Tiling the integers with translates of one finite set

A set is said to tile the integers if and only if the integers can be written as a disjoint union of translates of that set. We consider the problem of finding necessary and sufficient conditions for a finite set to tile the integers. For sets of prime power size, it was solved by D. Newman [J. Number Theory 9 (1977), 107--111]. We solve it for sets of size having at most two prime factors. The conditions are always sufficient, but it is unknown whether they are necessary for all finite sets.Comment: 12 page

J Regularization Improves Imbalanced Multiclass Segmentation

We propose a new loss formulation to further advance the multiclass segmentation of cluttered cells under weakly supervised conditions. When adding a Youden's J statistic regularization term to the cross entropy loss we improve the separation of touching and immediate cells, obtaining sharp segmentation boundaries with high adequacy. This regularization intrinsically supports class imbalance thus eliminating the necessity of explicitly using weights to balance training. Simulations demonstrate this capability and show how the regularization leads to correct results by helping advancing the optimization when cross entropy stagnates. We build upon our previous work on multiclass segmentation by adding yet another training class representing gaps between adjacent cells. This addition helps the classifier identify narrow gaps as background and no longer as touching regions. We present results of our methods for 2D and 3D images, from bright field images to confocal stacks containing different types of cells, and we show that they accurately segment individual cells after training with a limited number of images, some of which are poorly annotated

Tessellations and Pattern Formation in Plant Growth and Development

The shoot apical meristem (SAM) is a dome-shaped collection of cells at the apex of growing plants from which all above-ground tissue ultimately derives. In Arabidopsis thaliana (thale cress), a small flowering weed of the Brassicaceae family (related to mustard and cabbage), the SAM typically contains some three to five hundred cells that range from five to ten microns in diameter. These cells are organized into several distinct zones that maintain their topological and functional relationships throughout the life of the plant. As the plant grows, organs (primordia) form on its surface flanks in a phyllotactic pattern that develop into new shoots, leaves, and flowers. Cross-sections through the meristem reveal a pattern of polygonal tessellation that is suggestive of Voronoi diagrams derived from the centroids of cellular nuclei. In this chapter we explore some of the properties of these patterns within the meristem and explore the applicability of simple, standard mathematical models of their geometry.Comment: Originally presented at: "The World is a Jigsaw: Tessellations in the Sciences," Lorentz Center, Leiden, The Netherlands, March 200

LEAFY Controls Auxin Response Pathways in Floral Primordium Formation

The transcription factor LEAFY is a master regulator of flowering and of flower development. It acts as a component of a switch that mediates the transition from the vegetative to the reproductive phase of plant development. Auxin is a plant hormone with many different roles in plant growth, including the induction of new primordia of both leaves and flowers at the shoot apex. We report that LEAFY acts in part by controlling the auxin response pathway in new primordia. Therefore, regulation of flower development by transcriptional master regulators and hormonal control of morphogenesis appear to be interacting processes. We found that hormone perception not only controls but is also controlled by the transcriptional signals that create plant form

cis-acting sequences required for expression of the divergently transcribed Drosophila melanogaster Sgs-7 and Sgs-8 glue protein genes

The Sgs-7 and Sgs-8 glue genes at 68C are divergently transcribed and are separated by 475 bp. Fusion genes with Adh or lacZ coding sequences were constructed, and the expression of these genes, with different amounts of upstream sequences present, was tested by a transient expression procedure and by germ line transformation. A cis-acting element for both genes is located asymmetrically in the intergenic region between -211 and -43 bp relative to Sgs-7. It is required for correct expression of both genes. This element can confer the stage- and tissue-specific expression pattern of glue genes on a heterologous promoter. An 86-bp portion of the element, from -133 to -48 bp relative to Sgs-7, is shown to be capable of enhancing the expression of a truncated and therefore weakly expressed Sgs-3 fusion gene. Recently described common sequence motifs of glue gene regulatory elements (T. Todo, M. Roark, K. Vijay Raghavan, C. A. Mayeda, and E.M. Meyerowitz, Mol. Cell. Biol. 10:5991-6002, 1990) are located within this 86-bp region

The N-end rule pathway controls multiple functions during Arabidopsis shoot and leaf development

The ubiquitin-dependent N-end rule pathway relates the in vivo half-life of a protein to the identity of its N-terminal residue. This proteolytic system is present in all organisms examined and has been shown to have a multitude of functions in animals and fungi. In plants, however, the functional understanding of the N-end rule pathway is only beginning. The N-end rule has a hierarchic structure. Destabilizing activity of N-terminal Asp, Glu, and (oxidized) Cys requires their conjugation to Arg by an arginyl–tRNA–protein transferase (R-transferase). The resulting N-terminal Arg is recognized by the pathway's E3 ubiquitin ligases, called “N-recognins.” Here, we show that the Arabidopsis R-transferases AtATE1 and AtATE2 regulate various aspects of leaf and shoot development. We also show that the previously identified N-recognin PROTEOLYSIS6 (PRT6) mediates these R-transferase-dependent activities. We further demonstrate that the arginylation branch of the N-end rule pathway plays a role in repressing the meristem-promoting BREVIPEDICELLUS (BP) gene in developing leaves. BP expression is known to be excluded from Arabidopsis leaves by the activities of the ASYMMETRIC LEAVES1 (AS1) transcription factor complex and the phytohormone auxin. Our results suggest that AtATE1 and AtATE2 act redundantly with AS1, but independently of auxin, in the control of leaf development

J Regularization Improves Imbalanced Multiclass Segmentation

We propose a new loss formulation to further advance the multiclass segmentation of cluttered cells under weakly supervised conditions. When adding a Youden's J statistic regularization term to the cross entropy loss we improve the separation of touching and immediate cells, obtaining sharp segmentation boundaries with high adequacy. This regularization intrinsically supports class imbalance thus eliminating the necessity of explicitly using weights to balance training. Simulations demonstrate this capability and show how the regularization leads to correct results by helping advancing the optimization when cross entropy stagnates. We build upon our previous work on multiclass segmentation by adding yet another training class representing gaps between adjacent cells. This addition helps the classifier identify narrow gaps as background and no longer as touching regions. We present results of our methods for 2D and 3D images, from bright field images to confocal stacks containing different types of cells, and we show that they accurately segment individual cells after training with a limited number of images, some of which are poorly annotated

Alignment between PIN1 Polarity and Microtubule Orientation in the Shoot Apical Meristem Reveals a Tight Coupling between Morphogenesis and Auxin Transport

Morphogenesis during multicellular development is regulated by intercellular signaling molecules as well as by the mechanical properties of individual cells. In particular, normal patterns of organogenesis in plants require coordination between growth direction and growth magnitude. How this is achieved remains unclear. Here we show that in Arabidopsis thaliana, auxin patterning and cellular growth are linked through a correlated pattern of auxin efflux carrier localization and cortical microtubule orientation. Our experiments reveal that both PIN1 localization and microtubule array orientation are likely to respond to a shared upstream regulator that appears to be biomechanical in nature. Lastly, through mathematical modeling we show that such a biophysical coupling could mediate the feedback loop between auxin and its transport that underlies plant phyllotaxis

Applying valence framing to enhance the effect of information on transport-related carbon dioxide emissions

The provision of information about transport-related carbon dioxide (CO2) emissions to the traveler can be seen as an instrument to increase the likelihood of more sustainable choices being made by individuals. However, as transport-related CO2 emissions are largely seen as a 'social' cost rather than a 'private' cost to the individual, the behavioral engagement with and response to information on environmental effects of travel choices may be limited. It is argued that framing, studied in a range of contexts, can be used to enhance the evaluation of choice attributes and promote more sustainable choices. An experiment is reported that examines the effect of valence framing of amounts of CO2 emissions on the perceived differences between alternative amounts. Through the use of positive and negative terms, the information is framed to focus attention either on the potential of a travel mode to provide environmental benefit (positive frame) or on its potential to reduce environmental loss (negative frame). Survey participants' estimates of CO2 amounts were compared for positive and negative framing of the same information using an ordered logit model. The findings imply that negative framing is more effective than positive framing in highlighting differences between CO2 amounts of alternative travel modes and therefore is likely to influence travel-related choices. © 2012 Elsevier Ltd