Rational self-affine tiles

An integral self-affine tile is the solution of a set equation $\mathbf{A} \mathcal{T} = \bigcup_{d \in \mathcal{D}} (\mathcal{T} + d)$, where $\mathbf{A}$ is an $n \times n$ integer matrix and $\mathcal{D}$ is a finite subset of $\mathbb{Z}^n$. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices $\mathbf{A} \in \mathbb{Q}^{n \times n}$. We define rational self-affine tiles as compact subsets of the open subring $\mathbb{R}^n\times \prod_\mathfrak{p} K_\mathfrak{p}$ of the ad\'ele ring $\mathbb{A}_K$, where the factors of the (finite) product are certain $\mathfrak{p}$-adic completions of a number field $K$ that is defined in terms of the characteristic polynomial of $\mathbf{A}$. Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles. We also associate a second kind of tiles with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with $\mathbb{R}^n \times \prod_\mathfrak{p} \{0\} \simeq \mathbb{R}^n$. Although these intersection tiles have a complicated structure and are no longer self-affine, we are able to prove a tiling theorem for these tiles as well. For particular choices of digit sets, intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. Therefore, we gain new results for tilings associated with numeration systems

Rational self-affine tiles

An integral self-affine tile is the solution of a set equation $\mathbf{A} \mathcal{T} = \bigcup_{d \in \mathcal{D}} (\mathcal{T} + d)$, where $\mathbf{A}$ is an $n \times n$ integer matrix and $\mathcal{D}$ is a finite subset of $\mathbb{Z}^n$. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices $\mathbf{A} \in \mathbb{Q}^{n \times n}$. We define rational self-affine tiles as compact subsets of the open subring $\mathbb{R}^n\times \prod_\mathfrak{p} K_\mathfrak{p}$ of the adéle ring $\mathbb{A}_K$, where the factors of the (finite) product are certain $\mathfrak{p}$-adic completions of a number field $K$ that is defined in terms of the characteristic polynomial of $\mathbf{A}$. Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles. We also associate a second kind of tiles with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with $\mathbb{R}^n \times \prod_\mathfrak{p} \{0\} \simeq \mathbb{R}^n$. Although these intersection tiles have a complicated structure and are no longer self-affine, we are able to prove a tiling theorem for these tiles as well. For particular choices of digit sets, intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. Therefore, we gain new results for tilings associated with numeration systems

Similar dissection of sets

In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner's questions have been generalized and some of them are already solved. In the present paper, we solve more of his questions and treat them in a much more general context. Let $D\subset \mathbb{R}^d$ be a given set and let $f_1,...,f_k$ be injective continuous mappings. Does there exist a set $X$ such that $D = X \cup f_1(X) \cup ... \cup f_k(X)$ is satisfied with a non-overlapping union? We prove that such a set $X$ exists for certain choices of $D$ and $\{f_1,...,f_k\}$. The solutions $X$ often turn out to be attractors of iterated function systems with condensation in the sense of Barnsley. Coming back to Gardner's setting, we use our theory to prove that an equilateral triangle can be dissected in three similar copies whose areas have ratio $1:1:a$ for $a \ge (3+\sqrt{5})/2$

The fecal microbiome in dogs with acute diarrhea and idiopathic inflammatory bowel disease.

Recent molecular studies have revealed a highly complex bacterial assembly in the canine intestinal tract. There is mounting evidence that microbes play an important role in the pathogenesis of acute and chronic enteropathies of dogs, including idiopathic inflammatory bowel disease (IBD). The aim of this study was to characterize the bacterial microbiota in dogs with various gastrointestinal disorders. Fecal samples from healthy dogs (n = 32), dogs with acute non-hemorrhagic diarrhea (NHD; n = 12), dogs with acute hemorrhagic diarrhea (AHD; n = 13), and dogs with active (n = 9) and therapeutically controlled idiopathic IBD (n = 10) were analyzed by 454-pyrosequencing of the 16S rRNA gene and qPCR assays. Dogs with acute diarrhea, especially those with AHD, had the most profound alterations in their microbiome, as significant separations were observed on PCoA plots of unweighted Unifrac distances. Dogs with AHD had significant decreases in Blautia, Ruminococcaceae including Faecalibacterium, and Turicibacter spp., and significant increases in genus Sutterella and Clostridium perfringens when compared to healthy dogs. No significant separation on PCoA plots was observed for the dogs with IBD. Faecalibacterium spp. and Fusobacteria were, however, decreased in the dogs with clinically active IBD, but increased during time periods of clinically insignificant IBD, as defined by a clinical IBD activity index (CIBDAI). Results of this study revealed a bacterial dysbiosis in fecal samples of dogs with various GI disorders. The observed changes in the microbiome differed between acute and chronic disease states. The bacterial groups that were commonly decreased during diarrhea are considered to be important short-chain fatty acid producers and may be important for canine intestinal health. Future studies should correlate these observed phylogenetic differences with functional changes in the intestinal microbiome of dogs with defined disease phenotypes

Novel lipoprotein density profiling in healthy dogs of various breeds, healthy miniature schnauzers, and miniature schnauzers with hyperlipidemia

BACKGROUND: Despite the importance of abnormalities in lipoprotein metabolism in clinical canine medicine, the fact that most previously used methods for lipoprotein profiling are rather laborious and time-consuming has been a major obstacle to the wide clinical application and use of lipoprotein profiling in this species. The aim of the present study was to assess the feasibility of a continuous lipoprotein density profile (CLPDP) generated within a bismuth sodium ethylenediaminetetraacetic acid (NaBiEDTA) density gradient to characterize and compare the lipoprotein profiles of healthy dogs of various breeds, healthy Miniature Schnauzers, and Miniature Schnauzers with primary hypertriacylglycerolemia. A total of 35 healthy dogs of various breeds with serum triacylglycerol (TAG) and cholesterol concentrations within their respective reference intervals were selected for use as a reference population. Thirty-one Miniature Schnauzers with serum TAG and cholesterol concentrations within their respective reference intervals and 31 Miniature Schnauzers with hypertriacylglyceridemia were also included in the study. RESULTS: The results suggest that CLPDP using NaBiEDTA provides unique diagnostic information in addition to measurements of serum TAG and cholesterol concentrations and that it is a useful screening method for dogs with suspected lipoprotein metabolism disorders. Using the detailed and continuous density distribution information provided by the CLPDP, important differences in lipoprotein profiles can be detected even among dogs that have serum TAG and cholesterol concentrations within the reference interval. Miniature Schnauzers with serum TAG and cholesterol concentrations within the reference interval had significantly different lipoprotein profiles than dogs of various other breeds. In addition, it was further established that specific lipoprotein fractions are associated with hypertriacylglyceridemia in Miniature Schnauzers. CONCLUSIONS: The results of the present study suggest that density gradient ultracentrifugation using NaBiEDTA is a useful screening method for the study of lipoprotein profiles in dogs. Therefore, this method could potentially be used for diagnostic purposes for the separation of dogs suspected of having lipoprotein abnormalities from healthy dogs

S100A12 concentrations and myeloperoxidase activity in the intestinal mucosa of healthy dogs

Background: Relatively few laboratory markers have been evaluated for the detection or monitoring of intestinal inflammation in canine chronic enteropathies, including inflammatory bowel disease (IBD). Previous research found that the intestinal mucosal levels of S100A12 and myeloperoxidase (MPO), as biomarkers of gut inflammation, were elevated in human patients with IBD. To date, the S100A12 and MPO levels in intestinal mucosal samples from either healthy dogs or from dogs suffering from IBD remain unreported. Therefore, this study aimed to evaluate the mucosal S100A12 and MPO levels in four different parts of the intestine (duodenum, jejunum, ileum and colon) in 12 healthy laboratory Beagle dogs using the ELISA and spectrophotometric methods, respectively. Results: Based on histological examinations, the recorded findings for all the samples were considered normal. The mucosal concentration of S100A12 in the ileum was significantly higher than in all other segments of the intestine (p <0.05). MPO activity was significantly higher in the ileal, jejunal and duodenal than in colonic mucosal samples (p <0.05). Moreover, its concentration was higher in the jejunum than in the duodenum. Conclusions: This study showed that S100A12 and MPO are reliably detectable in canine intestinal mucosa. The assays used appeared to be sufficient to further evaluate the role of S100A12 and MPO in the pathogenesis of canine chronic enteropathies, including IBD. These biomarkers may play a role in the initial detection of gut inflammation suggesting the need for further investigations to confirm IBD or to differentiate between IBD subtypes. Understanding the role of S100A12 and MPO in the pathogenesis of chronic intestinal inflammation in future may result in an improved understanding of canine chronic intestinal inflammation.Peer reviewe

Assessment of the Variation Associated with Repeated Measurement of Gastrointestinal Transit Times and Assessment of the Effect of Oral Ranitidine on Gastrointestinal Transit Times Using a Wireless Motility Capsule System in Dogs

This study aimed to evaluate the variation associated with repeated measurement of gastrointestinal (GI) transit times and the effect of oral ranitidine on GI transit times in healthy dogs using a wireless motility capsule (WMC) system. Eight privately owned healthy adult dogs were enrolled, and one developed diarrhea and was removed from the study. For the first 3 repetitions, each dog was fed a standard meal followed by oral administration of a WMC. For the 4th repetition, each dog was given ranitidine hydrochloride (75 mg PO every 12 hours) prior to and during assessment of GI transit times. Mean between-subject coefficients of variation for gastric emptying time (GET), small and large bowel transit time (SLBTT), and total transit time (TTT) were 26.9%, 32.3%, and 19.6%, respectively. Mean within-subject coefficients of variation for GET, SLBTT, and TTT were 9.3%, 19.6%, and 15.9%, respectively. Median GET, SLBTT, and TTT without ranitidine were 719, 1,636, and 2,735 minutes, respectively. Median GET, SLBTT, and TTT with ranitidine were 757, 1,227, and 2,083 minutes, respectively. No significant differences in GI transit times were found between any of the 4 repetitions. Under these experimental conditions, no significant effects of oral ranitidine on GI transit times were observed

Serum canine pancreatic-specific lipase concentrations in dogs with naturally occurring Babesia rossi infection

Babesia rossi is the cause of a highly virulent multisystemic disease with a variable outcome, which is a reliable model of systemic inflammatory response syndrome (SIRS). The objective of this study was to determine the concentration of canine pancreatic-specific lipase (cPL) in a population of dogs with naturally acquired B. rossi infection. In addition, the associations between serum cPL and death and SIRS status were examined. An observational study recruited 87 dogs diagnosed with B. rossi infection and serum cPL concentrations were measured daily until discharge or death. The median concentration of serum cPL was 124.0 µg/L (interquartile range: 51.0 µg/L – 475.5 µg/L) on admission (n = 87) and 145.5 µg/L (62.3 µg/L – 434.0 µg/L) on day two of hospitalisation (n = 40). Twenty-four dogs (28%) had a serum cPL concentration within the diagnostic range for pancreatitis (> 400 µg/L) at admission with 13 dogs (32.5%) presenting as such on the second day of hospitalisation. The median concentration of serum cPL in dogs with SIRS was 158 µg/L (interquartile range: 52.5 µg/L – 571.5 µg/L; n = 53), which was significantly higher than in those without SIRS (75 µg/L; 50.3 µg/L – 131.8 µg/L; n = 32) (P = 0.018). This study demonstrated that an unexpectedly high number of dogs diagnosed with naturally acquired canine babesiosis had a serum cPL concentration within the diagnostic range for acute pancreatitis and a significantly higher serum cPL concentration was found in dogs that were classified as having SIRS