Kinds Of Minima Research Articles

Guiding iterative optimisation methods to a predefined kind of optima for unconstrained optimisation problems

One of the most fundamental issues in the field of mathematical optimisation is the convergence of an iterative optimisation method and by this we are referring to two things. First, will the method find an optimum and second, will this optimum be a local one or a global one? A recently proposed technique (Nikolakakou et al., 2015b) that is used in order to lessen the dependence a locally convergent iterative optimisation method has on the initial guess, is exploited in this paper. A way so that such a method may be guided to a predefined kind of minimum (local or global) is presented. Well known test functions were used for experimentation. Statistical analysis was conducted by applying a logistic regression classification model on data arisen from the numerical results which confirmed that iterative optimisation methods can be guided to a predefined kind of optimum.

Critical Cones for Sufficient Second Order Conditions in PDE Constrained Optimization

In this paper, we analyze optimal control problems governed by semilinear parabolic equations. Box constraints for the controls are imposed and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Unlike finite dimensional optimization or control problems involving Tikhonov regularization, second order sufficient optimality conditions for the control problems we deal with must be imposed in a cone larger than the one used to obtain necessary conditions. Different extensions of this cone have been proposed in the literature for different kinds of minima: strong or weak minimizers for optimal control problems. After a discussion on these extensions, we propose a new extended cone smaller than those considered until now. We prove that a second order condition based on this new cone is sufficient for a strong local minimum.

Minimum of the order parameter fluctuations of seismicity before major earthquakes in Japan

It has been shown that some dynamic features hidden in the time series of complex systems can be uncovered if we analyze them in a time domain called natural time χ. The order parameter of seismicity introduced in this time domain is the variance of χ weighted for normalized energy of each earthquake. Here, we analyze the Japan seismic catalog in natural time from January 1, 1984 to March 11, 2011, the day of the M9 Tohoku earthquake, by considering a sliding natural time window of fixed length comprised of the number of events that would occur in a few months. We find that the fluctuations of the order parameter of seismicity exhibit distinct minima a few months before all of the shallow earthquakes of magnitude 7.6 or larger that occurred during this 27-y period in the Japanese area. Among the minima, the minimum before the M9 Tohoku earthquake was the deepest. It appears that there are two kinds of minima, namely precursory and nonprecursory, to large earthquakes.

Atom-Cage Charge Transfer in Endohedral Metallofullerenes: Trapping Atoms Within a Sphere-Like Ridge of Avoided Crossings.

Endohedral fullerences have great potential for a variety of techological applications. Here we consider B@C60 and show that the amount of charge transfer from the semimetal boron atom to the cage is a strong function of the radial distance of the atom from the center of the fullerene, and it is controlled by multistate conical intersections whose associated ridge of avoided crossings has the topology of a Euclidean sphere. The potential energy surfaces of B@C60 are characterized by two kinds of local minima: those with a boron atom located in the geometric center of the fullerene, and those with a boron atom bound to the fullerene inner wall. At the lowest-energy minimum, at the center, the boron atom is neutral, whereas the transition to the wall is accompanied by an electron transfer from boron to the fullerene cage. The two kinds of minima are separated by a ridge of avoided crossings that forms a surface with a nearly spherical shape. The properties of such systems may be altered by controlling the populations of the two kinds of minima, for example, by application of an external field. Such switchable atom-cage charge transfer may find applications in novel molecular devices.

NH 3D + dominated proton spin-lattice relaxation in partly deuterated ammonium compounds

Proton spin-lattice relaxation is studied in partly deuterated ammonium compounds at low temperatures. A model is proposed for the NH 3D + related contribution, which increases the relaxation rate many times larger than in nondeuterated samples. The model introduces two kinds of level-crossing minima in T 1, where some tunnel frequency is equal to a separation between the proton Zeeman levels in the external magnetic field. One kind of minimum involves a CH 3-type tunnel splitting of NH 3D +, when the deuteron is stationary at anyone of the four threefold axes of the ammonium ion. The corresponding relaxation rate is expected to depend on the experimental pulse sequence. The other kind of minima (there could be six of them) result from the NH 3D + rotations moving the deuteron from one threefold axis to another and back, which make the otherwise motionally independent AA part of the magnetic dipolar interaction of NH 3D + time dependent. The involved tunnel splitting is equal to (2/3) times the difference between the CH 3-type tunnel splittings. In a level-crossing transition the tunnel energy is changed by that splitting, but the resulting energy imbalance is transferred fast to the lattice by spin-state preserving reverse deuteron jumps, removing any coupling to the tunnel energy reservoir. Thereafter another level-crossing transition is possible. Experiments on polycrystalline 4% and 10% deuterated ammonium hexachlorotellurate samples reveal two additional level-crossing minima in T 1 below 20 K at the proton resonance frequencies 25 MHz and about 28 MHz, which are not found in the nondeuterated sample. The minima show different characteristics relative to the experimental pulse sequence and also agree otherwise well with the predictions of the model.

Electronic and Geometrical Structure of the Sc[BO]+ Cation. An Ab Initio Investigation

The Sc[BO]+ molecular cation has been investigated theoretically by multireference CI methods. We have examined the potential energy surfaces of all states that correlate to the ground-state Sc+(3D) + BO(X2Σ+) fragments. Three kinds of minima were discovered corresponding to the linear Sc−BO+ and BOSc+ and to the bent BOSc+ configurations, with the bent structure being the global minimum for all states. Full potential energy curves are reported for all symmetries examined, i.e., 2A‘(3), 2A‘ ‘(2) for the bent and 2Δ, 2Π, 2Σ+, 4Δ, 4Π, and 4Σ+ for the linear geometries. The ground state is of 2A‘ ‘ symmetry with a BO−Sc binding energy De = 63.9 kcal/mol at an equilibrium geometry with B−OSc and BO−Sc bond lengths of 1.260 and 2.046 Å, respectively, and a BOSc angle of 90.7°.

Ab initio study of energy, structure and dynamics of the water–carbon dioxide complex

The supermolecular Moller–Plesset perturbation theory (MPPT) is applied to calculate and analyze selected portions of the potential-energy surface (PES) of the H2O⋯CO2 complex. Two kinds of minima have been found. The global minimum, which corresponds to the T-shaped structure with the C atom bonded to the O atom, and the local minimum for the H-bonded arrangement OCO⋯HOH. The global minimum was estimated to be 920 cm−1 deep at the fourth order of MPPT combined with the extended spdf-quality basis set supplemented with bond functions. At the same level of theory the optimal H-bonded structure is 357 cm−1 higher in energy, and reveals a small 10° departure from the collinear arrangement OCO⋯H–O. Both the T-shaped and H-bonded forms are primarily bound by the electrostatic term, which is twice as large as the dispersion component. One-dimensional sections of the potential-energy surface were subsequently used to calculate vibrational energy levels for the wagging motion of the water moiety in the T-shaped and H-bonded forms. Two-dimensional cuts of the PES along the intermolecular Jacobi coordinates, r and θ, were employed to simulate the dynamics of the stretch–bend coupling close to the minima.

Optical spectroscopy of Pb2+ in NaCl: Validity of a Jahn-Teller (Fukuda) model.

The optical spectroscopy (luminescence and decay-time measurements) of ${\mathrm{Pb}}^{2+}$ in doubly doped NaCl:${M}^{2+}$,${\mathrm{Pb}}^{2+}$ (M=Mg, Ca, Sr, Pb) has been studied. Two emission bands have been observed, one of them appearing at 310 nm in all systems whereas the position of the other band (at about 380 nm) is affected by the additional ion ${M}^{2+}$. These results have been interpreted in terms of a Jahn-Teller model which considers two kinds of minima in the adiabatic potential-energy surface (APES). The role of the ${M}^{2+}$ ion in the ${\mathrm{Pb}}^{2+}$APES is discussed.

Quadratic Jahn-Teller Effect in the Line Shapes of the A-Emission and A-Absorption Bands in KBr: Tl+-Type Phosphors

The adiabatic potential energy surfaces (APES's) of Tl + -type impurities in alkali halides, \(^{3}T_{1u}(\varGamma_{4}^{-})\) and \(^{3}A_{1u}(\varGamma_{1}^{-})\) have been calculated taking account of the spin-orbit interaction and both the linear and quadratic Jahn-Teller effect with the α 1 g and ε g vibrational modes. It has been found that the 3 T 1 u APES's have one kind of minima or the two kinds of minima, depending on whether we assume the linear or both the linear and quadratic Jahn-Teller interactions. A trap level 3 A 1 u lies just below both the minima. The line shapes of the A -emission and A -absorption bands in KBr: Tl + are calculated by using the Franck-Condon and Condon approximations. It has been found that the theoretical line shape of the A -emission band has the asymmetric doublet structure and reproduces the experimental emission bands A T and A X . This shows that the large energy differnce between the A X and A T bands is caused by the quadratic Jahn-Teller terms involving Q 1 (α 1 g ).

Luminescence of the Bi3+ Ion in Compounds LiLnO2 and NaLnO2 ( Ln = Sc , Y , La , Gd , Lu )

The luminescence properties of the Bi3+ ion as an activator in the series of compounds and vary considerably with the host lattice composition. The Stokes shift of the emission varies from 1500 cm−1 in the case of to 12,800 cm−1 in the case of . The large variation of the position of the emission band is ascribed to a Jahn‐Teller effect in the excited state of the Bi3+ ion resulting in different kinds of minima on the 3P APES. The internal pressure on the Bi3+ ion, caused by the differences in the ionic radii of the trivalent cations, can explain the variation in the relative depths of the minima. The Stokes shift of the emission is inversely proportional to the energy difference between the and the excited levels. For small Stokes shifts we observed vibrational structure in the spectra. The vibrational modes involved are ascribed to more or less localized modes of the octahedron.

The A-emission of KBr : Tl

The line shapes and relative efficiencies of A T and A X emission bands of KBr : Tl have been studied over a wide temperature range (10–500 K). The behavior of A T and A X emissions has been described phenomenologically in terms of the coexistence of two different kinds of minima on the adiabatic potential energy surface of the 3T 1u relaxed excited state. The line shape parameters and the A-emission temperature dependence show that T and X minima belong to distortions of different symmetry. A model considering the emission processes from T and X minima separately, in spaces of different distortions, is shown to account well for the experimental results. The emission at very low temperatures (below 50 K) was found to be sensitive to trace impurities.

Electronic spectra of potassium bromide containing thallium II. Emission spectrum excited at 2537 Å

The emission spectrum of KBr: Tl + excited at 2537 Å has been measured in the temperature range 15–296°K. At low temperatures the spectrum consists of a prominent band at 4.01 eV, a much smaller band at 3.40 eV and a very small band at 3.15 eV. The last does not appear to change much with temperature and so could not be measured accurately. The temperature-dependence of the two main bands is complex. Between 60 and 100°K the low-energy band increases sharply in intensity, while the high-energy band decreases correspondingly. Above 110°K the situation reverses and the low-energy band decreases in intensity while the high-energy band grows. Both bands closely approximate symmetric Gaussians. The temperature-dependence of the intensity of these two bands is well-explained qualitatively by the existence of two kinds of minima on the adiabatic potential energy surface for the A state. However, predictions of the temperature-dependence of the two emission bands based on calculated adiabatic potential energy surfaces are not in quantitative agreement with the experimental results. Possible reasons for this are our lack of knowledge of precise values for the parameters which enter into the theory, namely the spin-orbit coupling constant, the exchange integral, and the electron lattice coupling constant. The possible role of the 3A 1u state in emission is discussed briefly.

Two kinds of Jahn-Teller distortions in an octahedral complex with the [formula omitted] electronic states coupled to Eg vibrational modes

The adiabatic potential-energy surfaces (APES's) of the a 1 g t 1 u electronic configuration of an octahedral complex interacting with doubly degenerate E g modes have been calculated assuming intermediate coupling. The possibility of coexistence of two different kinds of minima (tetragonal and rhombic) on the 3T∗ 1u APES's is investigated by varying the parameters of the system. It is shown that two kinds of minima can coexist in a wide range of parameters values. Finally, it is suggested that these results may explain the double luminescence emission of KI: Tl-type phosphors excited in the A absorption band.

Intermolecular Forces and Rotational Phase Transition in the C2H2 Crystal

Abstract The potential energy of the acetylene crystal was calculated as a function of the rotational molecular orientations. The potential energy of this crystal has two kinds of minima, not a double minima; one arises from the quadrupole-quadrupole interaction, to which the high-temperature is due, while the other arises from the overlap repulsion interaction with which the low temperature structure is coincident. It was suggested that the mechanism of the phase transition in this crystal is a rotational transition between the two molecular orientations.

Jahn-Teller Effect on the Structure of the Emission Produced by Excitation in theABand of KI: Tl-Type Phosphors. Two Kinds of Minima on theΓ4−(T1u3)Adiabatic Potential-Energy Surface

Emission spectra, excitation spectra, and decay times of ${\mathrm{Ga}}^{+}$, ${\mathrm{In}}^{+}$, and ${\mathrm{Ti}}^{+}$ in KI, KBr, KCl, and NaCl have been investigated at 1.8-300\ifmmode^\circ\else\textdegree\fi{}K under excitation in the $A$-absorption band. There are three types of temperature variation of the emission spectra; these types are represented by KI:Tl, KBr:Tl, and KCl:Tl, respectively. In case of types (1) and (2), two emission bands, ${A}_{T}$ (high energy) and ${A}_{X}$ (low energy), are observed. The excitation spectrum for the ${A}_{T}$ emission is slightly different from that for the ${A}_{X}$ emission. The decay times of the ${A}_{X}$ emission are complex and temperature dependent, as exemplified in KI:Ga and KI:In. The ${A}_{T}$ emission is surely due to the inverse process of the $A$ absorption, ${{\ensuremath{\Gamma}}_{4}}^{\ensuremath{-}}(^{3}T_{1u})\ensuremath{\rightarrow}{{\ensuremath{\Gamma}}_{1}}^{+}(^{1}A_{1g})$. Four reasons are given why the ${A}_{X}$ emission is not due to the inverse process of the $D$ absorption as proposed by Illingworth and supported by Donahue and Teegarden. To explain the ${A}_{X}$ emission, the Jahn-Teller effect on the $^{3}T_{1u}$ state has been considered by assuming Russell-Saunders coupling. Since the correlation of polarization indicates the existence of tetragonal minima, adiabatic potential-energy surfaces (APES's) in the ${E}_{g}({Q}_{2}, {Q}_{3})$ configuration coordinate space have been obtained for the ${{\ensuremath{\Gamma}}_{4}}^{\ensuremath{-}}(^{3}T_{1u})$ and ${{\ensuremath{\Gamma}}_{1}}^{\ensuremath{-}}(^{3}T_{1u})$ states. When the spin-orbit interaction is of suitable value as compared with the Jahn-Teller (electron-lattice) interaction, the obtained APES's have two kinds of minima. Emission is considered to occur from these minima to the ground state, at least at low temperatures. The APES's can explain why there exist the three types of the emission spectra as well as the characteristic features of the emission spectra, excitation spectra, decay times, and polarization. The importance of these APES's for analyzing the stress, Stark, and Zeeman effects on the ${A}_{T}$ and ${A}_{X}$ emission is pointed out.