Constructive sheaf models of type theory

We provide a constructive version of the notion of sheaf models of univalent type theory. We start by relativizing existing constructive models of univalent type theory to presheaves over a base category. Any Grothendieck topology of the base category then gives rise to a family of left-exact modalities, and we recover a model of type theory by localizing the presheaf model with respect to this family of left-exact modalities. We provide then some examples.

Installation for Producing Mechanical Vibrations used in the Finishing of the Welded Joints Structure

The paper presents a constructive version of a mechanical vibration generator designed to be used at stress relieving of welded joints. This installation can be used on all brands of steels and welded joints, whether they have magnetic properties or not. In the generator construction a DC motor was used, in order to be able to adjust the speed more easily. The speed adjustment is made with a potentiometer, and the speed measurement is made with a tachometer. The tachometer measures the speeds based on the pulses given by a four-pointed cam rotating between the plates of a capacitor. The vibration system also contains a piezoelectric accelerograph, which is attached to the welded part subjected to vibration, in order to measure the level of vibration and the parameters of the vibration wave.

A Multibody Inertial Propulsion Drive with Symmetrically Placed Balls Rotating on Eccentric Trajectories

Starting with the last century, a lot of enthusiastic researchers have invested significant time and energy in proposing various drives capable to generate linear propulsion force. Regrettably, only a few of these devices passed the patent phase and have been practically materialized. The aim of this paper was to simulate the dynamic behavior of an inertial propulsion drive (IPD) developed by the authors, to demonstrate its functionality. The core of the IPD consists of two symmetric drivers that each performs rotation of eight steel balls on an eccentric path. We propose three solutions for the element which maintain the off-center trajectory of the balls. For the simulation, we used the multibody system approach and determine the evolution of the displacement, velocity, and power consumption. Further, we analyze the collisions between the elements of the system and the influence of this phenomenon on the dynamic behavior of the IPD. We found that collisions generate impact forces which affect the ball acceleration values achieved by simulation. We have concluded that the developed system is capable to generate linear movement. In addition, in terms of velocity and power consumption, the best constructive version of the retaining disk is that which has a cylindrical inner bore placed eccentric relative to the rotation center of the balls.

Computation of the unipotent Albanese map on elliptic and hyperelliptic curves

We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the p-adic de Rham period map $$j^{dr}_n$$ j n dr on elliptic and hyperelliptic curves over number fields via their universal unipotent connections $${\mathscr {U}}$$ U . Several algorithms forming part of the computation of finite level versions $$j^{dr}_n$$ j n dr of the unipotent Albanese maps are presented. The computation of the logarithmic extension of $${\mathscr {U}}$$ U in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a lemma of Hadian used in the computation of the Hodge filtration on $${\mathscr {U}}$$ U over affine elliptic and odd hyperelliptic curves. We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated p-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well.

Graphic Algorithms for Constructing a Quadric, Given Nine Points

The fundamental issue of constructing a nine-point quadric was frequently discussed by mathematicians in the 19th century. They failed to find a simple linear geometric dependence that would join ten points of a quadric (similar to Pascal's theorem, which joins six points of a conic section). Nevertheless, they found different algorithms for a geometrically accurate construction (using straightedge and compass or even using straightedge alone) of any number of points of a quadric that passes through nine given points. While the algorithms are quite complex, they can be implemented only with the help of computer graphics. The paper proposes a simplified computer-based realization of J.H. Engel’s well-known algorithm, which makes it possible to define the ninepoint quadric metric. The proposed graphics algorithm can be considered an alternative to the algebraic solution of the stated problem. The article discusses two well-known graphical algorithms for constructing a quadric (the Rohn — Papperitz algorithm and the J.H. Engel algorithm) and proposes a simplified version of the J.H. algorithm. For its constructive implementation using computer graphics. All algorithms allow you to determine the set of points and the set of flat sections of the surface of the second order, given by nine points. The Rohn — Papperitz algorithm, based on the spatial configuration of Desargues, is best suited for its implementation on an axonometric drawing using 3D computer graphics. Algorithm J.H. Engel allows you to solve a problem on the plane. The proposed simplified constructive version of the algorithm J.H. Engel is supplemented with an algorithm for constructing the principal axes and symmetry planes of a quadric, given by nine points. The construction cannot be performed with a compass and a ruler, since this task reduces to finding the intersection points of two second-order curves with one known general point (third degree task). For its constructive solution, a computer program is used that performs the drawing of a second order curve defined by an arbitrarily specified set of five points and tangents (both real and imaginary). The proposed graphic algorithm can be considered as an alternative to the algebraic solution of the problem.

АРМІЯ СУЧАСНОЇ ДЕРЖАВИ: ВИЗНАЧЕННЯ СТРАТЕГІЇ ДОСЛІДЖЕННЯ

The article contains an attempt to determine the strategy of research of the army of the modern state in the context of analyzing the evolution of the institutional approach in Western social and political discourse. The emphasis is on the content of changes that allowed institutional theory not only to transform into a non-institutional, but also to become the leading methodological basis for modern political research at all levels. It is about a change: in the categorical and conceptual apparatus of the research (for example, the new interpretation of the content of the main concepts (from the institute to the rational choice and the introduction into the scientific circulation of the concepts institutional space, institutional practice, rules of the game and etc.); principles of research; levels of research (identification of institutional, organizational and individual levels); subject field of research (first of all, introduction of individual behavior of actors into it); methods and procedures of analyze (use of tools of rational choice theory, structural and functional analysis, behavioral approach). It is substantiated that the perspective of using the main provisions of this approach to analyze the problem field of the modern army is based on such moments as: the existence of a fairly successful practice of considering the army as a specific political institution in modern scientific discourse; development of a new (more constructive) version of the interpretation of the concept of political institution; the formation, within the framework of neoinstitutionalism, of a basic research set, which allows considering collective, organizational, and mental structures and procedures, as well as individual behavior from a unified analytical position.

ABSTRACT INDUCTIVE AND CO-INDUCTIVE DEFINITIONS

In [G. Curi, On Tarski’s fixed point theorem. Proc. Amer. Math. Soc., 143 (2015), pp. 4439–4455], a notion of abstract inductive definition is formulated to extend Aczel’s theory of inductive definitions to the setting of complete lattices. In this article, after discussing a further extension of the theory to structures of much larger size than complete lattices, as the class of all sets or the class of ordinals, a similar generalization is carried out for the theory of co-inductive definitions on a set. As a corollary, a constructive version of the general form of Tarski’s fixed point theorem is derived.

Towards a Simply Typed CALculus for Semantic Knowledge Bases

This paper demonstrates how a constructive version of the description logic ALC can serve as a semantic type system for an extension of the simply typed $\lambda$-calculus to express computations in knowledge bases. This cALculus embodies a functional core language which provides static type checking of semantic information for programming with data whose structure is organised under a relational data model. The cALculus arises from a natural interpretation of the tableau rules for constructive ALC following the Curry-Howard-Isomorphism.