Abstract. In this paper, we consider a high accurate method for SIMP (Solid Isotropic Material with Penalization) type topology optimization problem. We adopt the gradient type method to solve our problem. In order to get high accurate solution of the partial difference equation in our algorithm, we use the lattice-free finite difference method which is a kind of the differential quadrature method. The library “exflib” for the scientific multiple-precision arithmetic is used in our calculation. By the numerical experiments, we check the effectiveness, stability, and convergency of our algorithm with the steepest descent and H¹ gradient method.

Abstract. The aim of this research is to find the fairest apportionment method．In this research，we estimate the biases of apportionment methods based on the Stolarsky mean，know as relaxed divisor methods．We use three bias measurements：the Balinski and Young measurement，the Ernst measurement and the B measurement which is a new measurement created by the researchers to compare with the other bias values．All three measurements are compared together，and the results show that our measurement produces similar results to the other two measurements．In addition，the Webster method gives the lowest bias value，compared to the other methods．

Abstract. We examine Hirayama’s numerical integration method for integrals over finite intervals, which is called the “hyperfunction method” in this paper. In the hyperfunction method, an integral is transformed into a complex integral on a closed contour and is approximated by the trapezoidal rule,which gives good results for integrals in the case that the integrands are periodic functions. Numerical examples show that the hyperfunction method is effective for integrals with strong end-point singularities. We also remark that the relation between the hyperfunction method and the hyperfuction theory.

Abstract. We generalize the extracellular matrix (ECM) degradation pathway network with 3 monomers to N monomers. Generalization of this network helps us to understand the network structure. All N(N+1) complexes in the network are classified to some groups. Each group behave as a unit. By this classification, we find N mass conservation laws between groups and aggregates N(N+1) complex ordinal differential equations (ODEs) to N group ODEs. The group ODEs can be solved explicitly and have asymptotically stable solutions．

Abstract. We discuss a sufficient condition which guarantees the existence and uniqueness of non-negative solutions to an initial value problem of non-autonomous ODE systems, which include a cardiac hypertrophy network model as one of the typical examples of ODE systems induced from biochemical reaction networks by applying the low of mass action, and show the non-negativity of the solutions to the model. Moreover, we define a dynamic equilibrium point of the cardiac hypertrophy network model, which is non-autonomous, and analyze its structure theoretically as well as investigating the convergence of global-in-time solutions to the dynamic equilibrium point using numerical simulations.

Abstract. We investigate a simple mathematical model for angiogenesis. From recent time-lapse imaging experiments on the dynamics of endothelial cells (ECs) in angiogenesis, we suppose that elongation and bifurcation of neogenetic vessel is determined by only the density of ECs near the tip, and introduce a model described by nonlinear simultaneous differential equations. We also incorporate proliferation of ECs and activation factor such as VEGF and show the exact solutions to that model and numerical simulations.