POINTS OF SMALL HEIGHT ON AFFINE VARIETIES DEFINED OVER FUNCTION FIELDS OF FINITE TRANSCENDENCE DEGREE

We provide a direct proof of a Bogomolov-type statement for affine varieties V defined over function fields K of finite transcendence degree over an arbitrary field k, generalising a previous result (obtained through a different approach) of the first author in the special case when K is a function field of transcendence degree $1$ . Furthermore, we obtain sharp lower bounds for the Weil height of the points in $V(\overline {K})$ , which are not contained in the largest subvariety $W\subseteq V$ defined over the constant field $\overline {k}$ .

Reliability improvement of electrically active defect investigations by analytical and experimental deep level transient: Fourier spectroscopy investigations

This article discusses the importance of analytical and experimental approaches in Deep level transient Fourier spectroscopy in terms of reliability, to support the current research and the utilization of this technique for complex investigations. An alternative evaluation approach is proposed and validated by relevant experiments. Attention is focused on a GaAs p-i-n structure, the undoped layer induced defect conduction type statement difficulty, accurate evaluation of a dual type majority-minority carrier defect complex and possible limitations of the DLTS experimental technique. Comprehensive evaluation is carried out and the method is discussed in detail. In comparison with reference data, higher precision of calculated activation energies, differences even lower as 10−3 order of magnitude, were achieved.

A Manin–Mumford Theorem for the Maximal Compact Subgroup of a Universal Vectorial Extension of a Product of Elliptic Curves

We study the intersection of an algebraic variety with the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves. For this intersection we show a Manin–Mumford-type statement. This answers some questions posed by Corvaja–Masser–Zannier, which arose in connection with their investigation of the intersection of an algebraic curve with the maximal compact subgroup of various algebraic groups. In particular they proved that these intersections are finite for universal vectorial extensions of elliptic curves. Using Khovanskii’s zero-estimates combined with a stratification result of Gabrielov–Vorobjov and recent work of the authors, we obtain effective bounds for this intersection that only depend on the degree of the algebraic variety and the dimension of the group. As a corollary, we obtain new uniform results of Manin–Mumford type for additive extensions of certain abelian varieties.

Crepant resolutions and open strings

In the present paper, we formulate a Crepant Resolution Correspondence for open Gromov–Witten invariants (OCRC) of toric Lagrangian branes inside Calabi–Yau 3-orbifolds by encoding the open theories into sections of Givental’s symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan–Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates–Corti–Iritani–Tseng and Ruan, we furthermore propose (1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and (2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold {A_{n}}-singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov–Witten theory of this family of targets.

Chapter 5. Functional-pragmatic text analysis

Drawing on Charles Bally’s distinction between dictum and modus, the final chapter deals with the pragmatic analysis of propositions identified at the previous step. Propositions can be expressed by an attribute linking together a number of objects or by a single word (implicit proposition). Implicit propositions should be made explicit, i.e. missing objects should be brought out. Each proposition is attributed a conceptual type. The authors posit four types of propositions, namely, fact, opinion, evaluation and expression of will. While the distinction between fact and opinion is common enough, evaluation emphasizes the emotional, or expressive, aspect of an utterance, and the expression of will has to do with persuasive speech acts. Next, relations between propositions, both local and distant, are established, which makes it possible to bring them together within a rhetorical text structure. The units of rhetorical structure are quite different from both the syntactic and communicative ones and vary depending on the its type (statement, persuasion, directive, etc.).

A Modular André–Oort Statement with Derivatives

In unpublished notes, Pila discussed some theory surrounding the modular function j and its derivatives. A focal point of these notes was the statement of two conjectures regarding j, j′ and j″: a Zilber–Pink-type statement incorporating j, j′ and j″, which was an extension of an apparently weaker conjecture of André–Oort type. In this paper, I first cover some background regarding j, j′ and j″, mostly covering the work already done by Pila. Then I use a seemingly novel adaptation of the o-minimal Pila–Zannier strategy to prove a weakened version of Pila's ‘Modular André–Oort with Derivatives’ conjecture. Under the assumption of a certain number-theoretic conjecture, the central theorem of the paper implies Pila's conjecture in full generality, as well as a more precise statement along the same lines.

PRIME SOLUTIONS TO POLYNOMIAL EQUATIONS IN MANY VARIABLES AND DIFFERING DEGREES

Let $\mathbf{f}=(f_{1},\ldots ,f_{R})$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations $f_{j}(x_{1},\ldots ,x_{n})=0~(1\leqslant j\leqslant R)$ satisfies a general local to global type statement, and has a solution where each coordinate is prime. In fact we obtain the asymptotic formula for number of such solutions, counted with a logarithmic weight, under these conditions. We prove the statement via the Hardy–Littlewood circle method. This is a generalization of the work of Cook and Magyar [‘Diophantine equations in the primes’, Invent. Math.198 (2014), 701–737], where they obtained the result when the polynomials of $\mathbf{f}$ all have the same degree. Hitherto, results of this type for systems of polynomial equations involving different degrees have been restricted to the diagonal case.

Maneuvering of Surface Vessels Using a Fuzzy Logic Controller

A fuzzy logic controller is developed for maneuvering control of surface vessels. The fuzzy controller uses a vessel's heading, yaw rate, distance from a reference point, and the velocity of the vessel relative to the reference point as inputs to generate the control outputs. The control outputs include rudder angle, increase in propeller thrust, and lateral bow thrust. The design of the fuzzy controller is simple and does not require a mathematical modeling of the complicated nonlinear system. The core of the fuzzy controller is a set of fuzzy associative memory (FAM) rules that correlate each group of fuzzy input sets to a fuzzy output set. A FAM rule is a logical if-then type statement based on one's sense of realism, experience, and expert knowledge. The effectiveness and robustness of the fuzzy controller are demonstrated through the numerical time-domain simulations of path tracking and dynamic positioning of a Mariner-class hull with use of nonlinear maneuvering equations of motions.