Characterizing Cohen-Macaulay local rings by Frobenius maps

Let be a commutative noetherian local ring of prime characteristic. Denote by the ring regarded as an -algebra through -times composition of the Frobenius map. Suppose that is F-finite, i.e., is a finitely generated -module. We prove that is Cohen-Macaulay if and only if the -modules have finite Cohen-Macaulay dimensions for infinitely many integers .

     

The best constant of the Moser-Trudinger inequality on

We consider the best constant of the Moser-Trudinger inequality on under a certain orthogonality condition. Applying Moser's calculation, we construct a counterexample to the sharper inequality with the condition.

     

Asymptotic behavior of solutions of forced nonlinear neutral difference equations

In this paper, we consider the asymptotic behavior of solutions of the forced nonlinear neutral difference equation $$\Delta \left[ {x(n) - \sum\limits_{i - 1}^m {p_i (n)x(n - k_i )} } \right] + \sum\limits_{j = 1}^s {q_j (n)f(x(n - l_j )) = r(n)} $$ with sign changing coefficients. Some sufficient conditions for every solution of (*) to tend to zero are established. The results extend and improve some known theorems in literature.     

A counterexample to Sorenson’s conjecture: The finitely generated case

It is shown that there exists a finitely generated, left amenable semigroup which is right cancellative but not left cancellative. This resolves negatively a question raised by M. Klawe.     

Asymptotic behavior of certain delay differential equations with forcing term

We study the asymptotic behavior of solutions of the following forced delay differential equation:

Some sufficient conditions that guarantee every solution of the equation to converge to zero are obtained. The results obtained are applied to some well-known delay differential ecological equations with forcing term.     

Alignment-free optical modules using solder-bump-bonding technique for free-space optical interconnections

We propose alignment-free optical modules using a solder-bump-bonding technique for constructing free-space optical interconnection systems without a special alignment procedure. Bonding pads for mounting an optoelectronic device chip are fabricated by exposing a photosensitive resin film to light traveling through the optical system of the module so that the image positions of the bonding pads on both image planes of the optical interconnection system are aligned with each other. A device chip is mounted by solder-bump bonding and is set at a proper position by the surface tension of molten solder. The effectiveness of the technique is verified by constructing alignment-free optical modules.     

Morita Invariant Properties of Twisted Poisson Manifolds

In this letter, we study some invariants of integrable twisted Poisson manifolds under Morita equivalence. We show that two twisted symplectic manifolds are Morita equivalent if and only if their fundamental groups are isomorphic. Moreover, the first cohomology group of twisted Poisson manifolds is invariant under Morita equivalence. Finally, we prove that if two twisted Poisson manifolds are Morita equivalent, then their representation categories are equivalent.      

The fluorescent reaction product in the determination of aromatic aldehydes with 2,2''-dithiobis(1-amino-naphthalene)

The fluorescent product formed from benzaldehyde or p-hydroxybenzaldehyde with the reagent in the presence of tri-n-butylphosphine, phosphite and sulphite has been characterized by elemental and structural analysis as the corresponding 2-arylnaphtho [1,2-d ] thiazole.     

Total synthesis of nagilactone F,biologically active nor-diterpenoid dilactone isolated from podocarpus nagi

Nagilactone F was synthesized from (+)-podocarpic acid of the established structure. This work constitutes the first total synthesis of the biologically active norditerpenoid dilactones in $\text{Podocarpus}$ plants.     

A short and stereoselective synthesis of the (±) Prelog-Djerassi lactonic acid

Reaction of meso-4-carbomethoxy-2-methylpentanal ($\text{1}$) with crotyltri-n-butyltin at ?78°C in the presence of 1 eq BF₃·OEt₂, followed by the lactonization with BF₃·OEt₂, gave 6-(1-methylallyl)-3,4,5,6-tetrahydro-3,5-dimethyl-2-pyranone ($\text{2a}$) with the correct stereochemistry (erythro, anti-Cram) in 92% yield, which was converted to the title compound ($\text{3}$) in 85% yield upon the ozonolytic cleavage of the double bond.