Quadratic Inequality Research Articles

Transportation cost inequality for backward stochastic differential equations

We prove that probability laws of a backward stochastic differential equation, satisfy a quadratic transportation cost inequality under the uniform metric. That is, a comparison of the Wasserstein distance from the law of the solution of the equation to any other absolutely continuous measure with finite relative entropy. From this we derive concentration properties of Lipschitz functions of the solution.

From Dissipativity Theory to Compositional Abstractions of Interconnected Stochastic Hybrid Systems

In this paper, we derive conditions under which compositional abstractions of networks of stochastic hybrid systems can be constructed using the interconnection topology and joint dissipativity-type properties of subsystems and their abstractions. In the proposed framework, the abstraction, itself a stochastic hybrid system (possibly with a lower dimension), can be used as a substitute of the original system in the controller design process. Moreover, we derive conditions for the construction of abstractions for a class of stochastic hybrid systems involving nonlinearities satisfying an incremental quadratic inequality. In this paper, unlike existing results, the stochastic noises and jumps in the concrete subsystem and its abstraction need not be the same. We provide examples with numerical simulations to illustrate the effectiveness of the proposed dissipativity-type compositional reasoning for interconnected stochastic hybrid systems.

PSEUDO SISWA DALAM MENYELESAIKAN SOAL BERTIPE HIGHER ORDER THINKING SKILLS BERDASARKAN AKTIVITAS PROBLEM SOLVING

Learning mathematics involves students not only to be able to ensure calculations but also be able to ensure reasoning and analyzing. Analytical skills are included in high order thinking skills (HOTS). Higher order thinking skills (HOTS) can be generated through problem solving activities. In solving a problems, it is possible that the way the student thinks is not suitable to the answer they get or it can be said that they are doing pseudo thinking. Pseudo thinking can be caused by the absence of reflection by students. This study aims to describe the student's occurring false-pseudo thinking in solving HOTS type problems on the quadratic inequality. Research subjects were taken based on purposive sampling, taking into account their communication abilities. The subjects in this study were high school students who had studied the material inequality squared. The data collection was done by using think-out-loud (TOL) method, the students were asked to express out loud what they were thinking when solving the problem which was given. From the results it can be explained that the process of the students’ false-pseudo thinking HOTS type problems on the quadratic inequality happen as follows: 1) It begins with the students’ mistake of making assumptions when they are understanding the problem 2) The incompleteness of the students’ thinking substructure in the process of understanding the problem, and 3) The incompleteness of the students’ thinking substructure in planning the solution.

Observer-Based Consensus for Positive Multiagent Systems With Directed Topology and Nonlinear Control Input

This paper studies the consensus problem for directed positive multiagent systems with nonlinear control input. The directed topology is supposed to be strongly connected. For the case of sector input nonlinearities, the non-negative global consensus conditions are presented by using a novel analysis method which directly considers the nonlinear input, and the feedback matrices are obtained with convex optimization method by solving the quadratic matrix inequalities subject to non-negative constraints. For the case of saturation-type sector input nonlinearities, the global consensus and local consensus results are analyzed, respectively. With the properties of Metzler matrix, the non-negative consensus results are further extended to the strongly connected and balanced positive multiagent systems subject to (saturation-type) sector input nonlinearities and parameter uncertainties. The results are simplified, and the feedback matrices can be given by solving iterative linear matrix inequality with non-negative constraints. Two simulation examples and a practical electrical circuit model are finally carried out to verify the proposed control schemes.

Positivity constraints on aQGC: carving out the physical parameter space

Searching for deviations in quartic gauge boson couplings (QGCs) is one of the main goals of the electroweak program at the LHC. We consider positivity bounds adapted to the Standard Model, and show that a set of positivity constraints on 18 anomalous QGC couplings can be derived, by requiring that the vector boson scattering amplitudes of specific channels and polarisations satisfy the fundamental principles of quantum field theory. We explicitly solve the positivity inequalities to remove their dependence on the polarisations of the external particles, and obtain 19 linear inequalities, 3 quadratic inequalities, and 1 quartic inequality that only involve the QGC parameters and the weak angle. These inequalities constrain the possible directions in which deviations from the standard QGC can occur, and can be used to guide future experimental searches. We study the morphology of the positivity bounds in the parameter space, and find that the allowed parameter space is carved out by the intersection of pyramids, prisms, and (approximately) cones. Altogether, they reduce the volume of the allowed parameter space to only 2.1% of the total. We also show the bounds for some benchmark cases, where one, two, or three operators, respectively, are turned on at a time, so as to facilitate a quick comparison with the experimental results.

Quadratic Matrix Inequality Approach to Robust Adaptive Beamforming for General-Rank Signal Model

The worst-case robust adaptive beamforming problem for general-rank signal model is considered. This is a nonconvex problem, and an approximate version of it (obtained by introducing a matrix decomposition on the presumed covariance matrix of the desired signal) has been well studied in the literature. Different from the existing literature, herein however the original beamforming problem is tackled. Resorting to the strong duality of linear conic programming, the robust adaptive beamforming problem for general-rank signal model is reformulated into an equivalent quadratic matrix inequality (QMI) problem. By employing a linear matrix inequality (LMI) relaxation technique, the QMI problem is turned into a convex semidefinite programming problem. Using the fact that there is often a positive gap between the QMI problem and its LMI relaxation, an approximate algorithm is proposed to solve the robust adaptive beamforming in the QMI form. Besides, several sufficient optimality conditions for the nonconvex QMI problem are built. To validate our results, simulation examples are presented, which also demonstrate the improved performance of the new robust beamformer in terms of the output signal-to-interference-plus-noise ratio.

An application of forward difference method in robust stability of discrete uncertainty system with delays

This paper deals with the problems of robust stability for a class of uncertainty parameter systems with delays. By using a new Lyapunov–Krasovskii functional, quadratic inequality, and Schur complement technique, two conditions are developed to guarantee the robust stability of a class of discrete systems with uncertainty parameters in terms of the linear matrix inequality (LMI). By applying the forward difference method and via a quadratic cost function, the criteria of LMI are obtained by the input control u(k)=0 and u(k)=Kx(k); meanwhile, the bounds of cost function are established. The feedback control gain is designed to ensure the robust stability of the closed-loop system. A numerical example and simulation figures are provided to illustrate the effectiveness and potential of the proposed techniques and results.

A quadratic matrix inequality based PID controller design for LPV systems

This paper develops a robust gain-scheduled proportional–integral–derivative (PID) controller design method for a linear-parameter-varying (LPV) system with parametric uncertainty. It is recognized in the literature that the robust fixed-order controller design can be formulated as a feasibility problem of a bilinear matrix inequality (BMI) constraint. Unfortunately, the search for a feasible solution of a BMI constraint is an NP hard problem in general. Previous researchers have applied a linearization method, such as a variable change technique or a congruence transformation, to transform the BMI into a LMI. The applicability of the linearization method depends on the specific structure of the problem at hand and cannot be generalized. This paper instead formulates the gain-scheduled PID controller design as a feasibility problem of a quadratic matrix inequality (QMI) constraint, which covers the BMI constraint as a special case. An augmented sequential LMI optimization method is proposed to search for a feasible solution of the QMI constraint iteratively. As an illustrative application, a vehicle lateral control problem is presented to demonstrate the applicability of the proposed algorithm to a real-world output feedback control design system.

Stability of nonlinear dynamical system motion under constantly acting perturbations

We consider the motion of a mechanical system with several degrees of freedom near zero of the phase space of states under conditions of constantly acting small perturbations. The generalized forces are represented in the dynamic equations by homogeneous forms of the first and the third degree with respect to the phase coordinates and by small time functions characterizing the constantly acting perturbations. It is assumed that there are no multiple eigenvalues of the matrix of the system linear part. For a definitely positive quadratic Lyapunov function, we define a differential inequality with Riccati differential comparison equation, together with an exponential differential inequality that is integrable in quadratures. When solving Riccati quadratic differential inequality, we assume one particular solution of Riccati equation to be known. As a result of integrating in quadratures of the exponential differential inequalities, the estimate of transient processes in the finite domain of the phase coordinates is obtained.

Stabilization for a class of rectangular descriptor systems via time delayed dynamic compensator

This paper focuses on the stabilization problem for a class of rectangular descriptor systems through dynamic compensation. Such class of systems may not be stabilized by delay-free dynamic compensators, while delayed dynamic compensator could achieve such purpose. We provide a design scheme of time-delayed dynamic compensator which makes the closed-loop system admissible. The design involves solving a quadratic matrix inequality, and consequently, we build a linear matrix inequality (LMI) based algorithm to compute compensator gains. We verify that, under certain circumstances for which delay-free dynamic compensators fail to stabilize, the proposed method works well. An illustrative example demonstrates the usefulness of the present scheme.

Lipschitz 1-connectedness for Some Solvable Lie Groups

AbstractA space X is said to be Lipschitz 1-connected if every Lipschitz loop 𝛾 : S 1 → X bounds a O (Lip(𝛾))-Lipschitz disk f : D 2 → X. A Lipschitz 1-connected space admits a quadratic isoperimetric inequality, but it is unknown whether the converse is true. Cornulier and Tessera showed that certain solvable Lie groups have quadratic isoperimetric inequalities, and we extend their result to show that these groups are Lipschitz 1-connected.

MIMO PID tuning for nonminimum phase systems: setting attainable limits for a stable behaviour

Abstract Nonminimum phase systems are presented in many industrial processes bringing some difficulties in their control scheme. In this paper we propose a methodology to tune multi-input-multi-output proportional-integral-derivative controllers that deals this particularity. The controller design employed here is based in an attainable trajectory determination, for the nonminimum phase system, that considers performance and robustness metrics. Based on that, two different tuning procedures are proposed: an optimization problem, in the frequency domain, that involves nonconvex quadratic matrix inequalities with a linear matrix inequality restriction and a time domain optimization of the simulated system. The methods are an alternative for tuning this kind of system leading the process to the best operational scenario presenting stability, robustness and limited control actions.

Talagrand concentration inequalities for stochastic heat-type equations under uniform distance

In this paper, we established a quadratic transportation cost inequality under the uniform/maximum norm for solutions of stochastic heat equations driven by multiplicative space-time white noise. The proof is based on a new inequality we obtained for the moments of the stochastic convolution with respect to space-time white noise, which is of independent interest. The solutions of such stochastic partial differential equations are typically not semimartingales on the state space.

Minimax and WLS Designs of Digital FIR Filters Using SOCP for Aliasing Errors Reduction in BI-DAC

This paper presented the optimal minimax and weighted least squares (WLS) methods for designing digital finite impulse response (FIR) filters to reduce the aliasing errors generated by the non-ideality of analog filters and mixers in bandwidth interleaving digital-to-analog converter (BI-DAC). To satisfy the given expected spurious free dynamic range (SFDR), we formulated these optimal designs of digital FIR filters in BI-DAC as a convex optimization problem-second-order cone programming (SOCP) that allowed the linear equality and convex quadratic inequality constraints including the magnitude flatness and the peak aliasing errors constraints to be merged. Furthermore, we derived the computational complexity of our presented optimal design. Several design examples were given to evaluate the performance of our presented unconstrained and constrained minimax and WLS designs using SOCP including their effectiveness and computational complexity. The simulation results showed that, in our presented unconstrained minimax and WLS designs using SOCP, the maximum distortion errors were all around 0.02 dB. The maximum aliasing errors were-73.9 and -80.5 dB, which satisfied the expected SFDR of a 12-bit BI-DAC system. In addition, we analyzed the influence of different values of the nonnegative weighting function on our presented unconstrained minimax and WLS designs using SOCP, and we found that there was a tradeoff among the nonnegative weighting function's value, and the distortion and aliasing errors. Moreover, when the constraints were imposed in our presented constrained minimax and WLS designs using SOCP in the selected frequency bands, the distortion errors were equal to zero and the aliasing errors were reduced below -110 dB, but the expense was that the larger distortion and aliasing errors achieved out of these selected frequency bands. Finally, we gave the computational complexity comparisons among our presented unconstrained and constrained minimax and WLS design using SOCP, we also compared the influence of the digital FIR filters' length on our presented designs' worst-case passband ripple and stopband roll-off, and we found that there was a tradeoff among the digital FIR filters' length, the passband ripple, the stopband roll-off, the computational complexity, and the actual hardware cost.

Secure Transmission for Heterogeneous Cellular Networks With Wireless Information and Power Transfer

In this paper, we investigate an artificial-noise-aided secure beamforming design for simultaneous wireless information and power transfer in a two-tier downlink heterogeneous cellular network, in which each energy receiver in a femtocell is seen as a potential eavesdropper to wiretap the confidential message intended for the information receiver. Our design objective is to maximize the secrecy rate at the information receiver, while satisfying the signal-to-interference-plus-noise ratio requirement of each macrouser and the energy harvesting and transmit power constraints. Both the scenarios of perfect and imperfect channel state information (CSI) are considered. With perfect CSI, the formulated optimization problem constitutes a difference of a convex function programming problem, which is hard to directly solve. To tackle this challenge, we transform it into a series of semidefinite programs by using successive convex approximation, and an iterative algorithm is proposed to arrive at a provably convergent solution. With imperfect CSI, we address robust secure beamforming relying on the worst-case design philosophy. To circumvent this predicament, we resort to the S-procedure to reformulate the robust quadratic matrix inequality (QMI) constraints and then obtain the linear matrix inequality representations for these QMIs. Numerical results are finally presented to demonstrate the performance of our proposed schemes in improving the secrecy rate of heterogeneous cellular networks.

2016년 국가수준 학업성취도 평가 결과에서 나타난 고등학생의 수학 학업 특성 분석1)

This study seeks to examine high school students’ academic characteristics from the 2016 National Assessment of Educational Achievement, which was published based on the 2009-revised mathematic curriculum and derive implications for teaching and learning. To do so, first, the test tool for mathematics in the 2016 National Assessment of Educational Achievement was introduced, and overall assessment results were examined based on achievement score, percentage of achievement levels, correct answer rate in the content areas, and representative test items per each achievement level. Second, specific assessment results for seven content areas were suggested. The results showed that those with ‘proficient level’(47.63%) accounted for the highest percentage of the students, while students with ‘basic level’ and ‘below-basic level’ accounted for 20%. Second, the results from examining representative test items per each achievement level showed that representative test items for advanced-level were evenly distributed, while representative test items for proficient level skewed towards topics ‘equations and inequalities’, ‘sets and propositions’, and ‘sequence’; for basic level, only ‘polynomials’ and ‘sets and propositions’ were observed. Third, an analysis of correct answer rate in the content areas revealed that the correctness rate of the ‘polynomial’ area was the highest and correctness rate in ‘functions’ area was the lowest. Fourth, in the ‘equations and inequalities’ area, ‘the relationship between quadratic functions and quadratic equations’ and ‘the relationship between quadratic functions and quadratic inequalities’ was more difficult for students than other achievement standards in equations and inequalities area. Fifth, within ‘Equations for geometric figures’ area, students scored low in ‘the positional relationship between a circle and a straight line’. Last, an analysis of essay type item asking for the reverse function of a composite function revealed that typical wrong answers occurred when symbols were used incorrectly to express the reverse function of a composite function, when the characteristics of reverse functions were not correctly applied, and when reverse functions were calculated inaccurately.

PENERAPAN STRATEGI QUESTIONS STUDENTS HAVE (QSH) DALAM PEMBELAJARAN LANGSUNG UNTUK MENINGKATKAN HASIL BELAJAR MATEMATIKA SISWA KELAS X1 SMA NEGERI 2 BANGKO KABUPATEN ROKAN HILIR

Students' success in learning math is largely determined by the achievement of the learning process and requires the participation of students in writing. This study aims to improve students' mathematics learning outcomes of grade X SMA public 2 Bangko Semester Odd Year 2016/2017 on the subject matter of Equation and Quadratic Inequality by implementing QSH strategy in direct learning. This research includes collaborative classroom action research. Implementation of the research conducted in SMA public 2 Bangko Rokan Hilir Regency in the odd semester of the academic year 2016/2017. The research subjects of class X2 students as many as 26 people. This study consists of two cycles based on Arikunto (2010). Technique of data collection using observation sheet and test result of learning. Data analysis technique with descriptive statistical analysis. Implementation of QSH strategy in direct learning can improve mathematics learning outcomes of students of class X SMA public 2 Bangko academic year 2016/2017, especially on the subject matter of equations and quadratic inequalities.

Quadratic transformation inequalities for Gaussian hypergeometric function

In the article, we present several quadratic transformation inequalities for Gaussian hypergeometric function and find the analogs of duplication inequalities for the generalized Grötzsch ring function.

Cutting stock problems with nondeterministic item lengths: a new approach to server consolidation

Based on an application in the field of server consolidation, we consider the one-dimensional cutting stock problem with nondeterministic item lengths. After a short introduction to the general topic we investigate the case of normally distributed item lengths in more detail. Within this framework, we present two lower bounds as well as two heuristics to obtain upper bounds, where the latter are either based on a related (ordinary) cutting stock problem or an adaptation of the first fit decreasing heuristic to the given stochastical context. For these approximation techniques, dominance relations are discussed, and theoretical performance results are stated. As a main contribution, we develop a characterization of feasible patterns by means of one linear and one quadratic inequality. Based on this, we derive two exact modeling approaches for the nondeterministic cutting stock problem, and provide results of numerical simulations.

Global optimal power flow over large-scale power transmission networks

Optimal power flow (OPF) over power transmission networks poses challenging large-scale nonlinear optimization problems, which involve a large number of quadratic equality and indefinite quadratic inequality constraints. These computationally intractable constraints are often expressed by linear constraints plus matrix additional rank-one constraints on the outer products of the voltage vectors. The existing convex relaxation technique, which drops the difficult rank-one constraints for tractable computation, cannot yield even a feasible point. We address these computationally difficult problems by an iterative procedure, which generates a sequence of improved points that converges to a rank-one solution. Each iteration calls a semi-definite program. Intensive simulations for the OPF problems over networks with a few thousands of buses are provided to demonstrate the efficiency of our approach. The suboptimal values of the OPF problems found by our computational procedure turn out to be the global optimal value with computational tolerance less than 0.01%.