Relative T -Injective Modules and Relative T -Flat Modules

Let T be a Wakamatsu tilting module. A module M is called (n, T)-copure injective (resp. (n, T)-copure flat) if ɛ _T ¹ (N, M) = 0 (resp. Γ₁ ^T (N, M) = 0) for any module N with T-injective dimension at most n (see Definition 2.2). In this paper, it is shown that M is (n, T)-copure injective if and only if M is the kernel of an I _n (T)-precover f: A → B with A ∈ Prod T. Also, some results on Prod T-syzygies are presented. For instance, it is shown that every nth Prod T-syzygy of every module, generated by T, is (n, T)-copure injective.     

Synthesis of Dialkyl 2‐(Alkylamino)‐4,9‐dihydro‐9‐oxocyclohepta[b]pyran‐3,4‐dicarboxylates

Dialkyl 2‐(alkylamino)‐4,9‐dihydro‐9‐oxocyclohepta[b]pyran‐3,4‐dicarboxylates are prepared in a one‐pot three‐component reaction of alkyl isocyanide, dialkyl acetylenedicarboxylate, and α‐tropolone (=2‐hydroxycyclohepta‐2,4,6‐trienone). The reaction proceeds smoothly at room temperature and under neutral conditions to afford tropolone derivatives in high yield.     

cis-Apa: a practical linker for the microwave-assisted preparation of cyclic pseudopeptides via RCM cyclative cleavage

A new linker cis-5-aminopent-3-enoic acid (cis-Apa) was prepared for the synthesis of cyclic pseudopeptides by cyclization-cleavage by using ring-closing methatesis (RCM). We developed a new synthetic pathway for the preparation of the cis-Apa linker that was tested in the cyclization-cleavage process of different RGD peptide sequences. Different macrocyclic peptidomimetics were prepared by using this integrated microwave-assisted method, showing that the readily available cis-Apa amino acid is well adapted as a linker in the cyclization-cleavage process.     

Cyclicity of the adjoint of weighted composition operators on the hilbert space of analytic functions

In this paper, we discuss the hypercyclicity, supercyclicity and cyclicity of the adjoint of a weighted composition operator on a Hilbert space of analytic functions.     

Nonlinear vibration mitigation of a flexible rotor shaft carrying a longitudinally dispositioned unbalanced rigid disc

Nonlinear Dynamics - This study is dedicated to investigate the nonlinear dynamics of a system composed of a flexible rotor shaft carrying a longitudinally dispositioned unbalanced rigid disc. The...     

Spectral collocation method and Darvishi’s preconditionings to solve the generalized Burgers–Huxley equation

In this paper, a numerical solution of the generalized Burgers–Huxley equation is presented. This is the application of spectral collocation method. To reduce roundoff error in this method we use Darvishi’s preconditionings. The numerical results obtained by this method have been compared with the exact solution. It can be seen that they are in a good agreement with each other, because errors are very small and figures of exact and numerical solutions are very similar.     

How many Langmuirs are required for monolayer formation?

In the present work, we provide the exact answer to the title question employing a probabilistic approach. The average number of Langmuirs L required for monolayer formation was found to be equal to (1/i), i.e., the armonic series up to the nth term, where n is the number of adsorption sites. This result is particularly useful when a reduced number of adsorption sites is considered, such as adsorption on small terraces of nanoscopic dimensions where the value of n could be in the range of a few thousands sites. In this case, the use of integrated equations derived from the mean-field approach would provide completely misleading results.