To demonstrate our preservation result for non-cofibrant operads, we develop a theory for when the category of commutative monoids in *M* inherits a model structure from *M* in which a map is a weak equivalence or fibration if and only if it is so in *M*. We then investigate properties of cofibrations of commutative monoids, functoriality of the passage from a commutative monoid *R* to the category of commutative *R*-algebras, rectification between *E*_{∞}-algebras and commutative monoids, and the relationship between commutative monoids and monoidal Bousfield localization functors. We recover numerous known examples and a few new examples of model categories in which commutative monoids inherit a model structure. We then work out when localization preserves commutative monoids and the commutative monoid axiom. Finally, we provide conditions so that a left Bousfield localization satisfies the monoid axiom.

We prove a uniform version of the conjecture in the case where the abelian varieties are elliptic curves with complex multiplication. In addition, we provide explicit bounds in cases where the number field has degree less than or equal to 100.

]]>Ash and Stevens proved that certain classes in *H ^{n}*(Γ,

We construct a general filtration *Fil ^{N}D*

Previous studies of the substrate specificity of the Streptomyces R61 DD-peptidase (LMMB) revealed that the enzyme shows specificity for the glycyl-L-aminopimelyl segment of its peptidoglycan. Following this strategy, the substrate specificity of LMMC and LMMA PBPs towards specific elements of peptidoglycan of their parent organism was investigated. Small peptidoglycan-mimetic peptides were synthesized and assessed as substrates of LMM PBPs by steady-state kinetics. Peptidoglycan-mimetic peptides containing the D-aminopimelate moiety were efficiently turned over by the LMMC enzymes, which have a well-defined and specific binding site for these peptides. Solubilized LMMA PBPs, on the other hand, were poorly reactive towards small peptidoglycan-mimetic peptides *in vitro*.

The LMMC PBP, *Bacillus subtilis* PBP4a, was found to have carboxypeptidase (hydrolyzing terminal D-alanyl-D-alanine) as well as endopeptidase (hydrolyzing cross-linked peptidoglycan) activity from steady-state kinetics studies with peptidoglycan-mimetic peptide substrates. A systematic investigation of the acyl donor and acceptor specificity of *B. subtilis* PBP4a towards amidated peptidoglycanmimetic peptides was conducted. This enzyme was found to have dual specificity, encompassing both carboxylate and carboxamide containing peptides. The results obtained from molecular modeling studies that were performed starting with the available crystal structures suggested that a neutral histidine residue, His 352, was able to favorably interact with both carboxylate and carboxamide termini of substrates, in agreement with the kinetics data.

The specificity of LMM PBPs for cross-linked pentapeptides in solution was investigated using two synthetic peptidoglycan-mimetic cyclic peptides and steady state kinetics. The larger 19-membered cyclic peptide was found to have higher reactivity than analogous acyclic peptides with two LMMC enzymes (*Actinomadura* R39 DD-peptidase and *Neisseria gonorrhoeae* PBP3) and an LMMA PBP (*Escherichia coli* PBP5) The conformational preference of the cyclic peptides, explored using CD, NMR and molecular modeling experiments, suggested that they are more ordered in solution than the acyclic analogues. The kinetics results were interpreted on the basis of a greater entropic advantage offered by cyclic peptides in bringing about effective catalysis and the enthalpic barrier in achieving the bioactive conformation.