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Let $\Lambda$ be a finite dimensional algebra with an action by a finite group $G$ and $A:= \Lambda *G$ the skew group algebra. One of our main results asserts that the canonical restriction-induction adjoint pair of the skew group algebra extension $\Lambda \subset A$ induces a poset isomorphism between the poset of $G$-stable support $\tau$-tilting modules over $\Lambda$ and that of $(\!\!\!\mod G)$-stable support $\tau$-tilting modules over $A$. We also establish a similar poset isomorphism of posets of appropriate classes of silting complexes over $\Lambda$ and $A$. These two results generalize and unify preceding results by Huang-Zhang, Breaz-Marcus-Modoi and the second and the third authors. Moreover, we give a practical condition under which $\tau$-tilting finiteness and silting discreteness of $\Lambda$ are inherited to those of $A$. As applications we study $\tau$-tilting theory and silting theory of the (generalized) preprojective algebras and the folded mesh algebras. Among other things, we determine the posets of support $\tau$-tilting modules and of silting complexes over preprojective algebra $\Pi(\Bbb{L}_{n})$ of type $\Bbb{L}_{n}$., Comment: 2nd version: minor corrections

The existence of a tilting or silting object is an important feature for an algebraic triangulated category since it gives an equivalence with the derived category of a ring. By applying tilting theory, we study Cohen-Macaulay representations of $\mathbb N$-graded Artin-Schelter Gorenstein algebras $A=\bigoplus_{i\in \mathbb N}A_i$, where do not assume that $A_0$ is a field. This is a large class of noncommutative Gorenstein rings containing Gorenstein orders. In this paper, we concentrate on the case where $A$ has dimension one. Under the assumptions that $A$ is ring-indecomposable and $A_0$ has finite global dimension, we show that the stable category $\operatorname{\underline{\mathsf{CM}}}_0^{\mathbb Z}A$ always admits an (explicitly constructed) silting object. We also show that $\operatorname{\underline{\mathsf{CM}}}_0^{\mathbb Z}A$ admits a tilting object if and only if either $A$ is Artin-Schelter regular or the average Gorenstein parameter $p^A_\mathrm{av} \in {\mathbb Q}$ of $A$ is non-positive. These results are far-reaching generalizations of the results of Buchweitz, Iyama, and Yamaura. We give two different proofs of the second result; one is based on Orlov-type semiorthogonal decompositions, and the other is based on a more direct calculation. We apply our results to a Gorenstein tiled order $A$ to prove that $\operatorname{\underline{\mathsf{CM}}}^{\mathbb Z}A$ is equivalent to the derived category of the incidence algebra of an (explicitly constructed) poset. We also apply our results and Koszul duality to prove that the derived category $\mathsf{D}^\mathrm{b}(\operatorname{\mathsf{qgr}} A)$ of a smooth noncommutative projective quadric hypersurface $\operatorname{\mathsf{qgr}} A$ admits an (explicitly constructed) tilting object, which contains the tilting object of $\operatorname{\underline{\mathsf{CM}}}^{\mathbb Z}A$ due to Smith and Van den Bergh as a direct summand., Comment: 46 pages

For a Noetherian $R$-algebra $\Lambda$, there is a canonical inclusion $\mathsf{tors}\Lambda\to\prod_{\mathfrak{p}\in \mathrm{Spec} R}\mathsf{tors}(\kappa(\mathfrak{p})\Lambda)$, and each element in the image satisfies a certain compatibility condition. We call $\Lambda$ compatible if the image coincides with the set of all compatible elements. For example, for a Dynkin quiver $Q$ and a commutative Noetherian ring $R$ containing a field, the path algebra $RQ$ is compatible. In this paper, we prove that $RQ$ is compatible when $Q$ is an extended Dynkin quiver and $R$ contains a field and is either a Dedekind domain or a Noetherian semilocal normal ring of dimension two., Comment: 13 pages

In representation theory of graded Iwanaga-Gorenstein algebras, tilting theory of the stable category $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ of graded Cohen-Macaulay modules plays a prominent role. In this paper we study the following two central problems of tilting theory of $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ in the case where $A$ is finite dimensional: (1) Does $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ have a tilting object? (2) Does the endomorphism algebras of tilting objects in $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ have finite global dimension? To the problem (2) we give the complete answer. We show that the endomorphism algebra of any tilting object in $\underline{\mathsf{CM}}^{\mathbb{Z}}A$ has finite global dimension. To the problem (1) we give a partial answer. For this purpose, first we introduce an invariant $g(A)$ for a finite dimensional graded algebra $A$. Then, we prove that in the case where $A$ is 1-Iwanaga-Gorenstein, an inequality for $g(A)$ gives a sufficient condition that a specific Cohen-Macaulay module $V$ becomes a tilting object in the stable category. As an application, we study the existence of tilting objects in $\underline{\mathsf{CM}}^{\mathbb{Z}}\Pi(Q)_w$ where $\Pi(Q)_w$ is the truncated preprojective algebra of a quiver $Q$ associated to $w\in W_Q$. We prove that if the underling graph of $Q$ is tree, then $\underline{\mathsf{CM}}^{\mathbb{Z}}\Pi(Q)_w$ has a tilting object., Comment: 18 pages, we revised the introduction

Deformed preprojective algebras are generalizations of the usual preprojective algebras introduced by Crawley-Boevey and Holland, which have applications to Kleinian singularities, the Deligne-Simpson problem, integrable systems and noncommutative geometry. In this paper we offer three contributions to the study of such algebras: (1) the 2-Calabi-Yau property; (2) the unification of the reflection functors of Crawley-Boevey and Holland with reflection functors for the usual preprojective algebras; and (3) the classification of tilting ideals in 2-Calabi-Yau algebras, and especially in deformed preprojective algebras for extended Dynkin quivers., Comment: The main changes are (1) the proof of the PBW property has been cut, as we can quote a theorem of He, Van Oystaeyen and Zang and (2) the proof of Theorem 1.7 has been expanded, filling a small gap in the corresponding theorem by Buan, Iyama, Reiten and Scott

The aim of this paper is to unify classification theories of torsion classes of finite dimensional algebras and commutative Noetherian rings. For a commutative Noetherian ring $R$ and a module-finite $R$-algebra $\Lambda$, we study the set $\mathsf{tors} \Lambda$ (respectively, $\mathsf{torf}\Lambda$) of torsion (respectively, torsionfree) classes of the category of finitely generated $\Lambda$-modules. We construct a bijection from $\mathsf{torf}\Lambda$ to $\prod_{\mathfrak{p}} \mathsf{torf}(\kappa(\mathfrak{p}) \otimes_R \Lambda )$, and an embedding $\Phi_{\rm t}$ from $\mathsf{tors} \Lambda$ to $\mathbb{T}_R(\Lambda):=\prod_{\mathfrak{p}} \mathsf{tors}(\kappa(\mathfrak{p}) \otimes_R \Lambda)$, where $\mathfrak{p}$ runs all prime ideals of $R$. When $\Lambda=R$, these give classifications of torsionfree classes, torsion classes and Serre subcategories of $\mathsf{mod} R$ due to Takahashi, Stanley-Wang and Gabriel. To give a description of $\mathrm{Im} \Phi_{\rm t}$, we introduce the notion of compatible elements in $\mathbb{T}_R(\Lambda)$, and prove that all elements in $\mathrm{Im} \Phi_{\rm t}$ are compatible. We give a sufficient condition on $(R, \Lambda)$ such that all compatible elements belong to $\mathrm{Im} \Phi_{\rm t}$ (we call $(R, \Lambda)$ compatible in this case). For example, if $R$ is semi-local and $\dim R \leq 1$, then $(R, \Lambda)$ is compatible. We also give a sufficient condition in terms of silting $\Lambda$-modules. As an application, for a Dynkin quiver $Q$, $(R, RQ)$ is compatible and we have a poset isomorphism $\mathsf{tors} RQ \simeq \mathrm{Hom}_{\rm poset}(\mathrm{Spec} R, \mathfrak{C}_Q)$ for the Cambrian lattice $\mathfrak{C}_Q$ of $Q$., Comment: 43 pages, the structure of Section 2 was modified, Subsection 3.6 was added

High-strain-rate shear tests were conducted on a three-layered bonded test piece comprising a central aluminum layer with PMMA resin layers bonded on both sides. Upon calculating the displacement field and the strain field using digital image correlation (DIC), the crack tip was located, and the fracture toughness was evaluated at the Aluminum/PMMA bonding interface. As a result of the DIC, it was possible to determine the process by which 1) the elastic stress wave propagated to the aluminum section, 2) the wave was transmitted to the PMMA section, and 3) the crack developed at the interface. The tip of the crack was identified using displacement distributions obtained using DIC. The fracture toughness of the interface was evaluated using the stress intensity factor. The true interfacial stress was calculated by correcting the strain value at the interface obtained using DIC. The distribution of the stress suggested that mode II fracture appears in the present test method when the crack is sufficiently shorter than the length of the bonding interface, and mode I and mode II fractures appear when the crack is longer in comparison. Although the value of the stress intensity factor was disturbed by the error of the DIC analysis, it was confirmed that the obtained values were similar regardless of the difference in the crack length, upon averaging the stress intensity factor values from the crack tips to the long-range with a ratio of 1 to the subset in DIC. As the obtained stress intensity factor value was similar to the values calculated in the related literature, it can be concluded that the method proposed in this study yields a reasonable stress intensity factor.

Lithium-ion secondary batteries have been used in a wide variety of purposes, such as for powering mobile devices and electric vehicles, but their performance should be improved. One of the factors that limits their performance is the non-uniformity of the chemical reaction in the process of charging and discharging. Many attempts have been made to elucidate the mechanism behind this reaction non-uniformity. In this paper, to detect non-uniformity in various physical properties from Co K-edge two-dimensional X-ray absorption spectroscopy (2D-XAS) images of lithium ion batteries, we propose a method that consists of one-sided orthogonal non-negative matrix factorization in combination with removal of the reference signal. The difference between X-ray absorption spectra acquired at different positions in the battery is very small. However, even in such a situation, our method can decompose the 2D-XAS data into different spatial domains and their corresponding absorption spectra. From the spectral decomposition of the obtained absorption spectra, we confirmed a transition-energy shift of the main peak as evidence for a change in the state of charge and also found spectral changes due to orbital hybridization in the decomposed spectral components.

We develop silting theory of a noetherian algebra $\Lambda$ over a commutative noetherian ring $R$. We study mutation theory of $2$-term silting complexes of $\Lambda$, and as a consequence, we see that mutation exists. As in the case of finite dimensional algebras, functorially finite torsion classes of $\Lambda$ bijectively correspond to silting $\Lambda$-modules, if $R$ is complete local. We show a reduction theorem of $2$-term silting complexes of $\Lambda$, and by using this theorem, we study torsion classes of the module category of $\Lambda$. When $R$ has Krull dimension one, we describe the set of torsion classes of $\Lambda$ explicitly by using the set of torsion classes of finite dimensional algebras., Comment: The title was changed from the first version. 25pages

A quasi-hereditary algebra is an Artin algebra together with a partial order on its set of isomorphism classes of simple modules which satisfies certain conditions. In this article we investigate all the possible choices that yield to quasi-hereditary structures on a given algebra, in particular we introduce and study what we call the poset of quasi-hereditary structures. Our techniques involve certain quiver decompositions and idempotent reductions. For a path algebra of Dynkin type $\mathbb{A}$, we provide a full classification of its quasi-hereditary structures. For types $\mathbb{D}$ and $\mathbb{E}$, we give a counting method for the number of quasi-hereditary structures. In the case of a hereditary incidence algebra, we present a necessary and sufficient condition for its poset of quasi-hereditary structures to be a lattice., Comment: 34 pages, 2 figures; typos corrected

In this paper, we study two-term tilting complexes for preprojective algebras of non-Dynkin type. We show that there exist two families of two-term tilting complexes, which are respectively parameterized by the elements of the corresponding Coxeter group. Moreover, we provide the complete classification in the case of affine type by showing that any two-term silting complex belongs one of them. For this purpose, we also discuss the Krull-Schmidt property for the homotopy category of finitely generated projective modules over a complete ring., Comment: v2. typo corrected, minor changes. 15 pages

We show that for the path algebra $A$ of an acyclic quiver, the singularity category of the derived category $\mathsf{D}^{\rm b}(\mathsf{mod}\,A)$ is triangle equivalent to the derived category of the functor category of $\underline{\mathsf{mod}}\,A$, that is, $\mathsf{D}_{\rm sg}(\mathsf{D}^{\rm b}(\mathsf{mod}\,A))\simeq \mathsf{D}^{\rm b}(\mathsf{mod}(\underline{\mathsf{mod}}\,A))$. This extends a result of Iyama-Oppermann for the path algebra $A$ of a Dynkin quiver. An important step is to establish a functor category analog of Happel's triangle equivalence for repetitive algebras., Comment: 21 pages

We study the stable category of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\bf{w})$ of this category associated with each reduced expression $\bf{w}$ of $w$ and give a sufficient condition on $\bf{w}$ such that $M(\bf{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\bf{w})$. Moreover, we compare it with a triangle equivalence given by Amiot-Reiten-Todorov for a cluster category., Comment: 22 pages, minor changes