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Abstract: Abstract In this paper, we introduce a Python library called mad-GP that enables users to more easily explore the design space of Gaussian process (GP) surrogate models for modeling potential energy surfaces (PESs). A user of mad-GP only needs to write down the functional form of the prior mean function (i.e., a prior guess for the PES) and kernel function (i.e., a constraint on the class of PESs), and the library handles all required derivative implementations via automatic differentiation (AD). We validate the design of mad-GP by applying it to perform geometry optimization of small molecules. In particular, we test the effectiveness of fitting GP surrogates to energies and/or forces, and perform a preliminary study on the use of non-constant priors and hierarchical kernels in GP PES surrogates. We find that GPs that fit forces perform comparably with GPs that fit both energies and forces, although force-only GPs are more robust for optimization because they do not require an additional step to be applied during optimization. We also confirm that constant mean functions and Matérn kernels work well as reported in the literature, although our tests also identify several other promising candidates (e.g., Coulomb matrices with three-times differentiable Matérn kernels). Our tests validate that AD is a viable method for performing geometry optimization with GP surrogate models on small molecules. PubDate: 2022-06-01
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Abstract: Abstract A new two-step method is introduced in the present paper. The new algorithm is conditionally P-Stable and an economical scheme. It has vanished phase-lag and it’s first to fourth derivatives. We use for the new scheme the symbol LOWPF5DECN2ST. We apply the newly introduced method to problems in Quantum Chemistry. The new scheme is defined as economical since it uses 4 function evaluations per step in order to achieve an algebraic order (AOR) of 10. PubDate: 2022-06-01
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Abstract: Abstract Special fermionic systems entered the realm of quantum chemistry in the seventies in the work of Borland and Dennis in the form of a toy model. This work was leading to a detailed study of the N-representability problem by Klyachko. The topic then has been reconsidered in the light of entanglement theory boiling down to the notion of entanglement polytopes. Recently building on certain properties of such special fermionic systems, a connection between the coupled cluster method and entanglement has been established. In this paper we show that precisely such a special class of systems also provides an interesting physical realization for structures related to the Lie algebras of exceptional groups. This result draws such exotic symmetry structures under the umbrella of entangled systems of physical relevance. PubDate: 2022-05-23
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Abstract: Abstract A practical electronic ground- and excited-state calculation method for lanthanide complexes is proposed by introducing frozen core potential (FCP) approximation to 4f electrons of a lanthanide atom ion (Ln \(^{3+}\) ). Based on the fact that the FCP method is formally equivalent to the elongation method, the 4f-frozen FCP calculations of Ln \(^{3+}\) complexes were successfully performed using the elongation method implemented in GAMESS quantum chemistry program. By comparing the 4f-frozen FCP calculation results of several lanthanide complexes with the results of the standard calculations, it was confirmed that the excitation energies by these calculations are comparable. Also, the SCF convergence and stability were significantly improved by the FCP approximation. We further propose a method to relax the rotational degrees of freedom for the frozen 4f orbitals. This relaxation slightly improves the accuracy of the excitation energies for f–f transitions. PubDate: 2022-05-22
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Abstract: Abstract In this work, we introduce and theoretically analyze various computational techniques to approximate the solutions of solve a fractional extension of a double condensate system. More precisely, the continuous model extends the well-known Gross–Pitaevskii equation to the fractional scenario, and considering two interacting condensates. The mathematical system considers two complex-valued regimes with coupling, and a mass and energy functions are associated to this model. Both are constant in time. Here, various discretizations are analyzed to solve this system. Some of them are able to preserve the mass and the energy, some are not. We discuss the existence of solutions, the consistency of the models, the stability and the convergence. Finally, from the computational point of view, some algorithms are simpler to code than others. In fact, those for which the mass and the energy are conserved are more difficult to implement. We discuss here pros and cons. PubDate: 2022-05-21
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Abstract: Abstract The arrangements of invariant tori that resemble rod packings with cubic symmetries are considered in three-dimensional solenoidal vector fields. To find them systematically, vector fields whose components are represented in the form of multiple Fourier series with finite terms are classified using magnetic groups. The maximal magnetic group compatible with each arrangement is specified on the assumption that the cores of the nested invariant tori are straight and located on the lines corresponding to the central axes of the rods packed. Desired rod-packing arrangements are demonstrated by selecting vector fields whose magnetic groups are the maximal ones and by drawing their integral curves that twine around invariant tori. In the demonstration of chiral arrangements, Beltrami flows (or force-free fields in plasma physics), which have the strongest chirality of all solenoidal vector fields satisfying the same vector Helmholtz equation, are used. As by-products, several chain-like arrangements of closed invariant tori were found. One of the chains consists of knotted invariant tori. In all vector fields (chiral or achiral) selected for the demonstration, the volume percentages of ordered regions formed by invariant tori in a unit cell were roughly measured with the aid of a supervised machine learning technique. PubDate: 2022-05-20
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Abstract: Abstract The topic of this study is in vitro proton magnetic resonance spectroscopy (MRS). The theme is on theoretical analysis of time signals encoded at a high magnetic field 14.1T, using a Bruker spectrometer, operating at a Larmor frequency of 600 MHz. The samples, dissolved in a D \({}_2\) O buffer, are from histopathologically analyzed ovarian cyst fluid from two patients. The benign and malignant diagnoses were serous cystadenoma and serous cystadenocarcinoma, respectively. It is of vital clinical importance to determine whether certain specific patterns, inferred from the analyzed/interpreted MRS data could be correlated with this and similar histopathologic findings for other patients. Encoded time signals contain the fingerprint of the examined sample, its metabolic content. Therefore, to detect the sought patterns from MRS data, the salient characteristics of a malignant tumor, implied by the diagnostically most relevant metabolites (including recognized cancer biomarkers, e.g. lactic acids, cholines, ...), need to be unambiguously identified by their significant departures from the associated control data of benign biomaterial, ovarian cyst fluid (serous cystadenoma) in the diagnostic problem under the present consideration. Such identifications are unfeasible by visualization in the domain of encoding (time domain). A direct inspection of the graphed waveforms of an encoded time signal would give no clue about its structure nor about the sample content. However, merely visualizing the plots of the equivalent, information-preserving spectral lineshape profiles in the frequency domain would make transparent at least some of clinically useful, discernible features of MRS data, a number of resonances assignable to the known and unknown metabolites. For instance, the size of each resonance (peak area) is proportional to the concentration of the given metabolite. This is a key quantitative measure, which could help differentiate a malignant from a benign specimen by reference to the normal standards. A number of metabolites (choline, alanine, lactate, threonine, \(\beta \) -hydroxybuturate, valine, isolecine, leucine, ...) have substantially different concentrations in the malignant compared with normal samples. Time signals can be processed by two substantially different categories of mathematical transforms, shape and parameter estimators. The former processors are alternatively called nonparametric estimators. They have been employed for envelopes in our recent study on this problem, which will presently be addressed with the prime focus on reconstructions of the corresponding components. Components and envelopes are partial and total shape spectra, respectively. The sum of all the component lineshapes (one per metabolite) yields the envelope nondegenerate spectrum representation of the entire sample. Presently, a deeper diagnostically valid insight is gained about the metabolic content of the scanned sample through the reported exact component spectra. The employed parameter estimators are the high-resolution, noise-suppressing nonderivative and derivative fast Padé transforms. Detailed are several critical achievements by the parametric Padé processing of direct clinical relevance. Importantly, all the accomplishments are shared by the nonparametric derivative Padé estimations. Three examples are highlighted here as follows. Confirmation of our recent nonparametric derivative detection of an unassigned metabolite (a singlet peak) co-resonating with free choline near chemical shift 3.19 ppm (parts per million). Therein, with the nonderivative envelope, only one compound peak usually appears and is conventionally assigned to a free choline singlet. However, such an oversight would yield about twice larger value of the true concentration of this key cancer biomarker. The concentration level of another cancer biomarker (lactate) is also overestimated by any nonparametric nonderivative envelope. In sharp contrast, the parametric nonderivative Padé estimation unequivocally detects six usually invisible resonances (assignable to other metabolites) beneath the lactate doublet, around chemical shift 1.41 ppm. At least two of the strongest among these invisible six resonances can be also identified in the nonparametric fourth derivative Padé envelope. Regularization of the spectral compound for the water residual (4.71 ppm), which deforms the neighboring resonance lineshapes and impacts adversely on the concentration assessments of other nearby metabolites. This is accomplished by the fourth derivative envelope (coincident with the components) whose narrowing of the widths, cutting off the long tails and the background flattening generate a quantifiable singlet of water. This can serve as a reliable calibration reference resonance. After such a localization, no distortion appears around water, so that even very near 4.71 ppm, several smaller resonances are detected (assignable to a multiplet of nitrogen acetyl asparate), totally invisible in the nonparametric nonderivative envelope. PubDate: 2022-05-20
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Abstract: Abstract Measures of delocalization in phase space are analyzed using Rényi entropies, especially two of which play an important role in characterizing extension and shape of distributions: the linear entropy related to the participation number and the Shannon-entropy. The difference of these two, termed as structural entropy, has been successfully applied in a large variety of physical situations and for various mathematical problems. A very similar quantity has coincidentally been used as a measure of complexity by some other authors. Hereby we show that various semiclassical phase space representations of quantum states can be well described by the structural entropy providing a transparent picture in relation to the thermodynamic description. Thermodynamic and quantum fluctuations are analytically treated for the special case of harmonic oscillators invoking the Einstein model of heat capacity. It is demonstrated that the thermal uncertainty relations are linked to the delocalization over the phase space. For respective limits of zero temperature implying quantum behavior or infinite temperature implying classical behavior we also show which quantities remain useful. As a byproduct the thermal extension of the phase space distribution can be calculated that is directly related to a decoherence parameter introduced by Zurek in a different context. PubDate: 2022-05-19
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Abstract: Abstract Local graph symmetry groups act in a non-identical fashion on just a proper (local) subset of a graph’s vertices, and consequent theorems for adjacency matrices simplify eigen-solutions. These theorems give a way to deal with a hierarchy of local sub-symmetries, such as are manifested by so-called “dendrimers”, which are (highly) branched polymers obtainable at a given generation number r from the polymer at the preceding generation number (r − 1) by connecting d copies of new branching monomer units to each end-unit of this preceding tree-like dendrimer, the initial generation number r = 1 consisting of a single monomer unit connected to d others. Our local symmetry methodology leads to an (essentially) analytic eigen-solution for the Bethe tree case, with the branching units just single sites—but further there result novel (qualitatively distinctive) features: eigenvector localization and eigenvalue clumping. Moreover, these novel characteristics persist for more general “dendrimers”, here considered and illustrated in the context of electronic structure of conjugated-carbon π-networks. The overall view here is of a systematic development and characterization for such dendrimer polymers paralleling some aspects of the standard development and characteristics for linear-chain benzenoid polymers—for instance, that of plotting eigen-energies as a function of symmetry. Clumping of eigen-spectra, and localization features in dendrimer eigenfunctions occur and are examined. PubDate: 2022-05-16
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Abstract: Abstract A Rayleigh–Ritz Method is suggested for solving linear Fredholm integral equations of the second kind numerically in a desired accuracy. To test the performance of the present approach, the classical one-dimensional Schrödinger equation \( -y^{\prime \prime }(x)+v(x)y(x)=\lambda y(x),\ x\in (-\infty ,\infty ) \) has been converted into an integral equation. For a regular problem, the unbounded interval is truncated to \(x\in [-\ell ,\ell ]\) , where \(\ell \) is regarded as a boundary parameter. Then, the resulting integral equation has been solved and the results are compared with the very well known eigenvalues of the Schrödinger equation with several types of potential functions v(x). It is shown that the eigenvalues recorded to about 15 significant figures are in excellent agreement with the results that exist in the literature. PubDate: 2022-04-27
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Abstract: Abstract In this paper we discuss the Laplace transform method for solving one dimensional Schrödinger equation in a semi infinite axis. As examples we discuss the delta potential, quantum bouncer, Coulomb-like potential and half harmonic potential. PubDate: 2022-04-23
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Abstract: Abstract Three sets of exact solutions of the time-dependent Schrödinger equation of a particle that is trapped in a spherical box with a moving boundary wall have been calculated analytically. For these solutions, some physical quantities such as time-dependent average energy, average force, disequilibrium, quantum similarity measures as well as quantum similarity index have been investigated. Moreover, these solutions are compared concerning these physical quantities. The time-correlation functions among these solutions are investigated. PubDate: 2022-04-23
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Abstract: Abstract We investigate several related measures of peripherality and centrality for vertices and edges in networks, including the Mostar index which was recently introduced as a measure of peripherality for both edges and networks. We refute a conjecture on the maximum possible Mostar index of bipartite graphs. We asymptotically answer another problem on the maximum difference between the Mostar index and the irregularity of trees. We also prove a number of extremal bounds and computational complexity results about the Mostar index, irregularity, and measures of peripherality and centrality. We discuss graphs where the Mostar index is not an accurate measure of peripherality. We construct a general family of graphs with the property that the Mostar index is strictly greater for edges that are closer to the center. We also investigate centrality and peripherality in two graphs which represent the SuperFast and MOZART-4 systems of atmospheric chemical reactions by computing various measures of peripherality and centrality for the vertices and edges in these graphs. For both of these graphs, we find that the Mostar index is closer to a measure of centrality than peripherality of the edges. We also introduce some new indices which perform well as measures of peripherality on the SuperFast and MOZART-4 graphs. PubDate: 2022-04-18
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Abstract: Abstract In this paper we show that the dynamics of a class of kinetic compartmental models with bounded capacities, monotone reaction rates and a strongly connected interconnection structure is persistent. The result is based on the chemical reaction network (CRN) and the corresponding Petri net representation of the system. For the persistence analysis, it is shown that all siphons in the Petri net of the studied model class can be characterized efficiently. Additionally, the existence and stability of equilibria are also analyzed building on the persistence and the theory of general compartmental systems. The obtained results can be applied in the analysis of general kinetic models based on the simple exclusion principle. PubDate: 2022-04-09
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Abstract: Abstract Large collections of molecules (chemical libraries) are nowadays routinely screened in the process of designing drugs for specific ailments. Chemical and structural similarities between these molecules can be quantified using molecular descriptors, and these similarities can in turn be used to represent any chemical library as an undirected network called a chemical space network (CSN). Here we study different CSNs using conventional graph measures as well as random matrix theory (RMT). For the conventional graph measures, we focus on the average degree, average path length, graph diameter, degree assortativity, transitivity, average clustering coefficient and modularity. For the RMT analyses, we examine the eigenvalue spectra of adjacency matrices constructed from the molecular similarities for different CSNs, and examine their local fluctuation properties, contrasting them with the predictions of RMT. Changes in the conventional graph measures and RMT statistics with the network structure are examined for three different chemical libraries by varying the edge density (fraction of the actual to the maximum possible number of edges) of the networks. It is found that the assortativity among the conventional graph measures, and long-range fluctuation statistics of RMT in eigenvalue space respond to the changes in global network structure as well as the chemical space. We expect that this investigation of the network characteristics of different kinds of chemical libraries will provide guidance in the design of high-throughput screening libraries for different drug design applications. PubDate: 2022-03-17
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Abstract: Abstract The Michaelis–Menten–Briggs–Haldane approximation and its extension, the total quasi-steady-state approximation (tQSSA) are famous assumptions for simplification of mathematical modeling of enzyme-substrate reactions. These approximations and their validity conditions are well studied for a single substrate reaction system. However, the extension of these studies for the tQSSA of the general case of multiple substrate reactions is yet to be performed precisely due to the consequent non-linear expressions for tQSSA. In this paper, we introduce a linearization method for equations governing the tQSSA of multiple substrate reactions to obtain an analytical solution for the evolution of concentrations of reactants that is valid throughout the whole time period. In addition, we provide the validity conditions of the tQSSA for multiple substrate reaction systems using the singular perturbation analysis method. PubDate: 2022-03-11 DOI: 10.1007/s10910-022-01339-6
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Abstract: Abstract A graphene nano-ribbon structure can be modelled by a 3-regular hexagonal grid. We convert this to a rectangular coordinate system in order to identify uniquely the position of either the \(\text {V}_2(5-8-5)\) double vacancy (DV) defect or the Stone–Wales SW(55–77) defect. This is done by using the lengths of the closed paths along the edges of the underlying graph. By sending a signal from one of the vertices and detecting the returning impulses one can observe experimentally the spectrum of the structure. Using the trace formula it is possible to determine the lengths of all closed paths (periodic orbits) starting and ending at the given vertex where a detector is placed. We present an algorithm which enables one to pinpoint the precise coordinates of a DV defect by using at most three reference points. Similarly we provide an algorithm for finding an SW defect. PubDate: 2022-03-11 DOI: 10.1007/s10910-022-01337-8
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Abstract: Abstract We are interested in the possibility of using Brandt groupoids in physical chemistry. They can, in particular, describe local symmetry and spatially limited motion of particles, in which the latter can move only a limited distance from their initial position. As an illustrative example, we consider the mutual exchange of the positions of the ligands in the coordination compounds. PubDate: 2022-03-11 DOI: 10.1007/s10910-022-01335-w
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Abstract: Abstract The exact analytic expressions of the autocorrelation function and Husimi distribution function for a Morse oscillator wave packet have been derived and we use them to see the evolution of the wave packet. The dynamics of Morse oscillator wave packets for the dimers ArXe, Be2 and Li2 have been discussed. Special emphasis has been given on the revival phenomenon of such wave packets. It is obtained that the exact revivals of wave packets for ArXe, Be2 and Li2 do not occur at the revival times (trev) but at the instances 3.5, 8.5 and 33.5 times and their simple multiple of trev respectively. PubDate: 2022-03-05 DOI: 10.1007/s10910-022-01336-9
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Abstract: Abstract This paper studies chemical kinetic systems which decompose into weakly reversible complex factorizable (CF) systems. Among power law kinetic systems, CF systems (denoted as PL-RDK systems) are those where branching reactions of a reactant complex have identical rows in the kinetic order matrix. Mass action and generalized mass action systems (GMAS) are well-known examples. Schmitz’s global carbon cycle model is a previously studied non-complex factorizable (NF) power law system (denoted as PL-NDK). We derive novel conditions for the existence of weakly reversible CF-decompositions and present an algorithm for verifying these conditions. We discuss methods for identifying independent decompositions, i.e., those where the stoichiometric subspaces of the subnetworks form a direct sum, as such decompositions relate positive equilibria sets of the subnetworks to that of the whole network. We then use the results to determine the positive equilibria sets of PL-NDK systems which admit an independent weakly reversible decomposition into PL-RDK systems of PLP type, i.e., the positive equilibria are log-parametrized, which is a broad generalization of a Deficiency Zero Theorem of Fortun et al. (MATCH Commun. Math. Comput. Chem. 81:621–638, 2019). PubDate: 2022-03-05 DOI: 10.1007/s10910-022-01332-z
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