Entropy in the category of perfect complexes with cohomology of finite length

Local and category-theoretical entropies associated with an endomorphism of finite length (i.e., with zero-dimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to category-theoretical entropy. The two entropies are shown to be equal when the ring is regular, and also for the Frobenius endomorphism of a complete local ring of positive characteristic.Furthermore, given a flat morphism of Cohen–Macaulay local rings endowed with compatible endomorphisms of finite length, it is shown that local entropy is “additive”. Finally, over a ring that is a homomorphic image of a regular local ring, a formula for local entropy in terms of an asymptotic partial Euler characteristic is given.     

Properties of abelian categories via recollements

A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an example where the converse does not hold and explore multiple sufficient conditions for it to hold.     

Traveling wave solutions in delayed higher dimensional lattice differential systems with partial monotonicity

In this paper, we consider the existence of traveling wave solutions in delayed higher dimensional lattice differential systems with partial monotonicity. By relaxing the monotonicity of the upper solutions and allowing it greater than positive equilibrium point, we establish the existence of traveling wave solutions by means of Schauder's fixed point theorem. And then, we apply our results to delayed competition‐cooperation systems on higher dimensional lattices.     

Restricted thresholding: recovering smoothness and preserving edges

Restricted non linear approximation is a generalization of the N‐term approximation in which a measure on the index set of the approximants controls the type, instead of the number, of elements in the approximation. Thresholding is a well‐known type of non linear approximation. We relate a generalized upper and lower Temlyakov property with the decreasing rate of the thresholding approximation. This relation is in the form of a characterization through some general discrete Lorentz spaces. Thus, not only we recover some results in the literature but find new ones. As an application of these results, we compress and reduce noise of some images with wavelets and shearlets and show, at least empirically, that the L²‐norm is not necessarily the best norm to measure the approximation error.     

关于6维近凯勒流形中二阶平行拉格朗日子流形的注记

证明6维严格近凯勒流形中的二阶平行拉格朗日子流形一定是全测地的,这推广了L.Vrancken等人文中的一个重要结果.特别地,得到了齐性近凯勒S³×S³中该类拉格朗日子流形的完全分类.     

Targeting the PI3K/AKT/mTOR Signaling Pathway in Lung Cancer: An Update Regarding Potential Drugs and Natural Products

Lung cancer is one of the most common cancers and has a high mortality rate. Due to its high incidence, the clinical management of the disease remains a major challenge. Several reports have documented a relationship between the phosphatidylinositol-3-kinase (PI3K)/ protein kinase B (AKT)/ mammalian target of rapamycin (mTOR) pathway and lung cancer. The recognition of this pathway as a notable therapeutic target in lung cancer is mainly due to its central involvement in the initiation and progression of the disease. Interest in using natural and synthetic medications to target these signaling pathways has increased in recent years, with promising results in vitro, in vivo, and in clinical trials. In this review, we focus on the current understanding of PI3K/AKT/mTOR signaling in tumor development. In addition to the signaling pathway, we highlighted the therapeutic potential of recently developed PI3K/AKT/mTOR inhibitors based on preclinical and clinical trials.     

Totally Permutable Products of Finite Groups Satisfying SC or PST

Finite groups G=AB factorized by two subgroups A and B such that every subgroup of A permutes with every subgroup of B are studied in this paper. The behaviour of such products with respect to the class of finite groups in which Sylow-permutability is transitive is analyzed.     

Characterization of polynomial decay rate for the solution of linear evolution equation

In this paper, we study the decay rate of solutions to strongly stable, but not exponentially stable linear evolution equations. It is known that the resolvent operator of such an equation must be unbounded on the imaginary axis. Our main result is an estimate of the decay rate when the unboundedness is of polynomial order. We then apply our main theorem to three strongly stable but not exponentially stable systems to obtain the decay rate, which is not available in the literature.Received: July 21, 2003