The Grothendieck groups and stable equivalences of mesh algebras

We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable categories have a structure of triangulated categories. Our main result determines the Grothendieck groups of these stable categories. As an application, we give an complete classification of the mesh algebras up to stable equivalences.Comment: 35 page

Semistable torsion classes and canonical decompositions in Grothendieck groups

We study two classes of torsion classes which generalize functorially finite torsion classes, that is, semistable torsion classes and morphism torsion classes. Semistable torsion classes are parametrized by the elements in the real Grothendieck group up to TF equivalence. We give a close connection between TF equivalence classes and the cones given by canonical decompositions of the spaces of projective presentations due to Derksen-Fei. More strongly, for $E$-tame algebras and hereditary algebras, we prove that TF equivalence classes containing lattice points are exactly the cones given by canonical decompositions. One of the key steps in our proof is a general description of semistable torsion classes in terms of morphism torsion classes. As an application of our results, we give an explicit description of TF equivalence classes of preprojective algebras of type $\widetilde{\mathbb{A}}$

MM-TF equivalences in the real Grothendieck groups

For an abelian length category $\mathcal{A}$ with only finitely many isoclasses of simple objects, we have the wall-chamber structure and the TF equivalence in the dual real Grothendeick group $K_0(\mathcal{A})_\mathbb{R}^*=\operatorname{Hom}_\mathbb{R}(K_0(\mathcal{A})_\mathbb{R},\mathbb{R})$, which are defined by semistable subcategories and semistable torsion pairs in $\mathcal{A}$ associated to elements $\theta \in K_0(\mathcal{A})_\mathbb{R}^*$. In this paper, we introduce the $M$-TF equivalence for each object $M \in \mathcal{A}$ as a systematic way to coarsen the TF equivalence. We show that the set $\Sigma(M)$ of the closures of $M$-TF equivalence classes is a finite complete fan in $K_0(\mathcal{A})_\mathbb{R}^*$, and that $\Sigma(M)$ is the normal fan of the Newton polytope $\mathrm{N}(M)$ in $K_0(\mathcal{A})_\mathbb{R}$.Comment: 18 pages, comments welcom

A Stochastic Variance Reduced Nesterov's Accelerated Quasi-Newton Method

Recently algorithms incorporating second order curvature information have become popular in training neural networks. The Nesterov's Accelerated Quasi-Newton (NAQ) method has shown to effectively accelerate the BFGS quasi-Newton method by incorporating the momentum term and Nesterov's accelerated gradient vector. A stochastic version of NAQ method was proposed for training of large-scale problems. However, this method incurs high stochastic variance noise. This paper proposes a stochastic variance reduced Nesterov's Accelerated Quasi-Newton method in full (SVR-NAQ) and limited (SVRLNAQ) memory forms. The performance of the proposed method is evaluated in Tensorflow on four benchmark problems - two regression and two classification problems respectively. The results show improved performance compared to conventional methods.Comment: Accepted in ICMLA 201

A Development of a Microscale Experiment of Stoichiometry by Active Learning in High School, Part 4 The calibration of the scales

この研究の目的は、高等学校新学習指導要領で謳われている主体的・対話的で深い学びを生徒に保障するための、一連のマイクロスケール実験の開発である。大阪府立長尾高等学校では、理科研究部を中心に、主に化学基礎の教科書の「化学反応の量的関係（炭酸カルシウムと塩酸との反応）」の実験に着目し、市販の安価な小型電子天秤（目量[最小表示単位]：0.01 g と0.001 g）を用いた、従来よりも少量かつ安価で実施可能なマイクロスケール実験（目量：0.01 g の小型電子天秤では1/4 スケール、目量：0.001 g の小型電子天秤では 1/6 スケール）を開発してきた。目量が0.001 g の小型電子天秤を用いた1/6 スケールの検証実験の際に、天秤の示度がまれに0.005 g 変動することがあった。2022 年度に入り、マイクロスケール実験の要である天秤の示度の変動の原因究明を開始した。小型軽量の天秤それ自体の問題と測定環境（天秤に実験で使える風防がなく、実験机が安定していないなど）の問題を切り分け、まず、前者である目量0.001 g の小型電子天秤（秤量 [計量できる最大値] 100 g）の較正作業に取り組んだ。。1/6 スケール実験では最大約60 g まで量るため、検定証印打刻済の精密分銅（0.1 g-50 g の9 種）を使用し、各分銅について10 回秤量し、その値の繰り返し性を確認した。4 台全ての天秤に関して、精密分銅の示度の平均値は分銅が表す質量の±0.001 g（±目量）の2 倍の範囲内にあり、ばらつきを示す標本標準偏差もその2 倍に収まっていた。その結果、本天秤は検証実験での使用に問題のなかったことがわかり、検証実験の妥当性が再確認された。次に、年度の後半に、小型天秤の秤量100 g までの使用を想定して再度繰り返し性、偏置誤差、さらに直線性を確認した。繰り返し性の調査では分銅100 g の示度の平均値は分銅が表す質量より 0.028 g 以上小さいものや、そのばらつきを示す標本標準偏差も目量の5 倍以上のものが複数あった。偏置誤差に関して4 台全てに問題はなかった。直線性では分銅100 g を載せたときに示度の安定に少し時間がかかったが、4 台全ての天秤に関して20 g-90 g までは問題がなかった。ここで扱った「化学反応の量的関係（炭酸カルシウムと塩酸との反応）」の実験は、新学習指導要領の下で検定された2022 年度から使用されている化学基礎の教科書にも記載されている。これら一連のマイクロスケール実験の開発にかかわる生徒の探究過程は、高等学校新学習指導要領で謳われている主体的・対話的で深い学びそのものであると確信する。The objective of this study is to improve the microscale experiment of stoichiometry in high school, which was developed in a previous work. We tried to make the microscale experimentalerrors smaller by active learning. Active learning is a new and key concept of the revised Courseof Study defined by the Ministry of Education, Culture, Sports, Science and Technology-Japan in2018. The developed microscale experiment using small electronic scales, an old type (readability:0.01 g) and a new type (readability: 0.001 g), which consume small quantities of reagents, includesthe reaction of calcium carbonate with hydrochloric acid to form carbon dioxide. These are twoscale-down models (1/4 scale and 1/6 scale) of the normal scale experiment described in a textbook of “Basic Chemistry” in the Course of Study mentioned above. The work of this Part 4 is to calibrate new small electronic scales (capacity: 100 g, readability: 0.001 g), with which we made the 1/6 scale experiment developed in the previous work Part 3. The reading of the new scales rarely changed by about 0.005 g. Entering the new term of 2022, the calibration of the scales, which include repeatability tests (checking every weight ten times), eccentricity tests and linearity tests (starting with zeroing the scales, then increasing the weights through all the increasing test points[20 g, 40 (= 20 + 10 + 10) g, 60 (= 50 + 10) g, 80 (= 50 + 20 + 10) g] to the maximum weight[100 g] and decreasing the weights through the decreasing test points back to no weight) was performed using stainless steel precision weights [0.1 g, 0.2 g, 0.5 g, 1 g, 2 g, 5 g, 10g, 20 g, 50 g, 100 g] made by MURAKAMI KOKI Co., Ltd. Japan. No problems were found in the eccentricity tests. All sample standard deviations of every mean of the nine readings of every precision weight [except for the 100 g weight] in the repeatability tests were within 0.002 (= 2 ×readability) g. We found no problems with nine readings of the weights [20 g-50 g]. In the linearity tests, all readings of the four test points mentioned before were right, except for the 100 g weight. In conclusion, it is reconfirmed that the 1/6 microscale experiment described above is valid. We are also convinced that the microscale experiment works well for active learning in high school chemistry classes.departmental bulletin pape