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Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view., Comment: Advanced Studies in Pure Mathematics 84, 2020, Various Aspects of Multiple Zeta Functions, in honor of Professor Kohji Matsumoto's 60th birthday, pp. 93-121

We implemented a Mathematica module of CGA which includes functions to denote CGA elements and compute several operations in CGA. We can draw the figure in 3D space which is corresponding to a CGA element. Our draw function is using Gr\"{o}bner Basis for simplifying equations of figures. It can be used for any dimensional figures. One of our motivations is to realize 3D origami system using our own CGA Library. We follow the 2D computational origami system E-Origami-System developed by Ida et.al. and formulated simple fold operations in 3D by using CGA points and motions. Then, we proved some geometric theorems about origami properties by computing CGA equation formulas.

Let K be a complex quadratic extension of and let denote the adeles of K. We find special values at all of the critical points of twisted tensor L-functions attached to cohomological cuspforms on GL2() and establish Galois equivariance of the values. To investigate the values, we determine the archimedean factors of a class of integral representations of these L–functions, thus proving a conjecture due to Ghate. We also investigate analytic properties of these L–functions, such as their functional equations.

The hypergeometric equations with polyhedral monodromy groups derive 3-integral-parameter families of polynomials.

The spectral problem for non-commutative harmonic oscillators is shown to be equivalent to solve Fuchsian ordinary differential equations with four regular singular points in a complex domain.

We introduce a new presentation of the two dimensional rigid transformation which is more concise and efficient than the standard matrix presentation. By modifying the ordinary dual number construction for the complex numbers, we define the ring of anti-commutative dual complex numbers, which parametrizes two dimensional rotation and translation all together. With this presentation, one can easily interpolate or blend two or more rigid transformations at a low computational cost. We developed a library for C++ with the MIT-licensed source code [13].

We give quotients V // G = Spec ( C [ V ] G ) for some prehomogeneous vector spaces ( G × H , V ).

We introduce multiple versions of L-functions for Witten zeta functions. We study their algebraic and analytic properties. Especially we investigate the existence of zeros at negative integers. These results strongly suggest the universal zero at -2. We look at their absolute limits also.

This is a survey on the non-commutative harmonic oscillator, which is a generalization of usual (scalar) harmonic oscillators to the system introduced by Parmeggiani and Wakayama. With the definitions and the basic properties, we summarize the positivity of several related operators with \(\mathfrak{sl}_{2}\) interpretations. We also mention some unsolved questions, in order to clarify the current status of the problems and expected further development.

We study the structure of the twisted homology groups attached to weighted lines in the plane with resonant singular points, and find possible intersection pairings. As a typical example, we treat homology groups attached to 2-dimensional Selberg-type integrals.

Commuting differential operators with symbols ∂ 1 2 + ∂ 2 2 , ∂ 1 2 ∂ 2 2 are related to the functional equation V 1 ( x 1 ) U + ( x 1 + x 2 ) + V 1 ( x 1 ) U − ( x 1 − x 2 ) + V 2 ( x 2 ) U + ( x 1 + x 2 ) − V 2 ( x 2 ) U − ( x 1 − x 2 ) = F + ( x 1 + x 2 ) + F − ( x 1 − x 2 ) + G 1 ( x 1 ) + G 2 ( x 2 ). We discuss several properties, such as a meromorphic extension of solutions and periodicity, of this functional equation and commuting differential operators.