On Structure of cluster algebras of geometric type I: In view of sub-seeds and seed homomorphisms

Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point of gluing of seeds. As an application, for (rooted) cluster algebras, we completely classify rooted cluster subalgebras and characterize rooted cluster quotient algebras in detail. Also, we build the relationship between the categorification of a rooted cluster algebra and that of its rooted cluster subalgebras. Note that cluster algebras of geometric type studied here are of the sign-skew-symmetric case.Comment: 41 page

Channel characterization for 1D molecular communication with two absorbing receivers

This letter develops a one-dimensional (1D) diffusion-based molecular communication system to analyze channel responses between a single transmitter (TX) and two fully-absorbing receivers (RXs). Incorporating molecular degradation in the environment, rigorous analytical formulas for i) the fraction of molecules absorbed, ii) the corresponding hitting rate, and iii) the asymptotic fraction of absorbed molecules as time approaches infinity at each RX are derived when an impulse of molecules are released at the TX. By using particle-based simulations, the derived analytical expressions are validated. Simulations also present the distance ranges of two RXs that do not impact molecular absorption of each other, and demonstrate that the mutual influence of two active RXs reduces with the increase in the degradation rate

Parameter Estimation in a Noisy 1D Environment via Two Absorbing Receivers

This paper investigates the estimation of different parameters, e.g., propagation distance and flow velocity, by utilizing two fully-absorbing receivers (RXs) in a one-dimensional (1D) environment. The time-varying number of absorbed molecules at each RX and the number of absorbed molecules in a time interval as time approaches infinity are derived. Noisy molecules in this environment, that are released by sources in addition to the transmitter, are also considered. A novel estimation method, namely difference estimation (DE), is proposed to eliminate the effect of noise by using the difference of received signals at the two RXs. For DE, the Cramer-Rao lower bound (CRLB) on the variance of estimation is derived. Independent maximum likelihood estimation is also considered at each RX as a benchmark to show the performance advantage of DE. Aided by particle-based simulation, the derived analytical results are verified. Furthermore, numerical results show that DE attains the CRLB and is less sensitive to the change of noise than independent estimation at each RX.Comment: 7 pages, 5 figures, accepted by Globecom 202

A note on the asymptotics for the randomly stopped weighted sums

Let {Xi , i ⩾ 1} be a sequence of identically distributed real-valued random variables with common distribution FX; let {θi , i ⩾ 1} be a sequence of identically distributed, nonnegative and nondegenerate at zero random variables; and let τ be a positive integer-valued counting random variable. Assume that {Xi , i ⩾ 1}, {θi , i ⩾ 1} and τ are mutually independent. In the presence of heavy-tailed Xi's, this paper investigates the asymptotic tail behavior for the maximum of randomly weighted sums Mτ = max1 ⩽ k ⩽ τ ∑ki = 1θi Xi under the condition that {θi , i ⩾ 1} satisfy a general dependence structure