Maximum Minimum Principle Research Articles

Polarizability, hardness and electrophilicity as global descriptors for intramolecular proton transfer reaction path

Potential energy (PE) curves for intramolecular proton transfer in the ground (GSIPT) and excited (ESIPT) states of 1-hydroxy-2-acetonaphthone (1H2AN) is studied using DFT-B3LYP/6-31+G(d,p) and TD-DFT/6-31+G(d,p) level of theory respectively. Our calculations suggest the non-viability of ground state intramolecular proton transfer in 1H2AN. Excited states PE calculations support the viability of ESIPT process in 1H2AN. Here, for the first time, polarizability, chemical hardness and electrophilicity are being used as global reactivity descriptors to locate the transition state for intermolecular proton transfer process. Both the minimum polarizability principle (MPP) and maximum hardness principle (MHP) are being obeyed along the intrinsic reaction co-ordinate (IRC) for intramolecular proton transfer process. We have also raised the decade old issue i.e. the use of O–H distance of enol tautomer as proton transfer co-ordinate instead of IRC. Our computation of these global reactivity descriptors along the proton transfer co-ordinate support intrinsic reaction coordinate (IRC) as the effective proton transfer co-ordinate, instead of variation of O–H distance of enol tautomer.

Alkylation of guanine by formononetin nitrogen mustard derivatives: A DFT study

We studied the thermodynamic and kinetic aspects of guanine alkylation by five derivatives of formononetin nitrogen mustard using density functional theory (DFT). Investigation of the reactivity pattern of the aziridinium ion intermediates as well as drug-guanine adducts were performed using Density functional reactivity theory (DFRT). Aziridinium ion formation was observed to be endothermic, in contrast, the drug-guanine adduct formation was exothermic. A significant interaction of the aziridinium ion with guanine has been observed. The results illustrate the applicability of maximum hardness principle (MHP) and minimum electrophilicity principle (MEP).

The inside–outside duality for scattering problems by inhomogeneous media

This paper investigates the relationship between interior transmission eigenvalues k0 > 0 and the accumulation point 1 of the eigenvalues of the scattering operator when k approaches k0. As is well known, the spectrum of is discrete, the eigenvalues μn(k) lie on the unit circle in and converge to 1 from one side depending on the sign of the contrast. Under certain (implicit) conditions on the contrast it is shown that interior transmission eigenvalues k0 can be characterized by the fact that one eigenvalue of converges to 1 from the opposite side if k tends to k0 from below. The proof uses the Cayley transform, Courant’s maximum–minimum principle, and the factorization of the far field operator. For constant contrasts that are positive and large enough or negative and small enough, we show that the conditions necessary to prove this characterization are satisfied at least for the smallest transmission eigenvalue.

Analysis of Three-Dimensional Cutting Process With Thin Shear Plane Model

A new and basic analytical model of three-dimensional cutting is proposed by assuming multiple thin shear planes with either the maximum shear stress or minimum energy principle. The three-dimensional cutting process with an arbitrarily shaped cutting edge in a flat rake face is formulated with simple vector equations in order to understand and quickly simulate the process. The cutting edge and workpiece profile are discretized and expressed by their position vectors. Two equations among three unknown vectors, which show the directions of shear, chip flow, and resultant cutting force, are derived from the geometric relations of velocities and forces. The last vector equation required to solve the three unknown vectors is obtained by applying either the maximum shear stress or minimum energy principle. It is confirmed that the directions and the cutting forces simulated by solving the proposed vector equations agree with experimentally measured data. Furthermore, the developed model is applied to consider the three-dimensional cutting mechanics, i.e., how the chip is formed in the three-dimensional cutting with compressive stress acting between the discrete chips, as an example.

S. Bernstein’s idea for bounding the gradient of solutions to the quasi-linear Dirichlet problem

S. Bernstein’s idea is outlined to estimate \({\| u_{x} \|_{0,0,\Omega}}\) of a solution u(x) to the quasi-linear elliptic differential equation, especially in the case of n > 2. Up to about 1956, investigations of nonlinear problems have been dealing with the case of n = 2. A large number of methods were developed, but none of them were applicable for n > 2. The maximum-minimum principle had been a powerful tool to find bounds, however in the case of n > 2 this tool is not available for the gradient of the solution. In 1956, Cordes [Math. Ann. 130 (1956), 278–312] gave estimates for the case n > 2. Later, Ladyzhenskaya and Ural’tseva [Linear and Quasilinear Elliptic Equations. Academic Press, New York, 1968] established estimates in the Sobolev space \({W^{2}_{2} (\Omega)}\) . In their line of reasoning they used the idea of S. Bernstein and investigated the transformed function υ(x) with u(x) = ϕ(υ(x)). In this paper, we give a more simpler proof of those results in the classical Banach space C2,α(Ω).

Possibility of Having HF-Doped Hydrogen Hydrates

A multiscale computational work is assessed to envisage the effect of HF doping on the stability, structure, and reactivity of clathrate hydrates. Interaction of molecular hydrogen with HF-doped clathrate hydrates is studied through ab initio molecular dynamics simulation with the aid of conceptual density functional theory. The viability of hydrogen encapsulation in HF-doped clathrate hydrates is justified in terms of the associated interaction energy, reaction enthalpy, hardness, and electrophilicity values. Calculated thermodynamic parameters are well-correlated with the maximum hardness principle and minimum electrophilicity principle for all cases. It is observed that HF doping significantly increases the stability of the studied clathrate hydrates and it also enhances the binding energy and reaction enthalpy for the encapsulation of hydrogen molecules.

On the Validity of the Maximum Hardness Principle and the Minimum Electrophilicity Principle during Chemical Reactions

Hardness and electrophilicity values for several molecules involved in different chemical reactions are calculated at various levels of theory and by using different basis sets. Effects of these aspects as well as different approximations to the calculation of those values vis-à-vis the validity of the maximum hardness and minimum electrophilicity principles are analyzed in the cases of some representative reactions. Among 101 studied exothermic reactions, 61.4% and 69.3% of the reactions are found to obey the maximum hardness and minimum electrophilicity principles, respectively, when hardness of products and reactants is expressed in terms of their geometric means. However, when we use arithmetic mean, the percentage reduces to some extent. When we express the hardness in terms of scaled hardness, the percentage obeying maximum hardness principle improves. We have observed that maximum hardness principle is more likely to fail in the cases of very hard species like F(-), H(2), CH(4), N(2), and OH appearing in the reactant side and in most cases of the association reactions. Most of the association reactions obey the minimum electrophilicity principle nicely. The best results (69.3%) for the maximum hardness and minimum electrophilicity principles reject the 50% null hypothesis at the 2% level of significance.

An Improved Spectral Clustering Algorithm Using Minimum Maximum Principle

In this paper a novel document clustering spectral algorithm is proposed, which uses a minimum maximum principle. Firstly the low dimensional embedding of documents is attained by eigenvalue decomposition, and then a minimum maximum principle is used to get the initial seeds for k-means algorithm. Finally, K-means algorithm is performed to get the clustering results. Experimental results show that the clustering results found by this method is better than traditional clustering algorithm.

Structure-stability diagrams and stability-reactivity landscapes: a conceptual DFT study

Electrophilicity and hardness have been shown to be adequate in constructing structure-stability diagrams. Maximum hardness principle and minimum electrophilicity principle provide a rough guide toward locating the domains of stability and reactivity in a fitness landscape. Bonding in solids, aromaticity, magic alkali clusters, bond—stretch isomers, multivalent superatoms, etc. have been analyzed within this purview.

Effect of external electric field on aziridinium ion intermediate: A DFT study

We have analyzed the effect of external electric field on the aziridinium ion intermediate of mustine drug molecule using conceptual density functional theory based reactivity descriptors. The aziridinium ion intermediate is formed during the alkylation of DNA by mustine, a member of the nitrogen mustard family. This species experiences a field exerted by polarity of the solvent and metal ions present in body fluids (in extra- and intra-cellular fluids, blood etc.). The stability and reactivity of the aziridinium ion is monitored by studying some density based reactivity descriptors in the presence of external electric fields. Further, shifting of the reactive center (i.e., the LUMO) of the drug intermediate with variation in the electric field was observed. In addition, the maximum hardness and minimum electrophilicity principles were analyzed.

Does structural variation in the aziridinium ion facilitate alkylation?

The reactivity and stability of the tricyclic aziridinium ion intermediate of the mustine drug molecule varies with the ∠ NCC bond angle (from 60° to 150°) of the tricyclic ring. As ∠ NCC bond angle increases, the tricyclic ring of the aziridinium ion opens up which leads to variation in its reactivity. We have observed shifting of the reactive center (i.e., the LUMO) of the drug intermediate with variation in the ∠ NCC bond angle in gas phase as well as in aqueous phase. It was also observed that the drug intermediate must undergo some structural changes before alkylating DNA. In addition, the maximum hardness principle and minimum electrophilicity principles were analyzed.

Prediction of lysine ubiquitination with mRMR feature selection and analysis

Ubiquitination, one of the most important post-translational modifications of proteins, occurs when ubiquitin (a small 76-amino acid protein) is attached to lysine on a target protein. It often commits the labeled protein to degradation and plays important roles in regulating many cellular processes implicated in a variety of diseases. Since ubiquitination is rapid and reversible, it is time-consuming and labor-intensive to identify ubiquitination sites using conventional experimental approaches. To efficiently discover lysine-ubiquitination sites, a sequence-based predictor of ubiquitination site was developed based on nearest neighbor algorithm. We used the maximum relevance and minimum redundancy principle to identify the key features and the incremental feature selection procedure to optimize the prediction engine. PSSM conservation scores, amino acid factors and disorder scores of the surrounding sequence formed the optimized 456 features. The Mathew's correlation coefficient (MCC) of our ubiquitination site predictor achieved 0.142 by jackknife cross-validation test on a large benchmark dataset. In independent test, the MCC of our method was 0.139, higher than the existing ubiquitination site predictor UbiPred and UbPred. The MCCs of UbiPred and UbPred on the same test set were 0.135 and 0.117, respectively. Our analysis shows that the conservation of amino acids at and around lysine plays an important role in ubiquitination site prediction. What's more, disorder and ubiquitination have a strong relevance. These findings might provide useful insights for studying the mechanisms of ubiquitination and modulating the ubiquitination pathway, potentially leading to potential therapeutic strategies in the future.

Edge-based compression of cartoon-like images with homogeneous diffusion

Edges provide semantically important image features. In this paper a lossy compression method for cartoon-like images is presented, which is based on edge information. Edges together with some adjacent grey/colour values are extracted and encoded using a classical edge detector, binary compression standards such as JBIG and state-of-the-art encoders such as PAQ. When decoding, information outside these encoded data is recovered by solving the Laplace equation, i.e. we inpaint with the steady state of a homogeneous diffusion process. For the discrete reconstruction problem, we prove existence and uniqueness and establish a maximum–minimum principle. Furthermore, we describe an efficient multigrid algorithm. The result is a simple codec that is able to encode and decode in real time. We show that for cartoon-like images this codec can outperform the JPEG standard and even its more advanced successor JPEG2000.

Computing short-time aircraft maneuvers using direct methods

This paper analyzes the applicability of direct methods to design optimal short-term spatial maneuvers for an unmanned vehicle in a faster than real-time scale. It starts by introducing different basic control schemes, which employ online trajectory generation. Next, it presents and analyzes the results obtained through two recently developed direct transcription (collocation) methods: the Gauss pseudospec-tral method and the Legendre-Gauss-Lobatto pseudosp ectral method. The achieved results are further compared with those found through the Pontryagin’s Maximum (Minimum) Principle, and the paper continues by providing another set of direct method simulations incorporating more realistic boundary conditions. Finally, the results obtained using the third direct method, based on inverse dynamics in the virtual domain, are presented and discussed.

Is an elementary reaction step really elementary? Theoretical decomposition of asynchronous concerted mechanisms

In this paper the primitive process concept is used to reproduce and explain the potential energy profile of an asynchronous concerted mechanism. The number of primitive processes constituting such a mechanism can be deduced from the study of the influence of the number of electrons on the potential energy profile of the elementary step. The study of a particular example indicates that the position of the transition states of the primitive processes constituting an asynchronous concerted mechanism seems not to be affected by the number of electrons in the system. This implies that the maximum hardness and minimum electrophilicity principles are valid for primitive processes but not for asynchronous concerted mechanisms. It appears that a potential energy profile with a shoulder before the transition state, and a reaction force profile with more than two extrema together with a hardness profile presenting a minimum shifted with respect to the transition state are thus indicators of the presence of an asynchronous concerted mechanism.

Mathematical Methods for Images and Surfaces

The last two decades have witnessed a greatly increasing interest in mathematical methods for images and surfaces that are originated from biomedical and biological applications. The research problems have necessitated the development of mathematical theories of wavelets, frames, harmonic analysis, geometric flows, differential geometry, topology statistical methods, machine learning, and so forth. This theoretical development is enhanced by the related numerical algorithm development and modeling development. Currently, mean curvature flow, Willmore flow, level set, generalized Laplace-Beltrami operator are commonly used mathematical techniques for the analysis of biomedical images and for the generation of biomolecular surfaces. Additionally, wavelets, frames, compressive sensing, and harmonic analysis are popular tools for biomedical visualization and image processing. Moreover, differential geometry, topology, and geometric measure theory are powerful approaches for the multiscale modeling of biomolecular structure, dynamics, and transport. Finally, persistently stable manifold, topological invariant, Euler characteristic, Frenet frame, and machine learning are vital to the dimensionality reduction of extremely massive biomedical and biomolecular data. These ideas have been successfully paired with current investigation and discovery in biomedical and biological systems. However, many mathematical challenges remain in image and surface analysis, such as the well-posedness of mathematical models under physical and biological constraints, maximum-minimum principle for high-order geometric evolution equations, numerical analysis of multiply coupled partial differential equations, effectiveness of approximation theory, and the modeling of complex biomolecular phenomena. This special issue was called to address mathematical difficulties and challenges in image and surface analysis. It consists of seventeen research papers.

Properties of Higher Order Nonlinear Diffusion Filtering

This paper provides a mathematical analysis of higher order variational methods and nonlinear diffusion filtering for image denoising. Besides the average grey value, it is shown that higher order diffusion filters preserve higher moments of the initial data. While a maximum-minimum principle in general does not hold for higher order filters, we derive stability in the 2-norm in the continuous and discrete setting. Considering the filters in terms of forward and backward diffusion, one can explain how not only the preservation, but also the enhancement of certain features in the given data is possible. Numerical results show the improved denoising capabilities of higher order filtering compared to the classical methods.

Non-negative mixed finite element formulations for a tensorial diffusion equation

We consider the tensorial diffusion equation, and address the discrete maximum–minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum–minimum principle) of mixed finite element formulations. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Raviart–Thomas, variational multiscale, and Galerkin/least-squares mixed formulations) do not produce non-negative solutions (that is, they do not satisfy the discrete maximum–minimum principle) on arbitrary meshes and for strongly anisotropic diffusivity coefficients. In this paper, we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques. These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart–Thomas spaces, and the second non-negative formulation is based on the variational multiscale formulation. For the former formulation we comment on the effect of adding the non-negative constraint on the local mass balance property of the Raviart–Thomas formulation. We perform numerical convergence analysis of the proposed optimization-based non-negative mixed formulations. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.

A direct relativistic four-component multiconfiguration self-consistent-field method for molecules

A new direct relativistic four-component Kramers-restricted multiconfiguration self-consistent-field (KR-MCSCF) code for molecules has been implemented. The program is based upon Kramers-paired spinors and a full implementation of the binary double groups (D(2h)(*) and subgroups). The underlying quaternion algebra for one-electron operators was extended to treat two-electron integrals and density matrices in an efficient and nonredundant way. The iterative procedure is direct with respect to both configurational and spinor variational parameters; this permits the use of large configuration expansions and many basis functions. The relativistic minimum-maximum principle is implemented in a second-order restricted-step optimization algorithm, which provides sharp and well-controlled convergence. This paper focuses on the necessary modifications of nonrelativistic MCSCF methodology to obtain a fully variational KR-MCSCF implementation. The general implementation also allows for the use of molecular integrals from a two-component relativistic Hamiltonian as, for example, the Douglas-Kroll-Hess variants. Several sample applications concern the determination of spectroscopic properties of heavy-element atoms and molecules, demonstrating the influence of spin-orbit coupling in MCSCF approaches to such systems and showing the potential of the new method.

Continuous and discrete parabolic operators and their qualitative properties

The basic requirement of numerical methods is convergence. However, from the practical point of view, it is generally not sufficient to construct convergent numerical methods for the solutions of partial differential equations. The qualitative adequateness of the methods is also an issue. The numerical solutions should mirror the characteristic properties of the original physical process that is modelled by the differential equation. In this paper, we give three important qualitative properties of parabolic partial differential equations: the maximum-minimum principle and its different versions, the non-negativity preservation and the maximum norm contractivity. The investigation of these properties is motivated by different physical principles. We formulate the analogues of the properties for general discrete operators and we analyse the conditions and the relations between the properties for both the continuous and the discrete operators. The approximation properties of the discrete operators are also analysed. The results of the paper are applied to the finite-difference solution methods of parabolic initial boundary-value problems.