Real Roots Research Articles

A Method for Locating the Real Roots of the Symbolic Quintic Equation Using Quadratic Equations

AbstractA method is proposed with which the locations of the roots of the monic symbolic quintic polynomial can be determined using the roots of two resolvent quadratic polynomials: and , whose coefficients are exactly those of the quintic polynomial. The different cases depend on the coefficients of and and on some specific relationships between them. The method is illustrated with the full analysis of one of the possible cases. Some of the roots of the symbolic quintic equation for this case have their isolation intervals determined and, as this cannot be done for all roots with the help of quadratic equations only, finite intervals containing 1 or 3 roots, or 0 or 2 roots, or, rarely, 0, or 2, or 4 roots of the quintic are identified. Knowledge of the stationary points of the quintic lifts this indeterminacy and allows finding the isolation interval of each root. Separately, using the complete root classification of the quintic, one can also lift this indeterminacy. The method also allows to see how variation of the individual coefficients of the quintic affect its roots. No root finding iterations or any numerical approximations are used and no equations of degree higher than two are solved.

Ultrananoporous supercapacitor with ionic liquid comprised of two-site cation: an Ising model study (II)

A four-state one-dimension discrete spin model with the nearest neighbor and partial next nearest neighbor lattice interactions is proposed for describing electrical double layer formed by ionic liquid in cylindrical pore that can accommodate only one row of particles. The model, in which cation is modeled as a dimer formed by two hard spheres (HSs): one is positively charged with diameter , while the other is neutral with diameter , and anion is modeled by charged HS with diameter , is designed to investigate effects of pore diameter , ion transfer free energy, ion size and electrical valence on behaviors of specific differential capacitance and specific energy storage and its saturation value . Relevant Kramers–Wannier transfer matrix is , which gives system partition function by numerically solving maximum real root of the relevant characteristic equation. Main findings are briefly described below. (i) The ion valence has little effect on the value, but does reduce the threshold voltage strength , beyond which the energy storage reaches . However, the ion valence (whether co- or counter-ion) raises both the and values as the relevant ion size decreases enough. (ii) Decreasing of the value weakly reduces the value, but raises the peak height and the value obviously; correlation between the value and value becomes stronger with the ions getting more pore hostile. (iii) With the counter-ion or co-ion charged site size decreasing, both the value and value rise. However, the effect of the co-ion charged site size is weakened as the co-ion gets more pore hostile, and the cation neutral site size influences on the value less than the cation charged site size does. (iv) Pore friendly co-ion or pore hostile counter-ion always helps in raising the value; the two have significant synergistic effect with the co-ion pore friendliness playing more important role than the counter-ion pore hostility.

Eliminating the White Supremacy Mindset from Global Health Education.

The term “decolonization” has been increasingly used to refer to the elimination of the colonial experience and its legacy. However, the use of this overarching term masks the real root of the problem. European countries, whose populations are majority white, used their assumed supremacy as justification for the colonization of current low- and middle-income countries (LMICs) where the majority of non-white people live. This clear overlap between geographic and skin color differences explains how the white supremacy ideology triggered European colonization. Therefore, calls to decolonize global health education must focus on the roots of colonization and fight for the elimination of white supremacy ideology that is one of the pillars of the current ills of our global health architecture. A step in this process acknowledging the expertise that emerges from LMICs, alongside challenging the traditional high-income country (HIC) hegemony over knowledge and strengthening universities in LMICs to provide quality medical and global health education. Additionally, we also need to reevaluate curricula, research selection, and design as well as partnerships. Students need to be equipped with the skills to question norms and contribute to the creation of equitable, mutually beneficial partnerships. This needs to accompanied by the adoption of transdisciplinary education to address critical societal challenges. By challenging the white supremacy ideology, we can shift the center of gravity in global health to respect the right to equal say in education and research according to the disease burden and the distribution of the world population.

Influence of needle working length and root canal curvature on irrigation: a computational fluid dynamics analysis based on a real tooth

BackgroundsTo compare the irrigation efficiency with different needle working length and different root canal curvature based on a real unshaped root canal using computational fluid dynamics (CFD) method.MethodsImages of the root canal of the maxillary first molar after being prepared to .04/15 were scanned using micro-CT, and then imported into the software for three-dimensional reconstruction. A palatal root canal with a curvature of 23.4° was selected as the experiment canal. The needle working length of the 30-G flat needle was 4.75 mm, 5 mm, 5.25 mm and 5.5 mm short of apical foramen respectively, the flow pattern, irrigation velocity, shear stress were compared. The modified curved canals with a curvature of 0°, 5°, 10°, 20° and 30°were reconstructed via software. The flat needle was replaced at the optical inserted depth, and key parameters of irrigation efficiency were analyzed.ResultsDecreased needle working length had a positive impact on irrigation efficiency. With the optimal needle working length, the replacement of the apical irrigation fluid, the effective velocity, and wall shear stress were significantly improved in more severely curved root canals. With the same needle working depth and analogous canal curvature, irrigation efficiency is higher in real canal than that of modified canal.ConclusionsShort needle working depth, large curvature and the anomalous inner wall of canals help to improve irrigation efficiency.

Proportional–integral–derivative stabilization of complex conjugate-order systems

Proportional–integral–derivative (PID) stabilization is an important control design strategy that provides the designer with all PID controller set which results control system stability absolutely. In this way, the designer has a wide range of freedom to obtain the controller that meets the desired criteria. This process is particularly advantageous in situations where it is difficult to clearly define the design criteria at the beginning of the design or to make a balanced decision among the design criteria. The main objective of this paper is to present for the first time a PID stabilization method for complex conjugate-order systems, which is a new type of system for the control community and has been little studied on. The method is based on obtaining of stability/instability regions using the D-decomposition method in the controller parameter space graphically. These regions are formed by stability boundaries that are defined as real root, infinite root and complex root boundaries. The stability of the regions is determined using generalized modified Mikhailov stability criterion that is a powerful stability tool of the system theory. The simulation results indicate that the presented stabilization method is effective and practically useful in the analysis and control of the complex conjugate-order systems.

Making Artificial Brains: Components, Topology, and Optimization.

Making Artificial Brains: Components, Topology, and Optimization.

Laguerre inequalities for discrete sequences

The Turán inequality, the Laguerre inequality and their m-rd generalizations have been proved to be closely relative with the Laguerre-Pólya class and Riemann hypothesis. Since these two inequalities are equivalent to log-concavity of the discrete sequences, we consider whether their generalizations hold for discrete sequences. Recently, Chen, Jia and Wang proved that the partition function satisfies the Turán inequality of order 2 and thus the 3-rd Jensen polynomials associated with the partition function have only real zeros. Griffin, Ono, Rolen and Zagier proved the n-th Jensen polynomials associated with the Maclaurin coefficients of the function in the Laguerre-Pólya class and the partition function have only real zeros except finite terms. In this paper, we show the Laguerre inequality of order 2 is true for the partition function, the overpartition function, the Bernoulli numbers, the derangement numbers, the Motzkin numbers, the Fine numbers, the Franel numbers and the Domb numbers.

Isolation Intervals of the Real Roots of the Parametric Cubic Equation and Improved Complete Root Classification

AbstractThe isolation intervals of the real roots of the real symbolic monic cubic polynomial are found in terms of simple functions of the coefficients of the polynomial (such as , , , , when is negative), and the roots of some auxiliary quadratic equations whose coefficients are also simple functions of the coefficients of the cubic. A much‐improved complete root classification, addressing the signs (together with giving the isolation intervals) of the individual roots, is also presented. No numerical approximations or root finding techniques are used. Instead of considering the discriminant of the cubic, criterion for the existence of a single real root or three real roots is found as conditions on the coefficients of the cubic, resulting from the roots of the auxiliary quadratic equations. It is also shown that, if a cubic equation has three real roots, then these lie in an interval such that , independent of . A detailed algorithm for applying the method for isolation of the roots of the cubic is also given and it is illustrated through examples, including the full mathematical analysis of the cubic equation associated with the Rayleigh elastic waves and finding the isolation intervals of its real roots.

The Siebeck–Marden–Northshield Theorem and the Real Roots of the Symbolic Cubic Equation

The isolation intervals of the real roots of the symbolic monic cubic polynomial x^3 + a x^2 + b x + c are determined, in terms of the coefficients of the polynomial, by solving the Siebeck–Marden–Northshield triangle—the equilateral triangle that projects onto the three real roots of the cubic polynomial and whose inscribed circle projects onto an interval with endpoints equal to stationary points of the polynomial.

Automatic reconstruction of three-dimensional root system architecture based on ground penetrating radar

Ground penetrating radar (GPR) is widely used in root detection because of its advantages of nondestructive and periodic detection. Previously, root point positioning and root diameter estimation based on GPR have been proved to be effective, but there is still little research on automatic reconstruction of root system architecture (RSA) based on GPR. Good RSA reconstruction can effectively reflect the real growth of the root and its adaptability to the environment, so the purpose of this study is to vividly reconstruct RSA from the two aspects of topological structure and geometric features. Therefore, we used 900 MHz tree radar to scan the roots of a 9-year-old ash tree in the field, extracted the spatial coordinates of the root points from the detection results, and estimated the diameter of the root points using BP neural network. Thus, RSA was automatically reconstructed by using the root topological structure and geometric features reconstruction algorithm. By comparing the RSA reconstructed by the algorithm with the simulated RSA of the real root system, the accuracy of the reconstruction algorithm was quantitatively evaluated. The results showed that for roots with a diameter greater than 1 cm, the root length similarity between the algorithm reconstructed RSA and the simulated RSA is 76.2%, and the root topology similarity is 81.4%. The high similarity between the two indicates that the reconstruction of RSA based on GPR is effective and feasible. Moreover, the proposed algorithm in this paper enriches the three-dimensional (3D) visualization method of RSA. According to the method, the induction and regulation during the growth of the root system can improve its utilization efficiency of soil nutrients and water.

On combinatorial properties and the zero distribution of certain Sheffer sequences

We present combinatorial and analytical results concerning a Sheffer sequence with a generating function of the form G(x,z)=Q(z)xQ(−z)1−x, where Q is a quadratic polynomial with real zeros. By using the properties of Riordan matrices we address combinatorial properties and interpretations of our Sheffer sequence of polynomials and their coefficients. We also show that apart from two exceptional zeros, the zeros of polynomials with large enough degree in such a Sheffer sequence lie on the line x=1/2+it.

Central Limit Theorem for the volume of the zero set of Kostlan-Shub-Smale random polynomial systems

We establish the Central Limit Theorem, as the degree goes to infinity, for the normalized volume of the zero set of a rectangular Kostlan–Shub–Smale random polynomial system. This paper is a continuation of Central Limit Theorem for the number of real roots of Kostlan–Shub–Smale random polynomial systems by the same authors in which the case of square systems was considered. Our main tools are Kac-Rice formula and an expansion of the volume of the level set into the Itô-Wiener Chaos.

Second Order Splitting Dynamics with Vanishing Damping for Additively Structured Monotone Inclusions

In the framework of a real Hilbert space, we address the problem of finding the zeros of the sum of a maximally monotone operator A and a cocoercive operator B. We study the asymptotic behaviour of the trajectories generated by a second order equation with vanishing damping, attached to this problem, and governed by a time-dependent forward–backward-type operator. This is a splitting system, as it only requires forward evaluations of B and backward evaluations of A. A proper tuning of the system parameters ensures the weak convergence of the trajectories to the set of zeros of A+B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A + B$$\\end{document}, as well as fast convergence of the velocities towards zero. A particular case of our system allows to derive fast convergence rates for the problem of minimizing the sum of a proper, convex and lower semicontinuous function and a smooth and convex function with Lipschitz continuous gradient. We illustrate the theoretical outcomes by numerical experiments.

Certified numerical real root isolation for bivariate nonlinear systems

In this paper, we present a new method for isolating real roots of a bivariate nonlinear system. It is a subdivision method based on analyzing the local geometrical properties of the given system. We propose the concept of opposite monotone system in a box and use it to determine the existence and the uniqueness of a simple real zero of the system in the box. We have implemented our method and the experiments show the effectivity and efficiency of our approach, especially for high degree sparse polynomial systems. This paper is an extended version of the ISSAC'19 conference paper (Cheng and Wen (2019)).

New Algorithm for 2-terminal Transmission Line Fault Location Integrating Voltage Phasor Feature and Phase Angle Jump Checking

The low precision and instability of fault location results of power system transmission lines is a major problem that affects the safe operation of power systems, which needs to be solved more effectively. With the increasing coverage of the synchronic Phasor Measurement Unit (PMU) in power systems, it has laid a good foundation for the research and application of high-precision fault location method based on 2-termnal synchronous measurement of phasor. Based on this, a new high-precision fault location algorithm integrating voltage phasor feature and phase-angle jump checking is proposed in this paper. Firstly, the synchronous fundamental positive sequence voltage and current phasors of the fault line are extracted; and then all the roots(including real and virtual roots) are calculated by using the condition that the fundamental positive sequence voltage phasor at the fault point is equal; finally the unique real root represents the actual fault point can be determined by using the feature that the check function of phase-angle will step-jump when crossing the fault point from left to right, and then the purpose of high precision fault location is realized. This algorithm has the characteristics of strong versatility, high precision and immunity to transition resistance. The results of simulation and example show the feasibility and effectiveness of the fault location algorithm the fault location algorithm proposed in this paper.

An integrated methodology for system-level early fault detection and isolation

Fault diagnosis is an indispensable technique to ensure the high-performance operation of a mechanical system during its life-cycle. Existing research works mainly limits in the diagnosis of single component or some certain components of interest. The systematic investigation of fault diagnosis for a full system is rare, making it inefficient to implement the method developed in the practice of a complete engineering system. In this paper, a system-level fault diagnosis methodology is proposed based on fault behaviour analysis, optimal sensor placement and intelligent data analytics for multiple fault detection and isolation in a complex machine system. A dynamical model of a mechanical system is firstly constructed using Bond graph, and the response characteristics of observable parameters under different faults are derived by analysing the functional relationships of system variables. A set partitioning theory-based sensor placement approach is then presented for designing a condition monitoring system with a quantity-optimal set of sensors for a desired performance of fault isolatability. Two multivariate statistic measures, i.e. Hotelling’s T2 and Q are calculated from the system operation parameters as the indices to detect the potential faults. The abnormal parameters are separated by constructing a contribution plot once a fault is reported. The separated abnormal parameters are then input to a Bayesian network model, and the real root cause of the abnormity is isolated by blending the observations with expert knowledge of diagnosis. The diesel engine lubrication system, which involves three common domains: mechanical, hydraulic and thermodynamic processes, is taken as an example to show the implementation and performance of proposed approach. This approach can be used to design a system-level fault diagnosis scheme that including various tasks such as fault behaviour analysis, sensor placement, data collection, data processing, information fusion in order to achieve an accurate fault isolation ultimately.

The Multivariate Schwartz--Zippel Lemma

Motivated by applications in combinatorial geometry, we consider the following question: Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ be an $m$-partition of a positive integer $n$, $S_i \subseteq \mathbb{C}^{\lambda_i}$ be finite sets, and let $S:=S_1 \times S_2 \times \cdots \times S_m \subset \mathbb{C}^n$ be the multigrid defined by $S_i$. Suppose $p$ is an $n$-variate degree $d$ polynomial. How many zeros does $p$ have on $S$? We first develop a multivariate generalization of the combinatorial nullstellensatz that certifies existence of a point $t \in S$ so that $p(t) \neq 0$. Then we show that a natural multivariate generalization of the DeMillo--Lipton--Schwartz--Zippel lemma holds, except for a special family of polynomials that we call $\lambda$-reducible. This yields a simultaneous generalization of the Szemerédi--Trotter theorem and the Schwartz--Zippel lemma into higher dimensions, and has applications in incidence geometry. Finally, we develop a symbolic algorithm that identifies certain $\lambda$-reducible polynomials. More precisely, our symbolic algorithm detects polynomials that include a Cartesian product of hypersurfaces in their zero set. It is likely that using Chow forms the algorithm can be generalized to handle arbitrary $\lambda$-reducible polynomials, which we leave as an open problem.

Estimation of Transport-Category Jet Airplane Maximum Range and Airspeed in the Presence of Transonic Wave Drag

One of the most difficult steps in estimating the cruise performance characteristics of high-subsonic transport-category turbofan-powered airplanes is the estimation of the transonic wave drag. Modern jet airplanes cruise most efficiently in the vicinity of the drag-divergence or drag-rise Mach numbers. In the initial design phase and later when the preliminary wind-tunnel and/or CFD computations and drag polars are known with increased accuracy, a method of estimating cruise performance is needed. In this study, a new semi-empirical transonic wave drag model using modified Lock’s equation was developed. For maximum range cruise estimations, an optimization criterion based on maximizing specific air range was used. The resulting nonlinear equations are of 12th- and 13th-order. Numerical Newton–Raphson nonlinear solvers were used to find real positive roots of such polynomials. The NR method was first tested for accuracy and convergence using known analytical solutions. A methodology for an initial guess was developed starting with the maximum-range cruise Mach without the wave-drag included. This guess resulted in fast quadratic convergence in all computations. Other novel features of this article include a new semi-empirical fuel-flow law, which was also extensively tested. Additionally, a semi-empirical turbofan thrust model usable for a wide range of bypass ratios and the entire flight envelope was developed. Such physics-based semi-empirical model can be used for a wide range of turbofans. The algorithm can be utilized to identify most beneficial input parameter values and combinations for the cruise flight phase. The model represents a powerful tool to estimate important cruise performance airspeeds located in the transonic regime. An intended application is in the conceptual development stages for early design optimizations of future airplanes. It is possible with additional effort to extend existing model capabilities to deal with supersonic transports optimal cruise parameters.

Jesus en las Americas: Exploring Latter-day Saint Latinx Politics and Culture

Jesus en las Americas: Exploring Latter-day Saint Latinx Politics and Culture

How many roots of a system of random Laurent polynomials are real?

Abstract We say that a zero of a Laurent polynomial that lies on the unit circle with centre is real. We also say that a Laurent polynomial that is real on this circle is real. In contrast with ordinary polynomials, it is known that for random real Laurent polynomials of increasing degree the average proportion of real roots tends to rather than to . We show that this phenomenon of the asymptotically nonvanishing proportion of real roots also holds for systems of Laurent polynomials of several variables. The corresponding asymptotic formula is obtained in terms of the mixed volumes of certain convex compact sets determining the growth of the system of polynomials. Bibliography: 11 titles.