Self-Similar Algebras with connections to Run-length Encoding and Rational Languages

A self-similar algebra $\left(\mathfrak{A}, \psi \right)$ is an associative algebra $\mathfrak{A}$ with a morphism of algebras $\psi: \mathfrak{A} \longrightarrow M_d \left( \mathfrak{A}\right)$, where $M_d \left( \mathfrak{A}\right)$ is the set of $d\times d$ matrices with coefficients from $\mathfrak{A}$. We study the connection between self-similar algebras with run-length encoding and rational languages. In particular, we provide a curious relationship between the eigenvalues of a sequence of matrices related to a specific self-similar algebra and the smooth words over a 2-letter alphabet. We also consider the language $L(s)$ of words $u$ in $(\Sigma\times \Sigma)^*$ where $\Sigma=\{0,1\}$ such that $s\cdot u$ is a unit in $\mathfrak{A}$. We prove that $L(s)$ is rational and provide an asymptotic formula for the number of words of a given length in $L(s)$

On the van der Waerden numbers w(2;3,t)

We present results and conjectures on the van der Waerden numbers w(2;3,t) and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39, where for t <= 30 we conjecture these lower bounds to be exact. The lower bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we present an improved conjecture. We also investigate regularities in the good partitions (certificates) to better understand the lower bounds. Motivated by such reglarities, we introduce *palindromic van der Waerden numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good partitions), defined as reading the same from both ends. Different from the situation for ordinary van der Waerden numbers, these "numbers" need actually to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide lower bounds, which we conjecture to be exact, for t <= 35. All computations are based on SAT solving, and we discuss the various relations between SAT solving and Ramsey theory. Especially we introduce a novel (open-source) SAT solver, the tawSolver, which performs best on the SAT instances studied here, and which is actually the original DLL-solver, but with an efficient implementation and a modern heuristic typical for look-ahead solvers (applying the theory developed in the SAT handbook article of the second author).Comment: Second version 25 pages, updates of numerical data, improved formulations, and extended discussions on SAT. Third version 42 pages, with SAT solver data (especially for new SAT solver) and improved representation. Fourth version 47 pages, with updates and added explanation

An implementation of the DPLL algorithm

The satisfiability problem (or SAT for short) is a central problem in several fields of computer science, including theoretical computer science, artificial intelligence, hardware design, and formal verification. Because of its inherent difficulty and widespread applications, this problem has been intensely being studied by mathematicians and computer scientists for the past few decades. For more than forty years, the Davis-Putnam-Logemann-Loveland (DPLL) backtrack-search algorithm has been immensely popular as a complete (it finds a solution if one exists; otherwise correctly says that no solution exists) and efficient procedure to solve the satisfiability problem. We have implemented an efficient variant of the DPLL algorithm. In this thesis, we discuss the details of our implementation of the DPLL algorithm as well as a mathematical application of our solver. We have proposed an improved variant of the DPLL algorithm and designed an efficient data structure for it. We have come up with an idea to make the unit-propagation faster than the known SAT solvers and to maintain the stack of changes efficiently. Our implementation performs well on most instances of the DIMACS benchmarks and it performs better than other SAT-solvers on a certain class of instances. We have implemented the solver in the C programming language and we discuss almost every detail of our implementation in the thesis. An interesting mathematical application of our solver is finding van der Waerden numbers, which are known to be very difficult to compute. Our solver performs the best on the class of instances corresponding to van der Waerden numbers. We have computed thirty of these numbers, which were previously unknown, using our solver

Some Results in Extremal Combinatorics

Extremal Combinatorics is one of the central and heavily contributed areas in discrete mathematics, and has seen an outstanding growth during the last few decades. In general, it deals with problems regarding determination and/or estimation of the maximum or the minimum size of a combinatorial structure that satisfies a certain combinatorial property. Problems in Extremal Combinatorics are often related to theoretical computer science, number theory, geometry, and information theory. In this thesis, we work on some well-known problems (and on their variants) in Extremal Combinatorics concerning the set of integers as the combinatorial structure. The van der Waerden number w(k;t_0,t_1,...,t_{k-1}) is the smallest positive integer n such that every k-colouring of 1, 2, . . . , n contains a monochromatic arithmetic progression of length t_j for some colour j in {0,1,...,k-1}. We have determined five new exact values with k=2 and conjectured several van der Waerden numbers of the form w(2;s,t), based on which we have formulated a polynomial upper-bound-conjecture of w(2; s, t) with fixed s. We have provided an efficient SAT encoding for van der Waerden numbers with k>=3 and computed three new van der Waerden numbers using that encoding. We have also devised an efficient problem-specific backtracking algorithm and computed twenty-five new van der Waerden numbers with k>=3 using that algorithm. We have proven some counting properties of arithmetic progressions and some unimodality properties of sequences regarding arithmetic progressions. We have generalized Szekeres’ conjecture on the size of the largest sub-sequence of 1, 2, . . . , n without an arithmetic progression of length k for specific k and n; and provided a construction for the lower bound corresponding to the generalized conjecture. A Strict Schur number S(h,k) is the smallest positive integer n such that every 2-colouring of 1,2,...,n has either a blue solution to x_1 +x_2 +···+x_{h-1} = x_h where x_1 < x_2 < ··· < x_h, or a red solution to x_1+x_2+···+x_{k-1} =x_k where x_1 <x_2 <···<x_k. We have proven the exact formula for S(3, k)

Effects of slaughter positions on catecholamine, blood biochemical and electroencephalogram changes in cattle restrained using a modified Mark IV box

The proper slaughter positioning of animals is among the most crucial factors in animal welfare. The lateral position in Halal slaughter is a technique used around the world by Muslims, with a few practicing the upright position. The literature on the effects of slaughter in upright versus lateral positions on pain and stress is scarce. Thus, this study was designed to evaluate the effects of slaughter positions on blood biochemical parameters, plasma catecholamines, and electroencephalographic (EEG) responses. Twenty Brahman crossbred steers were subjected to slaughter in either lateral recumbency (LP) (n = 10) or an upright position (UP) (n = 10). There was a significant increase in adrenaline (p < 0.0001) and noradrenaline (p < 0.05) at T2 compared to T1 in the animals of both groups. A significant difference (p < 0.0001) was observed in the median frequency (MF) and total power (Ptot) of EEG, parameters for pain and stress, between the animals slaughtered in the upright and the lateral position. However, MF and delta waves were significantly higher (p < 0.05) after slaughter in the UP group than in the LP group. The results demonstrate a lesser amount of stress and pain responses among the LP group

Natural selection shapes the evolution of SARS-CoV-2 Omicron in Bangladesh

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has evolved to give rise to a highly transmissive and immune-escaping variant of concern, known as Omicron. Many aspects of the evolution of SARS-CoV-2 and the driving forces behind the ongoing Omicron outbreaks remain unclear. Substitution at the receptor-binding domain (RBD) in the spike protein is one of the primary strategies of SARS-CoV-2 Omicron to hinder recognition by the host angiotensin-converting enzyme 2 (ACE2) receptor and avoid antibody-dependent defense activation. Here, we scanned for adaptive evolution within the SARS-CoV-2 Omicron genomes reported from Bangladesh in the public database GISAID (www.gisaid.org; dated 2 April 2023). The ratio of the non-synonymous (Ka) to synonymous (Ks) nucleotide substitution rate, denoted as ω, is an indicator of the selection pressure acting on protein-coding genes. A higher proportion of non-synonymous to synonymous substitutions (Ka/Ks or ω &gt; 1) indicates positive selection, while Ka/Ks or ω near zero indicates purifying selection. An equal amount of non-synonymous and synonymous substitutions (Ka/Ks or ω = 1) refers to neutrally evolving sites. We found evidence of adaptive evolution within the spike (S) gene of SARS-CoV-2 Omicron isolated from Bangladesh. In total, 22 codon sites of the S gene displayed a signature of positive selection. The data also highlighted that the receptor-binding motif within the RBD of the spike glycoprotein is a hotspot of adaptive evolution, where many of the codons had ω &gt; 1. Some of these adaptive sites at the RBD of the spike protein are known to be associated with increased viral fitness. The M gene and ORF6 have also experienced positive selection. These results suggest that although purifying selection is the dominant evolutionary force, positive Darwinian selection also plays a vital role in shaping the evolution of SARS-CoV-2 Omicron in Bangladesh

A clinical study of arrhythmias associated with acute coronary syndrome: a hospital based study of a high risk and previously undocumented population

Background: ACS represents a global epidemic. Arrhythmia in ACS is common. Careful investigation may lead to further improvement of prognosis. Retrospectively analyzed the year- round data of our center. Study was undertaken to analyze the incidence, frequency and type of arrhythmias in ACS. This is to aid timely intervention and to modify the outcome. Identification of the type of arrhythmia is of therapeutic and prognostic importance.Methods: This cross sectional analytical study was conducted in the Department of Cardiology, Apollo Hospitals Dhaka, from January 2019 to January 2020 with ACS patients. Enrolled consecutively and data analyzed.Results: There were 500 patients enrolled considering inclusion and exclusion criteria. Sample was subdivided into 3 groups on the type of ACS. Group-I with UA, Group-II with NSTE - ACS and Group-III with STE - ACS. Different types of arrhythmia noted. Types of arrhythmia were correlated with type of ACS. 500 patients included. Mean age 55.53±12.70, 71.6% male and 28.4% female. 60.4% hypertensive, 46.2% diabetic, 20.2% positive family history of CAD, 32.2% current smoker, 56.4% dyslipidaemic and 9.6% asthmatic. 31.2% UA, 39.2% NSTE-ACS and 29.6% STE-ACS. Type of arrhythmias noted. 22% sinus tachycardia, 20.2% sinus bradycardia, 9% atrial fibrillation, 5.2% ventricular ectopic, 4.8% supra ventricular ectopic, 2.8% bundle branch block, 2.2% atrio-ventricular block, 1% broad complex tachycardia, 0.4% narrow complex tachycardia, 0.2% sinus node dysfunction and 32.2% without any arrhythmia. Significant incidences of arrhythmia detected - respectively 29.8%, 39.2% and 31%, p<0.001.Conclusions: In conclusion, arrhythmias in ACS are common. More attention should be paid to improve their treatment and prognosis

Minimal information for studies of extracellular vesicles 2018 (MISEV2018):a position statement of the International Society for Extracellular Vesicles and update of the MISEV2014 guidelines

The last decade has seen a sharp increase in the number of scientific publications describing physiological and pathological functions of extracellular vesicles (EVs), a collective term covering various subtypes of cell-released, membranous structures, called exosomes, microvesicles, microparticles, ectosomes, oncosomes, apoptotic bodies, and many other names. However, specific issues arise when working with these entities, whose size and amount often make them difficult to obtain as relatively pure preparations, and to characterize properly. The International Society for Extracellular Vesicles (ISEV) proposed Minimal Information for Studies of Extracellular Vesicles (“MISEV”) guidelines for the field in 2014. We now update these “MISEV2014” guidelines based on evolution of the collective knowledge in the last four years. An important point to consider is that ascribing a specific function to EVs in general, or to subtypes of EVs, requires reporting of specific information beyond mere description of function in a crude, potentially contaminated, and heterogeneous preparation. For example, claims that exosomes are endowed with exquisite and specific activities remain difficult to support experimentally, given our still limited knowledge of their specific molecular machineries of biogenesis and release, as compared with other biophysically similar EVs. The MISEV2018 guidelines include tables and outlines of suggested protocols and steps to follow to document specific EV-associated functional activities. Finally, a checklist is provided with summaries of key points

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