On the entire functions from the Laguerre--P\'olya class having monotonic second quotients of Taylor coefficients

We investigate the famous Laguerre–Pólya class of entire functions and its subclass, the Laguerre–Pólya class of type I. The functions from these classes can be expressed in terms of the Hadamard Canonical Factorization (see Chapter 1, Definition 1.2 and 1.3). The prominent theorem by E. Laguerre and G. Pólya gives a complete description of the Laguerre–Pólya class and the Laguerre–Pólya class of type I, showing that these classes are the respective closures in the topology of uniform convergence on compact sets of the set of real polynomials having only real zeros (that is, the set of so-called hyperbolic polynomials) and the set of real polynomials having only real negative zeros. Both the Laguerre–Pólya class and the Laguerre–Pólya class of type I play an essential role in complex analysis. For the properties and characterizations of these classes, see, for example, [31] by A. Eremenko, [40] by I.I. Hirschman and D.V. Widder, [43] by S. Karlin, [57] by B.Ja. Levin, [66, Chapter 2] by N. Obreschkov, and [74] by G. Pólya and G. Szegö. In the thesis, we study entire functions with positive coefficients and with the monotonic sequence of their second quotients of Taylor coefficients. We find necessary and sufficient conditions under which such functions belong to the Laguerre–Pólya class (or the Laguerre–Pólya class of type I).:List of symbols Introduction 1 Background of research 1 1.1 The Laguerre–Pólya class .................... 1 1.2 The quotients of Taylor coefficients ............... 3 1.3 Hutchinson’s constant ...................... 4 1.4 Multiplier sequences ....................... 4 1.5 Apolar polynomials........................ 8 1.6 The partial theta function .................... 10 1.7 Decreasing second quotients ................... 13 1.8 Increasing second quotients ................... 14 2 A necessary condition for an entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre–Pólya class 15 2.1 Proof of Theorem 2.1....................... 16 2.2 The q-Kummer function ..................... 29 2.3 Proof of Theorem 2.10 ...................... 31 2.4 Proof of Theorem 2.11 ...................... 43 3 Closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre–Pólya I class 49 3.1 Proof of Statement 3.1 ...................... 50 3.2 Proof of Theorem 3.2....................... 53 3.3 Proof of Theorem 3.4....................... 61 3.4 Proof of Theorem 3.6....................... 66 4 Entire functions from the Laguerre–Pólya I class having the increasing second quotients of Taylor coefficients 69 4.1 Proof of Theorem 4.1....................... 70 4.2 Proof of Theorem 4.3....................... 76 5 Number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficients 81 5.1 Proof of Theorem 5.1....................... 82 5.2 Proof of Corollary 5.2....................... 88 5.3 Proof of Theorem 5.4....................... 88 6 Further questions 95 Acknowledgements 97 Selbständigkeitserklärung 101 Curriculum Vitae 103 Bibliography 10

A general framework for the realistic analysis of sorting and searching algorithms. Application to some popular algorithms

We describe a general framework for realistic analysis of sorting and searching algorithms, and we apply it to the average-case analysis of five basic algorithms: three sorting algorithms (QuickSort, InsertionSort, BubbleSort) and two selection algorithms (QuickMin and SelectionMin). Usually, the analysis deals with the mean number of key comparisons, but, here, we view keys as words produced by the same source, which are compared via their symbols in the lexicographic order. The "realistic" cost of the algorithm is now the total number of symbol comparisons performed by the algorithm, and, in this context, the average-case analysis aims to providee stimates for the mean number of symbol comparisons used by the algorithm. For sorting algorithms, and with respect to key comparisons, the average-case complexity of QuickSort is asymptotic to 2n log n, InsertionSort to n^2/4 and BubbleSort to n^2/2. With respect to symbol comparisons, we prove that their average-case complexity becomes Theta(n log^2n), Theta(n^2), Theta (n^2 log n). For selection algorithms, and with respect to key comparisons, the average-case complexity of QuickMin is asymptotic to 2n, of SelectionMin is n - 1. With respect to symbol comparisons, we prove that their average-case complexity remains Theta(n). In these five cases, we describe the dominant constants which exhibit the probabilistic behaviour of the source (namely, entropy, and various notions of coincidence) with respect to the algorithm

Incentives and Firm Behaviors

My research interests focus on the economic behavior, choices, and actions of organizations as well as individuals given their incentives, and analyze the consequences of such decisions to the financial health of firms and the macro economy. A firm is incentivized by the value investors place on its operations; while employees, particularly the management team, is incentivized by the private benefits the firm gives them. Understanding the impact of such incentives will help alleviate the classic agency costs in modern organizations. Stock illiquidity raises the cost of share ownership to outside investors. The sizable adverse price impact of trading increases transactions costs and stock volatility. The reduced gains from informed trading discourages the acquisition of private information and impedes the price discovery process. The first essay substantiates that shares of financially constrained firms are significantly more illiquid than shares of similar but financially unconstrained firms. Acting as buyers of last resort for their own shares, share repurchases by financially constrained firms enhance stock liquidity, which alleviates the cost of external financing and underinvestment. Increased stock liquidity improves information efficiency, inducing higher value-added from incremental capital investments. Further, higher stock liquidity lowers stock volatility and allows financially constrained firms to issue equity. In the second and third essays, I investigate whether the incentives given to the employees and the management team at banks contribute to the financial crisis. I provide evidence that CEO compensation is weakly related to bank risk measures and risky bank activities. However, when looking at banks with regards to their reward cultures, I find that during the 2008 crisis period, banks either at the high or low reward culture groups perform worse, and are more risky than banks in the average reward culture group. The reward culture score represents the common factor in incentives across all levels of the bank, from CEO, Vice Presidents to all other employees. The findings are consistent with the problems of adverse selection and moral hazard associated with incentive misalignment when incentives are too low or too high. This shows the importance of reward culture in understanding the role of performance-based compensation

Consequences of urban migration of adult children for the elderly left-behind in rural Vietnam

Following market liberalisation in Vietnam, concern has been raised about the elderly ‘left behind’ by the urban migration of adult children from rural areas. This study used qualitative methods to examine the consequences for these elderly and the strategies they adopt to cope. It found overall positive outcomes for rural aged parents with migrant children, as long as at least one family member remained nearby. However, elderly with no remaining family members suffered more

APPLICATION OF THE MODEL 5E TO DESIGN A TEACHING PLAN FOR THE TOPIC “WHAT ARE THE CHARACTERISTICS OF WATER?” IN 4TH GRADE SCIENCE SUBJECT OF VIETNAM IN STEM EDUCATIONAL ORIENTATION

STEM education is an educational model based on the interdisciplinary integration of science, technology, engineering and mathematics to solve real-life problems (Bentley et al., 2022; Kaleci & Korkmaz, 2018). When implementing STEM education in primary schools, there are many different approaches: discovery model, engineering design model, 5E, 6E, TRIAL model (Chacko et al., 2015; Kaleci & Korkmaz, 2018; Matsuura & Nakamura, 2021; Stohlmann et al., 2012). The selection of a model to apply appropriately is important, depending on the student, the actual teaching conditions, the teaching content, etc. The article analyzes the content of Science in primary school, teaching practice in Vietnam and illustrates the design of a specific teaching plan for grade 4 students on the basis of applying the model 5E to organize STEM education-oriented teaching in order to provide teachers with an approach when designing science teaching plans for primary school students.  Article visualizations

Investigation of botulism in free-range duck farming in the Mekong Delta, Vietnam

Background: One of the most common diseases in free-range ducks in the Mekong Delta is botulism. Botulism is a poultry disease caused by botulinum exotoxin of Clostridium botulinum. Aim: To evaluate the prevalence of botulism in free-range ducks in the Mekong Delta and the risk of infection by determining the presence of C. botulinum in the farming environment. Methods: Research was carried out on 200 duck flocks with 187,050 individuals raised freely in the fields in the provinces of the Mekong Delta, including An Giang, Can Tho, Hau Giang, and Kien Giang. The ducks were diagnosed with botulism based on clinical symptoms. To demonstrate the presence of botulinum neurotoxins and identify serotype, samples of serum and/or gut were analyzed by mouse bioassay. Samples of soil (n = 600), water (n = 600), crabs (n = 216), and snails (n = 400) were taken from the grazing regions for C. botulinum analysis by PCR assay. Results: There were 1.19% (2,235/187,050) free-range ducks in the Mekong Delta positive for botulism. Clinical symptoms of botulism including limberneck, drooping eyelids–enlarged pupils, and leg paralysis were prevalent across free-range ducks, with the frequency of 87.92% (1,965/2,235), 90.07% (2,013/2,235), and 79.78% (1,783/2,235), respectively. The lesions of pulmonary edema–hemorrhage, hemorrhagic liver, and gas-producing intestines were common, accounting for 86.19% (362/420), 95.48% (401/420), and 92.14% (387/420), respectively. Botulin toxin type C was found in a considerable number of serum samples, accounting for 40.48% (51/126). Meanwhile, the percentage of serum samples containing botulin toxin types E and D was 28.57% (36/126) and 25.40% (32/126), respectively. Clostridium botulinum was detected in the farming environment specifically 17.5% (105/600) in soil, 19.67% (118/600) in water, 8.33% (18/216) in crabs, and 3.00% (12/400) in snails. Conclusion: The free-range ducks in the Mekong Delta were at high risk of botulism because of the latent presence of C. botulinum in the farming environment