Duality Principle Research Articles

The Kechris–Pestov–Todorčević Correspondence from the Point of View of Category Theory

The Kechris–Pestov–Todorcevic correspondence (KPT-correspondence for short) is a surprising correspondence between model theory, combinatorics and topological dynamics. In this paper we present a categorical re-interpretation of (a part of) the KPT-correspondence with the aim of proving a dual statement. Our strategy is to take a “direct” result and then analyze the necessary infrastructure that makes the result true by providing a purely categorical proof of the categorical version of the result. We can then capitalize on the Duality Principle to obtain the dual statements almost for free. We believe that the dual version of the KPT-correspondence can not only provide the new insights into the interplay of combinatorial, model-theoretic and topological phenomena this correspondence binds together, but also explores the limits to which categorical treatment of combinatorial phenomena can take us.

Newton projection method as applied to assembly simulation

In this paper, we consider Newton projection method for solving the quadratic programming problem that emerges in simulation of joining process for assembly with compliant parts. This particular class of problems has specific features such as an ill-conditioned Hessian and a sparse matrix of constraints as well as a requirement for the large-scale computations. We use the projected Newton method with a quadratic rate of convergence and suggest some improvements to reduce the solving time: a method for solving the system of linear equations, so-called constraint recalculation method, and compare different approaches for step-size selection. We use the duality principle to formulate alternative forms of the minimization problem that, as a rule, can be solved faster. We describe how to solve the considered nonlinear minimization problem with the nonsmooth objective function by modifying Newton projection method and employing subgradients. In addition, we prove the convergence of the suggested algorithm. Finally, we compare Newton projection method with the other quadratic programming techniques on a number of assembly simulation problems.

A Novel Kalman Filter Formulation for Improving Tracking Performance of the Extended Kernel RLS

In this paper, two contributions are presented. Firstly, a novel variant of the Kalman filter is developed and used to improve the tracking performance of the extended kernel recursive least squares algorithm (Ex-KRLS). Without resorting to the Riccati equation, the proposed formulation of the Kalman filter relies on principles of optimization and convex duality to obtain a stable quadratic form. It allows for performance and stability efficiency by minimizing the estimation error with the manipulation of a single scalar variable. Secondly, the proposed Kalman filter formulation is embedded into the Ex-KRLS-KF algorithm, an extension of the Ex-KRLS, to further improve its tracking performance. For this purpose, the state model of the proposed Ex-KRLS-KF variant is constructed in the original state space, while the hidden state is then estimated using the proposed Kalman filter formulation. The standard KRLS algorithm learns the measurement model used in hidden state estimation. We show through a comprehensive set of computer experiments that the proposed hybrid algorithm has more flexible state and noise models than competing alternative algorithms. We evaluate the proposed algorithm in two benchmarking tasks (nonlinear Rayleigh multipath channel tracking and Lorenz system modeling) and compare its performance with those provided by the state-of-the-art algorithms.

On various Riesz-dual sequences for Schauder frames

In this paper, we introduce various definitions of R-duals, to be called R-duals of type I, II, which leads to a generalization of the duality principle in Banach spaces. A basic problem of interest in connection with the study of R-duals in Banach spaces is that of characterizing those R-duals which can essentially be regarded as M-basis. We give some conditions under which an R-dual sequence to be an M-basis for X.

Gabor duality theory for Morita equivalent C∗-algebras

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalence bimodules with some extra properties. For certain twisted group [Formula: see text]-algebras, the reformulation of the duality principle to the setting of Morita equivalence bimodules reduces to the well-known Gabor duality principle by localizing with respect to a trace. We may lift all results at the module level to matrix algebras and matrix modules, and in doing so, it is natural to introduce [Formula: see text]-matrix Gabor frames, which generalize multi-window super Gabor frames. We are also able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for [Formula: see text]-matrix Gabor frames.

Backward doubly stochastic Volterra integral equations and their applications

In this paper, we introduce a new class of equations called backward doubly stochastic Volterra integral equations (BDSVIEs, for short). First, the well-posedness of BDSVIEs in the sense of introduced M-solutions is established. Second, a comparison theorem of BDSVIEs is proved. As an application of the comparison theorem, we derive the existence of solutions of BDSVIEs with continuous coefficients. Third, a duality principle between linear forward doubly stochastic Volterra integral equations (FDSVIEs, for short) and BDSVIEs is obtained. Moreover, by virtue of the duality principle, a maximum principle of Pontryagin type is established for an optimal control problem of FDSVIEs.

Lower and upper bound limit analysis via the alternating direction method of multipliers

Abstract Computational limit analysis methods invariably lead to the need to solve a mathematical programming problem. The alternating direction method of multipliers (ADMM) is one versatile and robust technique to solve non-linear convex optimization problems that has recently found applications in a wide range of fields. Its solution scheme, based on an operator splitting algorithm, is not only easy to implement but also suitable to efficiently solve large-scale variational problems. Starting from the ADMM framework, we derive a strict upper bound finite element formulation using a two-(primal)-field approximation, one for the velocity field and the other for the plastic strain rate field. Next, following a similar approach, we develop a novel strict lower bound formulation. Here, the two-(primal)-field model is based on a redundant approximation of the stress field. Duality principles are then explored in order to unify these two formulations.The effectiveness of this approach is demonstrated on test problems and, to conclude, some considerations are made about the performance results.

New duality based generalized rank annihilation algorithm for determining analyte concentration with realistically estimated error level for practical data sets

A new duality based algorithm is developed to achieve the “second-order advantage” and determination of the standard component in the presence of uncalibrated components. For this purpose, the duality principle, as a powerful mathematical approach, is used to define the interference(s)’ subspace. Afterward, the concentration ratio, λ, i.e., the amount of the standard component (analyte) in the test sample to its amount in the standard has been calculated systematically. The provided estimations of λ yield reliable results for the standard component with adequate data-driven error intervals. Additionally, the geometrical visualizations beside algebraic formulas are used to show the quantitative analysis of this study. By illustrating the procedure in graphical plots, the details of the proposed method can be explained understandably. This contribution can also be considered as another attempt for the rehabilitation of the generalized rank annihilation concept.

Plasmonic Antennas with Electric, Magnetic, and Electromagnetic Hot Spots Based on Babinet’s Principle

We theoretically study plasmonic antennas featuring areas of extremely concentrated electric or magnetic field, known as hot spots. We combine two types of electric-magnetic complementarity to increase the degree of freedom for the design of the antennas: bow-tie and diabolo duality and Babinet's principle. We evaluate the figures of merit for different plasmon-enhanced optical spectroscopy methods: field enhancement, decay rate enhancement, and quality factor of the plasmon resonances. The role of Babinet's principle in interchanging electric and magnetic field hot spots and its consequences for practical antenna design are discussed. In particular, diabolo antennas exhibit slightly better performance than bow-ties in terms of larger field enhancement and larger Q factor. For specific resonance frequency, diabolo antennas are considerably smaller than bow-ties which makes them favourable for the integration into more complex devices but also makes their fabrication more demanding in terms of spatial resolution. Finally, we propose Babinet-type dimer antenna featuring electromagnetic hot spot with both the electric and magnetic field components treated on equal footing.

Manufacturing Pairs of Woven Frames Applying Duality Principle on Hilbert Spaces

Manufacturing Pairs of Woven Frames Applying Duality Principle on Hilbert Spaces

Hidden Convexities in Decentralized Coordination for the Distribution Networks

The modern power grid is undergoing unprecedented levels of transformations due to the rising prevalence of diverse power entities, cyber-enablement of grid components and energy deregulations. In this paper, we focus on distribution networks (DNs) to enable the seamless plug-and-play coordination of actuating cyber-enabled power entities for cost-effective and feasible system operations. The proposed distributed algorithm empowers individual cyber-physical agents residing in active power nodes with the ability to iteratively compute local actuation setpoints by exchanging information with neighbouring entities. The main contribution of this work is the identification of hidden convexities in the original non-convex optimal power flow (OPF) formulation for the DN via strategic decomposition and strong duality principles. These eliminate the need for OPF relaxations/approximations. Strong convergence and feasibility results are presented via theoretical analysis and practical simulation studies conducted on realistic systems.

Generalized Lie Algebraic Geometry in R3x SO(3) Configuration Space for SU(3) of Elementary Particles and for Wave-packing of Atomic Structure

In this paper, we show, by extending Lie’s original 1871 thesis “on philosophical reflections upon the nature of Cartesian geometry” based on “transformation by which surfaces that touch each other are turned into similar surfaces … between the Plucker line geometry and a geometry whose elements are the space’s spheres” to include toroidal deformation of the sphere, how an algebraic geometric principle of duality between points, lines and planes of 3..dimesional space provides a sufficiently general framework for realizations of Lie-algebra and its Lie-isotopic and Lie-admissible generalizations in solid state configuration space R3×SO(3) compatible with translational periodicity of 3–dimensional space lattice. The generalization provide not only representation of SU(3) symmetry of extended (string-like) elementary particles with complementary duality of leptons and baryons, but also dual wave-packet representation of atomic structure and the periodic table, highlighting the significance of the fact that Mendeleev originally moulded his two-dimensional rendering of the periodic system on the dual Sanskrit grammar/phonetics.

Euclidean Forward–Reverse Brascamp–Lieb Inequalities: Finiteness, Structure, and Extremals

A new proof is given for the fact that centered gaussian functions saturate the Euclidean forward-reverse Brascamp-Lieb inequalities, extending the Brascamp-Lieb and Barthe theorems. A duality principle for best constants is also developed, which generalizes the fact that the best constants in the Brascamp-Lieb and Barthe inequalities are equal. Finally, as the title hints, the main results concerning finiteness, structure and gaussian-extremizability for the Brascamp-Lieb inequality due to Bennett, Carbery, Christ and Tao are generalized to the setting of the forward-reverse Brascamp-Lieb inequality.

Rational extended thermodynamics of dense polyatomic gases incorporating molecular rotation and vibration.

The paper aims to construct a rational extended thermodynamics (RET) theory of dense polyatomic gases by taking into account the experimental evidence that the relaxation time of molecular rotation and that of molecular vibration are quite different from each other. For simplicity, we focus on gases with only one dissipative process due to bulk viscosity. In fact, in some polyatomic gases, the effect of bulk viscosity is much larger than that of shear viscosity and heat conductivity. The present theory includes the previous RET theory of dense gases with six fields as a particular case, and it also includes the RET theory of rarefied polyatomic gases with seven fields in the rarefied-gas limit. The closure is carried out by using the universal principles, that is, Galilean invariance and objectivity, entropy principle, and thermodynamic stability (convexity of entropy), where the duality principle connecting rarefied gases to dense gases also plays an important role. A detailed discussion is devoted to the expression of the production terms in the system of balance equations. As typical examples, we study a gas with virial equations of state and a van der Waals gas. Lastly the dispersion relation of a linear wave is derived, and its comparison with experimental data is made. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

Applications of matroids in electric network theory

Professor Masao Iri and his students had several important contributions to the theory of electric networks. He was one of the first scientists who recognized the importance of matroid theory in studying the qualitative problems of electric networks. The present paper summarizes his results in the unique solvability of linear active networks and in the existence of hybrid immittance description of linear multiports. Then his contribution to the proper understanding of the duality principle in electric network theory is presented. The last two sections are strongly related to the concept of genericity, leading to some results in the qualitatively reliable synthesis of linear multiports and to some considerations on how to store the data of linear multiports. These applications require a broad spectrum of matroidal tools, like the matroid partition, intersection and parity algorithms and concepts like matroid union, gammoids, transversal matroids, etc. The survey is extended by some more recent results showing that the influence of Professor Iri to the younger generation of researchers will be thriving for many years to come.

ПРАКТИЧНА ПРОФЕСІЙНО-ПЕДАГОГІЧНА ПІДГОТОВКА МАЙБУТНІХ УЧИТЕЛІВ ПРОФЕСІЙНИХ ШКІЛ У НІМЕЧЧИНІ: ОСОБЛИВОСТІ ДУАЛЬНОЇ МОДЕЛІ

A unique tool that helps to balance the imbalance between the growing needs of young people, the demands of the labor market and educational opportunities in the countries of the European Union is a dual model of vocational education, which has found its application in the field of higher education in Germany. This fact actualizes the need to study the practice of dual training in teacher education in this country. The article describes the structure and content of practical training of future vocational school teachers in Germany, identifies and analyzes the peculiarities of applying a dual approach to its organization. It has been found out that gaining the professional experience, forming necessary practical skills and competences is carried out in the process of industrial and school practices, pedagogical internships. An empirical foundation for higher education is formed during the industrial practice. The school practice is considered to be a form of gaining skills, not a field of activity. The pedagogical internship organized on the principle of duality of places of study helps to form the pedagogical activity competence of a beginner teacher. The unity and interpenetration of theoretical and practical pedagogical training are fully realized during the pedagogical internship. These components of practical vocational training make it possible to combine and arrange theoretical and practical components of studying, to synthesize scientific knowledge and practical skills, to form the own style of professional activity. The prospects of using the positive experience of applying the dual model of practical vocational training of teachers at vocational schools in Germany in the similar practice of pedagogical education in Ukraine are identified in the article: initiation of previous practical activity in manufacturing of a chosen profession as a necessary background for admission into the higher education institution; introduction of pedagogical internship; cooperation of higher education institutions with enterprises, institutions and organizations. Keywords: vocational school teacher, practical vocational training, industrial practice, school practice, pedagogical internship, duality, Germany.

Reversible Wideband Hybrid Model of Two-Winding Transformer Including the Core Nonlinearity and EMTP Implementation

Accurate high-frequency modeling of transformers is crucial for the analysis and simulations of the fast electromagnetic transients of power systems and smart grids. However, most high-frequency models do not include the effects of core nonlinearity and may lead to inaccurate results. This article proposes an improved wideband model considering core nonlinearity for single-phase two-winding transformers and its implementation in the Electromagnetic Transients Program (EMTP). The hybrid model is developed by merging the wideband admittance module and the duality-derived π module using filters with compensation. The duality-derived π module is developed from the direct application of the principle of duality. The wideband admittance module is developed based on discrete state equations using the technology of the modified vector fitting method. All parameters of the hybrid model are easy to be obtained because they are determined using terminal tests instead of the detail information of the transformer. Simulations and experiments, including inrush current, lightning excitation, and ferroresonance tests, show that the proposed hybrid model can accurately simulate both the low- and high-frequency transients of a single-phase two-winding transformer with a maximum difference of 5%.

Confessionele hermeneutiek. Vermittlungstheologie tussen belijden als opdracht en erfenis (1845-1855)

The reformed traditon features a tension between the historical confessions and the duty of confessing in response to God's revelation, which asks for a hermeneutics that understands the relation of contemporary Christians to ancient confessions. This article analyses the role of German theological movement of Vermittlungstheologie in this debate. After a sketch of the historical setting, the essay describes this movement’s theories on the matter of a common protestant confession (among others, the dual principle), of the necessity of differentiation, and the resulting practical consequences for the church the last aspect on the basis of the Acts of the Generalsynode of 1846. The article concludes with a brief discussion of this movement’s thought as a ‘confessional hermeneutics’.

Controlling a Linear MIMO System by a Measurement Vector Using Multilevel Decomposition

A variant of the synthesis method for controlling a linear MIMO system by an output vector is developed that provides the given spectrum of a multidimensional dynamic system in the state space. It is based on the principle of duality of the task of controlling a MIMO system by a state vector and the task of constructing a state observer. The algorithm, without any changes, is applicable for both continuous and discrete cases of describing the mathematical model of the MIMO system, has no restrictions on specifying the spectrum’s elements of the closed MIMO system, and allows us to obtain synthesis problems in analytical form and to parameterize the set of equivalent control laws with feedback. An example of an analytical synthesis of the control law of a hypothetical aircraft is given.

Financial Aspects of the Dual Form of Education

The article shows the role of the dual form of education in improving the quality of training of graduates of domestic educational institutions. The need to diversify the sources of funding for the dual form of higher education at the stages of its implementation is emphasized. The dual form of the organization of training by the motivating factor of diversification of sources of financing of education is defined. The content of the principle of duality and the peculiarities of its application in improving the professionalism of the graduate are revealed. The importance of two forms of student learning in different environments is emphasized. The concepts of «dual form of education acquisition» and «dual form of education organization» are described. The correctness of the application in scientific vocabulary and the practical application of the concepts «dual form of education acquisition» and «dual form of education organization» is emphasized. Foreign experience in financing higher education is analyzed. The multidimensionality in the training of specialists in free economic education with the use of academic capital and diversification of sources of funding for education in foreign practice is shown. The foreign experience of training specialists in working professions and financing vocational training is generalized. The benefits of direct and indirect funding of vocational education for the creation of additional jobs in the workplace have been identified. The German experience of formation of the Framework Curriculum by an educational institution and the Framework Plan of Practice with the participation of the employer is given. Emphasis is placed on the incentives and motivation of the mentor in teaching students in the workplace. The results of pilot projects on introduction of the dual form of the organization of training in domestic establishments on preparation of working trades are processed. It is stated a high level of employment of graduates of vocational schools and raise of savings due to increased practical training in the workplace. The peculiarities of realization of the dual form of organization of education and its financing in the Bar College of Transport and Construction are generalized. The necessity of taking into account the work schedule of employers when forming educational institutions’ curricula is proved. A comparison of the ratios of student learning in educational institutions and in workplaces with different models of the dual form of education is conducted. Calculations were made for the educational program with the distribution of loans between domestic institutions of higher education and employers. The main directions of development of educational and professional programs in institutions of higher education are offered. Keywords: forms of education, cost of education, theoretical training, practical training, base of practice, interests of an employer, educational-professional program, principle of duality, dual form of education acquisition, dual form of education organization, models of realization of dual form of education, institutions of higher education, institutions of professional education, financing of education, direct and indirect financing, financing of vocational training, financial support, diversification of sources of financial resources.