Positive Real Roots Research Articles

Bounds on set exit times of affine systems, using Linear Matrix Inequalities

Abstract Efficient computation of trajectories of switched affine systems becomes possible, if for any such hybrid system, we can manage to efficiently compute the sequence of switching times. Once the switching times have been computed, we can easily compute the trajectories between two successive switches as the solution of an affine ODE. Each switching time can be seen as a positive real root of an analytic function, thereby allowing for efficient computation by using root finding algorithms. These algorithms require a finite interval, within which to search for the switching time. In this paper, we study the problem of computing upper bounds on such switching times, and we restrict our attention to stable time-invariant affine systems. We provide semi-definite programming models to compute upper bounds on the time taken by the trajectories of an affine ODE to exit a set described as the intersection of a few generalized ellipsoids. Through numerical experiments, we show that the resulting bounds are tighter than bounds reported before, while requiring only a modest increase in computation time.

Sign conditions for the existence of at least one positive solution of a sparse polynomial system

We give sign conditions on the support and coefficients of a sparse system of d generalized polynomials in d variables that guarantee the existence of at least one positive real root, based on degree theory and Gale duality. In the case of integer exponents, we relate our sufficient conditions to algebraic conditions that emerged in the study of toric ideals.

Direct transformation from Cartesian into geodetic coordinates on a triaxial ellipsoid

This paper1 presents two new direct symbolic-numerical algorithms for the transformation of Cartesian coordinates into geodetic coordinates considering the general case of a triaxial reference ellipsoid. The problem in both algorithms is reduced to finding a real positive root of a sixth degree polynomial. The first approach consists of algebraic manipulations of the equations describing the geometry of the problem and the second one uses Gröbner bases. In order to perform numerical tests and accurately compare efficiency and reliability, our algorithms together with the iterative methods presented by M. Ligas (2012) and J. Feltens (2009) have been implemented in C++. The numerical tests have been accomplished by considering 10 celestial bodies, referenced in the available literature. The obtained results show that our algorithms improve the aforementioned, up-to-date reference, iterative methods, in terms of both efficiency and accuracy.

Motion around equilibrium points of an oblate body in the PR3BP with disc

This paper examines motion around equilibrium points (EPs) of an oblate body having a third body in the PR3BP with the disc. The problem is photogravitational, such that the third body gravitates under the influence of two primaries, which emit radiation pressure and the Poynting–Robertson (P–R) drag. Further, the primaries are enclosed by a disc. The mathematical analysis (equations of motion) and the locations of EPs have been studied. There exist six collinear EPs in the line joining the primaries. The first is an additional EP, which depends on parameters, $$\mu$$ and $$W_{i} \,(i = 1,2)$$ of the primaries, while the three are analogous to the collinear EPs of the classical R3BP. The last two are the Jiang–Yeh EPs, which exist due to the mass parameter and the disc. Further, two triangular EPs are found and are characterized by the disc, oblateness of the third body, radiation pressure, P–R drag and mass parameters of the primaries. The stability of the EPs is examined, and it is that they are unbounded due to either a positive root or a positive real part of the complex root. In particular, the numerical exploration is being performed using the binary Achird to compute numerically the locations/or positions of the EPs and the roots of their corresponding characteristic equations. In the case of the collinear point, the roots contain a positive real root and a complex root with positive real part. In view of the triangular points, the roots contain positive real parts of the complex roots. Hence, we conclude that for the R3BP when the primaries emit radiation pressure and P–R drag with disc and the third body is in shape of a sphere, eight EPs exist and are unstable due to radiational forces which reveal positive real part of the complex root.

UAV Optimal Guidance in Wind Fields Using ZEM/ZEV With Generalized Performance Index

This article presents an optimal guidance approach for unmanned aerial vehicle navigation between two given points in 3D considering the wind influence. The proposed cost function to be minimized involves the weighting of the travel time and the control energy. An analytical expression is derived for the optimal cost yielding a fourth-order polynomial, whose positive real roots correspond to the optimal travel times. The optimization problem is shown to be equivalent to the zero-effort-miss/zero-effort-velocity optimal guidance approach for the case of a constant wind acceleration. Case studies for rendezvous and intercept problems are shown through simulation examples for different wind conditions.

Статическая термоустойчивость пологой геометрически нерегулярной оболочки из ортотропного термочувствительного материала

Рассматривается пологая ортотропная геометрически нерегулярная оболочка (ГНО) постоянного кручения, термомеханические параметры которой линейно зависят от температуры. При достижении температуры определенного значения происходит скачкообразно смена формы равновесия, что вызывает изменение первоначальной геометрии оболочки. Эти значения температур называют критическими. Для практики значительный интерес представляют соотношения, связывающие критические температуры с геометрическими и термомеханическими параметрами ГНО. Решение задач статической термоустойчивости ГНО, как правило, начинается с анализа их исходного безмоментного состояния. Тангенциальные усилия, вызванные нагревом оболочки, определяются как решения системы сингулярных дифференциальных уравнений безмоментной термоупругости. Эти усилия содержатся в формах Брайена или Рейсснера в уравнениях статической термоустойчивости, и от их структуры существенно зависит успех в дальнейшем решении задачи. В работе решение сингулярной безмоментной термоупругости найдено в элементарных функциях. Уравнения моментной термоупругости, записанные в компонентах поля перемещений, методом функции перемещений сведены к одному сингулярному дифференциальному уравнению в частных производных восьмого порядка. Решение записывается в виде двойного тригонометрического ряда, коэффициенты которого на основании процедуры Галeркина определяются как решения линейной однородной алгебраической системы уравнений. Из равенства нулю определителя этой системы (в первом приближении) получено алгебраическое уравнение пятой степени для относительной критической температуры, наименьший положительный действительный корень которого и есть искомая температура. Проводится количественный анализ влияния геометрических и термомеханических параметров ГНО на величины критических температур.

Logcf: An Efficient Tool for Real Root Isolation

Computing upper bounds of the positive real roots of some polynomials is a key step of those real root isolation algorithms based on continued fraction expansion and Vincent's theorem. The authors give a new algorithm for computing an upper bound of positive roots in this paper. The complexity of the algorithm is O(n log(u+1)) additions and multiplications where u is the optimal upper bound satisfying Theorem 3.1 of this paper and n is the degree of the polynomial. The method together with some tricks have been implemented as a software package logcf using C language. Experiments on many benchmarks show that logcf is competitive with RootIntervals of Mathematica and the function realroot of Maple averagely and it is much faster than existing open source real root solvers in many test cases.

Tricritical point and solid/liquid/gas phase transition of higher dimensional AdS black hole in massive gravity

By considering the fifth order graviton term in massive gravity theory, we study the P–V critical behaviors of AdS black hole in d≥7 dimensional space–time, and find the solid/liquid/gas phase transition with the “common” tricritical point, which does not exist in absence of higher graviton terms. Moreover, we also find the number of positive real roots of critical radius equation plays a key role in classifying different kinds of phase transitions. If there is only one positive root of equation, the first order Van der Waals-like phase transition emerges. For two roots, the system could experience a zero-order reentrant phase transition. Notably, for three roots, the solid/liquid/gas phase transition can occur. In addition, because the critical radius equation is a (n−2)-order polynomial equation associated with the graviton terms, the critical phenomena of black holes in extended phase space depend crucially on the number n of the graviton terms in high dimensional massive gravity.

A COMBINATORIAL APPROACH TO DECIDE INITIAL ROOT VALUE FOR THE SOLUTION OF NON-LINEAR SYSTEM OF EQUATIONS

This study aims at developing the real root of an algebraic or a transcendental equation based on non-linear model and a combinatorial numerical approach for solving nonlinear equations. The introduction of Bisection, False-Position and Newton–Raphson methods to identify the real positive root of an equation which is provided here. The rate of convergence is fastest in Newton–Raphson method due to the quadratic convergence compared with the other two iterative methods. We have applied here the three iterative methods in solving non-linear equations in MATLAB environment and compared the outcomes of results as well as geometrically also. Finally, an illustrative algebraic example is discussed and the MATLAB outputs are provided to check the validity of the new approach.

Equilibrium states and growth of quasi-lattice ordered monoids

Each multiplicative real-valued homomorphism on a quasi-lattice ordered monoid gives rise to a quasi-periodic dynamics on the associated Toeplitz C*-algebra; here we study the KMS equilibrium states of the resulting C*-dynamical system. We show that, under a nondegeneracy assumption on the homomorphism, there is a critical inverse temperature $\beta_c$ such that at each inverse temperature $\beta \geq \beta_c$ there exists a unique KMS state. Strictly above $\beta_c$, the KMS states are generalised Gibbs states with density operators determined by analytic extension to the upper half-plane of the unitaries implementing the dynamics. These are faithful Type~I states. The critical value $\beta_c$ is the largest real pole of the partition function of the system and is related to the clique polynomial and skew-growth function of the monoid, relative to the degree map given by the logarithm of the multiplicative homomorphism. Motivated by the study of equilibrium states, we give a proof of the inversion formula for the growth series of a quasi-lattice ordered monoid in terms of the clique polynomial as in recent work of Albenque--Nadeau and McMullen for the finitely generated case, and in terms of the skew-growth series as in recent work of Saito. Specifically, we show that $e^{-\beta_c}$ is the smallest pole of the growth series and thus is the smallest positive real root of the clique polynomial. We use this to show that equilibrium states in the subcritical range can only occur at inverse temperatures that correspond to roots of the clique polynomial in the interval $(e^{-\beta_c},1)$, but we are not aware of any examples in which such roots exist.

Exoatmospheric Interception via Linear Quadratic Optimization

Linear quadratic optimization is applied to an ideal vehicle without kinematics linearization for exoatmospheric vector guidance interception. Adopting an idea from bounded control approach, a linear guidance law is presented, where time to go is calculated either open-loop or as the smallest positive real root of a quartic polynomial equation. In the later case, feeding the time to go back into the linear guidance law results in a nonlinear guidance law. As a consequence, a nonlinear closing speed is used, suitable even in case the range diverges for some limited period of time. Moreover, a dispersal surface is presented that separates “first-pass” interception (in which the range decreases throughout the flight) from “second-pass” interception (in which the range decreases, increases, and finally decreases).

Nondegenerate multistationarity in small reaction networks

Much attention has been focused in recent years on the following algebraic problem arising from applications: which chemical reaction networks, when taken with mass-action kinetics, admit multiple positive steady states? The interest behind this question is in steady states that are stable. As a step toward this difficult question, here we address the question of multiple nondegenerate positive steady states. Mathematically, this asks whether certain families of parametrized, real, sparse polynomial systems ever admit multiple positive real roots that are simple. Our main results settle this problem for certain types of small networks, and our techniques may point the way forward for larger networks.

On the stability of triangular points in the relativistic R3BP when the bigger primary is oblate and the smaller one radiating with application on Cen X-4 binary system

Abstract In the framework of the relativistic R3BP, we examine the effects of oblateness of the primary body and radiation pressure of the secondary on the positions and stability of the triangular points L4,5. It is found that the parameters involved all affect the positions and increase in any of the parameters leads to a reduction in the size of the region of stability. Thus establishing their destabilizing tendencies. The presence of positive real roots or positive real part in complex roots affirms the instability of L4,5 of the problem when applied to Cen X-4.

Dynamics of flexural gravity waves: from sea ice to Hawking radiation and analogue gravity.

The propagation of flexural gravity waves, routinely used to model wave interaction with sea ice, is studied, including the effect of compression and current. A number of significant and surprising properties are shown to exist. The occurrence of blocking above a critical value of compression is illustrated. This is analogous to propagation of surface gravity waves in the presence of opposing current and light wave propagation in the curved space-time near a black hole, therefore providing a novel system for studying analogue gravity. Between the blocking and buckling limit of the compressive force, the dispersion relation possesses three positive real roots, contrary to an earlier observation of having a single positive real root. Negative energy waves, in which the phase and group velocity point in opposite directions, are also shown to exist. In the presence of an opposing current and certain critical ranges of compressive force, the second blocking point shifts from the positive to the negative branch of the dispersion relation. Such a shift is known as the Hawking effect from the analogous behaviour in the theory of relativity which leads to Hawking radiation. The theory we develop is illustrated with simulations of linear waves in the time domain.

Optimal additional data layers amount determining for interconnect latency hiding scheme

The key component of parallel computing efficiency is the structure of data exchange between computing nodes. It is necessary to reduce the latency of data exchange to improve the efficiency of parallel computing. A B+2R algorithm for overlapping delays in the data exchange was offered for this purpose. Existing works do not offer a method for selecting the additionally transmitted data layer count R. We introduce the models of all systems affecting the parallel calculation time. It makes possible to apply the analytical optimization. We introduce a data transmission network and a parallel computing application models. We consider the cell calculation time is a constant value. The cell calculation time depends on the specific calculation parameters. We introduce an estimate of the computation time. Computation time depends on the additionally transmitted data layers count. Further we introduce the derivative of computation time equation. We use lowest positive real roots of the cubic equation. It’s possible that the final cubic equation hasn’t real positive roots. It’s mean that local structured grid calculation time is much bigger than network latency. In that case, it’s not recommended to use latency hiding schemes. Otherwise we recommend to use R equal to 1. Purpose of out research is to find optimal R. Optimal R value should lead to a calculation time equation minimum. The method proposed in the paper correspond to experimental result. Designed analytical model for B+2R algorithm makes possible to select optimal R value, which leads to the best calculation speedup.

Descartes’ Rule of Signs for Polynomial Systems Supported on Circuits

We give a multivariate version of Descartes' rule of signs to bound the number of positive real roots of a system of polynomial equations in n variables with n+2 monomials, in terms of the sign variation of a sequence associated both to the exponent vectors and the given coefficients. We show that our bound is sharp and is related to the signature of the circuit.

Imaginary Schur-Weyl duality

We study imaginary representations of the Khovanov-Lauda-Rouquier algebras of affine Lie type. Irreducible modules for such algebras arise as simple heads of standard modules. In order to define standard modules one needs to have a cuspidal system for a fixed convex preorder. A cuspidal system consists of irreducible cuspidal modules---one for each real positive root for the corresponding affine root system ${\tt X}_l^{(1)}$, as well as irreducible imaginary modules---one for each $l$-multipartition. We study imaginary modules by means of `imaginary Schur-Weyl duality'. We introduce an imaginary analogue of tensor space and the imaginary Schur algebra. We construct a projective generator for the imaginary Schur algebra, which yields a Morita equivalence between the imaginary and the classical Schur algebra. We construct imaginary analogues of Gelfand-Graev representations, Ringel duality and the Jacobi-Trudy formula.

An optimal control framework for the climb and descent economy modes of flight management systems

This paper proposes an optimal control framework for the climb and descent economy modes of a flight management system (FMS) yielding a solution that can be implemented in real-time flights below the drag divergence Mach number. The problem is formulated as the optimization of a functional that trades off the fuel- and time-related costs of a flight as a function of a (crew-supplied) parameter called the cost index. The work builds on previous research of the authors for the cruise phase and extends it to the climb and descent phases of flight. More specifically, for both climb and descent, it is found that suboptimal solutions can be obtained as the positive real roots of a fifth-degree polynomial lying inside the flight envelope, which can be found using fast-converging algorithms such as Newton's method. The main contributions of this work are threefold. First, the proposed method gives physical insight because there is an analytical expression for each coefficient of the polynomial. Second, this approach eliminates the need to have a performance database in the system, thus making its implementation faster in real-time. Third, the solution exhibits the same behavior of airborne FMS units as a function of the cost index, which is justified in this paper based on Bellman's principle of optimality. This justification is an important theoretical contribution of the paper. A validation of the approximate solution is obtained using the shooting method to compute the optimal trajectories and compare them against the proposed suboptimal solution. Simulation results show that, for an Airbus A320 model and for a Gulfstream-IV aircraft model, the relative error of the suboptimal trajectories when compared to the optimal trajectories is small for climb and descent trajectories, respectively.

Limit cycles bifurcating from the periodic annulus of the weight-homogeneous polynomial centers of weight-degree 2

We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of cubic polynomial differential centers when it is perturbed inside the class of all cubic polynomial differential systems. The family considered is the unique family of weight-homogeneous polynomial differential systems of weight-degree 2 with a center. The computations has been done with the help of the algebraic manipulator Mathematica.

Out-of-Plane Equilibrium Points in the Photogravitational CR3BP with Oblateness and P-R Drag

This paper investigates the motion of a test particle around the out-of-plane equilibrium points in the circular photogravitational restricted three-body problem when the effect of radiation pressure from the smaller primary and its Poynting-Robertson (P-R) drag are taken into account, and the bigger primary is modeled as an oblate spheroid. These points lie in the xz-plane almost directly above and below the center of the oblate primary. The equilibrium points are sought, and we observe that, there are two coordinate points L 6,7 which depend on the oblateness of the bigger primary, and the radiation pressure force and P-R drag of the smaller primary. The positions and linear stability of the problem are investigated both analytically and numerically for the binary system Cen X-4. The out-of-plane equilibrium points are found to be unstable in the sense of Lyapunov due to the presence of a positive real root.