with a much weaker condition ε=Ω(Δxm), where m(r,p)⩾2 is the optimal sensitivity order, regardless of critical points. Both theoretical results are confirmed numerically on smooth functions with arbitrary order of critical points. This is a highly desirable feature, as illustrated with the Lax problem and the Mach 3 shock-density wave interaction of one dimensional Euler equations, for a smaller ε allows a better essentially non-oscillatory shock capturing as it does not over-dominate over the size of βk. We also show that numerical oscillations can be further attenuated by increasing the power parameter 2⩽p⩽r-1, at the cost of increased numerical dissipation. Compact formulas ofβk for WENO schemes are also presented.

Critical points and points of the bifurcation of the rotating magnetized newtonian polytropes with a 1 ⩽ n ⩽ 1.6 index

The presence of critical points and the bifurcation points of rotating Newtonian polytropes with an index of 1 ⩽ n ⩽ 1.6 has been shown for the first time in this paper. The symbolic-numerical calculation error in metric L2 has reached the size of the 10−4 order. The approximate analytical solution to the problem to the abovementioned accuracy has been set forth. A critical value of the polytropes index n = nk = 1.51025 has been calculated which is the highest among the critical points and bifurcation points. The value nk corresponds to the infinitely slow polytropes rotation. Furthermore, in this paper the presence of the period jump at the bifurcation point Tb has been predicted and the relative value of this jump ΔTb/Tb ∼ B0in4/3 has been estimated.

Adsorption of gases on binary and multicomponent semiconductors of the ZnSe-CdTe system

Adsorption of carbon monoxide(II) and oxygen on powders and nanofilms of solid solutions and binary compounds of the ZnSe-CdTe system was studied volumetrically, and by piezoquartz microweighing and IR spectroscopy of multiple disturbed complete internal reflections. The mechanisms and principles of adsorption were established in dependence on the conditions of the habitus of an experimental sample and the composition of the system’s semiconductors, based on an analysis of IR spectra; the thermodynamic and kinetic characteristics of adsorption; experimental dependences αp = f(T), αT = f(P), and αT = f(t); and the acid-base and other physicochemical characteristics of adsorbents and the electron nature of adsorbate molecules. Conclusions drawn earlier as to the retention of local active centers on the surface of a diamond-like semiconductor (which are responsible for adsorption and catalytic processes) upon a change in the habitus of a sample were confirmed. Certain features in the behavior of solid solutions (ZnSe)x(CdTe)1 − x were revealed alongside commonalities with binary compounds (ZnSe, CdTe), testifying to the presence of critical points on “adsorption characteristics-composition” diagrams. The most active adsorbents (with respect to CO and O2) were discovered on the basis of these diagrams, which were used in creating highly sensitive and selective sensors.

Anisotropic dynamical spin-density response in quantum wells with spin-orbit interaction

The electric field-induced spin polarization of a two-dimensional electron gas with spin-orbit interaction is investigated in the frequency domain. We calculate the spin-polarizability tensor using the linear-response theory, taking into account the presence of both Rashba and Dresselhaus spin-orbit couplings. We consider quantum wells grown in the main crystallographic directions and concentrate on the high-frequency response in the collisionless regime. For an anisotropic spin splitting, it is shown that the spin-density response becomes dependent on the direction of the applied electric field and presents characteristic spectral features, in notable contrast to the case of an isotropic spin-orbit coupling. Such behavior is explained in terms of the nonisotropic momentum space available for electric-dipole transitions and the presence of critical points. This anisotropic dynamic response suggests new possibilities of spin manipulation which could find spintronic applications.

Analysis of Hydrogen Tunneling in an Enzyme Active Site using von Neumann Measurements.

We build on our earlier quantum wavepacket study of hydrogen transfer in the biological enzyme, soybean lipoxygenase-1, by using von Neumann quantum measurement theory to gain qualitative insights into the transfer event. We treat the enzyme active site as a measurement device which acts on the tunneling hydrogen nucleus via the potential it exerts at each configuration. A series of changing active site geometries during the tunneling process effects a sequential projection of the initial, reactant state onto the final, product state. We study this process using several different kinds of von Neumann measurements and show how a discrete sequence of such measurements not only progressively increases the projection of the hydrogen nuclear wavepacket onto the product side but also favors proton over deuteron transfer. Several qualitative features of the hydrogen tunneling problem found in wavepacket dynamics studies are also recovered here. These include the shift in the "transition state" towards the reactant as a result of nuclear quantization, greater participation of excited states in the case of deuterium, and presence of critical points along the reaction coordinate that facilitate hydrogen and deuterium transfer and coincide with surface crossings. To further "tailor" the dynamics, we construct a perturbation to the sequence of measurements, that is a perturbation to the dynamical sequence of active site geometry evolution, which leads us to insight on the existence of sensitive regions of the reaction profile where subtle changes to the dynamics of the active site can have an effect on the hydrogen and deuterium transfer process.

Global bifurcations at the onset of pulse self-replication

In this work, we carried out an extensive numerical exploration of the delicate global dynamics of pulse self-replication and analyzed the stability of singular homoclinic stationary solutions and their bifurcations in the one-dimensional Gray-Scott model. This stability analysis has several implications for understanding the recently discovered phenomena of self-replicating pulses. The solutions of the ordinary differential equation are organized around a codimension-2 global bifurcation from which two or N branches of homoclinic orbits originate, corresponding to solitary pulse solutions in the partial differential equation. A careful analysis of the bifurcation scenarios in the global bifurcation diagram suggests that the dynamics of the self-replicating system are related to a hierarchy structure of folding bifurcation branches in parameter space. The numerical simulation suggests that the Bogdanov-Takens points, together with the presence of critical points emanating from a particular codimension-2 homoclinic orbit, play a central role for the global bifurcation of periodic orbits, the homoclinic solutions, and the complex chaotic dynamics. The numerical simulation also reveals the existence of a modulating two-pulse or multipulse, which accompanies the procedure of pulse self-replication in reaction-diffusion systems.

Minimal rearrangements, strict convexity and critical points

An asymmetry estimate for extremal functions in Pólya–Szegö type inequalities for Dirichlet integrals is established in presence of critical points and nonstrictly convex integrands.

Arc-length procedures with BEM in physically nonlinear problems

Geometrically or physically nonlinear problems are often characterised by the presence of critical points with snapping behaviour in the structural response. These structural or material instabilities usually lead to inefficiency of standard numerical solution techniques. Special numerical procedures are, therefore, required to pass critical points. The authors mean to present an arc-length procedure combined with the Boundary Element Method (BEM). The arc-length methods are intended to enable solution algorithms to pass limit points. Particularly for snap-back behaviour, the arc-length methods are the only procedures, which enable to follow the equilibrium path. The interest is mainly devoted to softening models where snap-backs and snap-throughs usually occur. No BEM applications have been possible so far due to the lack of a procedure enforcing the arc-length constraint. This paper intends to overcome such a difficulty. The analysis beyond the collapse point is mainly justified by two facts: (1) the investigation concerns a structural component and, therefore, it may be desirable to incorporate the load/deflection response of this component within a further analysis of the complete structure; (2) it may be important to know not just the collapse load but whether or not this collapse is of a ductile or brittle form. The procedure can be easily applied both to plasticity and damage, but numerical results will be presented for 2D elastoplasticity. Accurate results will be obtained in the case of both hardening and softening plasticity. The procedure can be used in BEM applications on nonlocal continuum models of the integral type and it can be easily extended to elasto-plasto-dynamics and to buckling in elastoplasticity.

Experimental Electron Density of 1-Phenyl-4-nitroimidazole and Topological Analysis of C−H···O and C−H···N Hydrogen Bonds

The electron density of 1-phenyl-4-nitroimidazole was analyzed by X-ray diffraction at 0.55 A resolution. The topology of the bonding scheme within the molecule as well as of the weak intermolecular C-H...O and C-H...N hydrogen bonds has been investigated. The topological analysis and the deformation density peak heights on the covalent bonds of the imidazole ring are consistent with the usual chemical knowledge. The nitro group has an electron-withdrawing effect, as it is globally negatively charged and the electrostatic potential around it is slightly negative. This potential is significantly weaker than that generated by other oxygen-containing chemical entities, which is related to the hydrophobicity of the nitro group. The nitro group forms three weak C-H...O hydrogen bonds, and the unsubstituted imidazole nitrogen is also hydrogen-bonded to a C-H group, as confirmed by the presence of critical points in the topological analysis.

Statistical properties of equilibrium states for rational maps

Equilibrium states of rational maps for Hölder continuous potentials are not $\phi$-mixing, mainly due to the presence of critical points. Here we prove that for disks the normalized return times of arbitrary orders are, in the limit, Poisson distributed as the radius of the disks go to zero. The return times are normalized by the measure of the disks. We also show that rational maps are weakly Bernoulli with respect to the partition given by Denker and Urbanski.

Energy spectrum of (Sn0.65Pb0.35)0.95Ge0.05Te solid solutions

The paper reports the study on the resistivity ρ and thermoemf S of the (Sn0.65Pb0.35)0.95Ge0.05Te solid solution layers. The dependences of ρ and S on the hole concentrations in the range 3×1019–2×1021 cm−3 exhibit jumps in the resistivity and thermoemf minima at close hole concentrations p1≈9×1019 cm−3, p2≈2.5×1020 cm−3, and p3≈4.5×1020 cm−3. The observed jumps and minima suggest a complex structure of the valence band and the presence of critical points in the energy spectrum of holes. According to the data for SnTe, the critical points in the energy spectrum at the given hole concentrations are identified as the Σ-extremum, saddle point LΣ, and Δ-extremum, respectively.

Enhanced solution control for physically and geometrically non-linear problems. Part I?the subplane control approach

SUMMARY Geometrically or physically non-linear problems are often characterized by the presence of critical points with snapping behaviour in the structural response. These structural or material instabilities usually lead to ineciency of standard numerical solution techniques. Special numerical procedures are therefore required to pass critical points. This paper presents a solution technique which is based on a constraint equation that is dened on a subplane of the degrees-of-freedom (dof’s) hyperspace or a hyperspace constructed from specic functions of the degrees-of-freedom. This unied approach includes many existing methods which have been proposed by various authors. The entire computational process is driven from only one control function which is either a function of a number of degrees-of-freedom (local subplane method) or a single automatically weighted function that incorporates all dof’s directly or indirectly (weighted subplane method). The control function is generally computed in many points of the structure, which can be related to the nite element discretization. Each point corresponds to one subplane. In the local subplane method, the subplane with the control function that drives the load adaptation is selected automatically during the deformation process. Part I of this two-part series of papers fully elaborates the proposed solution strategy, including a fully automatic load control, i.e. load estimation, adaptation and correction. Part II presents a comparative analysis in which several choices for the control function in the subplane method are confronted with classical update algorithms. The comparison is carried out by means of a number of geometrically and physically non-linear examples. General conclusions are drawn with respect to the eciency and applicability of the subplane solution control method for the numerical analysis of engineering problems. Copyright ? 1999 John Wiley & Sons Ltd.