Branching network is one of the most universal phenomena in living or non-living systems, such as river systems and the bronchial trees of mammals. To topologically characterize the branching networks, the Branch Length Similarity (BLS) entropy was suggested and the statistical methods based on the entropy have been applied to the shape identification and pattern recognition. However, the mathematical properties of the BLS entropy have not still been explored in depth because of the lack of application and utilization requiring advanced mathematical understanding. Regarding the mathematical study, it was reported, as a theorem, that all BLS entropy values obtained for simple networks created by connecting pixels along the boundary of a shape are exactly unity when the shape has infinite resolution. In the present study, we extended the theorem to the network created by linking infinitely many nodes distributed on the bounded or unbounded domain in Rk for k ≥ 1. We proved that all BLS entropies of the nodes in the network go to one as the number of nodes, n, goes to infinite and its convergence rate is 1 - O(1= ln n), which was confirmed by the numerical tests.

The foraging territories of 2 subterranean termites, Coptotermes formosanus Shiraki and Reticulitermes flavipes (Kollar), were simulated using a model to explore how territorial intraspecific competition changes with 4 variables characterizing the formation of territory: the number of primary tunnels, $N_0$; the branching probability, $P_{branch}$; the number of territories, $N$; and the blocking probability, $P_{block}$. The blocking probability $P_{block}$ quantitatively describes the probability that a tunnel will be terminated when another tunnel is encountered; higher $P_{block}$ values indicate more likely termination. Higher tunnel-tunnel encounters led to denser tunnel networks. We defined a territory as a convex polygon containing a tunnel pattern and explored the effects of competition among termite colonies on territory size distribution at steady state attained after sufficient simulation time. At the beginning of the simulation, $N = 10, 20, \cdots, 100$ initial territory seeds were randomly distributed within a square area. In our previous study, we introduced an interference coefficient $\gamma$; to characterize territorial competition. Higher $\gamma$; values imply higher limitations on network growth. We theoretically derived $\gamma$; as a function of $P_{block}$ and $N$. In this study, we considered the constants in $\gamma$; as functions of $N_0$ and $P_{branch}$ so as to quantitatively examine the effect of tunnel structure on territorial competition. By applying statistical regression to the simulation data, we determined the generalized $\gamma$; functions for both species. Under competitive conditions, territory size is most strongly affected by $N_0$, while the outcome of territorial competition is most strongly affected by $N$, followed by $P_{block}$ and $N_0$.

This paper proves the large deviation principle for affine diffusion processes with initial values in the interior of the state space $\mathbb{R}^m_{+} \times \mathbb{R}^n$. We approach this problem in two different ways. In the first approach, we first prove the large deviation principle for finite dimensional distributions, and then use it to establish the sample path large deviation principle. For this approach, a more careful examination of the affine transform formula is required. The second approach exploits the exponential martingale method of Donati-Martin et al. for the squares of Ornstein-Uhlenbeck processes. We provide an application to importance sampling of affine diffusion models.

This paper introduces three different simulation algorithms of the shifted Poisson distribution. The first algorithm is the inverse transform method, the second is the rejection sampling, and the third is gamma-Poisson hierarchy sampling. Three algorithms have different regions of parameters at which they are efficient. We numerically compare those algorithms with different sets of parameters. As an application, we give a simulation method of the constant elasticity of variance model.

Recently, Andrews, Chan, Kim and Osburn introduced the even strings and the odd strings in the overpartitions. We show that their conjecture \[ A_k (n) \ge B_k (n) \] holds for large enough positive integers $n$, where $A_k (n)$ (resp. $B_k (n)$) is the number of odd (resp. even) strings along the overpartitions of $n$. We introduce $m$-strings and show how this new combinatorial object is related with another positivity conjecture of Andrews, Chan, Kim, and Osburn. Finally, we confirm that the positivity conjecture is also true for large enough integers.

In this note, we provide a new proof for the number of partitions of $n$ having subpartitions of length $\ell$ with gap $d$. Moreover, by generalizing the definition of a subpartition, we show what is counted by $q$-expansion \[ \prod_{n=1}^{\infty} \frac{1}{1-q^n} \sum_{n=0}^{\infty} (-1)^n q^{(an^2 + bn)/2} \] and how fast it grows. Moreover, we prove there is a special sign pattern for the coefficients of $q$-expansion \[ \prod_{n=1}^{\infty} \frac{1}{1-q^n} \left( 1 - 2 \sum_{n=0}^{\infty} (-1)^n q^{(an^2 + bn)/2} \right). \]

We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$ with the sectional curvature bounded from below by $−κ$ for $κ≥0.$ In the elliptic case, Wang and Zhang [24] recently extended the results of [5] to nonlinear elliptic equations in nondivergence form on such $M$, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on $M$. The Harnack inequality of nonnegative viscosity solutions to the elliptic equations is also proved.

Lipmaa et al. introduced a new security notion of designated verifier signature schemes, non-delegatability: neither a signer nor a designated verifier can delegate the signing rights to any third party without revealing their secret keys. In this paper, we classify designated verifier signature schemes into three types and then discuss delegatability of existing designated verifier signature schemes, strong designated verifier signature schemes and universal designated verifier signature schemes, and open research issues.

Wireless reprogramming in a wireless sensor network (WSN) involves the process of propagating a new code image or relevant command to sensor nodes. As a WSN is usually deployed in a hostile environment, secure reprogramming is a major concern. Recently, He et al. proposed a secure distributed reprogramming protocol, SDRP, based on an identity-based signature (IBS) scheme. Subsequently, they showed that SDRP is insecure against impersonation attacks due to the use of insecure IBS scheme. They then proposed a modified SDRP (MSDRP) based on Barreto et al.’s IBS scheme which is provably secure under a mathematically hard problem. Also, they proposed a DoS-resistant distributed code dissemination protocol, DiCode, based on a warrant-based proxy signature scheme. However, the two protocols are inefficient on sensor nodes: MSDRP requires a heavy pairing computation and DiCode requires two modular exponentiations with an RSA modulus n whose size is 1024 bits at an 80-bit security level to verify a signature. In this paper, we show that MSDRP with the implementation of eta T paring defined on E(F_{3^97} x E(F_{3^97}) ->E(F_3^{697}) is entirely broken. We then propose a new SDRP, S2DRP, based on a pairing-free IBS scheme to reduce the computational and communication overhead and give its performance results.

The vector decomposition problem has been considered as a hard problem, which is applicable to cryptography. Okamoto and Takashima proposed various types of public key cryptographic schemes based on the VDP. In this paper, we study the cryptographic implications of Okamoto-Takashima schemes with respect to the properties of public keys. In the public key cryptography, one public key is associated to a unique private key, and an action using the public key implicitly assumes that the corresponding private action can be done only with the corresponding private key.We formalize this security issue by introducing the notion of equivalent public keys. We show that equivalent public keys exist in the Okamoto-Takashima basic signature scheme and the homomorphic encryption scheme.We present a strong key substitution attack to their basic signature. We suggest how to prevent equivalent public keys and strong key substitution attack in their signature scheme. We point out that there are cases with no efficient methods to prevent equivalent public keys in their encryption scheme.