Real Roots Research Articles

Improving the Step Response of Flexible Systems in the Presence of Real Nonminimum Phase Zeros

Abstract It is well known that real nonminimum phase (RNMP) zeros impose a tradeoff between the settling time and undershoot in the step response of flexible systems. Existing methods to alleviate this tradeoff predominantly rely on various advanced control strategies without delving into a broader mechatronic approach that combines physical system and control system design. To address this gap, this article proposes a proportional viscous damping-based physical system design in combination with feedback control and prefilter design. First, the effect of proportional viscous damping on RNMP zeros of flexible systems is established to propose a damping strategy that pushes all the RNMP zeros further away from the imaginary axis. Then, a step-by-step mechatronic system design process is presented to apply this damping strategy along with a full-state feedback control strategy and prefilter to a multi-degree-of-freedom (DoF) flexible system. The application of this design process yields simultaneous improvement in the settling time and undershoot in the step response of this flexible system.

Mobilizing Intolerance: Teungku Dayah (Ulama) and Religious Discourse in Aceh, Indonesia

In 2018, Aceh was ranked 32nd in the index of tolerant cities in Indonesia conducted by the Setara Institute. The poor rating of religious tolerance in Aceh was due to an act of violence that occurred in Aceh in the previous periods, in which religious grounds were used. Ironically, this act was carried out by people who claim the authority of the Aswaja ideology in Aceh. The field research seeks to uncover the real root of the problem by interviewing religious authorities in Aceh, namely Teungku Dayah (ulama in Acehnese society), Santri (Muslim Students), and lecturers at Islamic higher education institutions. The results of the study showed that the acts of violence in Aceh stemmed from Teungku Dayah's inaccurate understanding of pluralism. Pluralism is understood and positioned as an ideology that contradicts and threatens the sustainability of the Aswaja's ideology in Aceh.

Essential Self-Adjointness of Even-Order, Strongly Singular, Homogeneous Half-Line Differential Operators

AbstractWe consider essential self-adjointness on the space $$C_0^{\infty }((0,\infty ))$$ C 0 ∞ ( ( 0 , ∞ ) ) of even-order, strongly singular, homogeneous differential operators associated with differential expressions of the type $$\begin{aligned} \tau _{2n}(c) = (-1)^n \frac{d^{2n}}{d x^{2n}} + \frac{c}{x^{2n}}, \quad x > 0, \; n \in {{\mathbb {N}}}, \; c \in {{\mathbb {R}}}, \end{aligned}$$ τ 2 n ( c ) = ( - 1 ) n d 2 n d x 2 n + c x 2 n , x > 0 , n ∈ N , c ∈ R , in $$L^2((0,\infty );dx)$$ L 2 ( ( 0 , ∞ ) ; d x ) . While the special case $$n=1$$ n = 1 is classical and it is well known that $$\tau _2(c)\big |_{C_0^{\infty }((0,\infty ))}$$ τ 2 ( c ) | C 0 ∞ ( ( 0 , ∞ ) ) is essentially self-adjoint if and only if $$c \ge 3/4$$ c ≥ 3 / 4 , the case $$n \in {{\mathbb {N}}}$$ n ∈ N , $$n \ge 2$$ n ≥ 2 , is far from obvious. In particular, it is not at all clear from the outset that $$\begin{aligned} \begin{aligned}&\textit{there exists }c_n \in {{\mathbb {R}}}, n \in {{\mathbb {N}}}\textit{, such that} \\&\quad \tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))} \, \textit{ is essentially self-adjoint}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (*)\\ {}&\quad \textit{ if and only if } c \ge c_n. \end{aligned} \end{aligned}$$ there exists c n ∈ R , n ∈ N , such that τ 2 n ( c ) | C 0 ∞ ( ( 0 , ∞ ) ) is essentially self - adjoint ( ∗ ) if and only if c ≥ c n . As one of the principal results of this paper we indeed establish the existence of $$c_n$$ c n , satisfying $$c_n \ge (4n-1)!!\big /2^{2n}$$ c n ≥ ( 4 n - 1 ) ! ! / 2 2 n , such that property (*) holds. In sharp contrast to the analogous lower semiboundedness question, $$\begin{aligned} \textit{for which values of }c\textit{ is }\tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))}{} \textit{ bounded from below?}, \end{aligned}$$ for which values of c is τ 2 n ( c ) | C 0 ∞ ( ( 0 , ∞ ) ) bounded from below ? , which permits the sharp (and explicit) answer $$c \ge [(2n -1)!!]^{2}\big /2^{2n}$$ c ≥ [ ( 2 n - 1 ) ! ! ] 2 / 2 2 n , $$n \in {{\mathbb {N}}}$$ n ∈ N , the answer for (*) is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly, $$\begin{aligned} c_{1}&= 3/4, \quad c_{2 }= 45, \quad c_{3 } = 2240 \big (214+7 \sqrt{1009}\,\big )\big /27, \end{aligned}$$ c 1 = 3 / 4 , c 2 = 45 , c 3 = 2240 ( 214 + 7 1009 ) / 27 , and remark that $$c_n$$ c n is the root of a polynomial of degree $$n-1$$ n - 1 . We demonstrate that for $$n=6,7$$ n = 6 , 7 , $$c_n$$ c n are algebraic numbers not expressible as radicals over $${{\mathbb {Q}}}$$ Q (and conjecture this is in fact true for general $$n \ge 6$$ n ≥ 6 ).

Real Roots of Hypergeometric Polynomials via Finite Free Convolution

Real Roots of Hypergeometric Polynomials via Finite Free Convolution

A hyperbolic Kac-Moody Calogero model

A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra AE3 with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincaré disk with a unique potential. Since the Weyl group of AE3 is a ℤ2 extension of the modular group PSL(2,ℤ), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of AE3, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincaré series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case we find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.

Foliar Application of Strontium for the Identification of Roots from Specific Wheat Plants

In the field of crop research, the study of roots involves many challenges, particularly the lack of effective methods for identifying crop roots. To deal with this problem, this study proposes a solution strategy: in applying strontium to crop leaves, the roots of different crops are distinguished. In this study, we applied strontium ions to the leaves of hydroponically grown wheat, and these strontium ions were partially absorbed by the leaves and transported to the roots. Therefore, the strontium concentration of roots was significantly increased through the foliar application of strontium. After mixing the treated wheat with the untreated wheat root, the greater the biomass of the wheat root applied with strontium ions, the higher the strontium concentration in the mixed root. Based on this phenomenon, we can establish the relationship between wheat root biomass and strontium concentration in mixed roots through linear fitting. Furthermore, we can also use the relationship between root biomass and root length to establish the correlation between strontium concentration in mixed roots and the root length of wheat with strontium ions. After measuring the strontium concentration of the mixed roots to be distinguished, the root biomass and root length of wheat applied with strontium ions can be calculated according to the equation obtained through linear fitting. The accuracy of this method was verified through a comparison with the actual value and the existing root staining method. The results show that the coefficient of determination (R2) of the root biomass estimation equation obtained through linear fitting reached 0.83, which is statistically significant (p < 0.01). The Pearson correlation coefficient with the measured value was more than 0.9, showing a very high correlation and significance. The root length estimation equation derived from the relationship between the root biomass, strontium concentration and root length was compared with the real root length value and the root length value obtained using the staining method. The Pearson correlation coefficient also exceeded 0.8 and reached a statistically significant level (p < 0.01). This study confirms that strontium ions can be absorbed and transported to roots through wheat leaves and successfully developed a new method for predicting the length of wheat roots, providing an effective new tool for wheat root research.

Rational Approximations for the Oscillatory Two-Parameter Mittag–Leffler Function

The two-parameter Mittag–Leffler function Eα,β is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When α>1, the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for α∈(0,1) often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for Eα,β(−t), t≥0, with α∈(1,2). This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants.

Алгоритмический поиск целых абелевых корней многочлена с целыми абелевыми коэффициентами

Алгоритмический поиск целых абелевых корней многочлена с целыми абелевыми коэффициентами

A novel approaches to 6th-order delay differential equations in toxic plant interactions and soil impact: beyond newton-raphson

This paper focuses on investigating a 6th-order delay differential equation root within the context of toxic interactions between competing plant populations and their impact on soil dynamics. The study introduces a novel approach for approximating solutions to nonlinear delay differential equations, drawing inspiration from the fundamental principles of Newton-Raphson’s method. This technique leverages the complex root theorem to ensure stability, enabling it to effectively handle widely dispersed roots within dynamic systems. Consequently, this approach holds considerable potential for a diverse array of applications. The analysis introduces time delay into a nonlinear dynamical system and explores the system’s threshold value. At this threshold, the dynamical system’s stability undergoes fluctuations, and observations of hopf bifurcation phenomena are made. The study also successfully identifies both real and complex roots of the dynamical system. Visualization of the dynamic system is accomplished using MATLAB-generated graphical representations. Moreover, this research’s implications extend to the realm of climate action and terrestrial ecosystems, underscoring its significance for promoting a sustainable environment and fostering healthy life on land.

The ultra log-concavity of Z-polynomials and γ-polynomials of uniform matroids

Proudfoot, Xu, and Young introduced the Z-polynomial for any matroid and conjectured the polynomial has only real roots. Recently, Ferroni, Nasr, and Vecchi introduced the γ-polynomial of a matroid. In this paper, we prove that both the Z-polynomials and γ-polynomials of uniform matroids are ultra log-concave, which due to Newton's inequality partially supports the real-rootedness conjecture. We also give an alternative formula for the γ-polynomials of uniform matroids. As an application, we use this formula to provide a new proof of the γ-positivity of sparse paving matroids.

On maximal totally real embeddings

Abstract We consider complex structures with totally real zero section of the tangent bundle. We assume that the complex structure tensor is real-analytic along the fibers of the tangent bundle. This assumption is quite natural in view of a well-known result by Bruhat and Whitney. We provide explicit integrability equations for such complex structures in terms of the fiberwise Taylor expansion. In a particular geometric case considered in the literature, we explicit much further the fiberwise Taylor expansion of the complex structure as well as the integrability equations.

Estimating genomic relationships of metafounders across and within breeds using maximum likelihood, pseudo-expectation–maximization maximum likelihood and increase of relationships

BackgroundThe theory of “metafounders” proposes a unified framework for relationships across base populations within breeds (e.g. unknown parent groups), and base populations across breeds (crosses) together with a sensible compatibility with genomic relationships. Considering metafounders might be advantageous in pedigree best linear unbiased prediction (BLUP) or single-step genomic BLUP. Existing methods to estimate relationships across metafounders Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\\Gamma}}$$\\end{document} are not well adapted to highly unbalanced data, genotyped individuals far from base populations, or many unknown parent groups (within breed per year of birth).MethodsWe derive likelihood methods to estimate Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\\Gamma}}$$\\end{document}. For a single metafounder, summary statistics of pedigree and genomic relationships allow deriving a cubic equation with the real root being the maximum likelihood (ML) estimate of Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\\Gamma}}$$\\end{document}. This equation is tested with Lacaune sheep data. For several metafounders, we split the first derivative of the complete likelihood in a term related to Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\\Gamma}}$$\\end{document}, and a second term related to Mendelian sampling variances. Approximating the first derivative by its first term results in a pseudo-EM algorithm that iteratively updates the estimate of Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\\Gamma}}$$\\end{document} by the corresponding block of the H-matrix. The method extends to complex situations with groups defined by year of birth, modelling the increase of Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\\Gamma}}$$\\end{document} using estimates of the rate of increase of inbreeding (ΔF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta F$$\\end{document}), resulting in an expanded Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\\Gamma}}$$\\end{document} and in a pseudo-EM+ΔF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta F$$\\end{document} algorithm. We compare these methods with the generalized least squares (GLS) method using simulated data: complex crosses of two breeds in equal or unsymmetrical proportions; and in two breeds, with 10 groups per year of birth within breed. We simulate genotyping in all generations or in the last ones.ResultsFor a single metafounder, the ML estimates of the Lacaune data corresponded to the maximum. For simulated data, when genotypes were spread across all generations, both GLS and pseudo-EM(+ΔF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta F$$\\end{document}) methods were accurate. With genotypes only available in the most recent generations, the GLS method was biased, whereas the pseudo-EM(+ΔF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta F$$\\end{document}) approach yielded more accurate and unbiased estimates.ConclusionsWe derived ML, pseudo-EM and pseudo-EM+ΔF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta F$$\\end{document} methods to estimate Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{\\Gamma}}$$\\end{document} in many realistic settings. Estimates are accurate in real and simulated data and have a low computational cost.

Transformer winding faults diagnosis using the transfer function of its high-frequency lumped equivalent model from the measured FRA data

The frequency response analysis (FRA) of the transformer is the data obtained in a wide frequency range that represents the transfer function (TF) that reflects the physical behavior of its coil. However, it can be considered one of the best tools for a better understanding of the electromagnetic characteristics of the winding. This article proposes the TF as a tool to locate and assess transformer winding failures. So, an autonomous algorithm has been developed to construct the TF in ‘s domain’ proposed in previous research. The TF has been determined from experimental measured FRA data from an isolated air-core winding, where the TF has been calculated for the axial defects (AD) realized on the winding discs. Besides, the diagnostic approach is based on the scaling factor (K) and the real zero (τ) of the TF that can be obtained from the measured equivalent capacitance, equivalent inductance, and the DC resistance of the winding in high and low-frequency ranges. However, the K, τ, and the slope of the equivalent inductance characteristics as a function of the winding defects degree have been used for locating and assessing the severity of the winding faults.

On unital absorbing extensions of C*-algebras of stable rank one and real rank zero

Suppose that B B is a separable stable C ∗ C^* -algebra with real rank zero, stable rank one and ( K 0 ( B ) , K 0 + ( B ) ) (\mathrm {K}_0(B), \mathrm {K}_0^+(B)) is weakly unperforated in the sense of Elliott [Internat. J. Math. 1 (1990), no. 4, pp. 361–380]. Let A A be a unital simple separable nuclear C ∗ \mathrm {C}^* -algebra. We show that B B has the corona factorization property and any unital extension of A A by B B is absorbing.

High-precision teeth reconstruction based on automatic multimodal fusion with CBCT and IOS

In digital orthodontic treatment, the high-precision reconstruction of complete teeth, encompassing both the crown and the actual root, plays a pivotal role. Current mainstream techniques, prioritizing the high resolution of intraoral scanned models (IOS), are confined to using IOS data for orthodontic treatments. However, the lack of root information in the IOS data may lead to complications such as dehiscence. In contrast, Cone Beam Computed Tomography (CBCT) data encompasses comprehensive dental information with roots. Nonetheless, the radiative character of CBCT scans renders patients unsuitable for repeated examinations in a short time. In addition, lower scanning precision of CBCT leads to suboptimal teeth segmentation outcomes, hindering the accurate representation of dental occlusal relationships. Therefore, in order to fully utilize the complementarity between dental multimodal data, we propose a method for high-precision 3D teeth model reconstruction based on IOS and CBCT, which mainly consists of global rigid registration and local nonrigid registration. Specifically, we extract the priori information of dental arch curves for coarse alignment to provide a good initial position for the Iterative Closest Point (ICP) algorithm, and design a conformal parameterization method for a single tooth to effectively obtain the point correspondence between IOS and CBCT crowns. The rough crown of the CBCT will gradually fit towards the IOS through iterative optimization of nonrigid registration. The experimental results show that our method robustly fuses the advantageous features of IOS and CBCT. The 3D teeth model reconstructed by our method contains the high-precision crown of IOS and the real root of CBCT, which can be effectively used in clinical orthodontic treatment.

Lithium-ion battery second life: pathways, challenges and outlook.

Net zero targets have resulted in a drive to decarbonise the transport sector worldwide through electrification. This has, in turn, led to an exponentially growing battery market and, conversely, increasing attention on how we can reduce the environmental impact of batteries and promote a more efficient circular economy to achieve real net zero. As these batteries reach the end of their first life, challenges arise as to how to collect and process them, in order to maximise their economical use before finally being recycled. Despite the growing body of work around this topic, the decision-making process on which pathways batteries could take is not yet well understood, and clear policies and standards to support implementation of processes and infrastructure are still lacking. Requirements and challenges behind recycling and second life applications are complex and continue being defined in industry and academia. Both pathways rely on cell collection, selection and processing, and are confronted with the complexities of pack disassembly, as well as a diversity of cell chemistries, state-of-health, size, and form factor. There are several opportunities to address these barriers, such as standardisation of battery design and reviewing the criteria for a battery's end-of-life. These revisions could potentially improve the overall sustainability of batteries, but may require policies to drive such transformation across the industry. The influence of policies in triggering a pattern of behaviour that favours one pathway over another are examined and suggestions are made for policy amendments that could support a second life pipeline, while encouraging the development of an efficient recycling industry. This review explains the different pathways that end-of-life EV batteries could follow, either immediate recycling or service in one of a variety of second life applications, before eventual recycling. The challenges and barriers to each pathway are discussed, taking into account their relative environmental and economic feasibility and competing advantages and disadvantages of each. The review identifies key areas where processes need to be simplified and decision criteria clearly defined, so that optimal pathways can be rapidly determined for each end-of-life battery.

SIMULATION OF TRANSIENT PROCESSES IN THE STABILIZATION SYSTEM OF DARRIEUS WIND TURBINE ROTOR ROTATIONS

In the work, the simulation of transient processes of the rotor speed system stabilization of vertical-axis wind turbines, controlled by changes of the blade length, is carried out. It is proved that real roots of the characteristic equation of a closed system with the proportional-integral law of feedback leads to a significantly longer length of transient processes than with complex-connected roots.

Total Cuntz semigroup, extension, and Elliott Conjecture with real rank zero

AbstractIn this paper, we exhibit two unital, separable, nuclear ‐algebras of stable rank one and real rank zero with the same ordered scaled total K‐theory, but they are not isomorphic with each other, which forms a counterexample to the Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of the total K‐theory. For the general setting, with a new invariant, the total Cuntz semigroup [2], we classify a large class of ‐algebras obtained from extensions. The total Cuntz semigroup, which distinguishes the algebras of our counterexample, could possibly classify all the ‐algebras of stable rank one and real rank zero.

In search of Newton-type inequalities

In this paper, we prove a number of results providing either necessary or sufficient conditions guaranteeing that the number of real roots of real polynomials of a given degree is either less or greater than a given number. We also provide counterexamples to two earlier conjectures refining Descartes rule of signs.

Real univariate polynomials with given signs of coefficients and simple real roots

We continue the study of different aspects of Descartes' rule of signs and discuss the connectedness of the sets of real degree $d$ univariate monic polynomials (i.~e. with leading coefficient $1$) with given numbers $\ell ^+$ and $\ell ^-$ of positive and negative real roots and given signs of the coefficients; the real roots are supposed all simple and the coefficients all non-vanishing. That is, we consider the space $\mathcal{P}^d:=\{ P:=x^d+a_1x^{d-1}+\dots +a_d\}$, $a_j\in \mathbb{R}^*=\mathbb{R}\setminus \{ 0\}$, the corresponding sign patterns $\sigma=(\sigma_1,\sigma_2,\dots, \sigma_d)$, where $\sigma_j=$sign$(a_j)$, and the sets $\mathcal{P}^d_{\sigma ,(\ell ^+,\ell ^-)}\subset \mathcal{P}^d$ of polynomials with given triples $(\sigma ,(\ell ^+,\ell ^-))$.We prove that for degree $d\leq 5$, all such sets are connected or empty. Most of the connected sets are contractible, i.~e. able to be reduced to one of their points by continuous deformation. Empty are exactly the sets with $d=4$, $\sigma =(-,-,-,+)$, $\ell^+=0$, $\ell ^-=2$, with $d=5$, $\sigma =(-,-,-,-,+)$, $\ell^+=0$, $\ell ^-=3$, and the ones obtained from them under the $\mathbb{Z}_2\times \mathbb{Z}_2$-actiondefined on the set of degree $d$ monic polynomials by its two generators which are two commuting involutions: $i_m\colon P(x)\mapsto (-1)^dP(-x)$ and $i_r\colon P(x)\mapsto x^dP(1/x)/P(0)$. We show that for arbitrary $d$, two following sets are contractible:1) the set of degree $d$ real monic polynomials having all coefficients positive and with exactly $n$ complex conjugate pairs of roots ($2n\leq d$);2) for $1\leq s\leq d$, the set of real degree $d$ monic polynomials with exactly $n$ conjugate pairs ($2n\leq d$) whose first $s$ coefficients are positive and the next $d+1-s$ ones are negative.For any degree $d\geq 6$, we give an example of a set $\mathcal{P}^d_{\sigma ,(\ell^+,\ell^-)}$ having $\Lambda (d)$ connected compo\-nents, where $\Lambda (d)\rightarrow \infty$ as $d\rightarrow \infty$.