1. |
Tomohiro Hayakawa, Tomoya Kamimura, Shizuo Kaji, Fumitoshi Matsuno, Autonomous Distributed System for Gait Generation for Single-Legged Modular Robots Connected in Various Configurations, *IEEE TRANSACTIONS ON ROBOTICS*, 10.1109/TRO.2020.2992983, 36, 5, 1491-1510, 2020.10, To date, many gait generation strategies have been designed for robots with leg configurations that model those of natural creatures. However, their leg configurations are limited to the $2 imes N$ type, such as hexapod or myriapod; hence, simultaneously, the potential ability of legged robots is implicitly limited. We consider single-legged modular robots that can be arranged to form a cluster with arbitrary 2-D leg configurations. By choosing configurations appropriately, these robots have the potential to perform several types of tasks, as is the case for reconfigurable modular robots. However, to use appropriate configurations for a given task, a unified gait generation system for various configurations of a cluster is required. In this article, we propose an autonomous distributed control system for each single-legged modular robot to collectively achieve static walking of the cluster with various leg configurations on planar ground. Moreover, our system is an autonomous distributed system with scalability and fault tolerance, in which each module determines the moving pattern of its foot through local communication without global information, such as the entire leg configuration of the cluster. We verified that several types of clusters achieved static walking using our system not only in dynamic simulations, but also in real robot experiments.. |

2. |
Jinichi Mori, Shizuo Kaji, Hiroki Kawai, Satoshi Kida, Masaharu Tsubokura, Masahiko Fukatsu, Kayo Harada, Hideyoshi Noji, Takayuki Ikezoe, Tomoya Maeda, Akira Matsuda, Assessment of dysplasia in bone marrow smear with convolutional neural network, *SCIENTIFIC REPORTS*, 10.1038/s41598-020-71752-x, 10, 1, 14734-14734, 2020.09, In this study, we developed the world's first artificial intelligence (AI) system that assesses the dysplasia of blood cells on bone marrow smears and presents the result of AI prediction for one of the most representative dysplasia-decreased granules (DG). We photographed field images from the bone marrow smears from patients with myelodysplastic syndrome (MDS) or non-MDS diseases and cropped each cell using an originally developed cell detector. Two morphologists labelled each cell. The degree of dysplasia was evaluated on a four-point scale: 0-3 (e.g., neutrophil with severely decreased granules were labelled DG3). We then constructed the classifier from the dataset of labelled images. The detector and classifier were based on a deep neural network pre-trained with natural images. We obtained 1797 labelled images, and the morphologists determined 134 DGs (DG1: 46, DG2: 77, DG3: 11). Subsequently, we performed a five-fold cross-validation to evaluate the performance of the classifier. For DG1-3 labelled by morphologists, the sensitivity, specificity, positive predictive value (PPV), negative predictive value (NPV), and accuracy were 91.0%, 97.7%, 76.3%, 99.3%, and 97.2%, respectively. When DG1 was excluded in the process, the sensitivity, specificity, PPV, NPV, and accuracy were 85.2%, 98.9%, 80.6%, and 99.2% and 98.2%, respectively.. |

3. |
Satoshi Kida, Shizuo Kaji, Kanabu Nawa, Toshikazu Imae, Takahiro Nakamoto, Sho Ozaki, Takeshi Ohta, Yuki Nozawa, Keiichi Nakagawa, Visual enhancement of Cone-beam CT by use of CycleGAN, *Medical physics*, 10.1002/mp.13963, 47, 3, 998-1010, 2020.03, Purpose: Cone-beam computed tomography (CBCT) offers advantages over conventional fan-beam CT in that it requires a shorter time and less exposure to obtain images. However, CBCT images suffer from low soft-tissue contrast, noise, and artifacts compared to conventional fan-beam CT images. Therefore, it is essential to improve the image quality of CBCT. Methods: In this paper, we propose a synthetic approach to translate CBCT images with deep neural networks. Our method requires only unpaired and unaligned CBCT images and planning fan-beam CT (PlanCT) images for training. The CBCT images and PlanCT images may be obtained from other patients as long as they are acquired with the same scanner settings. Once trained, three-dimensionally reconstructed CBCT images can be directly translated into high-quality PlanCT-like images. Results: We demonstrate the effectiveness of our method with images obtained from 20 prostate patients, and provide a statistical and visual comparison. The image quality of the translated images shows substantial improvement in voxel values, spatial uniformity, and artifact suppression compared to those of the original CBCT. The anatomical structures of the original CBCT images were also well preserved in the translated images. Conclusions: Our method produces visually PlanCT-like images from CBCT images while preserving anatomical structures.. |

4. |
Shizuo Kaji, Shintaro Kuroki, Eunjeong Lee, Dong Youp Suh, Flag bott manifolds of general lie type and their equivariant cohomology rings, *Homology, Homotopy and Applications*, 10.4310/HHA.2020.v22.n1.a21, 22, 1, 375-390, 2020.01, In this article we introduce flag Bott manifolds of general Lie type as the total spaces of iterated flag bundles. They generalize the notion of flag Bott manifolds and generalized Bott manifolds, and admit nice torus actions. We calculate the torus equivariant cohomology rings of flag Bott manifolds of general Lie type.. |

5. |
Shizuo Kaji, Kenji Kajiwara, Hyeongki Park, Linkage Mechanisms Governed by Integrable Deformations of Discrete Space Curves, *Nonlinear Systems and Their Remarkable Mathematical Structures. Volume 2, CRC Press*, https://doi.org/10.1201/9780429263743, 2019.12. |

6. |
Keigo Matsumoto, Nami Ogawa, Hiroyuki Inou, Shizuo Kaji, Yutaka Ishii, Michitaka Hirose, Polyvision 4D space manipulation through multiple projections, *SIGGRAPH Asia 2019 Emerging Technologies - International Conference on Computer Graphics and Interactive Techniques, SA 2019
SIGGRAPH Asia 2019 Emerging Technologies, SA 2019*, 10.1145/3355049.3360518, 36-37, 2019.11, Seeing is believing. Our novel virtual reality system, Polyvision, applies this old saying to the fourth dimension. Various shadows of an object in a four-dimensional (4D) space are simultaneously projected onto multiple three-dimensional (3D) screens created in a virtual environment to reveal its intricate shape. The understanding of high-dimensional shapes and data can essentially be enhanced when good visualization is complemented by interactive functionality. However, a method to implement an interface for handling complex 4D transformations in a user-friendly manner must be developed. Using our Polyvision system, the user can manipulate each shadow as if it were a 3D object in their hand. The user’s action on each projection is reflected to the original 4D object, and in turn its projections, in real time. While controlling the object’s orientation minutely on one shadow, the user can grasp its global structure from multiple changing projections. Our system has a wide variety of applications in visualization, education, mathematical research, and entertainment, as we demonstrate with a variety of 4D objects that appear in mathematics and data sciences.. |

7. |
Alexandre Derouet-Jourdan, Shizuo Kaji, Yoshihiro Mizoguchi, A linear algorithm for Brick Wang tiling, *Japan Journal of Industrial and Applied Mathematics*, 10.1007/s13160-019-00369-z, 36, 3, 749-761, 2019.09, The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand, Wang tiles are used as a tool to generate textures and patterns in computer graphics. In these applications, a set of tiles is normally chosen so that it tiles the plane or its sub-regions easily in many different ways. With computer graphics applications in mind, we introduce a class of such tileset, which we call sequentially permissive tilesets, and consider tiling problems with constrained boundary. We apply our methodology to a special set of Wang tiles, called Brick Wang tiles, introduced by Derouet-Jourdan et al. in 2016 to model wall patterns. We generalise their result by providing a linear algorithm to decide and solve the tiling problem for arbitrary planar regions with holes.. |

8. |
Shizuo Kaji, Satoshi Kida, Overview of image-to-image translation by use of deep neural networks denoising, super-resolution, modality conversion, and reconstruction in medical imaging, *Radiological physics and technology*, 10.1007/s12194-019-00520-y, 12, 3, 235-248, 2019.09, Since the advent of deep convolutional neural networks (DNNs), computer vision has seen an extremely rapid progress that has led to huge advances in medical imaging. Every year, many new methods are reported at conferences such as the International Conference on Medical Image Computing and Computer-Assisted Intervention and Machine Learning for Medical Image Reconstruction, or published online at the preprint server arXiv. There is a plethora of surveys on applications of neural networks in medical imaging (see [1] for a relatively recent comprehensive survey). This article does not aim to cover all aspects of the field, but focuses on a particular topic, image-to-image translation. Although the topic may not sound familiar, it turns out that many seemingly irrelevant applications can be understood as instances of image-to-image translation. Such applications include (1) noise reduction, (2) super-resolution, (3) image synthesis, and (4) reconstruction. The same underlying principles and algorithms work for various tasks. Our aim is to introduce some of the key ideas on this topic from a uniform viewpoint. We introduce core ideas and jargon that are specific to image processing by use of DNNs. Having an intuitive grasp of the core ideas of applications of neural networks in medical imaging and a knowledge of technical terms would be of great help to the reader for understanding the existing and future applications. Most of the recent applications which build on image-to-image translation are based on one of two fundamental architectures, called pix2pix and CycleGAN, depending on whether the available training data are paired or unpaired (see Sect. 1.3). We provide codes ([2, 3]) which implement these two architectures with various enhancements. Our codes are available online with use of the very permissive MIT license. We provide a hands-on tutorial for training a model for denoising based on our codes (see Sect. 6). We hope that this article, together with the codes, will provide both an overview and the details of the key algorithms and that it will serve as a basis for the development of new applications.. |

9. |
Suyoung Choi, Shizuo Kaji, Hanchul Park, The Cohomology Groups of Real Toric Varieties Associated with Weyl Chambers of Types C and D, *Proceedings of the Edinburgh Mathematical Society*, 10.1017/S001309151800086X, 62, 3, 861-874, 2019.08, Given a root system, the Weyl chambers in the co-weight lattice give rise to a real toric variety, called the real toric variety associated with the Weyl chambers. We compute the integral cohomology groups of real toric varieties associated with the Weyl chambers of type Cn and Dn, completing the computation for all classical types.. |

10. |
Shizuo Kaji, Toshiaki Maeno, Koji Nuida, Yasuhide Numata, Polynomial expressions of p-ary auction functions, *Journal of Mathematical Cryptology*, 10.1515/jmc-2018-0016, 13, 2, 69-80, 2019.06, One of the common ways to design secure multi-party computation is twofold: To realize secure fundamental operations and to decompose a target function to be securely computed into them. In the setting of fully homomorphic encryption, as well as some kinds of secret sharing, the fundamental operations are additions and multiplications in the base field such as the field F_{2} with two elements. Then the second decomposition part, which we study in this paper, is (in theory) equivalent to expressing the target function as a polynomial. It is known that any function over the finite prime field F_{p} has a unique polynomial expression of degree at most p-1 with respect to each input variable; however, there has been little study done concerning such minimal-degree polynomial expressions for practical functions. This paper aims at triggering intensive studies on this subject, by focusing on polynomial expressions of some auction-related functions such as the maximum/minimum and the index of the maximum/minimum value among input values.. |

11. |
Shizuo Kaji, Stephen Theriault, Suspension Splittings and Self-maps of Flag Manifolds, *Acta Mathematica Sinica, English Series*, 10.1007/s10114-019-8051-z, 35, 4, 445-462, 2019.04, If G is a compact connected Lie group and T is a maximal torus, we give a wedge decomposition of ΣG/T by identifying families of idempotents in cohomology. This is used to give new information on the self-maps of G/T.. |

12. |
S. Kaji, A. Derouet-Jourdan, H. Ochiai, Dappled tiling, *Mathematical Insights into Advanced Computer Graphics Techniques*, 10.1007/978-981-13-2850-3, 59-72, 2019.03. |

13. |
Soojin Cho, Suyoung Choi, Shizuo Kaji, Geometric representations of finite groups on real toric spaces, *Journal of the Korean Mathematical Society*, 10.4134/JKMS.j180646, 56, 5, 1265-1283, 2019.01, We develop a framework to construct geometric representations of finite groups G through the correspondence between real toric spaces X<sup>R</sup> and simplicial complexes with characteristic matrices. We give a combinatorial description of the G-module structure of the homology of X<sup>R</sup>. As applications, we make explicit computations of the Weyl group representations on the homology of real toric varieties associated to the Weyl chambers of type A and B, which show an interesting connection to the topology of posets. We also realize a certain kind of Foulkes representation geometrically as the homology of real toric varieties.. |

14. |
Shizuo Kaji, Representing a point and the diagonal as zero loci in flag manifolds, *Algebraic and Geometric Topology*, 10.2140/agt.2019.19.2061, 19, 4, 2061-2075, 2019.01, The zero locus of a generic section of a vector bundle over a manifold defines a submanifold. A classical problem in geometry asks to realise a specified submanifold in this way. We study two cases: a point in a generalised flag manifold and the diagonal in the direct product of two copies of a generalised flag manifold. These cases are particularly interesting since they are related to ordinary and equivariant Schubert polynomials, respectively.. |

15. |
Adrien Fauré, Shizuo Kaji, A circuit-preserving mapping from multilevel to Boolean dynamics, *Journal of Theoretical Biology*, 10.1016/j.jtbi.2017.12.013, 440, 71-79, 2018.03. |

16. |
Neşet Deniz Turgay, Shizuo Kaji, The mod 2 dual Steenrod algebra as a subalgebra of the mod 2 dual Leibniz-Hopf algebra, *Journal of Homotopy and Related Structures*, 10.1007/s40062-016-0163-x, 12, 3, 727-739, 2017.09. |

17. |
Suyoung Choi, Shizuo Kaji, Stephen Theriault, Homotopy decomposition of a suspended real toric space, *Boletin de la Sociedad Matematica Mexicana*, 10.1007/s40590-016-0090-1, 23, 1, 153-161, 2017.04. |

18. |
Ho Kyoung Ko, Hyung Won Kim, Shizuo Kaji, Suyoung Choi, Elementary school students who give up on learning mathematics: Correlations with non-cognitive learner characteristics, *J. Korea Soc. Math. Ed. Ser. C*, 20, 2, 143-151, 2017.04. |

19. |
Shizuo Kaji, Michihiro Sakai, Stephen Theriault, Counting the number of homotopy associative multiplications on certain H-spaces, *Topology and its Applications*, 10.1016/j.topol.2016.10.008, 214, 137-149, 2016.12. |

20. |
Shizuo Kaji, Hiroyuki Ochiai, A concise parametrization of affine transformation, *SIAM Journal on Imaging Sciences*, 10.1137/16M1056936, 9, 3, 1355-1373, 2016.09. |

21. |
S. Kaji, Tetrisation of triangular meshes and its application in shape blending, *Mathematical Progress in Expressive Image Synthesis III*, 10.1007/978-981-10-1076-7_2, 7-19, 2016.05. |

22. |
Shizuo Kaji, Akihiro Ohsita, Stephen Theriault, Mod p decompositions of the loop spaces of compact symmetric spaces, *Algebraic and Geometric Topology*, 10.2140/agt.2015.15.1771, 15, 3, 1771-1811, 2015.06. |

23. |
S. Kaji and G. Liu, Probe-type deformers, *Mathematical Progress in Expressive Image Synthesis I*, 10.1007/978-4-431-55483-7_6, 63-77, 2015.05. |

24. |
S. Kaji, Three presentations of torus equivariant cohomology of flag manifolds, *Perspectives and Developments in Mathematics, Proceedings of the International Mathematics Conference in honour of the 70th Birthday of Professor S. A. Ilori*, 37-54, 2015.05. |

25. |
Koji Nuida, Takuro Abe, Shizuo Kaji, Toshiaki Maeno, Yasuhide Numata, A mathematical problem for security analysis of hash functions and pseudorandom generators, *International Journal of Foundations of Computer Science*, 10.1142/S0129054115500100, 26, 2, 169-194, 2015.02. |

26. |
Shizuo Kaji, Weyl group symmetry on the gkm graph of a gkm manifold with an extended lie group action, *Osaka Journal of Mathematics*, 52, 1, 31-41, 2015.01. |

27. |
G. Matsuda, S. Kaji, H. Ochiai, Anti-commutative Dual Complex Numbers and 2D Rigid Transformation, *Mathematical Progress in Expressive Image Synthesis I*, 10.1007/978-4-431-55007-5_17, 131-138, 2014.05. |

28. |
Shizuo Kaji, S. Hirose, S. Sakata, Yoshihiro Mizoguchi, K. Anjyo, Mathematical analysis on affine maps for 2D shape interpolation, *11th ACM SIGGRAPH / Eurographics Symposium on Computer Animation, SCA 2012
Computer Animation 2012 - ACM SIGGRAPH / Eurographics Symposium Proceedings, SCA 2012*, 71-76, 2012.07. |

29. |
Shizuo Kaji, Equivariant Schubert calculus of Coxeter groups, *Proceedings of the Steklov Institute of Mathematics*, 10.1134/S0081543811080177, 275, 1, 239-250, 2011.12, We consider an equivariant extension for Hiller's Schubert calculus on the coinvariant ring of a finite Coxeter group.. |

30. |
Koji Nuida, Takuro Abe, Shizuo Kaji, Toshiaki Maeno, Yasuhide Numata, A mathematical problem for security analysis of hash functions and pseudorandom generators, *6th International Workshop on Security, IWSEC 2011
Advances in Information and Computer Security - 6th International Workshop, IWSEC 2011, Proceedings*, 10.1007/978-3-642-25141-2_10, 144-160, 2011.11, The aim of this paper is to emphasize the significance of a certain mathematical problem in research on information security. We point out that the mathematical problem, which we refer to as "Function Density Problem," has connections to the following two major cryptographic topics; security analysis of hash functions in the real world (like SHA-1), and construction of pseudorandom generators with some enhanced security property. We also provide a first example to show how a study of Function Density Problem can contribute to the progress of the above-mentioned two topics. Other potential applications of Function Density Problem to information security are also discussed.. |

31. |
S. Kaji, Schubert calculus, seen from torus equivariant topology, *Trends in Mathematics - New Series*, 12, 1, 71-89, 2010.05. |

32. |
Shizuo Kaji, Daisuke Kishimoto, Homotopy nilpotency in p-regular loop spaces, *Mathematische Zeitschrift*, 10.1007/s00209-008-0459-6, 264, 1, 209-224, 2010.01, We consider the problem how far from being homotopy commutative is a loop space having the homotopy type of the p-completion of a product of finite numbers of spheres. We determine the homotopy nilpotency of those loop spaces as an answer to this problem.. |

33. |
Hiroaki Hamanaka, Shizuo Kaji, Akira Kono, Samelson products in Sp (2), *Topology and its Applications*, 10.1016/j.topol.2008.02.008, 155, 11, 1207-1212, 2008.06, We calculate certain Samelson products of Sp (2). Using the result, we classify the homotopy types of the gauge groups of principal Sp (2) bundles over S^{8} and we also derive the homotopy commutativity of Sp (2) localized at 3.. |

34. |
Shizuo Kaji, Mod 2 cohomology of 2-compact groups of low rank, *Kyoto Journal of Mathematics*, 10.1215/kjm/1250281055, 47, 2, 441-450, 2007.01, We determine the mod 2 cohomology algebra over the Steenrod algebra A _{2} of the classifying space of loop groups LG where G = Spin(7), Spin(8), Spin(9), F_{4} and DI(4). Then we show they are isomorphic as algebras over A_{2} to the mod 2 cohomology of the 2-compact groups of type G.. |

35. |
Shizuo Kaji, Low rank cohomology of the classifying spaces of gauge groups over 3-manifolds, *Publications of the Research Institute for Mathematical Sciences*, 10.2977/prims/1166642116, 42, 2, 581-587, 2006.06, The purpose of this paper is to calculate the cohomology of the function space Map(M, BG) for degree less than or equal to 3, where G is a simply connected compact Lie group and M is a closed orientable 3-manifold. The calculation enables us to obtain a simple proof and an improvement of the result [4, Theorem 1.2].. |

36. |
Shizuo Kaji, On the nilpotency of rational H-spaces, *Journal of the Mathematical Society of Japan*, 10.2969/jmsj/1150287307, 57, 4, 1153-1165, 2005.10, In [BG], it is proved that the Whitehead length of a space Z is less than or equal to the nilpotency of ΩZ. As for rational spaces, those two invariants are equal. We show this for a 1-connected rational space Z by giving a way to calculate those invariants from a minimal model for Z. This also gives a way to calculate the nilpotency of an homotopy associative rational H-space.. |