arXiv subjects
Publications and source records attributed to Shinji Tanimoto.
Best simultaneous approximation (BSA) for finitely or infinitely many functions are considered under the uniform norm and other important norms. Characterization theorems for a BSA from a finite-dimensional subspace are obtained by a generalized minimax theorem. From the characterization theorem a strong unicity theorem is also deduced for a BSA.
Dimensional analysis provides many simple and useful tools for various situations in science. The objective of this paper is to investigate its relations to functions, i.e., the dimensions for functions that yield physical quantities and those for the arguments of functions. In particular, it is shown that there are three types of functions from a viewpoint of dimensional analysis. Relationships between Planck's constant and dimensional analysis are also considered.
The combinatorial theory for the set of parity alternating permutations is expounded. In view of the numbers of ascents and inversions, several enumerative aspects of the set are investigated. In particular, it is shown that signed Eulerian numbers have intimate relationships to the set.
A dual control problem is presented for the optimal stochastic control of a system governed by partial differential equations. Relationships between the optimal values of the original and the dual problems are investigated and two duality theorems are proved. The dual problem serves to provide upper bounds for the optimal and maximum value of the original one or even to give the optimal value.
A generalization of the Emden-Fowler equation is presented and its solutions are investigated. This paper is devoted to asymptotic behavior of its solutions. The procedure is entirely based on a previous paper by the author.
We propose a natural model of evolving weighted networks in which new links are not necessarily connected to new nodes. The model allows a newly added link to connect directly two nodes already present in the network. This is plausible in modeling many real-world networks. Such a link is called an inner link, while a link connected to a new node is called an outer link. In view of interrelations between inner and outer links, we investigate power-laws for the strength, degree and weight distributions of weighted complex networks. This model enables us to predict some features of weighted networks such as the worldwide airport network and the scientific collaboration network.
Bipartite networks are composed of two types of nodes and there are no links between nodes of the same type. Thus the study of epidemic spread and control on such networks is relevant to sexually transmitted diseases (STDs). When entire populations of two types cannot be immunized and the effect of immunization is not perfect, we have to consider the targeted immunization with immunization rates. We derive the epidemic thresholds of SIR and SIS models with immunization and illustrate the results with STDs on heterosexual contact networks.
We investigate the spread of diseases, computer viruses or information on complex networks and also immunization strategies to prevent or control the spread. When an entire population cannot be immunized and the effect of immunization is not perfect, we need the targeted immunization with immunization rate. Under such a circumstance we calculate epidemic thresholds for the SIR and SIS epidemic models. It is shown that, in scale-free networks, the targeted immunization is effective only if the immunization rate is equal to one. We analyze here epidemic spreading on directed complex networks, but similar results are also valid for undirected ones.
The spread of a disease, a computer virus or information is discussed in a directed complex network. We are concerned with a steady state of the spread for the SIR and SIS dynamic models. In a scale-free directed network it is shown that the threshold of its outbreak in both models approaches zero under a high correlation between nodal indegrees and outdegrees.
A model for directed networks is proposed and power laws for their in-degree and/or out-degree distributions are derived from the model. It is based on the Barabasi-Albert model and contains two parameters. The parameters serve as regulatory factors for the contributions of new nodes through in- and out-degrees. The model allows a newly added link to connect directly two nodes already present in the network. Such a link is called an inner link, while a link attached to a new node is called an outer link. Using relationships between inner and outer links, we investigate power laws for in- and out-degree distributions of directed networks. This model enables us to predict some interesting features of directed networks; in particular, the World Wide Web and the networks of citation and phone-call.
The degree distributions of many real world networks follow power-laws whose exponents tend to fall between two and three. Within the framework of the Barabasi-Albert model (BA model), we explain this empirical observation by a simple fact. To that end we propose a modified BA model with one parameter that serves as a regulatory factor for the growth rate of added links in scale-free networks. The regulatory factor has something to do with the obvious fact that one link has two nodes. The modified model also allows to connect nodes by newly added links that do not necessarily emanate from new nodes. Another related model using the master equation is also given, from which the same power-law degree distribution can be derived.
The arithmetic mean is the mean for addition and the geometric mean is that for multiplication. Then what kind of binary operation is associated with the arithmetic-geometric mean (AGM) due to C. F. Gauss? If it is possible to construct an arithmetic operation such that AGM is the mean for this operation, it can be regarded as an intermediate operation between addition and multiplication in view of the meaning of AGM. In this paper such an operation is introduced and several of its algebraic properties are proved.
In order to study signed Eulerian numbers, we introduce permutations of a particular type, called parity-alternate permutations, because they take even and odd entries alternately. The objective of this paper is twofold. The first is to derive several properties of those permutations, by subdividing them into even and odd ones. The second is to discuss relationships between those and signed Eulerian numbers. Divisibility properties by prime powers are also deduced for signed Eulerian numbers and several related numbers.
The numbers of even and odd permutations with a given ascent number are investigated using an operator that was previously introduced by the author. Their difference is called a signed Eulerian number. By means of the operator the recurrence relation for signed Eulerian numbers is deduced, which was obtained by an analytic method. Our approach is straightforward and enables us to deduce other properties including divisibility by prime powers.