Introduction: In this section we study a pre-dual for the Banach space of bounded analytic functions on a region which was introduced by Rubel and Shields [7 ].. Further introductory det[r]

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Since 1995, exponentiated distributions have been widely studied in statistics and numerous authors have developed various classes of these distributions. A good review of some of these models is presented by Pham and Lai (2007). The exponentiation of distributions is a mechanism that makes the **model** more flexible, Nadarajah and Kotz (2006) introduce four more exponentiated type distributions: the Exponentiated Gamma, Exponentiated Weibull, exponentiated Gumbel and the Exponentiated Fréchet distribution. There are also several authors presented exponentiated distributions, such as Barriga, Louzada and Cancho (2011) with the Complementary Exponential Power distribution which is the exponentiation of the Exponential Power distribution proposed by Smith and Bain (1975) denoted as Complementary Exponential Power distribution, Bakouch, Al-Zahrani, Al-Shomrani, Marchi and Louzada (2011) with the extension of the Lindley (EL) distribution and the Complementary Exponential Power distribution (CEP) introduced by Barriga, Louzada and Cancho (2011).

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Physical and Engineering science prediction and intuition are sometimes braked by technical constraints, e.g. progress in elementary particle studies depends on the power of modern accelerators. Here, Mathematics provides new ideas and develops new concepts that must then be confirmed experimentally. Particular top- ics that will be considered by this programme include Conformal and Quasiconformal Mappings, Potential **Theory**, Banach Spaces of Analytic Functions and their **applications** to the problems of Fluid Mechanics, Conformal Field **Theory**, Hamiltonian and Lagrangian Mechanics, and Signal Processing.

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Lagrange first thought of permutations as functions from a set to itself, but it was Cauchy who developed the basic theorems and notation for permutations. He was the first to use cycle notation. Augustin-Louis Cauchy (1789–1857) was born in Paris at the height of the French Revolution. His family soon left Paris for the village of Arcueil to escape the Reign of Terror. One of the family’s neighbors there was Pierre-Simon Laplace (1749– 1827), who encouraged him to seek a career in mathematics. Cauchy began his career as a mathematician by solving a problem in geometry given to him by Lagrange. Cauchy wrote over 800 papers on such diverse topics as differential equations, finite groups, applied mathematics, and **complex** **analysis**. He was one of the mathematicians responsible for making calculus rigorous. Perhaps more theorems and concepts in mathematics have the name Cauchy attached to them than that of any other mathematician.

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Less need be said about the exercises at the end of each section because exercises have always received more favorable publicity than have questions. Very often the diﬀerence between a question and an exercise is a matter of terminology. The abundance of exercises should help to give the student a good indication of how well the material in the section has been understood. The prerequisite is at least a shaky knowledge of advanced calculus. The ﬁrst nine chapters present a solid foundation for an introduction to **complex** variables. The last four chapters go into more advanced topics in some detail, in order to provide the groundwork necessary for students who wish to pursue further the general **theory** of **complex** **analysis**.

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Composition of piecewise-linear node-level workload models yields system-level models which are also piecewise-linear. Instead of using composition repeatedly on multiple node-level models, we can also directly create system-level models that capture the rela- tionship between the incoming workload of a node and the work- load at some node downstream to this node. We can extract the data on the outgoing edges of the downstream node and the in- coming edges of the ancestor node and then use MARS to fit a piecewise-linear function just like in the case of creating a node- level workload **model**. We compare the CDF of errors obtained by using system-level models built using this direct modeling ap- proach with that obtained by using composition to build these mod- els. Figures 11 and 12 plot our results on traces of the Market Data Dissemination application and the Stock Trade Processing applica- tion respectively. The CDF of prediction errors for models created using direct modeling closely follows the CDF of prediction errors obtained by using composition-based models. Composition based modeling, however, provides us with the added benefit of being able to reuse node-level models and can also account for node sat- uration, an aspect direct modeling can not capture.

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Erdos and Rényi (Erdös & Rényi, 1959; Erdös & Rényi, 1960) first studied the degree distribution of real systems introducing the random networks. In random networks, the degree is randomly distributed around an average value meaning that many nodes have a similar number of connections, i.e. the network is characterized by a high homogeneity. Random networks describe network features in a more realistic way with respect to regular networks, i.e. networks having a regular topology. The degree distribution of regular networks, in fact, is characterized by an absolute homogeneity, i.e. all nodes have the same degree. Later, Watts and Strogatz (Watts & Strogatz, 1998) defined the small world networks based on Milgram’s experiment (Milgram, 1967) on six degrees of separation of social networks, that is, to connect two nodes within a network requires at most six steps. They demonstrated the existence of the small world effect for the most part of real systems (WWW, social networks, etc.) starting from regular networks and replacing some of the links with others between different nodes, giving randomness to the network. Therefore, the small world networks present a certain level of homogeneity, which is lower than regular and higher than random networks. The degree distribution of small world networks is very similar to random networks; thus, a Poisson distribution of nodal degrees is assumed to cover from regular to random networks (Figure 1). In fact, for random networks, the Poisson distribution **model** shows that each pair of nodes is connected randomly with a probability p, which generates a network having a great number of nodes with similar degrees. For regular networks, the probability p is zero and tends to increase with increasing random connections in the network; for small world networks, the probability is greater than the null value of regular networks but rather lower than values of random networks.

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The most important paper, from which all subsequent disequilibrium econometric models have evolved, is that by Fair and Jaffee (1972), although the origins of many of the techniques they discuss may be traced back to Page (1955, 1957), Quandt (1958, 1960) and Goldfeld, Kelejian and Quandt (1971). While the paper by Fair and Jaffee has been shown to contain a number of errors, the ideas put forward have been generally accepted for **analysis** of markets in disequilibrium. It should be noted that some of the techniques discussed have a much wider application in economic **theory** to any models where, according to some (known or unknown) variation in certain economic variables, the parameters of a particular relationship switch from one value to another.1 On the other hand, are those techniques using some form of Walrasian price adjustment, either as an indicator of the regime that is operative, or as a separate equation with estimat able parameters, which are specifically applicable to the markets in disequilibrium **model**. It is the latter that are of main concern in this thesis and so discussion of the **theory** will be directed in such a way as to be always closely related to such models.

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Classical works concerned with the throughput and delay **analysis** of random access protocols (e.g., Aloha or CSMA) rely on strong assumptions. One is that the point process comprising of both newly generated and retransmitted (due to collisions) packets is a Poisson process (Abramson [1], Kleinrock and Tobagi [23], and more recently Yang and Yum [32]). A related assumption is that, at each source, packets arrive as a blocked Poisson process, in the sense that at most one packet can be backlogged at any source (Tobagi [29] or Beuerman and Coyle [3]); this **model** is related to the infinite source **model** in which each source generates a single packet during its lifetime (Lam [24]). Another related and simplifying assumption is to discard the buffered packets at the beginning of a transmission period for a source (Takagi and Kleinrock [27]).

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The assessment of fatigue failure in welded joints is a common problem in failure **analysis**, which in recent years has been greatly aided by the development of comprehensive standards such as the recent Eurocodes for welded steel and aluminium. However, cases often arise where the welded joint design is very different from anything found in the standards, and one must resort to a detailed **analysis** using FEA. But how to interpret the results? There are many different proposals regarding the creation of the **model** and its post-processing. One issue which arises is how accurately one must **model** the details of the weld. Cracking frequently starts at the point where the weld bead meets the base metal (fig.5): at this point the radius of curvature is small, but not zero, and if one models it to be zero then a singularity is created in the FE **model**, giving rise to stress values at that point which are meaningless. However, if using the TCD this is not important because we are not relying on the stress at that particular point, and provided the radius of curvature in the actual weld is less than L, we are allowed to use a zero radius in the **model** without affecting the results. For low and medium strength steels we found that the appropriate value of L is 0.43mm [9], so this simplification is allowable in most cases.

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As mentioned previously, the evidence **theory** formal- ism is close to the probability one, but it takes into account inconsistencies and incompleteness in a better way. The frame of discernment allows to define how many events concern the state {U p} or the state {Down} and how many events can not be affected to the previous cases. In addition, plausibility and belief functions help to compute measures that bound the real value of the probability of failure that is often preferred by reliability engineer in current reliability **analysis**. The precise value can be obtained by a transformation of belief from the credal level to the pignistic one.

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This paper reviewed evolution stages of websites and web-based **applications** according to their historical emergence, usage, complexity and integration of new aspects. The authors believe that this classification clarified the evolvement of internet from simple static websites to **complex** web **applications**. Individual stages of this development were discussed with connection to the evolutionary approach of general systems **theory**. Framework for websites **analysis** and evaluation was then proposed in a form of criteria list for each defined stage. By confronting these criteria with factual state of the evaluated website, we can define its maturity in the meaning of internet´s evolution and available possibilities and necessities, which are connected with individual evolution stages. Elaboration of this framework along with concrete computation of websites evolution index is planned to be subject of further studies.

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For the criminological data, fitted survival curves by risk group for the observed offence times with estimated risk scores Rsc under the weighted hazards model 8.13, using the weight fu[r]

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In this chapter the work of [11] has been extended to facilitate GSA for large scale PDE- constrained optimization problems. By coupling a C++ implementation and randomized eigen- value solver, a scalable software infrastructure has been developed to determine the sensitivity of the optimal control solution to changes in uncertain parameters. The framework is able to exploit low dimensional structure which is commonly found in high dimensional parameter spaces, and is scalable in both the complexity of the parameter space and the underlying PDE itself. Finding this low dimensional structure has many useful **applications**. For instance, performing uncertainty quantification and robust optimization (the ultimate goal) are challenging in high dimensions. Reducing the number of parameters may enable experimental and computation **analysis** which would otherwise be infeasible in the high dimensional parameter space. The sensitivities may also direct **model** development by identifying which **model** parameters (and hence corresponding physical effects) exhibit the greatest influence on the control strategy.

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the individual OLS estimates. 1 However, as in the error-component **model**, the Swamy estimator of the random coefficient covariance matrix is not necessarily nonnegative definite. We have investigated the consequences of this drawback in finite samples, in particular when testing hypotheses, in Chapter 2. In this Chapter, we propose a solution to the above mentioned problem by applying the EM algorithm. In particular, following the seminal papers of Dempster et al. (1977), and Patterson and Thompson (1971), we propose to estimate het- erogeneous panels by applying the EM algorithm to obtain tractable closed form solutions of restricted maximum likelihood (REML) estimates of both fixed and random components of the regression coefficients as well as the variance para- meters. The proposed estimation procedure is quite general, as we consider a broad framework which incorporates various panel data models as special case. Our approach yields an estimator of the average effects which is asymptotically related to both the GLS and the Mean Group estimator, and which performs relatively well in finite sample as shown in our limited Monte Carlo **analysis**. We also review some of the existing sampling and Bayesian methods commonly used to estimate heterogeneous panel data, to highlight similarities and differences with the EM-REML approach.

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b) Weaknesses, limitations and assumptions of Markowitz mean-variance However, research has shown that the Markowitz mean-variance has some weaknesses and several constraints. These limitations have taken centre stage of research. Researchers like: Fuerst (2008), Norton (2009), Ceria and Stubbs (2006), Goldfarb and Iyengar (2003), Jorion (1992), Konno and Suzuki (1995), Michaud (1989a) (1989b), Bowen (1984), Ravipti (2012) etc. discussed the weaknesses, limitations and assumptions in their works. Markowitz himself in Markowitz (1952) says that they tried to avoid mathematical proofs and could only get a geometrical presentation for 3 or 4 security cases. Therefore, these two are the main limitations of MV; the **model** did not allow n-security in a portfolio. Michaud (1989b), shows that the fundamental flaws of the mean-variance optimiser are its estimation error. It tends to overweigh those securities with a high estimate of return, negative correlations, and small variances or underweight those securities with a low rating of return, positive correlations, and large deviations. He pointed out that the statement; 'Optimizer, in general, produce a unique optimal portfolio for a given level of risk' is highly misleading. The ill-conditioning of the covariance matrix is yet another problem of MV; it makes the optimisation to be highly unstable by making a small change in the input assumption to lead to a significant difference in the solution. Konno and Suzuki (1995) in their research show that based on the assumption that investor is risk-averse; MV believes that the distribution of the return is multivariate normal, or the utility of the investor is a quadratic function of the rate of return. But unfortunately, they noticed that neither of the two holds in practice. Haung et al. (2008) in their words say that when probability distributions of security returns are asymmetric, variance becomes a deficient measure of investment risk because the selected portfolio based on variation may have a potential danger to sacrifice too much-expected return in eliminating both low and high return extremes.

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In reflection, the proposed numerical method turns out to be very closely related to taking harmonic averages of observable traces. Instead of approximating the harmonic averages of a generic (hence typically aperiodic) system, the averages were computed of a “near-by” periodic system. For periodic systems, these averages are effectively reducible to the Discrete Fourier Transform. In the end, only smooth weighted sums of these averages turn out to have any “spectral meaning”. It is generally hard to assess how a finite-state **model** approximation of a dynamical system, in combination with an infinite series truncation, may affect the computation of time averages. This dissertation has shown that, in the special case of measure-preserving systems, these concerns are fully addressed if one uses a periodic approximation.

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SAMPLE SELECTION BIAS. Two decisions were about households to include in this **analysis**, each of which may have implications for the results. The first was made on strictly technical grounds and can be demonstrated to have no effect on the results. Recall that in order to include variables measuring a household’s migrants and former members living nearby, the sample had to be reduced to so-called “old” households – those that had at least one member present during a previous wave of the study – because only these households were asked about migrants. All three migration-related variables proved to have no association with either outcome in the modeling, but this was not known a priori, and so the original models are presented, rather than stripped-down models reflecting knowledge gained in the modeling process itself. But as a check against selection bias arising from this particular decision models were re-run without the migration variables on the entire sample of “old” and “new” households with not one coefficient losing or gaining significance and only very minor alterations in the magnitude of coefficients. Sensitivity tests were also run with all “new” households simply coded as having zero migrants and leaving these three variables in the **model** with the same result. 9 Thus, I am confident that no significant threat to the internal or external validity of the results arises from presenting the original models based only on the sample of “old” households as initially specified.

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reporting of pregnancy losses and misreporting of time of marriage and first conception. The second may arise "because the interview date curtailed the observed childbearing experience... This means that short marriage durations with a conception select for quick conceptions, thereby increasing mean fecundability" (p.76). It seemed that, due to under-reporting of foetal losses, memory bias produced some of the observed fecundability decline after the third year of marriage. Further, truncation bias seemed not to be serious when truncation occurred after three years: one would expect that more subfecund women would be included in the **analysis** as marriage duration increased but the pattern of fecundability decline with ascending marriage duration still persisted when those women who took over three years to conceive had been excluded. Jain decided, as a result, that "the mean fecundability for women married for 3-8 years would provide a closer estimate of fecundability for Taiwanese women than that based on all women" (p.78). He saw the two effects as compensatory for that period and estimated mean fecundability for women in this sample to be closer to 0.195 than 0.163. The higher value agrees well with 0.190 (Henry, Indiana polis) and 0.180 at marriage (Westoff, FGIMA)^. A value of 0.2

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A further descriptive biomechanical study was undertaken by Mason et al. (1989) to develop an ‘optimal’ **model** of the bowling technique (i.e., one that maximises ball release speed but minimises the likelihood of injury), which was to be used as a basis for teaching young fast bowlers. Fifteen fast-medium bowlers (x = 32.4 m.s'1) from the Australian Institute of Sport Cricket Academy were filmed from the front and side using two phase-locked high-speed (100 Hz) cine cameras. A force platform measured ground reaction forces at front foot impact, a series of light gates positioned four metres apart down the length of the bowler’s run-up was used to determine horizontal speed during different intervals of the run-up and a radar gun was used to measure ball release speed. The trial performed by each bowler yielding the highest release speed was selected for **analysis**. The results of this study indicated that 14 of the bowlers analysed adopted side-on actions and only one bowler adopted a front-on action. Although the exact classification criteria were not disclosed, this finding seemed to contradict the previously reported results of Elliott et al (1986) indicating that fast bowling techniques were increasingly becoming more front-on. The mean run-up speed during the 16-12 metre, 12-8 metre, 8-4 metre and 4-0 metre intervals before the

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