Tilting mutation of weakly symmetric algebras and stable equivalence

We consider tilting mutations of a weakly symmetric algebra at a subset of simple modules, as recently introduced by T. Aihara. These mutations are defined as the endomorphism rings of certain tilting complexes of length 1. Starting from a weakly symmetric algebra A, presented by a quiver with relations, we give a detailed description of the quiver and relations of the algebra obtained by mutating at a single loopless vertex of the quiver of A. In this form the mutation procedure appears similar to, although significantly more complicated than, the mutation procedure of Derksen, Weyman and Zelevinsky for quivers with potentials. By definition, weakly symmetric algebras connected by a sequence of tilting mutations are derived equivalent, and hence stably equivalent. The second aim of this article is to study these stable equivalences via a result of Okuyama describing the images of the simple modules. As an application we answer a question of Asashiba on the derived Picard groups of a class of self-injective algebras of finite representation type. We conclude by introducing a mutation procedure for maximal systems of orthogonal bricks in a triangulated category, which is motivated by the effect that a tilting mutation has on the set of simple modules in the stable category.Comment: Description and proof of mutated algebra made more rigorous (Prop. 3.1 and 4.2). Okuyama's Lemma incorporated: Theorem 4.1 is now Corollary 5.1, and proof is omitted. To appear in Algebras and Representation Theor

Canonical time-frequency, time-scale, and frequency-scale representations of time-varying channels

Mobile communication channels are often modeled as linear time-varying filters or, equivalently, as time-frequency integral operators with finite support in time and frequency. Such a characterization inherently assumes the signals are narrowband and may not be appropriate for wideband signals. In this paper time-scale characterizations are examined that are useful in wideband time-varying channels, for which a time-scale integral operator is physically justifiable. A review of these time-frequency and time-scale characterizations is presented. Both the time-frequency and time-scale integral operators have a two-dimensional discrete characterization which motivates the design of time-frequency or time-scale rake receivers. These receivers have taps for both time and frequency (or time and scale) shifts of the transmitted signal. A general theory of these characterizations which generates, as specific cases, the discrete time-frequency and time-scale models is presented here. The interpretation of these models, namely, that they can be seen to arise from processing assumptions on the transmit and receive waveforms is discussed. Out of this discussion a third model arises: a frequency-scale continuous channel model with an associated discrete frequency-scale characterization.Comment: To appear in Communications in Information and Systems - special issue in honor of Thomas Kailath's seventieth birthda

On Berenstein-Douglas-Seiberg Duality

I review the proposal of Berenstein-Douglas for a completely general definition of Seiberg duality. To give evidence for their conjecture I present the first example of a physical dual pair and explicitly check that it satisfies the requirements. Then I explicitly show that a pair of toric dual quivers is also dual according to their proposal. All these computations go beyond tilting modules, and really work in the derived category. I introduce all necessary mathematics where needed.Comment: 22 pages, LaTe

Power-Weighted Divergences for Relative Attenuation and Delay Estimation

Power-weighted estimators have recently been proposed for relative attenuation and delay estimation in blind source separation. Their provenance lies in the observation that speech is approximately windowed-disjoint orthogonal (WDO) in the time-frequency (TF) domain; it has been reported that using WDO, derived from TF representations of speech, improves mixing parameter estimation. We show that power-weighted relative attenuation and delay estimators can be derived from a particular case of a weighted Bregman divergence. We then propose a wider class of estimators, which we tune to give better parameter estimates for speech

Progression of RNA-sequencing to single-cell applications

New methods enable new discoveries. My time as a PhD student has run in parallel with the maturation of the RNA-seq method, and I have used it to discover basic properties of gene expression and transcriptomes. My part has been bioinformatics – the computer analysis of biological data. RNA-seq quantifies gene expression for all genes in one experiment, allowing discoveries without prior knowledge, as opposed to single-gene hypothesis testing. When I started my PhD, this was done by microarray followed by qRT-PCR validation, which can be arduous. In contrast to microarrays, RNA-seq quantifies expression with little ambiguity of which gene each expression value corresponds to, and in absolute terms. But at the time, data analysis of RNA-seq was full of unknowns and there were little software available. Nowadays, partly the result of my work, the data analysis is much less complicated, and RNA-seq can be performed on diminutive samples, down to single cells, which was not viable using microarrays. My first study (Paper I) used one of the very first RNA-seq datasets to study general features of transcriptomes, such as mean mRNA length (~1,500 nt) and the number of genes expressed per tissue (~13,000). I also found special features of some tissues: the liver transcriptome is dominated by a few highly expressed gene, brain expresses especially long mRNAs and testis expresses many more genes than other tissues. Following this tissue RNA-seq study, I evaluated a new library preparation method for single-cell RNA-seq (Paper III), developed before the prevalence of single-cell RNA-seq. I used technical replicates to show that the method was accurate and reliable for the more highly expressed genes at single-cell RNA levels, and with input RNA amounts corresponding to >50 cells it produced as good quality data as bulk RNA-seq. Then the method was applied on melanoma cells isolated from human blood, and I listed surface antigen genes that distinguished these circulating tumour cells from other cells in the blood. This single-cell RNA-seq method was then applied on pre-implantation embryo cells (Paper IV). Using first-generation crosses between two mouse strains, I could separate the expression from the maternal and the paternal copies of the genes. I found that 12-24% of the genes express only one of their two copies in any given cell, in a random manner that affects almost all the expressed genes. I also found that the two copies are expressed independently from each other. Finally, I studied Sox transcription factors during neural development (Paper II), combining RNA-seq and microarray data for different cell types with ChIP-seq data for transcription factor binding and histone modifications. I found that Sox proteins bind to the enhancers active in the stem cells where the Sox proteins are active, but also to enhancers specific to subsequent cells in ii development. I also found that different Sox factors bind to much the same enhancers, and that they can induce histone modifications. In conclusion, my work has advanced the RNA-seq method and increased the understanding of transcriptional regulation and output