Duality Principle Research Articles

Simultaneous construction of dual Borgen plots. II: Algorithmic enhancement for applications to noisy spectral data

AbstractBorgen plots are low‐dimensional representations of the set of all nonnegative factorizations of spectral data matrices. Classical Borgen plots are limited to nonnegative data and can be constructed for the spectral factor or for the concentration profiles. In the first part of this paper, a simultaneous construction of the two dual Borgen plots is presented, which intensively exploits the underlying duality principles. The second part introduces algorithmic enhancements which make the simultaneous Borgen plot construction possible for noisy experimental data matrices which can contain small negative matrix entries. The new method is tested for FT‐IR spectral data from the Rhodium catalyzed hydroformylation process. The results are compared to those by the FACPACK‐implementation of the polygon inflation method.

Investigation of an Egyptian Mummy board by Using Clinical Multi-slice Computed Tomography

ABSTRACTA modern approach to the diagnosis of artistic heritage makes use of non-destructive techniques which avoid damage to artworks or signs of interaction with the analytical technique used. For this reason, today, the employment of X-rays to study artistic heritage is quite widespread. Computed tomography (CT) Imaging was first used as specialized medical diagnostic tool in the early 1970s. Today, CT scanners have become more common, and non-destructive three-dimensional imaging has found an increasing number of uses in different areas such as materials research, geology, archaeology, and museum conservation. In this work, a new generation of clinical CT (based on the Dual Energy and multi-slice acquisition principle) is used to investigate an Egyptian wooden mummy board with a total acquisition time of 93.23 seconds and a voxel size of 0.078 × 0.078 × 1.5 mm. The use of two distinct techniques of image post-processing, the two-dimensional multi-planar reconstruction (MPR) and the three-dimensional volume rendering technique (VRT) allowed us to: evaluate the state of conservation of the mummy board, determine how the wooden object was implemented, recognize a nineteenth-century intervention of consolidation, and advance the hypothesis that the wooden planks were reused. In particular, we highlight the use of the cardiovascular and pulmonary filter with VRT reconstruction, which allows a virtual elimination of different parts of the mummy board to investigate those areas otherwise accessible only by removing parts of the board and thus damaging the work.

Cassirer and the Structural Turn in Modern Geometry

The paper investigates Ernst Cassirer’s structuralist account of geometrical knowledge developed in his Substanzbegriff und Funktionsbegriff (1910). The aim here is twofold. First, to give a closer study of several developments in projective geometry that form the direct background for Cassirer’s philosophical remarks on geometrical concept formation. Specifically, the paper will survey different attempts to justify the principle of duality in projective geometry as well as Felix Klein’s generalization of the use of geometrical transformations in his Erlangen program. The second aim is to analyze the specific character of Cassirer’s geometrical structuralism formulated in 1910 as well as in subsequent writings. As will be argued, his account of modern geometry is best described as a “methodological structuralism”, that is, as a view mainly concerned with the role of structural methods in modern mathematical practice.

A Duality Theory for Non-convex Problems in the Calculus of Variations

We present a new duality theory for non-convex variational problems, under possibly mixed Dirichlet and Neumann boundary conditions. The dual problem reads nicely as a linear programming problem, and our main result states that there is no duality gap. Further, we provide necessary and sufficient optimality conditions, and we show that our duality principle can be reformulated as a min–max result which is quite useful for numerical implementations. As an example, we illustrate the application of our method to a celebrated free boundary problem. The results were announced in Bouchitte and Fragala (C R Math Acad Sci Paris 353(4):375–379, 2015).

Research of optical flow aided MEMS navigation based on convex optimization and ROF denoising

Because of its own advantages, micro air vehicles can accomplish reconnaissance / surveillance tasks in low altitude and indoor flight environments, with inertial / satellite integrated navigation system. As a useful supplement to its navigation system, optical flow can assist INS to update position and attitude information in some occasions without GPS assistent. Considering the low calculation precision of optical flow in unconstant illumination condition, optical flow navigation method based on ROF denoising is studied, convex optimization theory and dual principle is discussed, and image decomposition method in the light texture changes is researched on. The obtained texture part is not affected by illumination variations, and used to calculate the movement of field information of the image. Then, the movement information is further calculated for position and attitude of micro air vehichles. The experimental results show that the accuracy of optical flow calculation is improved by decomposing and denoising.

Семантичні властивості п’ятизначних логік

A new class of program-oriented logical formalisms – propositional five-valued logics and five-valued logics of quasiary predicates is proposed and studied in the article. The quasiarity of predicates means that their arity is not fixed. A special feature of such five-valued logics is the use of special truth values, indicating errors or uncertainties in various systems, in particular, software systems. The article gives an ontological justification of five-valued logics and examines their examples. For the propositional level, an algebra of truth values is constructed and its semantic properties are studied. For the predicate level, the five-valued logic of quasiary predicates is constructed, its semantic properties are studied, the relation of equivalence of formulas and the relation of logical consequence are introduced, the principle of duality is proved and various normal forms are considered. Further research is focused on constructing algorithms for checking satisfiability and refutability of formulas and constructing various types of calculi for the introduced logics. Problems in programming 2018; 1: 22-35

Design of State-Feedback Controllers for Discrete-Time Delayed Systems Using the Principle of Duality

Abstract A technique for the synthesis of state-feedback gains, capable of stabilizing discrete-time systems affected by time-varying uncertain delays, is proposed in this paper. The technique extends another synthesis condition presented in the literature, which is based on the application of a Lyapunov-Krasovskii function to generate a set of linear matrix inequalities (LMIs) and equality constraints. The method proposed in this paper is based on the dual representation of the delayed system, resulting in a LMI condition free of the equality constraints, thus reducing the complexity of the technique. A simulation example illustrates the validity of the proposed method.

Design and implementation of a model predictive observer for AHRS

A GPS-aided Inertial Navigation System (GAINS) is used to determine the orientation? position and velocity of ground and aerial vehicles. The data measured by Inertial Navigation System (INS) and GPS are commonly integrated through an Extended Kalman Filter (EKF). Since the EKF requires linearized models and complete knowledge of predefined stochastic noises? the estimation performance of this filter is attenuated by unmodeled nonlinearity and bias uncertainties of MEMS inertial sensors. The Attitude Heading Reference System (AHRS) is applied based on the quaternion and Euler angles methods. A moving horizon-based estimator such as Model Predictive Observer (MPO) enables us to approximate and estimate linear systems affected by unknown uncertainties. The main objective of this research is to present a new MPO method based on the duality principle between controller and observer of dynamic systems and its implementation in AHRS mode of a low-cost INS aided by a GPS. Asymptotic stability of the proposed MPO is proven by applying Lyapunov's direct method. The field test of a GAINS is performed by a ground vehicle to assess the long-time performance of the MPO method compared with the EKF. Both the EKF and MPO estimators are applied in AHRS mode of the MEMS GAINS for the purpose of real-time performance comparison. Furthermore? we use flight test data of the GAINS for evaluation of the estimation filters. The proposed MPO based on both the Euler angles and quaternion methods yields better estimation performances compared to the classic EKF.

Duality of Complex Systems Built from Higher‐Order Elements

The duality of nonlinear systems built from higher‐order two‐terminal Chua’s elements and independent voltage and current sources is analyzed. Two different approaches are now being generalized for circuits with higher‐order elements: the classical duality principle, hitherto restricted to circuits built from R‐C‐L elements, and Chua’s duality of memristive circuits. The so‐called storeyed structure of fundamental elements is used as an integrating platform of both approaches. It is shown that the combination of associated flip‐type and shift‐type transformations of the circuit elements can generate dual networks with interesting features. The regularities of the duality can be used for modeling, hardware emulation, or synthesis of systems built from elements that are not commonly available, such as memristors, via classical dual elements.

On Lorentzian spaces of constant sectional curvature

We investigate Osserman-like conditions for Lorentzian curvature tensors that imply constant sectional curvature. It is known that Osserman (moreover zwei-stein) Lorentzian manifolds have constant sectional curvature. We prove that some generalizations of the Rakic duality principle (Lorentzian totally Jacobi-dual or four-dimensional Lorentzian Jacobi-dual) imply constant sectional curvature. Moreover, any four-dimensional Jacobi-dual algebraic curvature tensor such that the Jacobi operator for some nonnull vector is diagonalizable, is Osserman. Additionally, any Lorentzian algebraic curvature tensor such that the reduced Jacobi operator for all nonnull vectors has a single eigenvalue has a constant sectional curvature.

Sawyer Duality Principle in Grand Lebesgue Spaces

The Sawyer duality principle is obtained for grand Lebesgue spaces on the unit interval, and the Hardy operators are shown to be bounded in these spaces.

Принципы бухгалтерского учета Томаса Джонса

Th e article considers two key provisions of the accounting theory of Th omas Jones, set out in the textbook «Th e principles and practice of book-keeping» (1841). Th e fi rst provision is the explanation of a double entry as a formal, artifi cial technique, which is based on the principle of duality. Th e second provision is the equality of the fi nancial result revealed by the accounts of the company’s property account (Primary Accounts) and the fi nancial result under the accounts of income and expenses (Secondary Accounts).

Extreme Sparse Multinomial Logistic Regression: A Fast and Robust Framework for Hyperspectral Image Classification

Although sparse multinomial logistic regression (SMLR) has provided a useful tool for sparse classification, it suffers from inefficacy in dealing with high dimensional features and manually set initial regressor values. This has significantly constrained its applications for hyperspectral image (HSI) classification. In order to tackle these two drawbacks, an extreme sparse multinomial logistic regression (ESMLR) is proposed for effective classification of HSI. First, the HSI dataset is projected to a new feature space with randomly generated weight and bias. Second, an optimization model is established by the Lagrange multiplier method and the dual principle to automatically determine a good initial regressor for SMLR via minimizing the training error and the regressor value. Furthermore, the extended multi-attribute profiles (EMAPs) are utilized for extracting both the spectral and spatial features. A combinational linear multiple features learning (MFL) method is proposed to further enhance the features extracted by ESMLR and EMAPs. Finally, the logistic regression via the variable splitting and the augmented Lagrangian (LORSAL) is adopted in the proposed framework for reducing the computational time. Experiments are conducted on two well-known HSI datasets, namely the Indian Pines dataset and the Pavia University dataset, which have shown the fast and robust performance of the proposed ESMLR framework.

A theory of duality in Euclidean geometry

The principle of duality is well established in projective geometry but can hardly be found in the literature on Euclidean geometry where it is “more a principle of analogy than a scientific principle with a logical foundation” (cp. Sommerville, The Elements of Non-Euclidean Geometry. The Open Court, London, 1919). We close this gap and develop a theory of duality in Euclidean geometry. Following Hilbert’s Grundlagen der Geometrie we consider the incidence, order and metric structure of a Euclidean plane and show (a) that there is a large class of theorems of Euclidean incidence geometry which allow a dualization (b) that Hilbert’s order structure can be introduced in a Euclidean plane in a self-dual way and (c) that appropriate definitions of metric notions (e.g., of an angle, a segment or a circle) lead to Euclidean theorems with meaningful dual versions. This shows that duality in Euclidean geometry is not a collection of isolated phenomena but corresponds to a rich and coherent theory.

Resource allocation in multicell systems with coordinated beamforming and partial data cooperation: a study on the effect of cooperation on achievable performance

The rise in wireless data traffic requires innovative interference management techniques to meet the demand. Transmit cooperation among multiple base stations has been proposed as a solution to this challenge and improve throughput. This paper proposes novel resource allocation algorithms with two different cooperation paradigms, viz. the cooperative beamforming and partial data cooperation. The well-known duality principle, which has been used to solve uplink and downlink optimisation problems in previous literature, is shown to exist for multicell systems with both coordinated beamforming and the novel paradigm termed as partial data cooperation. Using the generalised version of duality, this paper solves relevant resource allocation problems in both cases. The proposed power allocation problems are shown to outperform some of the existing optimisation techniques in terms of throughput while having significant conceptual and theoretical advantage.

Asymptotic asset pricing and bubbles

We define the concept of asymptotic superreplication, and prove a duality principle of asset pricing for sequences of financial markets (e.g., weakly converging financial markets and large financial markets) based on contiguous sequences of equivalent local martingale measures. This provides a pricing mechanism to calculate the fundamental value of a financial asset in the asymptotic market. We introduce the notion of asymptotic bubbles by showing that this fundamental value can be strictly lower than the current price of the asset. In the case of weakly converging markets, we show that this fundamental value is equal to an expectation of the terminal value of the asset in the weak-limit market. From a practical perspective, we relate the asymptotic superreplication price to a limit of quantile-hedging prices. This shows that even when a price process is a true martingale, it can have properties similar to a bubble, up to a set of small probability. For practical applications, we give examples of weakly converging discrete-time models (e.g. some GARCH models) and large financial models that present bubbles.

Exact controllability of a micro beam with boundary bending moment

ABSTRACTIn this paper, we show the exact controllability of a micro beam with boundary bending moment at time T > 0, which gives a confirmative answer to the open problem proposed by P. Guzmán and J. Zhu. The semi-group approach is adopted in investigation. The asymptotic spectral expression of the micro beam system is presented to verify an observability inequality for the dual system. By the duality principle, the exact controllability of the system is then established.

Optimal Control of Evolution Macro-Hybrid Mixed Variational Inclusions

Optimal control problems governed by primal and dual evolution macro-hybrid mixed variational state inclusions, in reflexive Banach spaces, are studied. This is a spatially localized macro-hybrid variational version of our mixed optimal control theory published in Appl. Math. Optim. 68, 2013, 445-473, where the solvability analysis of the state systems is given in terms of duality principles, and the mixed optimality analysis is performed via a perturbation conjugate duality method. Applications to nonlinear constrained problems from mechanics exemplify the theory.

Variational principles for shakedown analysis of materials with internal rotation

Based on the extended lower bound shakedown theorem for materials with internal rotation, variational principles for elastic and plastic shakedown are presented considering the indeterminate and modified couple-stress theory, proposed by Yang et al. (Int J Solids Struct 39(10):2731–2743, 2002), combined with the first strain gradient plasticity of Fleck and Hutchinson (J Mech Phys Solids 41(12):1825–1857, 1993). Using the concepts of convex analysis, a dual principle and the optimum conditions are derived. A numerical method based on the Newton’s method is derived from the plastic shakedown variational principle to find the load amplification factor. Two illustrative numerical examples are presented: a cantilever beam subjected to a constant axial load and to a variable external bending moment and a cylindrical bar subjected to an axial load and to a variable external torque. In both cases, the external torque and external bending moment vary arbitrarily between prescribed symmetric bounds. The results are compared with those obtained considering classical plastic shakedown theory in order to see the influence of the length scale on the load amplification factor.

A local duality principle in derived categories of commutative Noetherian rings

Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors γW with supports in arbitrary subsets W of SpecR. If W is a specialization-closed subset, then γW coincides with the right derived functor RΓW of the section functor ΓW with support in W. We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for γW with W being an arbitrary subset.