A coding problem for pairs of subsets

Let $X$ be an $n$--element finite set, $0<k\leq n/2$ an integer. Suppose that $\{A_1,A_2\}$ and $\{B_1,B_2\}$ are pairs of disjoint $k$-element subsets of $X$ (that is, $|A_1|=|A_2|=|B_1|=|B_2|=k$, $A_1\cap A_2=\emptyset$, $B_1\cap B_2=\emptyset$). Define the distance of these pairs by $d(\{A_1,A_2\} ,\{B_1,B_2\})=\min \{|A_1-B_1|+|A_2-B_2|, |A_1-B_2|+|A_2-B_1|\}$. This is the minimum number of elements of $A_1\cup A_2$ one has to move to obtain the other pair $\{B_1,B_2\}$. Let $C(n,k,d)$ be the maximum size of a family of pairs of disjoint subsets, such that the distance of any two pairs is at least $d$. Here we establish a conjecture of Brightwell and Katona concerning an asymptotic formula for $C(n,k,d)$ for $k,d$ are fixed and $n\to \infty$. Also, we find the exact value of $C(n,k,d)$ in an infinite number of cases, by using special difference sets of integers. Finally, the questions discussed above are put into a more general context and a number of coding theory type problems are proposed.Comment: 11 pages (minor changes, and new citations added

Effects of the fatty acid amide hydrolase inhibitor URB597 on coping behavior under challenging conditions in mice

RATIONALE: Recent evidence suggests that in addition to controlling emotional behavior in general, endocannabinoid signaling is engaged in shaping behavioral responses to challenges. This important function of endocannabinoids is still poorly understood. OBJECTIVES: Here we investigated the impact of blockade of fatty acid amide hydrolase (FAAH), the degrading enzyme of anandamide on behavioral responses induced by challenges of different intensity. METHODS: Mice treated with FAAH inhibitor URB597 were either manually restrained on their backs (back test) or received foot-shocks. RESULTS: The behavior of mice showed bimodal distribution in the back test: they either predominantly showed escape attempts or equally distributed time between passivity and escape. URB597 increased escapes in animals with low escape scores. No effects were noticed in mice showing high escape scores, which is likely due to a ceiling effect. We hypothesized that stronger stressors would wash out individual differences in coping; therefore, we exposed mice to foot-shocks that decreased locomotion and increased freezing in all mice. URB597 ameliorated both responses. The re-exposure of mice to the shock cage 14 days later without delivering shocks or treatment was followed by reduced and fragmented sleep as shown by electrophysiological recordings. Surprisingly, sleep was more disturbed after the reminder than after shocks in rats receiving vehicle before foot-shocks. These reminder-induced disturbances were abolished by URB597 administered before shocks. CONCLUSIONS: These findings suggest that FAAH blockade has an important role in the selection of behavioral responses under challenging conditions and-judging from its long-term effects-that it influences the cognitive appraisal of the challenge

Atrasentan and renal events in patients with type 2 diabetes and chronic kidney disease (SONAR): a double-blind, randomised, placebo-controlled trial

Background: Short-term treatment for people with type 2 diabetes using a low dose of the selective endothelin A receptor antagonist atrasentan reduces albuminuria without causing significant sodium retention. We report the long-term effects of treatment with atrasentan on major renal outcomes. Methods: We did this double-blind, randomised, placebo-controlled trial at 689 sites in 41 countries. We enrolled adults aged 18–85 years with type 2 diabetes, estimated glomerular filtration rate (eGFR)25–75 mL/min per 1·73 m 2 of body surface area, and a urine albumin-to-creatinine ratio (UACR)of 300–5000 mg/g who had received maximum labelled or tolerated renin–angiotensin system inhibition for at least 4 weeks. Participants were given atrasentan 0·75 mg orally daily during an enrichment period before random group assignment. Those with a UACR decrease of at least 30% with no substantial fluid retention during the enrichment period (responders)were included in the double-blind treatment period. Responders were randomly assigned to receive either atrasentan 0·75 mg orally daily or placebo. All patients and investigators were masked to treatment assignment. The primary endpoint was a composite of doubling of serum creatinine (sustained for ≥30 days)or end-stage kidney disease (eGFR <15 mL/min per 1·73 m 2 sustained for ≥90 days, chronic dialysis for ≥90 days, kidney transplantation, or death from kidney failure)in the intention-to-treat population of all responders. Safety was assessed in all patients who received at least one dose of their assigned study treatment. The study is registered with ClinicalTrials.gov, number NCT01858532. Findings: Between May 17, 2013, and July 13, 2017, 11 087 patients were screened; 5117 entered the enrichment period, and 4711 completed the enrichment period. Of these, 2648 patients were responders and were randomly assigned to the atrasentan group (n=1325)or placebo group (n=1323). Median follow-up was 2·2 years (IQR 1·4–2·9). 79 (6·0%)of 1325 patients in the atrasentan group and 105 (7·9%)of 1323 in the placebo group had a primary composite renal endpoint event (hazard ratio [HR]0·65 [95% CI 0·49–0·88]; p=0·0047). Fluid retention and anaemia adverse events, which have been previously attributed to endothelin receptor antagonists, were more frequent in the atrasentan group than in the placebo group. Hospital admission for heart failure occurred in 47 (3·5%)of 1325 patients in the atrasentan group and 34 (2·6%)of 1323 patients in the placebo group (HR 1·33 [95% CI 0·85–2·07]; p=0·208). 58 (4·4%)patients in the atrasentan group and 52 (3·9%)in the placebo group died (HR 1·09 [95% CI 0·75–1·59]; p=0·65). Interpretation: Atrasentan reduced the risk of renal events in patients with diabetes and chronic kidney disease who were selected to optimise efficacy and safety. These data support a potential role for selective endothelin receptor antagonists in protecting renal function in patients with type 2 diabetes at high risk of developing end-stage kidney disease. Funding: AbbVie

On reverse-free codes and permutations

A set F of ordered k-tuples of distinct elements of an n-set is pairwise reverse free if it does not contain two ordered k-tuples with the same pair of elements in the same pair of coordinates in reverse order. Let F(n, k) be the maximum size of a pairwise reverse-free set. In this paper we focus on the case of 3-tuples and prove limF(n, 3)/(n/3) = 5/4, more exactly, 5/14n 3 - 1/2n 2 - O(nlogn) &gt; F(n, 3) ≥ 5/24 n 3 - 1/2n 2 + 5/8n, and here equality holds when n is a power of 3. Many problems remain open. © 2010 Societ y for Industrial and Applied Mathematics

Synthesis and Characterization of Calcium Phosphate Materials Derived from Eggshells from Different Poultry with and without the Eggshell Membrane

Calcium phosphate materials such as hydroxyapatite (HA) or tricalcium phosphate (β-TCP) are highly attractive due to their multitude of applications in bone replacement as well as their environmental and ecological credentials. In this research, quail, hen, duck, and pigeon eggshells were used as a calcium source to obtain calcium phosphate materials via the environmentally friendly wet synthesis. Using the eggshells with the organic membrane, the biphasic calcium phosphate materials composed mainly of HA were obtained. The second mineral phase was β-TCP in the case of using quail, hen, and pigeon eggshells and octacalcium phosphate (OCP) in the case of duck eggshells. The HA content in the obtained materials depended on the amount of membrane in the eggshells and decreased in the order of pigeon, duck, hen, and quail eggshells. The eggshell membrane removal from the eggshells caused the reduced content of HA and the presence of the more soluble β-TCP or OCP phase in the obtained materials. The calcium ions release profile in the PBS buffer indicates the potential biomedical application of these materials