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The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Mustaţă, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case. [ABSTRACT FROM AUTHOR]

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl's book: for the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings, and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with SG⊆S being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring SG is a pure subring of S, equivalently, SG is a direct summand of S as an SG-module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion SG⊆S is pure. It turns out that if SG⊆S is pure, then either the invariant ring SG is regular, or the group G is linearly reductive. [ABSTRACT FROM AUTHOR]

For multiplicities arising from a family of ideals we provide a general approach to transformation rules for a ring extension that is étale in codimension one. Our result can be applied to bound the size of the local étale fundamental group of a singularity in terms of F-signature, recovering a recent result of Carvajal-Rojas, Schwede, and Tucker, and differential signature, providing the 1st characteristic-free effective bound. [ABSTRACT FROM AUTHOR]

Enzyme discovery for use in the manufacture of chemicals, requiring high stereoselectivities, continues to be an important avenue of research. Here, a sequence directed metagenomics approach is described to identify short chain carbonyl reductases. PCR from a metagenomic template generated 37 enzymes, with an average 25% sequence identity, twelve of which showed interesting activities in initial screens. Six of the most productive enzymes were then tested against a panel of 21 substrates, including bulkier substrates that have been noted as challenging in biocatalytic reductions. Two enzymes were selected for further studies with the Wieland Miescher ketone. Notably, enzyme SDR‐17, when co‐expressed with a co‐factor recycling system produced the anti‐(4aR,5S) isomer in excellent isolated yields of 89% and 99% e.e. These results demonstrate the viability of a sequence directed metagenomics approach for the identification of multiple homologous sequences with low similarity, that can yield highly stereoselective enzymes with applicability in industrial biocatalysis. [ABSTRACT FROM AUTHOR]

In their work on differential operators in positive characteristic, Smith and Van den Bergh define and study the derived functors of differential operators; these arise naturally as obstructions to differential operators reducing to positive characteristic. In this paper, we provide formulas for the ring of differential operators as well as these derived functors of differential operators. We apply these descriptions to show that differential operators behave well under reduction to positive characteristic under certain hypotheses. We show that these functors also detect a number of interesting properties of singularities. [ABSTRACT FROM AUTHOR]

Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; in each of the cases that they considered, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of linearly reductive groups over the complex numbers, Smith and Van den Bergh asked if differential operators on the corresponding rings of positive prime characteristic lift to characteristic zero differential operators. We prove that, in general, this is not the case for determinantal hypersurfaces, as well as for Pfaffian and symmetric determinantal hypersurfaces. We also prove that, with very few exceptions, these hypersurfaces—and, more generally, classical invariant rings—do not admit a mod p 2 lift of the Frobenius endomorphism. [ABSTRACT FROM AUTHOR]

Neural rings and ideals, as introduced by Curto, Itskov, Veliz-Cuba, and Youngs, are useful algebraic tools for organizing the combinatorial information of neural codes. However, they are neither graded nor local, and consequently, many algebraic approaches to understanding such rings fail. In this paper, we introduce an operation, called "polarization," that allows us to relate neural ideals with squarefree monomial ideals, which are very well studied and known for their nice behavior in commutative algebra. We use this notion of polarization to construct free resolutions of neural ideals. We also relate the structure of polarizations of neural ideals to combinatorial properties of codes. [ABSTRACT FROM AUTHOR]

The manufacturing of mRNA vaccines relies on cell-free based systems that are easily scalable and flexible compared with the traditional vaccine manufacturing processes. Typically, standard processes yield 2 to 5 g L−1 of mRNA, with recent process optimisations increasing yields to 12 g L−1. However, increasing yields can lead to an increase in the production of unwanted by-products, namely dsRNA. It is therefore imperative to reduce dsRNA to residual levels in order to avoid intensive purification steps, enabling cost-effective manufacturing processes. In this work, we exploit sequence modifications downstream of the T7 RNA polymerase promoter to increase mRNA yields whilst simultaneously minimising dsRNA. In particular, transcription performance was optimised by modifying the sequence downstream of the T7 promoter with additional AT-rich sequences. We have identified variants that were able to produce higher amounts of mRNA (up to 14 g L−1) in 45 min of reaction. These variants exhibited up to a 30% reduction in dsRNA byproduct levels compared to a wildtype T7 promoter, and have similar EGFP protein expression. The results show that optimising the non-coding regions can have an impact on mRNA production yields and quality, reducing overall manufacturing costs. [ABSTRACT FROM AUTHOR]

Transaminase enzymes have significant potential for the sustainable synthesis of amines using mild aqueous reaction conditions. Here a metagenomics mining strategy has been used for new transaminase enzyme discovery. Starting from oral cavity microbiome samples, DNA sequencing and bioinformatics analyses were performed. Subsequent in silico mining of a library of contiguous reads built from the sequencing data identified 11 putative Class III transaminases which were cloned and overexpressed. Several screening protocols were used and three enzymes selected of interest due to activities towards substrates covering a wide structural diversity. Transamination of functionalized cinnamaldehydes was then investigated for the production of valuable amine building blocks. [ABSTRACT FROM AUTHOR]

The j-multiplicity plays an important role in the intersection theory of Stuckrad-Vogel cycles, while recent developments confirm the connections between the multiplicity and equisingularity theory. In this paper, we establish, under some constraints, a relationship between the j-multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the j-multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the j- and ϵ-multiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory. [ABSTRACT FROM AUTHOR]

The study of separating invariants is a recent trend in invariant theory. For a finite group acting linearly on a vector space, a separating set is a set of invariants whose elements separate the orbits of G . In some ways, separating sets often exhibit better behavior than generating sets for the ring of invariants. We investigate the least possible cardinality of a separating set for a given G -action. Our main result is a lower bound that generalizes the classical result of Serre that if the ring of invariants is polynomial then the group action must be generated by pseudoreflections. We find these bounds to be sharp in a wide range of examples. [ABSTRACT FROM AUTHOR]

Methyltransferases are gaining traction as a method to achieve greener and regioselective methylations of a wide array of substrates. In this work, a halide methyltransferase biomethylation cascade was successfully adapted with the incorporation of a methionine adenosyltransferase to generate the S‐adenosyl‐L‐methionine (SAM) methylation cofactor in situ from less costly ATP and L‐methionine, instead of directly adding SAM to start the cascade. Furthermore, this cascade was applied to achieve novel catechol and tetrahydroisoquinoline methylations with the O‐methyltransferases RnCOMT, MxSafC and NpN4OMT with high conversions and regioselectivities. In addition, the cascade was successfully trialled with the less toxic methyl group donor, methyl tosylate. [ABSTRACT FROM AUTHOR]

The F -signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong F -regularity. However, it is very difficult to compute. Motivated by different aspects of the F -signature, we define a numerical invariant for rings of characteristic zero or p > 0 that exhibits many of the useful properties of the F -signature. We also compute many examples of this invariant, including cases where the F -signature is not known. We also obtain a number of results on symbolic powers and Bernstein-Sato polynomials. [ABSTRACT FROM AUTHOR]

This paper investigates the existence and properties of a Bernstein–Sato functional equation in nonregular settings. In particular, we construct D-modules in which such formal equations can be studied. The existence of the Bernstein–Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of V-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings. [ABSTRACT FROM AUTHOR]

In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the n-th symbolic power of a given prime ideal consists of the elements that vanish up to order n on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use p-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of p-derivations to commutative algebra. [ABSTRACT FROM AUTHOR]

Abstract A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we develop algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the Stanley–Reisner ideal. Using the neural ideal together with its standard generating set, the canonical form , we provide algebraic signatures of certain families of codes that are non-convex. We connect these signatures to the precise conditions on the arrangement of sets that prevent the codes from being convex. Finally, we also provide algebraic signatures for some families of codes that are convex, including the class of intersection-complete codes. These results allow us to detect convexity and non-convexity in a variety of situations, and point to some interesting open questions. [ABSTRACT FROM AUTHOR]

Transaminases remain one of the most promising biocatalysts for use in chiral amine synthesis, however their industrial implementation has been hampered by their general instability towards, for example, high amine donor concentrations and organic solvent content. Herein we describe the identification, cloning and screening of 29 novel transaminases from a household drain metagenome. The most promising enzymes were fully characterised and the effects of pH, temperature, amine donor concentration and co-solvent determined. Several enzymes demonstrated good substrate tolerance as well as an unprecedented robustness for a wild-type transaminase. One enzyme in particular readily accepted IPA as an amine donor giving the same conversion with 2–50 equivalents, as well as being tolerant to a number of co-solvents, and operational in up to 50% DMSO – a characteristic as yet unobserved in a wild-type transaminase. This work highlights the value of using metagenomics for biocatalyst discovery from niche environments, and here has led to the identification of one of the most robust native transaminases described to date, with respect to IPA and DMSO tolerance. [ABSTRACT FROM AUTHOR]