New Pythagorean Entropy Measure with Application in Multi-Criteria Decision Analysis

The purpose of this paper is to propose a new Pythagorean fuzzy entropy for Pythagorean fuzzy sets, which is a continuation of the Pythagorean fuzzy entropy of intuitionistic sets. The Pythagorean fuzzy set continues the intuitionistic fuzzy set with the additional advantage that it is well equipped to overcome its imperfections. Its entropy determines the quantity of information in the Pythagorean fuzzy set. Thus, the proposed entropy provides a new flexible tool that is particularly useful in complex multi-criteria problems where uncertain data and inaccurate information are considered. The performance of the introduced method is illustrated in a real-life case study, including a multi-criteria company selection problem. In this example, we provide a numerical illustration to distinguish the entropy measure proposed from some existing entropies used for Pythagorean fuzzy sets and intuitionistic fuzzy sets. Statistical illustrations show that the proposed entropy measures are reliable for demonstrating the degree of fuzziness of both Pythagorean fuzzy set (PFS) and intuitionistic fuzzy sets (IFS). In addition, a multi-criteria decision-making method complex proportional assessment (COPRAS) was also proposed with weights calculated based on the proposed new entropy measure. Finally, to validate the reliability of the results obtained using the proposed entropy, a comparative analysis was performed with a set of carefully selected reference methods containing other generally used entropy measurement methods. The illustrated numerical example proves that the calculation results of the proposed new method are similar to those of several other up-to-date methods.     

Stabilization of the Low‐Spin State in a Mononuclear Iron(II) Complex and High‐Temperature Cooperative Spin Crossover Mediated by Hydrogen Bonding

The tetrapyridyl ligand bbpya (bbpya=N,N‐bis(2,2′‐bipyrid‐6‐yl)amine) and its mononuclear coordination compound [Fe(bbpya)(NCS)₂] ( 1 ) were prepared. According to magnetic susceptibility, differential scanning calorimetry fitted to Sorai’s domain model, and powder X‐ray diffraction measurements, 1 is low‐spin at room temperature, and it exhibits spin crossover (SCO) at an exceptionally high transition temperature of T_1/2=418 K. Although the SCO of compound 1 spans a temperature range of more than 150 K, it is characterized by a wide (21 K) and dissymmetric hysteresis cycle, which suggests cooperativity. The crystal structure of the LS phase of compound 1 shows strong N?H???S intermolecular H‐bonding interactions that explain, at least in part, the cooperative SCO behavior observed for complex 1 . DFT and CASPT2 calculations under vacuum demonstrate that the bbpya ligand generates a stronger ligand field around the iron(II) core than its analogue bapbpy (N,N′‐di(pyrid‐2‐yl)‐2,2′‐bipyridine‐6,6′‐diamine); this stabilizes the LS state and destabilizes the HS state in 1 compared with [Fe(bapbpy)(NCS)₂] ( 2 ). Periodic DFT calculations suggest that crystal‐packing effects are significant for compound 2 , in which they destabilize the HS state by about 1500 cm^?1. The much lower transition temperature found for the SCO of 2 compared to 1 appears to be due to the combined effects of the different ligand field strengths and crystal packing.     

Exact Spectrum of Anomalous Dimensions of Planar <Emphasis Type="Italic">N</Emphasis> = 4 Supersymmetric Yang–Mills Theory: TBA and excited states

Using the thermodynamic Bethe ansatz method we derive an infinite set of integral non-linear equations for the spectrum of states/operators in AdS/CFT. The Y-system conjectured in Gromov et al. (Integrability for the Full Spectrum of Planar AdS/CFT. arXiv:0901.3753 [hep-th]) for the spectrum of all operators in planar N = 4 SYM theory follows from these equations. In particular, we present the integral TBA type equations for the spectrum of all operators within the sl(2) sector. We prove that all the kernels and free terms entering these TBA equations are real and have nice fusion properties in the relevant mirror kinematics. We find the analog of DHM formula for the dressing kernel in the mirror kinematics.     

On the role of longitudinal momenta in high energy hadron-hadron scattering

We demonstrate a new method to calculate inelastic scattering cross-sections, which in contrary to the Regge-based methods takes into account the energy momentum conservation law. It is shown that the main contribution to integral expressing inelastic scattering cross-sections does not come from the multi-Regge domain. In particular, accounting for the longitudinal momenta contribution to virtualities is sufficient and results in the new mechanism of cross-section growth. The reasons for taking into account the sufficiently high number of interference contributions are shown and the approximate method for this purpose is developed. By fitting single free parameter of the model achieved a qualitative agreement of the total and inelastic cross sections with experimental data.     

N-(R-Cyclopropyl) Trifluoroacetimidoyl Phosphonates

Abstract

A convenient synthetic approach for previously unknown N-(R-cyclopropyl)trifluoroacetimidoyl phosphonates 5a,b (R?H, CF₃) was developed on the basis of the reaction of respective trifluoroacetimidoyl chlorides with triethyl phosphite. It was shown that imidoyl phosphonates 5a,b exist as equilibrium mixture of Z/E isomers (Z:E ～92:8). Activation parameters of Z–E isomerization were evaluated by ¹⁹F NMR spectroscopy. Catalytic hydrogenation of 5a,b can serve as a convenient method for the synthesis of trifluoroethylaminophosphonates with a rigid N-cyclopropyl group.     

Non-coercive Lyapunov functions for infinite-dimensional systems

We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Finally, we derive new non-coercive converse Lyapunov theorems and give some examples showing the necessity of our assumptions.     

Arylidene Pyruvic Acids Motif in the Synthesis of New 2H,5H-Chromeno[4′,3′:4,5]thiopyrano[2,3-d]thiazoles via Tandem Hetero-Diels–Alder-Hemiacetal Reaction

We have developed a tandem hetero-Diels–Alder-hemiacetal reaction using arylidene pyruvic acids with 5-(ortho-hydroxybenzylidene)-substituted 4-thioxo-2-thiazolidinones, leading to 6-hydroxy-2-oxo-5-phenyl-3,5a,6,11b-tetrahydro-2H,5H-chromeno[4′,3′:4,5]thiopyrano[2,3-d][1,3]thiazole-6-carboxylic acids. The stereochemistry of cycloaddition was confirmed by NMR spectra and a single-crystal x-ray diffraction analysis.