Lecture Notes taken by Nilufer Koldan In this lecture I will describe one of the most significant achievements of the second half of the XX century – the Atiyah-Singer index theorem. I will also discuss some more recent developments in the area as well as some open problems. 1. The definition of the index 1.1. The finite-dimensional case. Let A : V 1 → V 2 be a linear map between finite dimensi...

In this paper, new definitions of a fuzzy basis and a fuzzy dimension for a fuzzy vector space are presented. A fuzzy basis for a fuzzy vector space (E, μ) is a fuzzy set β on E. The cardinality of a fuzzy basis β is called the fuzzy dimension of (E, μ). The fuzzy dimension of a finite dimensional fuzzy vector space is a fuzzy natural number. For a fuzzy vector space, any two fuzzy bases have t...

We introduce the concept of the ghost dimension gh. dim. R of a ring R. This is the longest nontrivial chain of maps in the derived category emanating from a perfect complex such that each map is zero on homology. We show that gh. dim. R ≤ w. dim. R, with equality if R is coherent or w. dim. R = 1.

BACKGROUND The objective of this study was to demonstrate the anti-skin cancer and chemopreventive potential of 1,1-bis(3'-indolyl)-1-(p-chlorophenyl methane) (DIM-D) using an in vitro model. METHODS In vitro cell cytotoxicity and viability assays were carried out in A431 human epidermoid carcinoma cell line and normal human epidermal keratinocytes (NHEK) respectively by crystal violet staini...

Functionally distinct chromatin domains are delineated by distinct posttranslational modifications of histones, and in some organisms by differences in DNA methylation. Proper establishment and maintenance of chromatin domains is critical but not well understood. We previously demonstrated that heterochromatin in the filamentous fungus Neurospora crassa is marked by cytosine methylation directe...

If S is an infinite sequence over a finite alphabet Σ and β is a probability measure on Σ, then the dimension of S with respect to β, written dim(S), is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension dim(S) when β is the uniform probability measure. This paper shows that dim(S) and its dual Dim(S), the strong dimension of S with respec...

We have identified the function of the Arabidopsis DIMINUTO/DWARF1 (DIM/DWF1) gene by analyzing the dim mutant, a severe dwarf with greatly reduced fertility. Both the mutant phenotype and gene expression could be rescued by the addition of exogenous brassinolide. Analysis of endogenous sterols demonstrated that dim accumulates 24-methylenecholesterol but is deficient in campesterol, an early p...

Drug-induced myopathy (DIM) is a rare side effect; however, the consequence could be fatal. There are few reports to systematically assess the underlying mechanism of DIM. In this study, we curated the comprehensive DIM drug list based on structured labeling products (SPLs) and carried out the analysis based on chemical structure space, drug protein interaction, side effect space, and transcrip...

This study employed cultured human primary hepatocytes to investigate the ability of the putative chemopreventive phytochemicals curcumin (CUR), 3,3'-diindolylmethane (DIM), isoxanthohumol (IXN), or 8-prenylnaringenin (8PN) to reduce DNA adduct formation of the hepatocarcinogen aflatoxin B1 (AFB). Following 48 h of pretreatment, DIM and 8PN significantly increased AFB-DNA adduct levels, whereas...