Exact Bound Research Articles

The Exact Bound of the Number of Limit Cycles Bifurcating from the Global Nilpotent Center with a Nonlinear Switching Curve

In this paper, we study the bifurcations of a class of planar Hamiltonian systems having a global nilpotent center under the perturbations of polynomials of degree [Formula: see text] with a nonlinear switching curve. We obtain the exact bound of the number of limit cycles bifurcating from the period annulus if the first order Melnikov function is not identically zero. We also give some examples to illustrate our results.

H∞ Control Observer-Based for Discrete-Time Singularly Perturbed Systems With Nonlinear Disturbances

This paper focuses on the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H\infty $ </tex-math></inline-formula> observer-based control problem for discrete-time singularly perturbed systems (DTSPSs) with nonlinear disturbances. The main contributions of this paper include three aspects: First, a proper observer is constructed. A sufficient condition in terms of linear matrix inequality (LMI) and Lyapunov function is proposed such that the resulting observer error system is asymptotically stable with a prescribed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> norm bound for sufficiently small values of the perturbation parameter. Second, based on the input-to-state stability (ISS) property, an observer-based feedback controller is designed such that the resulting closed-loop system is ISS with respect to the observer error. Meanwhile, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H\infty $ </tex-math></inline-formula> performance index is also satisfied. Then, a workable way for solving the exact upper bound is also given. Finally, two numerical examples are given to demonstrate the validity of the developed method.

Improved frequency sweeping technique and stability analysis of the second-order consensus protocol with distributed delays

The consensus problem of a general class of second-order multi-agent system with distributed delays is investigated from a new perspective. We propose a consensus protocol with distributed delays to enhance the delay robustness of the system. First, we establish a spectral equivalence of such a system and the one with multiple discrete delays from the stability viewpoint. We then employ a factorization procedure to reduce the computational complexity. Next, we uniquely use the Dixon resultant theory to determine the exact upper bound of the imaginary spectra, leading to the kernel and offspring hypersurfaces (KOH) in the delay domain. Furthermore, the associated standing roots of the system are analyzed to determine the precise stationary root boundaries, which constitute the potential stability changing curves together with the KOH. Finally, we resort to the Cluster Treatment of Characteristic Roots (CTCR) paradigm to examine the exact and complete stability of the system.

2-Distance Strong b-coloring of Perfect 𝜟-ary Tree

A 2-distance 𝑏-coloring is a 2-distance coloring in which every color class contains a vertex which has a neighbor in every other color class. A 2-distance strong 𝑏-coloring (2𝑠𝑏- coloring) is a 2-distance coloring in which every color class contains a vertex 𝑢 such that there is a vertex 𝑣 in every other color class satisfying the condition that the distance between 𝑢 and 𝑣 is at most 2. The 2-distance 𝑏-chromatic number 𝜒2𝑏(𝐺) (2𝑏-number) is the largest integer 𝑘 such that 𝐺 admits a 2-distance 𝑏-coloring with 𝑘 colors and the 2-distance strong bchromatic number 𝜒2𝑠𝑏(𝐺) (2𝑠𝑏-number) is the maximum k such that 𝐺 admits a 2𝑠𝑏-coloring with 𝑘 colors. A tree with a special vertex called the root is called a rooted tree. A perfect 𝛥- ary tree, is a rooted tree in which all internal vertices are of degree 𝛥 and all pendant vertices are at the same level. In this paper, the exact bound of the 2𝑠𝑏-number of perfect 𝛥-ary tree are obtained.

On large time behavior of solutions of higher order evolution inequalities with fast diffusion

We obtain stabilization conditions and large time estimates for weak solutions of the inequality∑|α|=m∂αaα(x,t,u)−ut≥f(x,t)g(u)in Ω×(0,∞), where Ω is a non-empty open subset of Rn, m,n≥1, and aα are Caratheodory functions such that|aα(x,t,ζ)|≤Aζp,|α|=m, with some constants A>0 and 0<p<1 for almost all (x,t)∈Ω×(0,∞) and for all ζ∈[0,∞). For solutions of homogeneous differential inequalities, we give an exact universal upper bound.

A new algorithm for solving uncapacitated transportation problem with interval-defined demands and suppliers capacities

The uncapacitated transportation problem (UTP) deals with minimizing the transportation costs related to the delivery of a homogeneous product from multi-suppliers to multi-consumers. The application of the UTP can be extended to other areas of operations research, including inventory control, personnel assignment, signature matching, product distribution with uncertainty, multi-period production and inventory planning, employment scheduling, and cash management. Such a UTP with interval-defined demands and suppliers capacities (UTPIDS) is investigated in this paper. In UTPIDS, the demands and suppliers capacities may not be known exactly but vary within an interval due to variation in the economic conditions of the global economy. Following the variation, the minimal total cost of the transportation can also be varied within an interval and thus, the cost bounds can be obtained. Here, although the lower bound solution can be attained methodologically, the correct estimation of the worst case realization (the exact upper bound) on the minimal total transportation cost of the UTPIDS is an NP-hard problem. So, the decision-makers seek for minimizing the transportation costs and they are interested in the estimation of the worst case realization on these minimal costs for better decision making especially, for proper investment and return. In literature very few approaches are available to find this estimation of the worst case realization with some shortcomings. First, we demonstrate that the available heuristic methods fail to obtain the correct estimation of the worst case realization always. In this situation, development of a better heuristic method to find the better near optimal estimation of the worst case realization on the minimal total costs of the UTPIDS is desirable. Then this paper provides a new polynomial time algorithm that runs in O (N2) time (N, higher of the numbers of source and destination nodes) for better estimation. A comparative assessment on solutions of available benchmark instances, some randomly generated numerical example problems and a real-world application shows promising performance of the current technique. So, our new finding would definitely be benefited to practitioners, academics and decision makers who deal with such type of decision making instances.

Numerical Investigation of Fractional-Order Differential Equations via φ -Haar-Wavelet Method

In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ -Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ -Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ -fractional integral of the Haar-wavelet by utilizing the φ -fractional integral operator. We also extended our method to nonlinear φ -FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ -FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.

An explicit formulation of the macroscopic strength criterion for porous media with pressure and Lode angle dependent matrix under axisymmetric loading

This paper aims to propose an explicit formulation of the macroscopic strength criterion for porous media with spherical voids. The matrix is assumed rigid and perfectly plastic with yield surface described by the three-parameter strength criterion, which is Lode angle and pressure dependent and capable of accounting for distinct values of the uniaxial tensile strength, uniaxial compressive strength (UCS) and equal biaxial compressive strength (eBCS). An exact upper bound of the macroscopic strength is derived for porous media subjected to purely hydrostatic loading. Besides, an estimate of the macroscopic strength profile of porous media under axisymmetric loading is obtained in parametric form. Moreover, a heuristic strength criterion in explicit form is further developed by examining limit cases of the parametric strength criterion. The developed strength criteria are assessed by finite-element based numerical solutions. Compared with the parametric strength criterion which involves cumbersome functions, the heuristic one is convenient for practical applications. For specific values of the matrix's strength surface, the proposed heuristic strength criterion can recover the well-known Gurson criterion. The present work also addresses the effect of the ratio of matrix's eBCS to UCS on the macroscopic strength of porous media. For matrix with distinct values of eBCS and UCS, neglecting the difference between eBCS and UCS would result in an underestimation of the macroscopic strength, especially when the pressure is large.

Quantization synchronization of chaotic neural networks with time delay under event-triggered strategy.

This paper shows solicitude for the quantization synchronization of delayed chaotic master and slave neural networks under an dynamic event-triggered strategy. In virtue of a generalized Halanay-type inequality, a theoretical criterion for quasi-synchronization of master and slave neural networks is derived. Meanwhile, we can obtain an exact upper bound of synchronization error by using this criterion. Compared with output feedback controller with event triggering and quantization, the case where the controller only affected by quantization is also considered. Then, we exclude the Zeno behavior of the event-triggered controller. A sufficient criterion for the existence of the quantized output feedback controllers is also provided. A numerical example is cited to illustrate the efficiency of our theoretical criteria. In addition, some experiments of secure image communication are conducted under quasi-synchronization of master and slave neural networks.

Performance Analysis for Exact and Upper Bound Capacity in DF Energy Harvesting Full-Duplex with Hybrid TPSR Protocol

In this paper, we investigate the full-duplex (FD) decode-and-forward (DF) cooperative relaying system, whereas the relay node can harvest energy from radiofrequency (RF) signals of the source and then utilize the harvested energy to transfer the information to the destination. Specifically, a hybrid time-power switching-based relaying method is adopted, which leverages the benefits of time-switching relaying (TSR) and power-splitting relaying (PSR) protocols. While energy harvesting (EH) helps to reduce the limited energy at the relay, full-duplex is one of the most important techniques to enhance the spectrum efficiency by its capacity of transmitting and receiving signals simultaneously. Based on the proposed system model, the performance of the proposed relaying system in terms of the ergodic capacity (EC) is analyzed. Specifically, we derive the exact closed form for upper bound EC by applying some special function mathematics. Then, the Monte Carlo simulations are performed to validate the mathematical analysis and numerical results.

On equilibrium two-phase microstructures at plane strain

Stress-induced phase transformations in linear elastic solids are considered for the case of plane strain on the basis of a previously developed procedure that includes finding the optimal composite microstructures which provide the exact lower bound of the energy and the equilibrium new phase volume fraction that satisfies the thermodynamic equilibrium condition. Stress–strain diagrams are constructed for various deformation paths on which phase transformations are controlled by average strains or by different components of average strain and average stresses. Strain hardening and strain softening effects are demonstrated on the path of the transformation. In the case of mixed strain-stress control, regimes for which the minimizing volume fraction does not correspond to any optimal microstructure, or optimal microstructures which could not satisfy the thermodynamic equilibrium condition with respect to the new phase volume fraction, are presented. It is shown that this leads to the incompleteness of a gradual phase transformation, jump-like transformation behavior, and jump in stress. An indifferent equilibrium regime is also discussed.

Bit Security Estimation Using Various Information-Theoretic Measures

In this paper, we propose various quantitative information-theoretic bit security reduction measures that correlate the statistical difference between two probability distributions with the security level gap for two cryptographic schemes. We derive tighter relations between information-theoretic measures for quantifying the precision and guarantee the security level of the cryptographic scheme implemented over a precision-restricted environment. Further, the generalized versions of previous security reductions are devised by relaxing the constraints on the upper bounds of the information-theoretic measures. This makes it possible to estimate bit security more reliable and improves the security level. We also estimate the effects on the security level when the $\kappa $ -bit secure original scheme is implemented on a $p$ -bit precision system. In previous studies, $p$ was fixed as $\frac {\kappa }2$ ; however, the proposed schemes are generalized such that the security level $\kappa $ and precision $p$ can vary independently. This results in a significant difference. Moreover, previous results cannot provide the exact lower bound of the security level for $p\ne {\frac {\kappa }{2}}$ . However, the proposed results can provide the exact lower bound of the estimation value of the security level as long as the precision $p$ satisfies certain conditions. We provide diverse types of security reduction formulas for the six types of information-theoretic measures. The proposed schemes can provide information-theoretic guidelines regarding the difference between the security levels of two identical cryptographic schemes when extracting randomness from two different probability distributions. In particular, the proposed schemes can be used to quantitatively estimate the effect of the statistical difference between the ideal and real distributions on the security level.

Exact Semidefinite Programming Bounds for Packing Problems

In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. This algorithm...

INEQUALITIES FOR ALGEBRAIC POLYNOMIALS ON AN ELLIPSE

The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci \(\pm 1\) of the derivative of an algebraic polynomial with real coefficients normalized on the segment \([- 1,1]\).

Exact Results and Bounds for the Joint Tail and Moments of the Recurrence Times in a Renewal Process

The best known result about the joint distribution of the backward and forward recurrence times in a renewal process concerns the asymptotic behavior for the tail of that bivariate distribution. In the present paper we study the joint behavior of the recurrence times at a fixed time point t, and we obtain both exact results and bounds for their joint tail behavior. We also obtain results about the joint moments of these two random variables and we show in particular that the expectation of the product between the two recurrence times increases with time when the interarrival distribution has a decreasing failure rate. The results are illustrated by some numerical examples.

Wafer sojourn time fluctuation analysis for time-constrained dual-arm multi-cluster tools with activity time variation

ABSTRACT In semiconductor manufacturing systems, a time-constrained multi-cluster tool should be scheduled such that a wafer stays in a process chamber in a given time range to satisfy a wafer residency time constraint. In practice, activity time is subject to variation. It could lead to some fluctuation of wafer residency time in a process chamber. Hence, it is crucial to analyze how wafer residency time varies with activity time variation. This issue is especially challenging for multi-cluster tools. This work focuses on determining the exact upper bound of wafer sojourn time delay resulted from activity time variation for dual-arm multi-cluster tools. After discussing their dynamic behaviours, it presents a two-level real-time operational architecture and a real-time control policy. Based on them, this work derives for the first time an efficient algorithm to calculate the exact upper bound of wafer sojourn time delay in a process chamber. As a result, engineers can test whether a given schedule is feasible. Several examples of industrial significance are used to demonstrate the application of the proposed approach.

Lightweight Iterative MDS Matrices: How Small Can We Go?

As perfect building blocks for the diffusion layers of many symmetric-key primitives, the construction of MDS matrices with lightweight circuits has received much attention from the symmetric-key community. One promising way of realizing low-cost MDS matrices is based on the iterative construction: a low-cost matrix becomes MDS after rising it to a certain power. To be more specific, if At is MDS, then one can implement A instead of At to achieve the MDS property at the expense of an increased latency with t clock cycles. In this work, we identify the exact lower bound of the number of nonzero blocks for a 4 × 4 block matrix to be potentially iterative-MDS. Subsequently, we show that the theoretically lightest 4 × 4 iterative MDS block matrix (whose entries or blocks are 4 × 4 binary matrices) with minimal nonzero blocks costs at least 3 XOR gates, and a concrete example achieving the 3-XOR bound is provided. Moreover, we prove that there is no hope for previous constructions (GFS, LFS, DSI, and spares DSI) to beat this bound. Since the circuit latency is another important factor, we also consider the lower bound of the number of iterations for certain iterative MDS matrices. Guided by these bounds and based on the ideas employed to identify them, we explore the design space of lightweight iterative MDS matrices with other dimensions and report on improved results. Whenever we are unable to find better results, we try to determine the bound of the optimal solution. As a result, the optimality of some previous results is proved.

EXACT BOUND ON THE NUMBER OF LIMIT CYCLES ARISING FROM A PERIODIC ANNULUS BOUNDED BY A SYMMETRIC HETEROCLINIC LOOP

In this paper, the bound on the number of limit cycles by Poincaré bifurcation in a small perturbation of some seventh-degree Hamiltonian system is concerned. The lower and upper bounds on the number of limit cycles have been obtained in two previous works, however, the sharp bound is still unknown. We will employ some new techniques to determine which is the exact bound between $ 3 $ and $ 4 $. The asymptotic expansions are used to determine the four vertexes of a tetrahedron, and the sharp bound can be reached when the parameters belong to this tetrahedron.

Exact Converging Bounds for Stochastic Dual Dynamic Programming via Fenchel Duality

The Stochastic Dual Dynamic Programming (SDDP) algorithm has become one of the main tools to address convex multistage stochastic optimal control problem. Recently a large amount of work has been devoted to improve the convergence speed of the algorithm through cut-selection and regularization, or to extend the field of applications to non-linear, integer or risk-averse problems. However one of the main downside of the algorithm remains the difficulty to give an upper bound of the optimal value, usually estimated through Monte Carlo methods and therefore difficult to use in the algorithm stopping criterion. In this paper we present a dual SDDP algorithm that yields a converging exact upper bound for the optimal value of the optimization problem. Incidently we show how to compute an alternative control policy based on an inner approximation of Bellman value functions instead of the outer approximation given by the standard SDDP algorithm. We illustrate the approach on an energy production problem involving zones of production and transportation links between the zones. The numerical experiments we carry out on this example show the effectiveness of the method.

Stability Analysis for the Modification Method Under the a Priori Strategy of the PTSP

We propose in this paper a new formulation for the stability of the Traveling Salesman Problem (TSP) compared with its probabilistic version, the Probabilistic Traveling Salesman Problem (PTSP). It is a real extension of the TSP, where the number of customers to be served each time is a random variable. That is only a subset of customers will need its services, moreover this subset varies from day to day. From the literature, several methods of resolution of the TSP have been proposed. In order to use these methods as they are for the PTSP came the idea of the study of stability. It is interested in finding cases where the solution for the TSP is also for the PTSP. First we survey and comment a number of easy TSPs. We also present via the notion of “Master tour” the stable problems in the TSP-PTSP context. An exact Branch and Bound is used for recognizing the TSPs that are not stable. Finally, we propose a new modification method, -different from the usual method of PTSP-, called the method of Taxi Driver, it takes the structure of the tour into consideration.