Z-scan studies on porphyrin derivative

Received: 10 January 1996 / Revised version: 15 June 1996     

Metal Derivatives of Heterocyclic Thioamides: Synthesis and Crystal Structures of Copper Complexes with 1‐Methyl‐1,3‐imidazoline‐2‐thione and 1,3‐Imidazoline‐2‐thione

Reactions of copper(I) halides (Cl, Br, I) with 1‐methyl‐1, 3‐imidazoline‐2‐thione (mimzSH) in 1 : 2 molar ratio yielded sulfur‐bridged dinuclear [Cu₂X₂(μ‐S‐mimzSH)₂(η¹‐S‐mimzSH)₂] (X = I, 1 , Br, 2 ; Cl, 3 ) complexes. Copper(I) iodide with 1,3‐imidazoline‐2‐thione (imzSH₂) and Ph₃P in 1 : 1 : 1 molar ratio has also formed a sulfur‐bridged dinuclear [Cu₂I₂(μ‐S‐imzSH₂)₂(PPh₃)₂] ( 4 ) complex. The central Cu(μ‐S)₂Cu cores form parallelograms with unequal Cu–S bond distances {2.324(2), 2.454(3) Å} ( 1 ); {2.3118(6), 2.5098(6) Å} ( 2 ); {2.3075(4), 2.5218(4) Å} ( 3 ); {2.3711(8), 2.4473(8) Å} ( 4 ). The Cu···Cu separations, 2.759–2.877Å in complexes 1 – 3 are much shorter than 3.3446Å in complex 4 . The weak intermolecular interactions {H₂CH···S^# ( 2 ); CH···Cl^# ( 3 ); NH···I^# ( 4 )} between dimeric units in complexes 2 – 4 lead to the formation of linear 1D polymers.     

Canonical commutation relation for operators with bounded spectrum

A solution is obtained for the problem of the canonical commutation relation of the angular momentum operator and the phase.     

Quantum mechanics in finite dimensions

We explicitly compute, following the method of Weyl, the commutator [Q, P] of the position operatorQ and the momentum operatorP of a particle when the dimension of the space on which they act is finite with a discrete spectrum; and we show that in the limit of a continuous spectrum with the dimension going to infinity this reduces to the usual relation of Heisenberg.     

On the classical solution of SO(3) gauge theory and ’t Hooft-Polyakov monopole

It is shown that the ansatz for the asymptotic (r → ∞) gauge fields used by ’t Hooft in the study of monopoles in SO(3) electroweak theory is not unique.     

Reverse lexicographic and lexicographic shifting

A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, Δ_lex—an operation that transforms a monomial ideal of S = K[x_i: i ∈ ℕ] that is finitely generated in each degree into a squarefree strongly stable ideal—is defined and studied. It is proved that (in contrast to the reverse lexicographic case) a squarefree strongly stable ideal I ⊂ S is fixed by lexicographic shifting if and only if I is a universal squarefree lexsegment ideal (abbreviated USLI) of S. Moreover, in the case when I is finitely generated and is not a USLI, it is verified that all the ideals in the sequence } are distinct. The limit ideal is well defined and is a USLI that depends only on a certain analog of the Hilbert function of I. Research partially supported by NSF grants DMS 0070571 and DMS 0100141.     

Effect of gravity on subject-specific human lung deformation

A biomechanical model of human lung is developed and used to investigate the effect of gravity on lung deformation. The lung is assumed to behave as a poro-elastic medium with spatially dependent elastic property. Finite element analysis is performed on a three-dimensional (3D) lung geometry reconstructed from a four-dimensional Computed Tomography (4DCT) scan dataset of human patient. The spatially dependent Young’s modulus (YM) values are estimated using inverse analysis from a linear elastic deformation model. The predicted deformation of selected landmarks is monitored with and without gravity, and compared with data obtained from 4DCT registration. The results show that gravity indeed significantly affects the magnitude and distribution of lung deformation with the maximum displacement enhanced by 54% in the direction of gravity, for the conditions investigated. In summary, the accuracy of predicted deformation is improved through incorporation of gravity in the biomechanical model of lung.     

The order of starlikeness of some classes of strongly starlike functions

In this paper we consider the order of strongly starlikeness for functions in Silverman’s class 𝒢_b and in the generalized Mocanu’s class M (α, β1, β2). Some previous results are improved.     

A linear operator associated with the Mittag-Leffer function and related conformal mappings

In the present paper, we introduce a linear operator associated with the Mittag-Leffler function. Some convolution properties of meromorphic functions involving this operator are given.