The Normal Form of Poincaré and Birkhoff

This thesis treats normal forms of matrices over rings of Ore polynomials. The whole thesis is divided in three parts: First, Ore polynomials are described and basic facts about them are recalled. This part also includes integro-differential operators as an extended example. Second, in the main part we present oneand two-sided normal forms of matrices. More precisely, we deal with the Popov normal form, Hermite normal form and the Jacobson normal form. In the last part, we explore an application of matrix normal forms to a problem in control theory. Below, we describe each of the parts in more detail. Ore polynomials, sometimes called skew polynomials, arise from the work of Oystein Ore in [Ore33]. They are a generalisation of the usual polynomials with almost all of their properties with the main exception being that the multiplication in not necessarily commutative: Neither need the coefficients commute with each other, nor does the indeterminate have to commute with them. Ore polynomials can be used to model differential or difference operators. For example, the famous Weyl algebra can be considered to be an Ore polynomial ring. As an example, we model integro-differential operators with polynomial coefficients using Ore polynomials. This part is based on our ISSAC 2009 paper [RRM09]. We arrive at a construction which is similar to the Weyl algebra in the purely differential case. In the main part, we consider normal forms of matrices. These make it possible to express systems of linear equations involving operators and to determine the properties of these systems such as, for example, solvability. We first consider normal forms with respect to row-operations. The coefficient domain here is a skew field. We treat row-reduction, the Hermite normal form, the Popov normal form and shifted Popov normal forms. We draw a connection between these normal forms to Grobner bases over modules. As an application of this connection, we present a modified FGLM algorithm for converting matrices from one normal form into another. Parts of this were presented at ACA 2010 and in [Mid10]. We also consider the Jacobson normal form which is a normal form with respect to simultaneous rowand column-operations. Here, we restrict ourselves to differential operators over a commutative coefficient domain. We present a modular algorithm for computing a Jacobson normal form which is based on cyclic vectors and which is guaranteed to succeed in characteristic zero, but under certain conditions also yields a result in positive characteristic. The last part deals with a topic from control theory. We examine linear time-varying differential systems with delays for differential flatness and π-flatness where we use an idea from [MCL10]. For this, we apply the one-sided normal forms from the main part instead of the originally proposed

Different extensive form games with the same reduced normal form can have different information sets and subgames. This generates a tension between a belief in the strategic relevance of information sets and subgames and a belief in the sufficiency of the reduced normal form. We identify a property of extensive form information sets and subgames which we term strategic independence. Strategic independence is captured by the reduced normal form, and can be used to define normal form information sets and subgames. We prove a close relationship between these normal form structures and their extensive form namesakes. Using these structures, we are able to motivate and implement solution concepts corresponding to subgame perfection, sequential equilibrium, and forward induction entirely in the reduced normal form, and show close relations between their implications in the normal and extensive form.

Carbohydrate utilization and nitrogen requirement of Trichophyton gypseum were studied and differences in the nutritional physiology between the normal and pleomorphic forms of this fungus were demonstrated. The pleomorphic form showed better growth and more efficient assimilation on monosaccharides than the normal form. Dior polysaccharides generally failed to support appreciable growth of either form. The pleomorphic form grew far more readily on single amino acid than the normal form, and reproduced even on ammonium nitrogen to some extent which the normal form could not utilize at all.

In this chapter I examine the traditional position in game theory concerning the adequacy of the normal form representations of extensive form games. According to this traditional view, an extensive form game can be reduced to a normal form game without loss of strategically relevant information about the structure of the game. Many game theorists have already raised questions about such a reduction from a game in extensive form to strategic form by considering solution concepts that explicitly use the extensive form structure of the game. In the same vein, I evaluate the traditional view of the equivalence of extensive and normal form games in a systematic way. Within the traditional game-theoretic framework, I show that two extensive form games with the same reduced normal form may lead to different outcomes, and thus are not strategically equivalent. In addition, I demonstrate that the reduced normal form of an extensive form game cannot sufficiently represent certain reasonable strategic reasoning in the game of extensive form.

When normal and extensive form decisions differ

The relationship between isolated teratozoospermia and clinical pregnancy after in vitro fertilization with or without intracytoplasmic sperm injection: a systematic review and meta-analysis

Chapter 2 - Normalization

Conditionals are defeasible rules of the form If A then usually B, and they play a central role in many approaches to nonmonotonic reasoning. Normal forms of conditional knowledge bases consisting of a set of such conditionals are useful to create, process, and compare the knowledge represented by them. In this article, we propose several new normal forms for conditional knowledge bases. Compared to the previously introduced antecedent normal form, the reduced antecedent normal form (RANF) represents conditional knowledge with significantly fewer conditionals by taking nonmonotonic entailments licenced by system P into account. The renaming normal form(ρNF) addresses equivalences among conditional knowledge bases induced by renamings of the underlying signature. Combining the concept of renaming normal form with other normal forms yields the renaming antecedent normal form (ρ ANF) and the renaming reduced antecedent normal form (ρ RANF). For all newly introduced normal forms, we show their key properties regarding, existence, uniqueness, model equivalence, and inferential equivalence, and we develop algorithms transforming every conditional knowledge base into an equivalent knowledge base being in the respective normal form. For the most succinct normal form, the ρ RANF, we present an algorithm KBρra systematically generating knowledge bases over a given signature in ρ RANF. We show that the generated knowledge bases are consistent, pairwise not antecedentwise equivalent, and pairwise not equivalent under signature renaming. Furthermore, the algorithm is complete in the sense that, when taking signature renamings and model equivalence into account, every consistent knowledge base is generated. Observing that normalizing the set of all knowledge bases over a signature Σ to ρ RANF yields exactly the same result as KBρra (Σ), highlights the interrelationship between normal form transformations on the one hand and systematically generating knowledge bases in normal form on the other hand.

In couples with infertility, abnormal strict morphology of 0% normal forms (NF) is a criterion to proceed rapidly to in vitro fertilization (IVF). Since no data currently exist, we investigated the outcomes for men with 0% NF to determine reproductive success without the use of assisted reproductive technologies (ART). A cohort of 24 men with 0% NF were identified (2010–2013) with 27 randomly selected men with ≥4% NF as controls. Patient charts were reviewed with men contacted and administered an Institutional Review Board (IRB)-approved telephone questionnaire to ascertain outcomes. After a median follow-up time of 2.5 years, 29.2% of men with 0% NF did not require ART for their first pregnancy (controls = 55.6%, P ≤ 0.05). When all pregnancies were analyzed together, men with 0% NF achieved twenty pregnancies of which 75% did not require IVF (controls = thirty pregnancies; 76.7% did not require IVF). The average age of men and female partners was similar between men with 0% NF and ≥4% NF. All men had normal follicle-stimulating hormone (FSH), testosterone, prolactin, sex hormone-binding globulin (SHBG), and estradiol. Although, global semen parameters were worse in men with 0% NF, when a first pregnancy was a natural conception (NC), 100% of men with 0% NF (n = 7/7) and 37.5% of controls (n = 3/8) went on to have a subsequent pregnancy via NC. Men with 0% NF conceived without IVF in 29.2% of cases compared to 55.6% of controls. Strict morphology should not be used to predict fertilization, pregnancy, or live birth potential. In men with 0% NF, alternative modalities should be considered before immediate IVF.

Post-processed semen specimens with deficits in strict morphology may not require intracytoplasmic sperm injection (ICSI) to achieve good reproductive outcomes

This work presents a method of verifying normal form constraints on a relational database. The problem under study is related to the study of database semantics by Bernstein and Beeri [1, 2] in which these authors studied normalization of relational schemas and later, algorithmic design of such schemas. Our approach is to formulate the normal forms as a set of Van Wijngaarden grammars (W-grammars).W-grammars (Cleaveland [3]) were developed to be used as a tool in the formal specification of programming languages, in particular, Algol 68. A W-grammar is composed of two sets of rules, meta-productions and hyper-rules. In our work meta-productions are used to generate elements of a relation. These elements are then placed in hyper-rule thus creating the rules that verify the static and dynamic semantics of normal forms. Using where clauses in W-grammar provides an additional method of enforcing relational database semantics [4].The normal forms under consideration are Qualified Normal Form, Third and Fourth Normal Forms. The Qualified Normal Form definition is: A relation R is in QNF iff it is in 1NF and there exist FD,s between its attributes.We have divided the Third Normal Form defined by Date [5] into three categories: Binary, Ternary, Non-Ternary; each of these categories can consist of individual attributes (columns) or combined attributes which are called super attributes. For example, a binary relation can contain two regular attribute or two super attributes. Using the binary, ternary, or non-ternary categories, the Third Normal Form has been formulated as a W-grammar. Using the Fourth Normal Form definition described by Ullman [6] and by Maier [7], a W-grammar formulation of 4NF can be given.These W-grammars are used to enforce the static and dynamic semantics of normal forms through the preservation of functional dependencies. The static W-grammar models the attribute level and the relations between attributes in normal forms. The dynamic W-grammars represent not only the attribute level and the relation between them but also consider the tuples of the relation and the relation of the elements of the tuples to one another. It is useful in the dynamic case to think of the attributes and their relationships forming a program and the tuples forming the data upon which the 'program' operates. It may be seen that given the appropriate W-grammar, static and dynamic semantics of the normal forms mentioned above can be verified.The most important result of our work is the discovery that for the particular W-grammars considered, parsers may be found that are essentially LR(1). (See for example Aho and Ullman [8] for a comprehensive theory of LR parsing.) This is not totally surprising since similar results have been obtained elsewhere in connection with a syntactic theory of data structures (Mejstrick [9]). This result comes about because, by means of a transformation, the original W-grammar with its attendant where clauses is mapped to an LR(1) grammar with an augmented terminal alphabet. The new terminal symbols call for further lexical processing with an added time complexity of 0(n) where n is the size of the augmented alphabet. The transformed grammar presents some interesting lexical analysis problems. Their solution requires two types of token recognizers, the in-string token recognizer and the out-string recognizer. For EX: given an input string *letter r1 symbol, 1br symbol, attribute al$ The in-string recognizer delineates the small segments of the string by moving the 'dot mark' from the beginning to the end of the segment string. These small segment strings are strings which are bound at one end or the other by commas. The out-string token recognizer will move the dot mark by using the results of the in-string recognizer thus recognizing the entire input string.Our investigations may lead to a method which would allow a database administrator to test the correctness of a proposed schema without possessing a deep knowledge of database theory.

Improvement of morphological deficiencies according to the degree of teratozoospermia: Is it possible to reach a comparable to normozoospermic semen specimen?

Lipid composition of the normal and flight forms of adult cowpea weevils, Callosobruchus maculatus

Boolean Symbolic Objects were introduced by Diday (1988) and since that time a large number of applications have been defined, using these objects, but relatively few of them take constraints on the variables into account. Even in this case, when the graph of dependencies becomes too large, the computational time becomes huge because dependencies are treated in a combinatorial way. We present a method inspired by the technique used in relational data bases (Codd 1972) leading to a decomposition of symbolic objects into a Normal Symbolic Form which allows an easier calculation, however huge the graph of dependencies rules may be. We will apply our method to distance computation following a method due to De Carvalho and inspired by Ichino (1994) but the normal form we present in this paper could be used for other purposes. In our first trials we obtained a 90% reduction of the computational time. In the present text we will only deal with nominal boolean Symbolic Objects, but the method could be used with other kinds of symbolic objects.

In his seminal paper on the relational data model, E. F. Codd formulated a number of design principles for a relational database (Codd, 1970). These principles were originally formalised in terms of three normal forms: first normal form, second normal form and third normal form. The process of transforming a database design through these three normal forms is known as normalisation. By the mid-1970s third normal form was shown to have certain inadequacies and a stronger normal form, known as Boyce-Codd normal form, was introduced (Codd, 1974). Subsequently Fagin introduced fourth normal form and indeed fifth normal form (Fagin 1977, 1979). In this chapter we consider Codd 's original ideas on normalisation whilst also describing a graphic technique used for designing fully normalised schema. We particularly emphasise the use of normalisation as a bottom-up technique for database design.