The L_(p) Shephard problem on entropy of log-concave functions

this paper,we introduce the L_(p) Shephard problem on entropy of log-concave functions,a comparison problem:whether ∏_(p)f≤∏_(p)g implies that Ent(f)≥Ent(g),for 1≤p<n,and Ent(f)≤Ent(g),for n<p,where ∏_(p)f is the L_(p) projection body of a log-concave function f.Our results give a partial answer to this problem.

A genome-wide RNAi screen identifies genes regulating the formation of P bodies in C. elegans and their functions in NMD and RNAi

Cytoplasmic processing bodies,termed P bodies,are involved in diverse post-transcriptional processes including mRNA decay,nonsense-mediated RNA decay(NMD),RNAi,miRNA-mediated translational repression and storage of translationally silenced mRNAs.Regulation of the formation of P bodies in the context of multicellular organisms is poorly understood.Here we describe a systematic RNAi screen in C.elegans that identified 224 genes with diverse cellular functions whose inactivations result in a dramatic increase in the number of P bodies.83 of these genes form a complex functional interaction network regulating NMD.We demonstrate that NMD interfaces with many cellular processes including translation,ubiquitin-mediated protein degradation,intracellular trafficking and cytoskeleton structure.We also uncover an extensive link between translation and RNAi,with different steps in protein synthesis appearing to have distinct effects on RNAi efficiency.Moreover,the intracellular vesicular trafficking network plays an important role in the regulation of RNAi.A subset of genes enhancing P body formation also regulate the formation of stress granules in C.elegans.Our study offers insights into the cellular mechanisms that regulate the formation of P bodies and also provides a framework for system-level understanding of NMD and RNAi in the context of the development of multicellular organisms.

L_p-mixed intersection bodies

In this paper the author first introduce a new concept of Lp-dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of Lp- intersection body to Lp-mixed intersection body. Inequalities for Lp-dual mixed volumes of Lp-mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities.

Brunn-Minkowski Inequalities of General L_(p)-Intersection Bodies

In this paper, we study the extremum inequalities of general L_(p)-intersection bodies. In addition, associating with the L_(q)-radial combination and Lq-harmonic Blaschke combination, we establish the Brunn-Minkowski type inequalities of general Lp-intersection bodies for dual quermassintegrals, respectively. As applications, inequalities of volume are derived.

Busemann-Petty Type Problem for the General L_(p)-Centroid Bodies

Lutwak showed the Busemann-Petty type problem(also called the Shephard type problem)for the centroid bodies.Grinberg and Zhang gave an affirmation and a negative form of the Busemann-Petty type problem for the L_(p)-centroid bodies.In this paper,we obtain an affirmation form and two negative forms of the BusemannPetty type problem for the general L_(p)-centroid bodies.

Inequalities on Complex L_(p) Centroid Bodies

Based on the notion of the complex L_(p)centroid body,we establish Brunn-Minkowski type inequalities and monotonicity inequalities for complex L_(p)centroid bodies in this article.Moreover,we obtain the affirmative form of Shephard type problem for the complex L_(p)centroid bodies and its negative form.