An Evidence-Based Review of Probiotics and Prebiotics

Probiotics and prebiotics have a variety of beneficial effects on the host’s health. Extensive studies have established probiotic strains such as Lactobacillus and Bifidobacterium, and further the concept of next-generation probiotics has been advocated. Clinical trials and mechanism of action research have demonstrated that the gut microbiota and host health are inextricably linked, and that probiotics can benefit intestinal-related disorders such as inflammatory bowel disease by controlling the gut microbiota. Accordingly, the host’s gut microbiota has the greatest direct effect on the efficiency of probiotics and prebiotics. Due to the highly individualized gut microbiota, supplementation with probiotics and prebiotics must take the host’s gut microbiota into account. Personalized and specific interventions, as well as the development of next-generation probiotics, will be the new focus of research

The Sylvester Theorem and the Rogers-Ramanujan Identities over Totally Real Number Fields

In this paper, we prove two identities on the partition of a totally positive algebraic integer over a totally real number field which are the generalization of the Sylvester Theorem and that of the Rogers-Ramanujan Identities. Additionally, we give an another version of generalized Rogers-Ramanujan Identities

The Euler-Glaisher Theorem over Totally Real Number Fields

In this paper, we study the partition theory over totally real number fields. Let $K$ be a totally real number field. A partition of a totally positive algebraic integer $\delta$ over $K$ is $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ for some totally positive integers $\lambda_i$ such that $\delta=\lambda_1+\lambda_2+\cdots+\lambda_r$. We find an identity to explain the number of partitions of $\delta$ whose parts do not belong to a given ideal $\mathfrak a$. We obtain a generalization of the Euler-Glaisher Theorem over totally real number fields as a corollary. We also prove that the number of solutions to the equation $\delta=x_1+2x_2+\cdots+nx_n$ with $x_i$ totally positive or $0$ is equal to that of chain partitions of $\delta$. A chain partition of $\delta$ is a partition $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ of $\delta$ such that $\lambda_{i+1}-\lambda_i$ is totally positive or $0$