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To a strongly 퐺-graded algebra 퐴 with 1-component 퐵, we associate the group Picent gr ⁢ (A) of isomorphism classes of invertible 퐺-graded (A , A) -bimodules over the centralizer of 퐵 in 퐴. Our main result is a Picent version of the Beattie–del Río exact sequence, involving Dade's group G ⁢ [ B ] , which relates Picent gr ⁢ (A) , Picent ⁢ (B) , and group cohomology. [ABSTRACT FROM AUTHOR]

In this paper, we describe the split exact sequences of skew left braces. We define a free action of the second cohomology group of a skew left brace H by Ann (I) on Ext α (H , I) and show that this action becomes transitive if I is a trivial skew brace. We also generalize the Well's type exact sequence for extensions by the trivial skew brace. [ABSTRACT FROM AUTHOR]

We prove a four-term exact sequence of surface orbifold pure braid groups for all genus ≥1, 2-dimensional orientable orbifolds with cone points. This also corrects and generalizes our earlier result (genus zero case) in [7] and [8]. [ABSTRACT FROM AUTHOR]

Given a non-necessarily commutative unital ring R and a unital partial representation Θ of a group G into the Picard semigroup PicS (R) of the isomorphism classes of partially invertible R -bimodules, we construct an abelian group C (Θ / R) formed by the isomorphism classes of partial generalized crossed products related to Θ and identify an appropriate second partial cohomology group of G with a naturally defined subgroup C 0 (Θ / R) of C (Θ / R). Then we use the obtained results to give an analogue of the Chase-Harrison-Rosenberg exact sequence associated with an extension of non-necessarily commutative rings R ⊆ S with the same unity and a unital partial representation G → S R (S) of an arbitrary group G into the monoid S R (S) of the R -subbimodules of S. This generalizes the works by Kanzaki and Miyashita. [ABSTRACT FROM AUTHOR]

In Crew and Spirkl (2020), the authors generalize Stanley's chromatic symmetric function (Stanley, 1995) to vertex-weighted graphs. In this paper we find a categorification of their new invariant extending the definition of chromatic symmetric homology to vertex-weighted graphs. We prove the existence of a deletion–contraction long exact sequence for chromatic symmetric homology which gives a useful computational tool and allow us to answer two questions left open in Chandler et al. (2019). In particular, we prove that, for a graph G with n vertices, the maximal index with nonzero homology is not greater that n − 1. Moreover, we show that the homology is non-trivial for all the indices between the minimum and the maximum with this property. [ABSTRACT FROM AUTHOR]

This study is motivated by the recently published papers on normal injective and normal projective hypermodules. We provide a new characterization of the normal injective and normal projective hypermodules by using the splitting of the short exact sequences of hypermodules. After presenting some of their fundamental properties, we show that if a hypermodule is normal projective, then every exact sequence ending with it is splitting. Moreover, if a hypermodule is normal injective, then every exact sequence starting with it is splitting as well. Finally, we investigate the relationships between semisimple, simple, normal injective, and normal projective hypermodules. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and Nil(R) denote the set of nilpotent elements of R. In this paper, we investigate two generalizations of exact sequences, ϕ -exact sequences and N -exact sequences. With the help of two generalizations of exact sequences, we generalize the concept of homology modules. We show that they behave in a way similar to the classical ones. [ABSTRACT FROM AUTHOR]

Recently, Zhao, Wang, and Pu introduced and studied new concepts of nonnilcommutative diagrams and nonnil-projective modules. They proved that an R-module that is nonnil-isomorphic to a projective module is nonnil-projective, and they proposed the following problem: Is every nonnil-projective module nonnil-isomorphic to some projective module? In this paper, we delve into some new properties of nonnil-commutative diagrams and answer this problem in the affirmative. [ABSTRACT FROM AUTHOR]

Let X be a 2-connected and 6-dimensional CW-complex such that H3(X) ⊗ Z2 = 0. This paper aims to describe the group E(X) of the self-homotopy equivalences of X modulo its normal subgroup E*(X) of the elements that induce the identity on the homology groups. Making use of the Whitehead exact sequence of X, denoted by WES(X), we define the group GS(X) of G-automorphisms of WES(X) and we prove that E(X)/E*(X) ≌ ΓS(X). [ABSTRACT FROM AUTHOR]

We consider sections of the étale homotopy exact sequence of a hyperbolic curve over a number field. We prove that two sections whose restrictions to decomposition groups are conjugate on a set of valuations of density one are globally conjugate, which establishes the local-global principle for the conjugacy classes of sections. In fact, we obtain this result as a corollary of a more general property concerning sections of the étale homotopy exact sequence, the finite covering property, which we prove as our main result. [ABSTRACT FROM AUTHOR]

In recent years, averaging operators on Lie algebras (also called embedding tensors in the physics literature) and associated tensor hierarchies have formed an efficient tool for constructing supergravity and higher gauge theories. A Lie algebra with an averaging operator is called an averaging Lie algebra. In the present paper, we introduce 2-term averaging L ∞ -algebras and give characterizations of some particular classes of such homotopy algebras. Next, we study non-abelian extensions of an averaging Lie algebra by another averaging Lie algebra. We define the second non-abelian cohomology group to classify the equivalence classes of such non-abelian extensions. Next, given a non-abelian extension of averaging Lie algebras, we show that the obstruction for a pair of averaging Lie algebra automorphisms to be inducible can be seen as the image of a suitable Wells map. Finally, we discuss the Wells short exact sequence in the above context. [ABSTRACT FROM AUTHOR]

Davydov–Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. In this paper we provide new tools for the computation of the DY cohomology for finite tensor categories and exact functors between them. The key point is to realize DY cohomology as relative Ext groups. In particular, we prove that the infinitesimal deformations of a tensor category C are classified by the 3-rd self-extension group of the tensor unit of the Drinfeld center Z (C) relative to C . From classical results on relative homological algebra we get a long exact sequence for DY cohomology and a Yoneda product for which we provide an explicit formula. Using the long exact sequence and duality, we obtain a dimension formula for the cohomology groups based solely on relatively projective covers which reduces a problem in homological algebra to a problem in representation theory, e.g. calculating the space of invariants in a certain object of Z (C) . Thanks to the Yoneda product, we also develop a method for computing DY cocycles explicitly which are needed for applications in the deformation theory. We apply these tools to the category of finite-dimensional modules over a finite-dimensional Hopf algebra. We study in detail the examples of the bosonization of exterior algebras Λ C k ⋊ C [ Z 2 ] , the Taft algebras and the small quantum group of sl 2 at a root of unity. [ABSTRACT FROM AUTHOR]

Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie triple systems. In this paper, we consider the inducibility problem for automorphisms of Lie-Yamaguti algebra extensions. More precisely, given an abelian extension [Display omitted] of a Lie-Yamaguti algebra L , we are interested in finding the pairs (ϕ , ψ) ∈ Aut (V) × Aut (L) , which are inducible by an automorphism in Aut (L ˜). We connect the inducibility problem to the (2 , 3) -cohomology of Lie-Yamaguti algebra. In particular, we show that the obstruction for a pair of automorphisms in Aut (V) × Aut (L) to be inducible lies in the (2 , 3) -cohomology group H (2 , 3) (L , V). We develop the Wells exact sequence for Lie-Yamaguti algebra extensions, which relates the space of derivations, automorphism groups, and (2 , 3) -cohomology groups of Lie-Yamaguti algebras. As an application, we describe certain automorphism groups of semi-direct product Lie-Yamaguti algebras. In a sequel, we apply our results to discuss inducibility problem for nilpotent Lie-Yamaguti algebras of index 2. We give examples of infinite families of such nilpotent Lie-Yamaguti algebras and characterize the inducible pairs of automorphisms for extensions arising from these examples. Finally, we write an algorithm to find out all the inducible pairs of automorphisms for extensions arising from nilpotent Lie-Yamaguti algebras of index 2. [ABSTRACT FROM AUTHOR]

The aim of this paper is defining the concepts of Noetherian and Artinian hoops by using the filter of hoop in the partial order set of all the filters of hoops and inclusion relation and find some equivalent definitions for this notion. We translate some important results from theory of rings to the case of hoop and their characterizations are established. The relation between short exact sequence on Noetherian and Artinian hoop studied and by using short exact sequence we prove that the Cartesian product of two hoops is Noetherian (Artinian) if and only if each one is a Noetherian (Artinian). By using the notion of filter in hoops, we define the notion of composition series and prove any V-hoop is Noetherian and Artinian if and only if it has composition series. Finally, Chinese Remainder theorem in hoop and the relation between maximal filter and Noetherian (Artinian) hoop are investigated. [ABSTRACT FROM AUTHOR]

Background: MicroRNA isoforms (isomiRs), tRNA-derived fragments (tRFs), and rRNA-derived fragments (rRFs) represent most of the small non-coding RNAs (sncRNAs) found in cells. Members of these three classes modulate messenger RNA (mRNA) and protein abundance and are dysregulated in diseases. Experimental studies to date have assumed that the subcellular distribution of these molecules is well-understood, independent of cell type, and the same for all isoforms of a sncRNA. Results: We tested these assumptions by investigating the subcellular distribution of isomiRs, tRFs, and rRFs in biological replicates from three cell lines from the same tissue and same-sex donors that model the same cancer subtype. In each cell line, we profiled the isomiRs, tRFs, and rRFs in the nucleus, cytoplasm, whole mitochondrion (MT), mitoplast (MP), and whole cell. Using a rigorous mathematical model we developed, we accounted for cross-fraction contamination and technical errors and adjusted the measured abundances accordingly. Analyses of the adjusted abundances show that isomiRs, tRFs, and rRFs exhibit complex patterns of subcellular distributions. These patterns depend on each sncRNA's exact sequence and the cell type. Even in the same cell line, isoforms of the same sncRNA whose sequences differ by a few nucleotides (nts) can have different subcellular distributions. Conclusions: SncRNAs with similar sequences have different subcellular distributions within and across cell lines, suggesting that each isoform could have a different function. Future computational and experimental studies of isomiRs, tRFs, and rRFs will need to distinguish among each molecule's various isoforms and account for differences in each isoform's subcellular distribution in the cell line at hand. While the findings add to a growing body of evidence that isomiRs, tRFs, rRFs, tRNAs, and rRNAs follow complex intracellular trafficking rules, further investigation is needed to exclude alternative explanations for the observed subcellular distribution of sncRNAs. [ABSTRACT FROM AUTHOR]

This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functors ˇH𝑛 on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor ˇH𝑛 def taking values in the category GPC of groups with a Polish cover (a category first introduced in this work's predecessor), followed by (ii) a forgetful functor from GPC to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of d-spheres or d-tori for any 𝑑 ≥ 1, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors ˇH𝑛 def to show that a seminal problem in the development of algebraic topology - namely, Borsuk and Eilenberg's 1936 problem of classifying, up to homotopy, the maps from a solenoid complement 𝑆3\Σ to the 2-sphere - is essentially hyperfinite but not smooth. Fundamental to our analysis is the fact that the Čech cohomology functors 𝑋 ↦→ ˇH𝑛(𝑋;𝐺) admit two main formulations: a more combinatorial one and a more homotopical formulation as the group [𝑋, 𝑃] of homotopy classes of maps from X to a polyhedral 𝐾(𝐺, 𝑛) space P. We describe the Borel-definable content of each of these formulations and prove a definable version of Huber's theorem reconciling the two. In the course of this work, we record definable versions of Urysohn's Lemma and the simplicial approximation and homotopy extension theorems, along with a definable Milnor-type short exact sequence decomposition of the space Map(𝑋, 𝑃) in terms of its subset of phantom maps; relatedly, we provide a topological characterization of this set for any locally compact Polish space X and polyhedron P. In aggregate, this work may be more broadly construed as laying foundations for the descriptive set theoretic study of the homotopy relation on such spaces Map(𝑋, 𝑃), a relation which, together with the more combinatorial incarnation of ˇH𝑛, embodies a substantial variety of classification problems arising throughout mathematics. We show, in particular, that if P is a polyhedral H-group, then this relation is both Borel and idealistic. In consequence, [𝑋, 𝑃] falls in the category of definable groups, an extension of the category GPC introduced herein for its regularity properties, which facilitate several of the aforementioned computations. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called ϕ -u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On ϕ -exact sequence and ϕ -projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of ϕ -flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On ϕ -Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on ϕ -Dedekind rings and ϕ -Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly ϕ -flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the ϕ -injective dimension, ϕ -projective dimension and ϕ -flat dimension for modules, and also the ϕ -(weak) global dimension of rings. Then, using these dimensions, we characterize several ϕ -rings (ϕ -Prüfer, ϕ -chained, ϕ -von Neumann, etc). Finally, we study the ϕ -(weak) global dimension of the trivial ring extensions defined by some conditions. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with a nonzero identity and M an R -module. Set ϕ -tor  (M) = { x ∈ M | s x = 0  for some  s ∈ R \ Nil (R) } , if ϕ - tor (M) = M then M is called a ϕ -torsion module. An R -module M is said to be ϕ -flat, if 0 → A ⊗ R M → B ⊗ R M → C ⊗ R M → 0 is an exact R -sequence, for any exact sequence of R -modules 0 → A → B → C → 0 , where C is ϕ -torsion. In this paper, we study some new properties of ϕ -flat modules. Then we introduce and study the class of ϕ - P -flat modules which is a generalization of ϕ -flat modules and P -flat modules. Finally, we give some new characterizations of the ϕ -von Neumann regular ring and its transfer to various contexts of constructions such as the amalgamation of rings along an ideal and trivial ring extension. [ABSTRACT FROM AUTHOR]

We investigate sections of the arithmetic fundamental group $\pi _1(X)$ where X is either a smooth affinoid p-adic curve , or a formal germ of a p-adic curve , and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, if X admits a compactification Y , and the exact sequence of $\pi _1(X)$ splits , then $\text {index} (Y)=1$. We also exhibit a necessary and sufficient condition for a section of $\pi _1(X)$ to arise from a rational point of Y. One of the key ingredients in our investigation is the fact, we prove in this paper in case X is affinoid, that the Picard group of X is finite. [ABSTRACT FROM AUTHOR]

Let X be a compact Riemann surface of genus g ≥ 3 . Let L = (L , [. ,. ] , ♯) be a holomorphic Lie algebroid over X of rank one and degree (L) < 2 − 2 g . We consider the moduli space of holomorphic L λ -connections over X, where λ ∈ C . We compute the Picard group of the moduli space of L λ -connections by constructing an explicit smooth compactification of the moduli space of those L λ -connections whose underlying vector bundle is stable such that the complement is a smooth divisor. We also show that the automorphism group of the moduli space of L λ -connections fits into a short exact sequence that involves the automorphism group of the moduli space of stable vector bundle over X. For λ = 1, we get Lie algebroid de Rham moduli space of L -connections and we determine its Chow group. Communicated by Manuel Reyes [ABSTRACT FROM AUTHOR]

Let K/F be a finite extension of number fields and S be a finite set of primes of F, including all the Archimedean ones. In this paper, using some results of González-Avilés (J Reine Angew Math 613:75–97, 2007), we generalize the notions of the relative Pólya group Po (K / F) (Chabert in J Number Theory 203:360–375, 2019; Maarefparvar and Rajaei in J Number Theory 207:367-384, 2020) and the Ostrowski quotient Ost (K / F) (Shahoseini et al. in Pac J Math 321(2):415–429, 2022) to their S-versions. Using this approach, we obtain generalizations of some well-known results on the S-capitulation map, including an S-version of Hilbert’s Theorem 94. [ABSTRACT FROM AUTHOR]

Let Γ be a congruence subgroup of level N in GL n (Z). Let P be a maximal Q -parabolic subgroup of GL n / Q , with unipotent radical U , and let Q = (P ∩ Γ) / (U ∩ Γ). Let p > dim Q ⁡ (U (Q)) + 1 be a prime number that does not divide N. Let M be a (U , p) -admissible Γ-module. Consider the Lyndon-Hochschild-Serre spectral sequence arising from the exact sequence 1 → U ∩ Γ → P ∩ Γ → Q → 1 , which abuts to H ⁎ (P ∩ Γ , M). We show that if M is a trivial U ∩ Γ -module, then certain classes in the E 2 page survive to E ∞. We use this to obtain information about classes in H ⁎ (P ∩ Γ , M) even if M is not a trivial U ∩ Γ -module. This information will be used in future work to prove a Serre-type conjecture for sums of two irreducible Galois representations. [ABSTRACT FROM AUTHOR]

The article studies bundle towers , , with fiber , where are compact smooth nilmanifolds and is a group of polynomial transformations of the line . The focus is on the well-known problem of calculating cohomology rings with rational coefficients of manifolds . Using the canonical bigradation in the de Rham complex of manifolds , we introduce the concept of polynomial Eulerian characteristic and calculate it for these manifolds. [ABSTRACT FROM AUTHOR]

Let be an abelian category with enough projectives. In the category of -complexes, we introduce generalized Tate cohomology with respect to subcategories, and show that there is an Avramov–Martsinkovsky type exact sequence connecting the absolute cohomology, relative cohomology and generalized Tate cohomology. [ABSTRACT FROM AUTHOR]

We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context - or equivalently for a strongly stratifying ideal - we show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. One of the main consequences is that Han's conjecture holds for an algebra admitting a strongly (co-)stratifying chain whose steps verify Han's conjecture. If Han's conjecture is true for local algebras and an algebra Λ admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for Λ. [ABSTRACT FROM AUTHOR]

In this paper, we will compute the non-abelian cohomology of the universal complete curve in positive characteristic. This extends Hain's result on the non-abelian cohomology of generic curves in characteristic zero to positive characteristics. Furthermore, we will prove that the exact sequence of etale fundamental groups of the universal n-punctured curve in positive characteristic does not split. [ABSTRACT FROM AUTHOR]

In this paper, we give a new description of the relative topological structure group of PL manifolds with boundary, and put the relative L -group into an exact sequence of groups. Then we define the relative index of PL manifolds with boundary, and show that this relative index will induce an additive map from the relative L -group to the K -theory of the relative Roe algebra. [ABSTRACT FROM AUTHOR]

· CE-DAD-ESI-MS enables for short peptide sequences identification and quantitation. · A simple quadrupole MS allows for crude peptide product purity determination. · Peptide impurities as small as 1.3 % were evidenced and identified in 0.5 mg.ml−1 sample solutions in water solvent. Nanostructures formed by the self-assembling peptide building blocks are attractive materials for the design of theranostic objects due to their intrinsic biocompatibility, accessible surface chemistry as well as cavitary morphology. Short peptide synthesis and modification are straightforward and give access to a great diversity of sequences, making them very versatile building blocks allowing for the design of thoroughly controlled self-assembled nanostructures. In this work, we developed a new CE-DAD-ESI-MS method to characterize short synthetic amphiphilic peptides in terms of exact sequence and purity level in the low 0.1 mg.mL−1 range, without sample treatment. This study was conducted using a model sequence, described to have pH sensitive self-assembling property. Peptide samples obtained from different synthesis processes (batch or flow, purified or not) were thus separated by capillary zone electrophoresis (CZE). The associated dual UV and MS detection mode allowed to evidence the exact sequence together with the presence of impurities, identified as truncated or non-deprotected sequences, and to quantify their relative proportion in the peptide mixture. Our results demonstrate that the developed CE-DAD-ESI-MS method could be directly applied to the characterization of crude synthetic peptide products, in parallel with the optimization of peptide synthetic pathway to obtain controlled sequences with high synthetic yield and purity, which is crucial for further design of robust peptide based self-assembled nanoarchitectures. [ABSTRACT FROM AUTHOR]

In this paper, we consider some properties of commutative diagrams of Rees short exact sequences, and we also investigate the sufficient and necessary condition under which the induced sequences by functors − ⊗ M for the left S-act M. The main conclusions extend some known results. Further, we investigate preenvelopes and precovers in the category 퓔S of Rees short exact sequences of right S-acts. [ABSTRACT FROM AUTHOR]

An abelian extension of a Novikov algebra by a module is a short exact sequence which induces exactly the same module structure as prescribed. By applying the cohomology of Chevalley-Eilenberg type we show that all abelian extensions are classified by the subspace of the second cohomology group given by quasi-associative bilinear forms. [ABSTRACT FROM AUTHOR]

Taxus is a genus of coniferous shrubs and trees, commonly known as the yews, in the family Taxaceae. All species of yew contain taxine alkaloids, which are ascribed as the toxic principles. Anecdotally, free ranging ruminants such as antelope, deer, elk, and moose have been regarded as tolerant to yew. Herein several cases of intoxication of deer, elk, and moose by yew from the state of Utah in the winter of 2022–2023 are documented. Ingestion of yew was documented by three means among the poisoned cervids; plant fragments consistent with yew were visually observed in the rumen contents, chemical analysis, and subsequent detection of the taxines from rumen and liver contents, and identification of exact sequence variants identified as Taxus species from DNA metabarcoding. Undoubtedly, the record snowfall in Utah during the winter of 2022–2023 contributed to these poisonings. [Display omitted] • Taxus (Yew) is acutely toxic to wild ungulates including deer, elk, and moose. • Taxines were detected in the rumen contents of poisoned wildlife. • Ingestion of yew was documented by DNA metabarcoding. [ABSTRACT FROM AUTHOR]

There is a natural connection between the third homology of SL 2 (A) and the refined Bloch group RB (A) of a commutative ring A. In this article we investigate this connection and as the main result we show that if A is a universal GE 2 -domain such that − 1 ∈ A × 2 , then we have the exact sequence H 3 (SM 2 (A) , Z) → H 3 (SL 2 (A) , Z) → RB (A) → 0 , where SM 2 (A) is the group of monomial matrices in SL 2 (A). Moreover, we show that RP 1 (A) , the refined scissors congruence group of A , is naturally isomorphic to the relative homology group H 3 (SL 2 (A) , SM 2 (A) ; Z). [ABSTRACT FROM AUTHOR]

Let A = F q [ t ] be the polynomial ring over a finite field F q and let ϕ and ψ be A -Drinfeld modules. In this paper we consider the group Ext 1 (ϕ , ψ) with the Baer addition. We show that if rank ϕ > rank ψ then Ex t 1 (ϕ , ψ) has the structure of a t -module. We give complete algorithm describing this structure. We generalize this to the cases: Ex t 1 (Φ , ψ) where Φ is a t -module and ψ is a Drinfeld module and Ex t 1 (Φ , C ⊗ e) where Φ is a t -module and C ⊗ e is the e -th tensor product of Carlitz module. We also establish duality between Ext groups for t -modules and the corresponding adjoint t σ -modules. Finally, we prove the existence of " Hom − Ext " six-term exact sequences for t -modules and dual t -motives. As the category of t -modules is only additive (not abelian) this result is nontrivial. [ABSTRACT FROM AUTHOR]

Our objectives were to 1) determine diet selection of goats and cattle grazing woody plant-encroached tallgrass prairie, and, 2) determine the effect of the presence of goats on cattle diet selection. In 2019, pastures (n = 6) were allocated to 1 of 2 experimental grazing treatments. Pastures (n = 3) grazed by cattle only (crossbred Angus yearling heifers) were moderately grazed year-round at 4.2 ha per heifer. Pastures (n = 3) grazed by cattle (similar heifers) and goats (Boer x Spanish crossbred mature does) were moderately grazed year-round at 4.7 ha per heifer and 2.3 ha per goat simultaneously. Each pasture was burned in 6 patches (2 patches per year) to facilitate range management objectives. Every 35 d from May 2022 until June 2023, fecal samples were collected from all animals and stored frozen and later composited into 2 composites per animal species per pasture. Composites were then analyzed by a commercial lab with DNA barcoding and high-throughput sequencing. For statistical analysis, the top 50 Exact Sequence Variants (ESVs), representing species of forage, across all treatments were used for analysis. Each ESV was georeferenced to eliminate plants that did not grow in the study area, and assigned to a representative functional group: graminoid, forb, legume, and woody plant. Analysis of variance and Tukey's post-hoc separation of estimated marginal means were performed to assess the effects of season, grazer species, and the interaction. The interaction of season and grazer species was significant (P < 0.01) for the percentage of diet selected from each functional group of plants. As expected, goats selected a greater percentage of their diet as woody plants in summer (57%) and winter (74%) compared with cattle (27%). Cattle selected more grass in winter (34%) than goats (11%). Goats selected more forbs in spring (33%) than cattle (10%). However, goats selected a lesser percentage of their diet as legumes in summer (16%) than cattle (58%), which was unexpected. Overall, these results indicate that both species of grazers select a diverse diet from these diverse pastures, and selection is dynamic over time. Goat presence in a pasture does not seem to drastically alter cattle selection, actually complementing cattle in these pastures where goats can be used to target species cattle do not prefer. [ABSTRACT FROM AUTHOR]

The second cohomology group of a left skew brace with coefficients in a trivial left brace with non-trivial actions is defined, its connection with extensions of a left skew brace by a trivial brace is established and a Wells' like exact sequence relating the second cohomology group with inducible automorphisms of an extension of left skew braces is constructed. [ABSTRACT FROM AUTHOR]