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To study the set of torsion classes of a finite dimensional basic algebra over a field, we use a decomposition, called sign-decomposition, parameterized by elements of { ± 1 } n where n is the number of simple modules. If A is an algebra with radical square zero, then for each ϵ ∈ { ± 1 } n there is a hereditary algebra A ϵ ! with radical square zero and a bijection between the set of torsion classes of A associated to ϵ and the set of faithful torsion classes of A ϵ !. Furthermore, this bijection preserves the property of being functorially finite. From a point of view of tilting theory, it implies that there is a bijection between the set of isomorphism classes of basic two-term silting complexes for A associated to ϵ and the set of isomorphism classes of basic tilting A ϵ ! -modules. As an application, we prove that the number of two-term tilting complexes over Brauer line algebras (respectively, Brauer cycle algebras) having n edges is ( 2 n n ) (respectively, 2 2 n − 1 if n is odd, and ∞ if n is even). [ABSTRACT FROM AUTHOR]

The categorical treatment of fuzzy modules presents some problems, due to the well known fact that the category of fuzzy modules is not abelian, and even not normal. Our aim is to give a representation of the category of fuzzy modules inside a generalized category of modules, in fact, a functor category, Mod − P , which is a Grothendieck category. To do that, first we consider the preadditive category P , defined by the interval P = (0 , 1 ] , to build a torsionfree class J in Mod − P , and a hereditary torsion theory in Mod − P , to finally identify equivalence classes of fuzzy submodules of a module M with F-pair, which are pair (G , F) , of decreasing gradual submodules of M, where G belongs to J , satisfying G = F d , and ∪ α F (α) is a disjoint union of F (1) and F (α) \ G (α) , where α is running in (0 , 1 ] . [ABSTRACT FROM AUTHOR]

Let k be a number field, p ≥ 2 a prime and S ≠ ∅ a set of finite places of k. We call K / k a totally S -ramified cyclic p -tower if Gal (K / k) ≃ ℤ / p N ℤ and if S is totally ramified. Using analogues of Chevalley's formula [G. Gras, Invariant generalized ideal classes—Structure theorems for p-class groups in p-extensions, Proc. Math. Sci.127(1) (2017) 1–34], we give an elementary proof of a stability theorem (Theorem 3.1) for generalized p -class groups n of the layers k n ⊆ K : let λ = max (0 , # S − 1 − ρ) given in Definition 1.1 (ρ = r 1 (k) + r 2 (k) − 1 for ordinary class groups); then # n = # 0 ⋅ p λ ⋅ n for all n ∈ [ 0 , N ] , if and only if # 1 = # 0 ⋅ p λ . This improves the case λ = 0 of [T. Fukuda, Remarks on ℤ p -extensions of number fields, Proc. Japan Acad. Ser. A70(8) (1994) 264–266; J. Li, Y. Ouyang, Y. Xu and S. Zhang, l-Class groups of fields in Kummer towers, Publ. Mat.  66(1) (2022) 235–267; Y. Mizusawa, K. Yamamoto, On p -class groups of relative cyclic p -extensions, Arch. Math.  117(3) (2021) 253–260] whose techniques are based on Iwasawa's theory or Galois theory of pro- p -groups. We deduce promising capitulation properties of 0 in the tower giving Conjecture 4.1. Finally, we apply our principles to the torsion groups n of abelian p -ramification theory. Numerical examples are given with PARI programs. [ABSTRACT FROM AUTHOR]

We construct families of elliptic curves defined over number fields and containing torsion groups Z / M 1 Z × Z / M 2 Z where (M 1 , M 2) belongs to { (1 , 11) , (1, 14), (1, 15), (2, 10), (2, 12), (3, 9), (4, 8), (6 , 6) } (i.e., when the corresponding modular curve X 1 (M 1 , M 2) has genus 1). We provide formulae for the curves and give examples of number fields for which the corresponding elliptic curves have non-zero ranks, giving explicit generators using D. Simon's program whenever possible. The reductions of these curves can be used to speed up ECM for factoring numbers with special properties, a typical example being (factors of) Cunningham numbers b n - 1 such that M 1 ∣ n . We explain how to find points of potentially large orders on the reduction, if we accept to use quadratic twists. [ABSTRACT FROM AUTHOR]

In this paper, we consider the quadratic Metric-Affine Gauge gravity Lagrangian, which contains all the algebraic invariants up to quadratic order in torsion, nonmetricity and curvature. The goal is to collect the known exact solutions for this theory that describe the propagation of a gravitational wave and some useful properties. We concentrate on solutions whose connections are different from the Levi-Civita one and provide also the extension of already known axial torsion solutions to more general Lagrangians. In the end, we will briefly discuss colliding waves and Riemannian solutions. [ABSTRACT FROM AUTHOR]

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We give a classification of the cuspidal automorphic representations attached to rational elliptic curves with a non‐trivial torsion point of odd order. Such elliptic curves are parameterizable, and in this paper, we find the necessary and sufficient conditions on the parameters to determine when split or non‐split multiplicative reduction occurs. Using this and the known results on when additive reduction occurs for these parametrized curves, we classify the automorphic representations in terms of the parameters. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the cosmological dynamics in a spatially flat Friedmann–Lemaître–Robertson–Walker geometry in scalar-tensor and scalar-torsion theories where the nonminimally coupled scalar field is a complex field. We derive the cosmological field equations and we make use of dimensionless variables in order to determine the stationary points and determine their stability properties. The physical properties of the stationary points are discussed while we find that the two-different theories, scalar-tensor and scalar-torsion theories, share many common features in terms of the evolution of the physical variables in the background space. [ABSTRACT FROM AUTHOR]

By using the Mellin integral transformation, we obtain the solutions of axisymmetric problems of the theory of elasticity for a cone, namely, of the first and mixed boundary-value problems and the problem of torsion. It is shown that, in the cases where one of the boundary conditions on the entire surface of the cone or on a part of its surface is inhomogeneous and possesses the inversion symmetry and the other boundary condition is homogeneous, some components of the solution also have the inversion symmetry. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1 1= 0 and M be an R-module. Suppose that S R is a multiplicatively closed set of R. Recently Sevim et al. in [19] introduced the notion of an S -prime submodule which is a generalization of a prime submodule and used them to characterize certain classes of rings/modules such as prime submodules, simple modules, torsion free modules, S -Noetherian modules and etc. Afterwards, in [2], Anderson et al. defined the concepts of S -multiplication modules and S -cyclic modules which are S -versions of multiplication and cyclic modules and extended many results on multiplication and cyclic modules to S -multiplication and S -cyclic modules. Here, in this article, we introduce and study S -comultiplication modules which are the dual notion of S -multiplication module. We also characterize certain classes of rings/modules such as comultiplication modules, S -second submodules, S -prime ideals and S -cyclic modules in terms of S -comultiplication modules. Moreover, we prove S -version of the dual Nakayama's Lemma. [ABSTRACT FROM AUTHOR]

Let Fo be an absolutely algebraic field of characteristic p > 0 and k an infinite cardinal. It is shown that there exists a field F such that F* = F0 ® (®KQ) with Br(F) = {0}. Let L be an algebraic closure of F. Then for any finite subextension K of L/F, we have K* = T(K*) ® (®KQ), where T(K*) is the group of torsion elements of K*. In addition, Br(K) = {0} and [K: F] = [T(K*) U{0}: Fo]. [ABSTRACT FROM AUTHOR]

Let R be a ring graded by a group G and n > 1 an integer. We introduce the notion of n-FCP-gr-projective R-modules and by using of special finitely copresented graded modules, we investigate that (1) there exist some equivalent characterizations of n-FCP-gr-projective modules and graded right modules of n-FCP-gr-projective dimension at most k over n-gr-cocoherent rings, (2) R is right n-gr-cocoherent if and only if for every short exact sequence 0 A B C 0of graded right R-modules, where B and C are n-FCP-gr-projective, it follows that A is n-FCP-gr-projective if and only if (gr-FCPn, gr-FCP) is a hereditary cotorsion theory, where gr-FCPn denotes the class of n-FCP-gr-projective right modules. Then we examine some of the conditions equivalent to that each right R-module is n-FCP-gr-projective. [ABSTRACT FROM AUTHOR]

In this paper, we study the cotorsion dimensions and cotorsion triples in the recollements of abelian categories. The main results are that recollements induce new (resp. complete hereditary) cotorsion triples from the middle category and that the cotorsion dimensions are bounded under certain conditions. As an application, the cotorsion triples in the recollements of module categories with respect to triangular matrix algebras are recovered. [ABSTRACT FROM AUTHOR]

We consider the most general quadratic metric-affine gravity setup in the presence of generic matter sources with non-vanishing hypermomentum. The gravitational action consists of all 17 quadratic invariants (both parity even and odd) in torsion and non-metricity as well as their mixings, along with the terms that are linear in the curvature namely the Ricci scalar and the totally antisymmetric Riemann piece. Adding also a matter sector to the latter we first obtain the field equations for the generalized quadratic theory. Then, using a recent theorem, we successfully find the exact form of the affine connection under some quite general non-degeneracy conditions. Having obtained the exact and unique solution of the affine connection we subsequently derive the closed forms of spacetime torsion and non-metricity and also recast the metric field equations into a GR form with modified source terms that are quadratic in the hypermomentum and linear in its derivatives. We also study the vacuum quadratic theory and prove that in this instance, or more generally for vanishing hypermomentum, the connection becomes the Levi-Civita one. Therefore, we also find exactly to what does the quadratic vacuum theory correspond to. Finally, we generalize our result even further and also discuss the physical consequences and applications of our study. [ABSTRACT FROM AUTHOR]

Mordell curves over a number field K are elliptic curves of the form y² = x³ + c, where c ∈ K \ {0}. Let p ≥ 5 be a prime number, K a number field such that [K : ℚ] ∈ {2p, 3p}. We classify all the possible torsion subgroups E(K)tors for all Mordell curves E defined over ℚ when [K : ℚ] ∈ {2p, 3p}. [ABSTRACT FROM AUTHOR]

Let Λ be an Artin algebra. In 2014, Adachi, Iyama, and Reiten proved that the torsion functorially finite classes in mod (Λ) can be described by the τ-tilting theory. The aim of this paper is to introduce the notion of F-torsion class in mod (Λ) , where F is an additive subfunctor of Ext Λ 1 , and to characterize when these classes are preenveloping and F-preenveloping. In order to do that, we introduce the notion of F-presilting Λ-module. The latter is both a generalization of τ-rigid and F-tilting in mod (Λ). [ABSTRACT FROM AUTHOR]

This work considers the modeling of torsion in elastic shafts accounting for the non-uniform warping of the cross sections along them. In this paper, we present an analysis of (1) the original formulation of Timoshenko–Wagner–Kappus–Vlasov, consisting of the underlying Saint-Venant torsion but with a non-constant rate of twist defining the warping magnitude, and (2) the alternative formulation first considered by Reissner–Benscoter–Vlasov involving an independent field for the warping amplitude. The theoretical results presented here characterize the kinematically constrained character of the first of these formulations, noting in the process the anomalies resulting from the full restrainment of the warping at a cross section in this setting. New explicit expressions for the warping shear stress and other features are obtained in this context. The second formulation relaxes the constraint between warping and twisting, with the analyses presented here identifying explicitly for the first time how its enforcement can be achieved in a limit process controlled by a parameter depending on the cross-sectional geometry. Hence, it avoids the anomalies of the first formulation, but at the price to involve local stresses not in equilibrium, a situation that does not seem to be much present in the literature. [ABSTRACT FROM AUTHOR]

Let q be a power of the prime number p , let K = F q (t) , and let r ⩾ 2 be an integer. For points a , b ∈ K which are F q -linearly independent, we show that there exist positive constants N 0 and c 0 such that for each integer ℓ ⩾ N 0 and for each generator τ of F q ℓ / F q , we have that for all except N 0 values λ ∈ F q ‾ , the corresponding specializations a (τ) and b (τ) cannot have orders of degrees less than c 0 log ⁡ log ⁡ ℓ as torsion points for the Drinfeld module Φ (τ , λ) : F q [ T ] ⟶ End F q ‾ (G a) (where G a is the additive group scheme), given by Φ T (τ , λ) (x) = τ x + λ x q + x q r . [ABSTRACT FROM AUTHOR]

Let R be a ring, Q a small k -preadditive category, and let Q , R Mod be the category of k -linear functors from Q to the left R -module category R Mod. Given a cotorsion pair (, ℬ) in R Mod , we construct four cotorsion pairs (Γ () , Γ () ⊥) , (⊥ Γ (ℬ) , Γ (ℬ)) , (Θ () , Θ () ⊥) and (⊥ ξ (ℬ) , ξ (ℬ)) in Q , R Mod and investigate when these cotorsion pairs are hereditary and complete. Moreover, under few assumptions, we show that there exist a recollement of homotopy categories induced by these cotorsion pairs. Finally, some applications are given in the category of N -periodic chain complexes of R -modules. [ABSTRACT FROM AUTHOR]

In [H. Rashidi, A. Golchin and H. Mohammadzadeh saany, On G P W -flat acts, Categ. Gen. Algebr. Struct. Appl. 12(1) (2020) 175–197], the study of G P W -flatness property of right acts A S over a monoid S that can be described by means of when the functor A S ⊗ S -preserves some pullbacks is initiated. In this paper, we extend these results to S -posets and present equivalent description of G P W -po-flatness of S -posets. We show that G P W -flatness does not imply torsion freeness in S -posets and give some general properties and a characterization of pomonoids for which some other properties of their posets imply this condition. [ABSTRACT FROM AUTHOR]

Given a finite dimensional algebra A over a field k , and a finite acyclic quiver Q , let Λ = A ⊗ k k Q / I , where kQ is the path algebra of Q over k and I is a monomial ideal. This paper is devoted to studying the separated monic representations over the bounded quiver (Q , I) over a subcategory X of A -mod, denoted by smon (Q , I , X). We show that (X , Y) is a (complete) hereditary cotorsion pair in A -mod if and only if (smon (Q , I , X) , rep (Q , I , Y)) is a (complete) hereditary cotorsion pair in Λ-mod. We also show that A is left weakly Gorenstein if and only if so is Λ. Provided that k Q / I is non-semisimple, the category ▪ of semi-Gorenstein-projective Λ-modules coincides with the category of separated monic representations ▪ if and only if A is left weakly Gorenstein. [ABSTRACT FROM AUTHOR]

We introduce image-cokernel-extension-closed (ICE-closed) subcategories of module categories. This class unifies both torsion classes and wide subcategories. We show that ICE-closed subcategories over the path algebra of Dynkin type are in bijection with basic rigid modules and that ICE-closed subcategories are precisely torsion classes in some wide subcategories. We also study natural maps from ICE-closed subcategories to torsion classes and wide subcategories in terms of rigid modules. Finally, we prove that the number of ICE-closed subcategories does not depend on the orientation of the quiver, and give an explicit formula for each Dynkin type, which is equal to the large Schröder number for type A. [ABSTRACT FROM AUTHOR]

We generalize some results of Greither and Popescu to a geometric Galois cover X → Y which appears naturally for example in extensions generated by p n -torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz–Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in [1]) with Goss ζ -function. [ABSTRACT FROM AUTHOR]

In this paper, we present a full-reference quality assessment metric based on the information of visual saliency. The saliency information is provided under the form of degrees associated to each vertex of the surface mesh. From these degrees, statistical attributes reflecting the structures of the reference and distorted meshes are computed. These are used by four comparisons functions genetically optimized that quantify the structure differences between a reference and a distorted mesh. We also present a statistical comparison study of six full-reference quality assessment metrics for 3D meshes. We compare the objective metrics results with humans subjective scores of quality considering the 3D meshes in one hand and the distorsion types in the other hand. Also, we show which metrics are statistically superior to their counterparts. For these comparisons we use the Spearman Rank Ordered Correlation Coefficient and the hypothetic test of Student (ttest). To attest the pertinence of the proposed approach, a comparison with a ground truth saliency and an application associated to the assessment of the visual rendering of smoothing algorithms are presented. Experimental results show that the proposed metric is very competitive with the state-of-the-art. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to introduce the so-called ideal approximation theory into higher homological algebra. To this end, we introduce some important notions from approximation theory into the theory of n-exact categories and prove some results. In particular, the higher version of notions such as ideal cotorsion pairs, phantom ideals, Salce's Lemma and Wakamatsu's Lemma for ideals are introduced and studied. Our results motivate the definitions and show that n-exact categories are the appropriate context for the study of higher ideal approximation theory. [ABSTRACT FROM AUTHOR]

We propose a new, infinite class of brackets generalizing the Frölicher–Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of operators. We also prove that the vanishing of a higher-level Nijenhuis torsion of an operator field is a sufficient condition for the integrability of its eigen-distributions. This result (which does not require any knowledge of the spectral properties of the operator) generalizes the celebrated Haantjes theorem. The same vanishing condition also guarantees that the operator can be written, in a local chart, in a block-diagonal form. [ABSTRACT FROM AUTHOR]

A novel theory of torsion of thin walled beams ("shear deformable beams") of arbitrary open cross-sections with influence of shear (TTTS) is presented. The theory is based on the classical Vlasov's theory of thin-walled beams of open cross-section, as well on the Timoshenko's beam bending theory. The theory is valid for general thin-walled open cross-section shapes, for distributed and concentrated transverse torsion loads and for isotropic materials. Four parameters due to shear warping are introduced. Closed-form analytical solutions for three-dimensional expressions of the normal and shear stresses are obtained. Under general transverse loads, reduced to moments of torsion with respect to the cross-section principal pole (by Vlasov's theory), for arbitrary cross-sections, the beam will be subjected to torsion with influence of shear and in addition to bending due to shear in the beam principal planes and to tension/compression due to shear. The theory is tested by comparing it to the classical Vlasov's thin-walled beam theory and to the finite element method using shell elements, as well as to some results in literatures. [ABSTRACT FROM AUTHOR]

Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra 𝓐 on X. Then we study the moduli scheme of torsion free 𝓐-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface. [ABSTRACT FROM AUTHOR]

We describe new Chow-weight (co)homology theories on the category DM^{\mathrm {eff}}_{gm}(k,R) of effective geometric Voevodsky motives (R is the coefficient ring). These theories are interesting "modifications" of motivic homology; Chow-weight homology detects whether a motive M\in ObjDM^{\mathrm {eff}}_{gm}(k,R) is r-effective (i.e., belongs to the rth Tate twist DM^{\mathrm {eff}}_{gm}(k,R)(r) of effective motives), bounds the weights of M (in the sense of the Chow weight structure defined by the first author), and detects the effectivity of "the lower weight pieces" of M. Moreover, we calculate the connectivity of M (in the sense of Voevodsky's homotopy t-structure, i.e., we study motivic homology) and prove that the exponents of the higher motivic homology groups (of an "integral" motive) are finite whenever these groups are torsion. We apply the latter statement to the study of higher Chow groups of arbitrary varieties. These motivic properties of M have plenty of applications. They are closely related to the (co)homology of M; in particular, if the Chow groups of a variety X vanish up to dimension r-1 then the highest Deligne weight factors of the (singular or étale) cohomology of X with compact support are r-effective. Our results yield vast generalizations of the so-called "decomposition of the diagonal" theorems, and we re-prove and extend some of earlier statements of this sort. [ABSTRACT FROM AUTHOR]

Let M be a torsion-free module over an integral domain D with quotient field K. We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module M is a v -multiplication module if and only if (M) v is a maximal v -submodule and ( − 1 ) v 1 = D for every minimal prime ideal of D. If M is a finitely generated Krull module, then M 1 = ∩ M is a Krull module and v -multiplication module. It is also shown that the following three conditions are equivalent: M is completely integrally closed, M [ x ] is completely integrally closed, and M [ [ x ] ] is completely integrally closed. [ABSTRACT FROM AUTHOR]

Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group WGK is connected, then a certain category of rational G-spectra "at K" has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this expository note, I present some of the key features of the lattice of torsion classes of a finite-dimensional algebra, focussing in particular on its complete semidistributivity and consequences thereof. This is intended to serve as an introduction to recent work by Barnard–Carroll–Zhu and Demonet–Iyama–Reading–Reiten–Thomas. [ABSTRACT FROM AUTHOR]

It is known that the order of a finite quandle annihilates its reduced quandle homology and the torsion subgroup of its rack homology if the quandle is quasigroup. However, this does not hold in general if a quandle is connected. In this paper, we prove that under a certain condition, the reduced quandle homology and the torsion subgroup of the rack homology of a connected quandle are annihilated by the order of the inner automorphism group of the quandle. [ABSTRACT FROM AUTHOR]

Let G be a compact connected Lie group and K a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of G and K is invertible in a given principal ideal domain k. It is known that in this case the cohomology of the homogeneous space G/K with coefficients in k and the torsion product of H∗(BK) and k over H∗(BG) are isomorphic as k‐modules. We show that this isomorphism is multiplicative and natural in the pair (G,K) provided that 2 is invertible in k. The proof uses homotopy Gerstenhaber algebras in an essential way. In particular, we show that the normalized singular cochains on the classifying space of a torus are formal as a homotopy Gerstenhaber algebra. [ABSTRACT FROM AUTHOR]

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsion. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group Γ implies that Γ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition. [ABSTRACT FROM AUTHOR]

We compute the mod p homology growth of residual sequences of finite index normal subgroups of right-angled Artin groups. We find examples where this differs from the rational homology growth, which implies the homology of subgroups in the sequence has lots of torsion. More precisely, the homology torsion grows exponentially in the index of the subgroup. For odd primes p, we construct closed locally CAT(0) manifolds with nonzero mod p homology growth outside the middle dimension. These examples show that Singer's conjecture on rational homology growth and Lück's conjecture on torsion homology growth are incompatible with each other, so at least one of them must be wrong. [ABSTRACT FROM AUTHOR]

Adachi et al. have introduced support τ-tilting pairs from the viewpoint of mutation. Let (A, T, B) be a tilting triple. In this article, it is proved that there is a bijection between the left mutations of a basic support τ-tilting module T and the indecomposable splitting projective objects of Fac T , in the sense of Auslander and Smalø. If T is a separating and splitting tilting module, then there is a bijection between the basic τ-rigid pairs of A and of B. This bijection preserves the number of indecomposable direct summands of basic τ-rigid pairs, and hence restricts to a bijection between the basic support τ-tilting pairs of A and of B. Moreover, it restricts to a bijection between the basic partial tilting pairs of A and of B, and a bijection between the basic support tilting pairs of A and of B, in the sense of Ingalls and Thomas. This bijection also preserves mutations of support τ-tilting pairs, thus, the underlying graphs of the support τ-tilting quivers of A and of B are the same, but in general their orientations are not the same, and a sufficient and necessary condition so that left mutations are preserved is given. [ABSTRACT FROM AUTHOR]

Our first result provides a new characterization of Auslander algebras using a property of hereditary torsion pairs. The second result shows an Auslander algebra Λ is left or right glued if and only if Λ is representation-finite. Finally, our third result shows the module category of any Auslander algebra contains a tilting module with a particular property, which we call the hereditary property. Applications of this property are investigated. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Beijing University of Technology is the property of Journal of Beijing University of Technology, Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we explicitly classify the minimal discriminants of all elliptic curves E / Q with a non-trivial torsion subgroup. This is done by considering various parameterized families of elliptic curves with the property that they parameterize all elliptic curves E / Q with a non-trivial torsion point. We follow this by giving admissible change of variables, which give a global minimal model for E. We also provide necessary and sufficient conditions on the parameters of these families to determine the primes at which E has additive reduction. In addition, we use these parameterized families to give new proofs of results due to Frey and Flexor-Oesterlé pertaining to the primes at which an elliptic curve over a number field K with a non-trivial K-torsion point can have additive reduction. [ABSTRACT FROM AUTHOR]

Let (R , m , k) be an equicharacteristic one-dimensional complete local domain over an algebraically closed field k of characteristic 0. R. Berger conjectured that R is regular if and only if the universally finite module of differentials Ω R is a torsion-free R module. We give new cases of this conjecture by extending works of Güttes (Arch Math 54:499–510, 1990) and Cortiñas et al. (Math Z 228:569–588, 1998). This is obtained by constructing a new subring S of Hom R (m , m) and constructing enough torsion in Ω S , enabling us to pull back a nontrivial torsion to Ω R . [ABSTRACT FROM AUTHOR]

By reformulating and extending results of Elkies, we prove some results on Q -curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees ℓ that a Q -curve without CM may have are those degrees that are already possible over Q itself (in particular, ℓ ≤ 37 ), and we show the existence of a bound on the degrees of cyclic isogenies between Q -curves depending only on the degree of the field. We also prove that the only possible torsion groups of Q -curves over number fields of degree not divisible by a prime ℓ ≤ 7 are the 15 groups that appear as torsion groups of elliptic curves over Q . Complementing these theoretical results, we give an algorithm for establishing whether any given elliptic curve E is a Q -curve, that involves working only over Q (j (E)) . [ABSTRACT FROM AUTHOR]

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal of q [ T ] , the question essentially asks whether, up to isogeny, a Drinfeld module ϕ over q (T) contains a rational -torsion point if the reduction of ϕ at almost all primes of q [ T ] contains a rational -torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2 , but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive. [ABSTRACT FROM AUTHOR]

We introduce the notion of hereditary G-compactness (with respect to interpretation). We provide a sufficient condition for a poset to not be hereditarily G-compact, which we use to show that any linear order is not hereditarily G-compact. Assuming that a long-standing conjecture about unstable NIP theories holds, this implies that an NIP theory is hereditarily G-compact if and only if it is stable (and by a result of Simon, this holds unconditionally for ℵ 0 -categorical theories). We show that if G is definable over A in a hereditarily G-compact theory, then G A 00 = G A 000 . We also include a brief survey of sufficient conditions for G-compactness, with particular focus on those which can be used to prove or disprove hereditary G-compactness for some (classes of) theories. [ABSTRACT FROM AUTHOR]

This review paper is a continuation of two previous review papers devoted to properties of modules over discrete valuation domains. The first part of this work was published in the volumes 121 (Part I, Chaps. 1–4, Secs. 1–22) and 122 (Part II, Chaps. 5–8, Secs. 23–39) of the series Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory (English translation: Journal of Mathematical Sciences (New York), vol. 145, No. 4, pp. 4997–5117 (2007), and vol. 151, No. 5, pp. 3225–3371 (2008)). In this review paper, we preserve the numeration of chapters and sections of parts I and II. The first part consists of Chapter 9 "Appendix," Secs. 40–42. In Sec. 40, we consider p-adic torsion-free modules with isomorphic automorphism groups. Section 41 is devoted to torsion-free modules over a complete discrete valuation domain with isomorphic radicals of their endomorphism rings. The volume of the paper does not allow us to provide the proofs of all the results that appeared after publication of the previous parts and directly related to the issues under consideration in it. In the final Sec. 42, we describe some of these new results. [ABSTRACT FROM AUTHOR]

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. This extends the work in Part I to arbitrary semialgebraic sets. All previous algorithms proposed for this problem have doubly exponential complexity. [ABSTRACT FROM AUTHOR]

We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff's Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form "A poset L is a finite semidistributive lattice if and only if there exists a set with some additional structure, such that L is isomorphic to the admissible subsets of ordered by inclusion; in this case, and its additional structure are uniquely determined by L." The additional structure on is a combinatorial abstraction of the notion of torsion pairs from representation theory and has geometric meaning in the case of posets of regions of hyperplane arrangements. We show how the FTFSDL clarifies many constructions in lattice theory, such as canonical join representations and passing to quotients, and how the semidistributive property interacts with other major classes of lattices. Many of our results also apply to infinite lattices. [ABSTRACT FROM AUTHOR]

Let X (resp. Y) be a curve of genus 1 (resp. 2) over a base field k whose characteristic does not equal 2. We give criteria for the existence of a curve Z over k whose Jacobian is up to twist (2,2,2)-isogenous to the products of the Jacobians of X and Y. Moreover, we give algorithms to construct the curve Z once equations for X and Y are given. The first of these is based on interpolation methods involving numerical results over C that are proved to be correct over general fields a posteriori, whereas the second involves the use of hyperplane sections of the Kummer variety of Y whose desingularization is isomorphic to X. As an application, we find a twist of a Jacobian over Q that admits a rational 70-torsion point. [ABSTRACT FROM AUTHOR]