Problems and Results on Additive Properties of General Sequences IV

Let A={a1,a2,⋯}, a1<a2<⋯, be an infinite sequence of positive integers. One defines its counting function as A(n)=Card{a∈A;a≤n} (n=0,1,2,⋯), and the representation functions R1(n), R2(n), R3(n) (n=0,1,2,⋯), as being the number of representations of n in the form: (1) n=a+a′, a∈A, a′∈A, (2) n=a+a′, a<a′, a∈A, a′∈A, (3) n=a+a′, a≤a′, a∈A, a′∈A, respectively. Of course, for any n≥0, R1(n)=R2(n)+R3(n), and R3(n) is equal either to R2(n)+1, if n is even and n/2 belongs to A, or to R2(n), otherwise. In the first three parts of this series of papers [Erdős and Sárközy, Part I, Pacific J. Math. 118 (1985), no. 2, 347–357; MR0789175 (86j:11015); Part II, Acta Math. Hungar. 48 (1986), 201–211; MR0858398 (88c:11016); Part III, the authors, Studia Sci. Math. Hungar. 22 (1987), no. 1, 53–63], regularity properties of the asymptotic behavior of the function R1 were studied. In Parts IV and V the authors study monotonicity properties of the three functions R1,R2,R3. In Part IV, they prove first that the function R1 is monotone increasing from a certain point on (i.e., there exists an n0 withR1(n+1)≥R1(n) for n≥n0) if and only if the sequence A contains all the integers from a certain point on. The proof uses elementary but complex considerations on counting functions. Secondly, they show that R2 has a different behavior, by exhibiting a class of sequences A satisfying A(n <n−cn1/3 for all large n and such that R2 is monotone increasing from some point onwards. The third result proved in Part IV is that if A(n)=o(n/logn) then the functions R2 and R3 cannot be monotone increasing from a certain point on. Here, the proof is based on analytic properties of the generating function f(z)=∑a∈Aza (|z|<1), corresponding to the sequence A. Part V treats the monotonicity of R3. The main result is as follows: If (4) limn→+∞(n−A(n))/logn=+∞, then lim supN→+∞∑k=1N(R3(2k)−R3(2k+1))=+∞. (Thus, roughly speaking, ai+aj assumes more even values than odd ones.) This theorem implies, firstly, that under hypothesis (4), which is weaker than A(n)=o(n/logn), R3 cannot be monotone increasing from a certain point on, and, secondly, that if A is an infinite "Sidon sequence'' (also called a "B2-sequence'', i.e., a sequence such that R3(n)≤1 for all n), then there are infinitely many integers k such that 2k can be represented in the form 2k=a+a′, a∈A, a′∈A, but 2k+1=a+a′, a∈A, a′∈A, is impossible. Part V finishes with the construction of a sequence showing that the main result is almost best possible. The proofs in Part V are of the same nature as those in Part IV

Monochromatic bounded degree subgraph partitions

Abstract Let F = {F1,F2,...} be a sequence of graphs such that Fn is a graph on n vertices with maximum degree at most Δ. We show that there exists an absolute constant C such that the vertices of any 2-edge-colored complete graph can be partitioned into at most 2CΔlogΔ vertex disjoint monochromatic copies of graphs from F. If each Fn is bipartite, then we can improve this bound to 2CΔ; this result is optimal up to the constant C. © 2015 Elsevier B.V

Isolated hypercholesterolemia leads to steatosis in the liver without affecting the pancreas

Abstract Background Lipid accumulation in the liver and pancreas is primarily caused by combined hyperlipidemia. However, the effect of isolated hypercholesterolemia without hypertriglyceridemia is not fully described. Therefore, our aim was to investigate whether hypercholesterolemia alone leads to alterations both in hepatic and pancreatic lipid panel and histology in rats. Methods Male Wistar rats were fed with 2% cholesterol +0.25% cholate-supplemented diet or standard chow for 12 weeks. Blood was collected at weeks 0, 4, 8 and 12 to measure serum cholesterol and triglyceride levels. At week 12, both the pancreas and the liver were isolated for further histological and biochemical analysis. Hepatic and plasma fatty acid composition was assessed by gas chromatography. Expression of mRNA of major enzymes involved in saturated/unsaturated fatty acid synthesis was analyzed by qPCR. In separate experiments serum enzyme activities and insulin levels were measured at week 9. Results At week 12, rats fed with 2% cholesterol +0.25% cholate-supplemented diet were characterized by elevated serum cholesterol (4.09 ± 0.20 vs. 2.89 ± 0.22 mmol/L, *p < 0.05) while triglyceride (2.27 ± 0.05 vs. 2.03 ± 0.03 mmol/L) and glucose levels (5.32 ± 0.14 vs. 5.23 ± 0.10 mmol/L) remained unchanged. Isolated hypercholesterolemia increased hepatic lipid accumulation, hepatic cholesterol (5.86 ± 0.22 vs. 1.60 ± 0.15 ng/g tissue, *p < 0.05) and triglyceride contents (19.28 ± 1.42 vs. 6.78 ± 0.71 ng/g tissue, *p < 0.05), and hepatic nitrotyrosine level (4.07 ± 0.52 vs. 2.59 ± 0.31 ng/mg protein, *p < 0.05). The histology and tissue lipid content of the pancreas was not affected. Serum total protein level, alanine aminotransferase (ALT) and aspartate aminotransferase (AST) activities remained unchanged in response to isolated hypercholesterolemia while serum alkaline phosphatase activity (ALP) significantly increased. Plasma insulin levels did not change in response to isolated hypercholesterolemia suggesting an intact endocrine function of the pancreas. Isolated hypercholesterolemia caused a significantly increased hepatic and serum fatty acid level associated with a marked alteration of fatty acid composition. Hepatic expression of Δ9-desaturase (SCD1) was increased 4.92×, while expression of Δ5-desaturase and Δ6-desaturase were decreased (0.447× and 0.577×, respectively) due to isolated hypercholesterolemia. Conclusions Isolated hypercholesterolemia leads to hepatic steatosis and marked alterations in the hepatic lipid profile without affecting the pancreas. Altered fatty acid profile might mediate harmful effects of cholesterol in the liver

Small ball probability, Inverse theorems, and applications

Let $\xi$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\R^d$. The random sum $$S_A := a_1 \xi_1 + ... + a_n \xi_n$$ where $\xi_i$ are iid copies of $\xi$ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.Comment: 47 page

Mechanisms and Modulation of Oxidative/Nitrative Stress in Type 4 Cardio-Renal Syndrome and Renal Sarcopenia

Chronic kidney disease (CKD) is a public health problem and a recognized risk factor for cardiovascular diseases (CVD). CKD could amplify the progression of chronic heart failure leading to the development of type 4 cardio-renal syndrome (T4CRS). The severity and persistence of heart failure are strongly associated with mortality risk in T4CRS. CKD is also a catabolic state leading to renal sarcopenia which is characterized by the loss of skeletal muscle strength and physical function. Renal sarcopenia also promotes the development of CVD and increases the mortality in CKD patients. In turn, heart failure developed in T4CRS could result in chronic muscle hypoperfusion and metabolic disturbances leading to or aggravating the renal sarcopenia. The interplay of multiple factors (e.g., comorbidities, over-activated renin-angiotensin-aldosterone system [RAAS], sympathetic nervous system [SNS], oxidative/nitrative stress, inflammation, etc.) may result in the progression of T4CRS and renal sarcopenia. Among these factors, oxidative/nitrative stress plays a crucial role in the complex pathomechanism and interrelationship between T4CRS and renal sarcopenia. In the heart and skeletal muscle, mitochondria, nicotinamide adenine dinucleotide phosphate (NADPH) oxidases, uncoupled nitric oxide synthase (NOS) and xanthine oxidase are major ROS sources producing superoxide anion (O2·−) and/or hydrogen peroxide (H2O2). O2·− reacts with nitric oxide (NO) forming peroxynitrite (ONOO−) which is a highly reactive nitrogen species (RNS). High levels of ROS/RNS cause lipid peroxidation, DNA damage, interacts with both DNA repair enzymes and transcription factors, leads to the oxidation/nitration of key proteins involved in contractility, calcium handling, metabolism, antioxidant defense mechanisms, etc. It also activates the inflammatory response, stress signals inducing cardiac hypertrophy, fibrosis, or cell death via different mechanisms (e.g., apoptosis, necrosis) and dysregulates autophagy. Therefore, the thorough understanding of the mechanisms which lead to perturbations in oxidative/nitrative metabolism and its relationship with pro-inflammatory, hypertrophic, fibrotic, cell death and other pathways would help to develop strategies to counteract systemic and tissue oxidative/nitrative stress in T4CRS and renal sarcopenia. In this review, we also focus on the effects of some well-known and novel pharmaceuticals, nutraceuticals, and physical exercise on cardiac and skeletal muscle oxidative/nitrative stress in T4CRS and renal sarcopenia