This thesis is devoted to a number of structural and behavioural problems in linear multivariable control system theory. The first problem addresses the subject of determination of the finite and infinite frequency structure of a rational matrix. A novel method is proposed that determines the finite and infinite frequency structure of any rational matrix. Some neat and numerically stable algorithms are developed to implement this method. The second problem concerns the resol vent decompositions of a regular polynomial matrix and solutions of regular polynomial matrix descriptions (PMDs). Regarding these fundamental is'sues, three contributions are made therein. Firstly, based on a general resolvent decomposition a complete solution of regular PMDs is presented that takes into account both the non-zero initial conditions of the pseudo state and the non-zero initial conditions of the input. Secondly, two special resolvent decompositions are proposed, both of which are applied to formulate the solution of the regular PMDs. The first one is formulated in terms of the finite, infinite, and the generalised infinite Jordan pairs, which is a refinement of the results given by Gohberg et al. [74] and Vardulakis [25]. The second resolvent decomposition is proposed on the Weierstrass canonical form of the generalised companion matrix of the polynomial matrix. Thirdly, a new characterization of the impulsive free initial conditions of regular PMDs is given and the relationship between the finite and infinite frequency structure of a regular polynomial matrix and its generalised companion matrix is determined. In the third problem a generalization of the chain-scattering representation for general plants is presented. Through the notion of input-output consistency, the conditions under which the generalised chain-scattering representation and the dual generalised chain-scattering representation exist are proposed. Some algebraic system properties of the GCSRs and DGCSRs are studied. The fourth problem is devoted to a new notion of realization of behaviour. We introduce a notion realization of behavior which is shown to be a generalization of the classical concept of a realization of transfer function. By using this approach, the input-output structures of the generalized chain-scattering representations and the dual generalized chain-scattering representations are investigated in a behavioral theory context. The last problem is devoted to the subjects of system wellposedness and internal stability. We present certain generalisations to the classical concepts of wellposedness and internal stability. The input consistency and output uniqueness of the closed-loop system in the standard control feedback configurations are discussed. Based on this, a number of notions are introduced such as fully internal wellposedness, externally internal wellposedness, and externally internal stability, which characterize the rich input-output and stability features of the general control systems in a general setting. On the basis of these notions the extended JL control problem is defined in a general setting.

Fluorescent protein labelling is a powerful tool for the sensitive visualization of proteins in living cells, allowing the elucidation of their localization, trafficking and ultimately their cellular function. We have developed a novel labelling technique based on the genetic fusion of a protein of interest to a small helical peptide sequence containing two Cys residues (dC10). This tag can undergo an efficient reaction with small fluorogenic labelling agents composed of a fluorophore and a dimaleimide core (dM10) that confers high reaction specificity, and quenches the latent fluorescence through photo-induced electron transfer, until both of its maleimide groups have formed robust covalent bonds with the tag Cys thiol groups. Our initial efforts at intracellular protein labelling demonstrated the importance of the selectivity of the labelling reaction, which is dependent on the reactivity of the dC10 tag. To that end, we re-engineered the dC10 tag through rational protein design. Mutant libraries were prepared through combinatorial mutation at specific positions of the helical tag sequence, and screened for their fluorogenic reactivity. In this way, we identified a novel sequence for a next-generation dC10 tag that confers 10-fold greater selectivity that we then applied to in cellulo labelling. Subsequent mechanistic studies revealed the basis for this dramatic increase in reactivity. Current applications of this powerful labelling technique, including the site-specific chelation of lanthanide ions for NMR spectroscopy and site-specific covalent heavy-atom labelling for X-ray crystallography, will also be discussed.

The purpose of this paper is to construct the real number system. The foundation upon which the real number system will be constructed will be the system of counting numbers.

Abstract available in the Journal of Adolescent Health.

We give algorithms to compute Coleman integrals on superelliptic curves over unramified extensions of the p-adics, and apply these via Chabauty methods to determine the set of rational points on such curves. We also determine the solution to an explicit instance of the Shafarevich conjecture by finding all elliptic curves with good reduction outside of the first 6 primes, subject to a heuristic. We use a combination of non-abelian Chabauty and the Mordell--Weil sieve to determine the rational points on several quotient modular curves, and therefore classify pairs of elliptic curves over the rationals with 67-, 73-, and 107-isogenies. We give methods to explicitly compute Coleman integrals on modular curves using a canonical lift of Frobenius and canonical local coordinates in each residue disk, and discuss the problem of computing the Weil pairing in finite rings.

The motto of the Expo 2015 in Milan, is "Feeding the planet, energy for life." The main idea of the design of the pavilion is Food – energy for life. The attempt of the thesis lies in pointing on rational eating and a healthy lifestyle. Not only is the reason the influence on an individual, but the influence on a whole society. The thesis is devoted to a solution of a thematic function of the pavilion, a way of presentation, a detailed solution of the pavilion construction and subsequent recycling of the pavilion.

Health insurance has become essential in mediating the extremely high costs of childbirth; however, it has been neglected in fertility research. This study examines health insurance coverage as an economic determinant of fertility and includes discussions of the cost of raising children, contraception and childbirth as well as opportunity costs to illustrate that the effect of health insurance may differ by resources, such as income and education. Using data from the 2001 Survey of Income and Program Participation, I analyze the effects of insurance on the probability of pregnancy and birth with Discrete-Time Hazard Models. Results show that health insurance is a powerful indicator of fertility. Married women with insurance coverage, especially private insurance coverage provided by someone else's plan, coverage in own name or public insurance, have increased expected odds of fertility compared to uninsured women.

A classroom teaching experiment was conducted in a semester-long undergraduate mathematics content course for elementary education majors. Preservice elementary teachers' development of rational number understanding was documented through the social and psychological perspectives. In addition, social and sociomathematical norms were documented as part of the classroom structure. A hypothetical learning trajectory and instructional sequence were created from a combination of previous research with children and adults. Transcripts from each class session were analyzed to determine the social and sociomathematical norms as well as the classroom mathematical practices. The social norms established included a) explaining and justifying solutions and solution processes, b) making sense of others' explanations and justifications, c) questioning others when misunderstandings occur, and d) helping others. The sociomathematical norms established included determining what constitutes a) an acceptable solution and b) a different solution. The classroom mathematical practices established included ideas related to a) defining fractions, b) defining the whole, c) partitioning, d) unitizing, e) finding equivalent fractions, f) comparing and ordering fractions, g) adding and subtracting fractions, and h) multiplying fractions. The analysis of individual students' contributions included analyzing the transcripts to determine the ways in which individuals participated in the establishment of the practices. Individuals contributed to the practices by a) introducing ideas and b) sustaining ideas. The transcripts and student work samples were analyzed to determine the ways in which the social classroom environment impacted student learning. Student learning was affected when a) ideas were rejected and b) ideas were accepted. As a result of the data analysis, the hypothetical learning trajectory was refined to include four phases of learning instead of five. In addition, the instructional sequence was refined to include more focus on ratios. Two activities, the number line and between activities, were suggested to be deleted because they did not contribute to students' development.

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