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In this article, we introduce two multivalued contractive mappings within the framework of Hausdorff controlled metric spaces, employing concepts of admissibility and C$$ C $$‐class functions. The first mapping is a (β∗,ζ)$$ \left({\beta}_{\ast },\zeta \right) $$‐generalized contractive multivalued mapping, and the second is a (β,W(ζ,ξ))$$ \left({\beta}_{,}\mathcal{W}\left(\zeta, \xi \right)\right) $$‐multivalued mapping. We establish conditions guaranteeing the existence of fixed points for these mappings. To support our theoretical findings, we provide a numerical example, demonstrating the independence of the contractive conditions for the mappings. Furthermore, we apply our results to a specific problem in polynomial equations. These findings complement several existing results in the literature. [ABSTRACT FROM AUTHOR]

Copyright of Automatisierungstechnik is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In algebra, polynomials are expressions in which a sum is multiplied by its coefficients involving one or more variables such as a0xn + a1xn−1 + a2xn−2 + ... + an. This paper proposes a convolutional neural network model that takes a picture of an equation as an input and outputs the value of the unknown variable "x" for that equation. The proposed model can solve cubic, quadratic, and quintic equations using an enhanced Inception V4 Convolutional Neural Network(CNN) model. The proposed model is implemented as a web application that allows the user to enter a picture of the handwritten equation and returns the recognized equation. Three datasets were used to collect handwritten arithmetic equations. MathNet was used to contain handwritten arithmetic symbols, the MNIST dataset was used to collect handwritten digits, and EMNIST was used to collect alphabets used as variables in polynomial equations. The proposed model dataset contains 16 classes, including 10 numerals, 3 mathematical operators, and 3 variables of image size 64 × 64 pixels. [ABSTRACT FROM AUTHOR]

In this article, we constructed a derivative-free family of iterative techniques for extracting simultaneously all the distinct roots of a non-linear polynomial equation. Convergence analysis is discussed to show that the proposed family of iterative method has fifth order convergence. Nonlinear test models including fractional conversion, predator-prey, chemical reactor and beam designing models are included. Also many other interesting results concerning symmetric problems with application of group symmetry are also described. The simultaneous iterative scheme is applied starting with the initial estimates to get the exact roots within the given tolerance. The proposed iterative scheme requires less function evaluations and computation time as compared to existing classical methods. Dynamical planes are exhibited in CAS-MATLAB (R2011B) to show how the simultaneous iterative approach outperforms single roots finding methods that might confine the divergence zone in terms of global convergence. Furthermore, convergence domains, namely basins of attraction that are symmetrical through fractal-like edges, are analyzed using the graphical tool. Numerical results and residual graphs are presented in detail for the simultaneous iterative method. An extensive study has been made for the newly developed simultaneous iterative scheme, which is found to be efficient, robust and authentic in its domain.

In this article, we constructed a derivative-free family of iterative techniques for extracting simultaneously all the distinct roots of a non-linear polynomial equation. Convergence analysis is discussed to show that the proposed family of iterative method has fifth order convergence. Nonlinear test models including fractional conversion, predator-prey, chemical reactor and beam designing models are included. Also many other interesting results concerning symmetric problems with application of group symmetry are also described. The simultaneous iterative scheme is applied starting with the initial estimates to get the exact roots within the given tolerance. The proposed iterative scheme requires less function evaluations and computation time as compared to existing classical methods. Dynamical planes are exhibited in CAS-MATLAB (R2011B) to show how the simultaneous iterative approach outperforms single roots finding methods that might confine the divergence zone in terms of global convergence. Furthermore, convergence domains, namely basins of attraction that are symmetrical through fractal-like edges, are analyzed using the graphical tool. Numerical results and residual graphs are presented in detail for the simultaneous iterative method. An extensive study has been made for the newly developed simultaneous iterative scheme, which is found to be efficient, robust and authentic in its domain. [ABSTRACT FROM AUTHOR]

Initial conditions can have a substantial impact on the behavior of iterative root-finding techniques for nonlinear equations. By allowing complex starting points and complex roots, it is possible to examine the basins of attraction in the complex plane in order to compare the performance of various iterative techniques. In this paper, a one-parameter family of third-order root-finding methods is studied by varying its parameter A within −2.0 and 2.4 and applying it to a polynomial equation of high degree (degree 25). This family includes the Euler–Chebyshev’s (A = 0), Halley’s (A = 1) and BSC (A = 2) techniques. According to the results, the one-parameter family provides the best performance for values near A = 1, which equals to the Halley’s method.

In the present paper, we consider a special class of third-order polynomial evolutionary equations. These equations, via Lie theory admit the same one-parameter point transformations which leave the equations invariant. Reductions with these invariant functions lead to highly nonlinear third-order ordinary differential equations. We use a power series to establish interesting solutions to the reduced equations, whereby recurrence relations occur and convergence of the series may be tested. Finally, the conserved vectors of the class are constructed. [ABSTRACT FROM AUTHOR]

This article explores the accessibility of symbolic computations, such as using the Wolfram Mathematica environment, in promoting the shift from informal experimentation to formal mathematical justifications. We investigate the accuracy of computational results from mathematical software in the context of a certain summation in trigonometry. In particular, the key issue addressed here is the calculated sum ∑ n = 0 44 tan ⁡ 1 + 4 n ° . This paper utilizes Wolfram Mathematica to handle the irrational numbers in the sum more accurately, which it achieves by representing them symbolically rather than using numerical approximations. Can we rely on the calculated result from Wolfram, especially if almost all the addends are irrational, or must the students eventually prove it mathematically? It is clear that the problem can be solved using software; however, the nature of the result raises questions about its correctness, and this inherent informality can encourage a few students to seek viable mathematical proofs. In this way, a balance is reached between formal and informal mathematics. [ABSTRACT FROM AUTHOR]

Solving polynomial eigenvalue problems with eigenvector nonlinearities (PEPv) is an interesting computational challenge, outside the reach of the well-developed methods for nonlinear eigenvalue problems. We present a natural generalization of these methods which leads to a contour integration approach for computing all eigenvalues of a PEPv in a compact region of the complex plane. Our methods can be used to solve any suitably generic system of polynomial or rational function equations. [ABSTRACT FROM AUTHOR]

Copyright of @limentech: Ciencia y Tecnología Alimentaria is the property of Journal @limentech, University of Pamplona and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Abstract In the present paper, we consider a special class of third-order polynomial evolutionary equations. These equations, via Lie theory admit the same one-parameter point transformations which leave the equations invariant. Reductions with these invariant functions lead to highly nonlinear third-order ordinary differential equations. We use a power series to establish interesting solutions to the reduced equations, whereby recurrence relations occur and convergence of the series may be tested. Finally, the conserved vectors of the class are constructed.

We apply polynomial approach to the design of optimal low-order controllers for linear stationary SISO systems described by differential and algebraic equations. The method of critical root diagrams and root polynomials is used to design such controllers in classical control systems. An unstable control plant is given by an improper transfer fraction with a sextic numerator and a quartic denominator. For it, stabilizing PI controllers are tuned; the optimal controller according to the relative stability is chosen among them. Calculation of the impulse response confirms its astatism and lack of impulse. The method is the same as for classical control systems, however, the resulting polynomial systems of equations turn out to be higher in degree and more difficult to solve numerically. [ABSTRACT FROM AUTHOR]

The assessment of labor progress is germane to every woman in labor. Two labor disorders—arrest of dilation and arrest of descent—are the primary indications for surgery in close to 50% of all intrapartum cesarean deliveries and are often contributing indications for cesarean deliveries for fetal heart rate abnormalities. Beginning in 1954, the assessment of labor progress was transformed by Friedman. He published a series of seminal works describing the relationship between cervical dilation, station of the presenting part, and time. He proposed nomenclature for the classification of labor disorders. Generations of obstetricians used this terminology and normal labor curves to determine expected rates of dilation and fetal descent and to decide when intervention was required. The analysis of labor progress presents many mathematical challenges. Clinical measurements of dilation and station are imprecise and prone to variation, especially for inexperienced observers. Many interrelated factors influence how the cervix dilates and how the fetus descends. There is substantial variability in when data collection begins and in the frequency of examinations. Statistical methods to account for these issues have advanced considerably in recent decades. In parallel, there is growing recognition among clinicians of the limitations of using time alone to assess progress in cervical dilation in labor. There is wide variation in the patterns of dilation over time and most labors do not follow an average dilation curve. Reliable assessment of labor progression is important because uncertainty leads to both over-use and under-use of cesarean delivery and neither of these extremes are desirable. This review traces the evolution of labor curves, describes how limitations are being addressed to reduce uncertainty and to improve the assessment of labor progression using modern statistical techniques and multi-dimensional data, and discusses the implications for obstetrical practice. [ABSTRACT FROM AUTHOR]

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root and we compute its multiplicity structure. More precisely, given a polynomial system f = (f 1 , ... , f N) ∈ C [ x 1 , ... , x n ] N , we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so-called inverse system that describes the multiplicity structure at the root. We use α -theory test to certify the quadratic convergence, and to give bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria. [ABSTRACT FROM AUTHOR]

This article explores the accessibility of symbolic computations, such as using the Wolfram Mathematica environment, in promoting the shift from informal experimentation to formal mathematical justifications. We investigate the accuracy of computational results from mathematical software in the context of a certain summation in trigonometry. In particular, the key issue addressed here is the calculated sum ∑n=044tan⁡1+4n°. This paper utilizes Wolfram Mathematica to handle the irrational numbers in the sum more accurately, which it achieves by representing them symbolically rather than using numerical approximations. Can we rely on the calculated result from Wolfram, especially if almost all the addends are irrational, or must the students eventually prove it mathematically? It is clear that the problem can be solved using software; however, the nature of the result raises questions about its correctness, and this inherent informality can encourage a few students to seek viable mathematical proofs. In this way, a balance is reached between formal and informal mathematics.

Polynomial through the origin equations based on the Bouguer—Lambert and Beer laws were proposed for the accurate representation of positive and negative deviations from linearity and absorption flattening. Cubic and higher order equations of absorbance on concentration, thickness, and molar absorptivity do not provide explicit inverse equations which are required to determine the concentration, thickness, and molar absorptivity. Quadratic equations provide explicit inverse equations. The proposed quadratic equations are and for positive and negative deviations from linearity of the Bouguer—Lambert law, and for positive and negative deviations from linearity of the Beer law, and for absorption flattening, where is absorbance, and are linear and nonlinear absorption coefficients, is molar concentration, is thickness, and are linear and nonlinear molar absorptivities, and are linear and nonlinear coefficients. Concentration, thickness, and molar absorptivity are determined from inverse equations. [ABSTRACT FROM AUTHOR]

Let F q be the finite field of q elements, where q = p r is a power of the prime p, and β 1 , β 2 , ⋯ , β r be an ordered basis of F q over F p . For ξ = ∑ i = 1 r x i β i , x i ∈ F p , we define the Thue–Morse or sum-of-digits function T (ξ) on F q by T (ξ) = ∑ i = 1 r x i. For a given pattern length s with 1 ≤ s ≤ q , a vector α = (α 1 , ... , α s) ∈ F q s with different coordinates α j 1 ≠ α j 2 , 1 ≤ j 1 < j 2 ≤ s , a polynomial f (X) ∈ F q [ X ] of degree d and a vector c = (c 1 , ... , c s) ∈ F p s we put T (c , α , f) = { ξ ∈ F q : T (f (ξ + α i)) = c i , i = 1 , ... , s }. In this paper we will see that under some natural conditions, the size of T (c , α , f) is asymptotically the same for all c and α in both cases, p → ∞ and r → ∞ , respectively. More precisely, we have | T (c , α , f) | - p r - s ≤ (d - 1) q 1 / 2 under certain conditions on d, q and s. For monomials of large degree we improve this bound as well as we find conditions on d, q and s for which this bound is not true. In particular, if 1 ≤ d < p we have the dichotomy that the bound is valid if s ≤ d and for s ≥ d + 1 there are vectors c and α with T (c , α , f) = ∅ so that the bound fails for sufficiently large r. The case s = 1 was studied before by Dartyge and Sárközy. [ABSTRACT FROM AUTHOR]

For feed producers who suffer from high intolerance to production costs, the only way to cope with the condition is to avoid devices that drive up costs. Extruded feed processed from a clay-slate dryer through a fluidized bed could be used to make fish feed. The aim of the study was optimise the process conditions on the clay-slate fluidized bed dryer operating at a commercial production of fish feed using the response surface methodology. The fish feed composition were processed at bed height (50-200 mm), drying air temperature (60–120°C), airflow velocity (0.66-0.70 m/s), drying time (10–90 min) and extrudates size (4–8 mm). Product quality parameters such as moisture ratio and dryer efficiency were determined and analyzed. Second-order polynomial equations, containing all the process variables, were used to measured process. Moisture ratio was influenced mostly by linear relationship temperature and drying time. The temperature and the quadratic temperature conditions significantly affected the efficiency of the dryer. For the fluidized bed drying of extruded fish feed, optimal conditions were set for the bed height of 185.76 mm, a temperature of 97.2°C, an air flow rate of 0.67, a drying time of 65.36 min and an extrudate size of 7.40 mm recommended. At these conditions the moisture ratio and efficiency were 0.86 and 74.39, respectively. The influence of the various components of the fluidized bed dryer on the drying rate must be better understood so that control systems can be developed to take full advantage of this technology.

Homotopy function is used in conjunction with homotopy continuation methods (HCM) and a classical numerical method to approximate the roots of nonlinear algebraic equations, particularly in solving polynomial equations. In this paper, we develop a new function, known as quadratic parameter homotopy function, and apply it to solve several non- application equations and several application equations. This homotopy function is used in two homotopy methods i.e., Newton-HCM and Ostrowski-HCM. The results obtained indicate that the proposed homotopy function performs better than the existing homotopy functions.

We study the robustness of the steady states of a class of systems of autonomous ordinary differential equations (ODEs), having as a central example those arising from (bio)chemical reaction networks. More precisely, we study under what conditions the steady states of the system are contained in a parallel translate of a coordinate hyperplane. To this end, we focus mainly on ODEs consisting of generalized polynomials and make use of algebraic and geometric tools to relate the local and global structure of the set of steady states. Specifically, we consider the local property termed zero sensitivity at a coordinate xi, which means that the tangent space is contained in a hyperplane of the form xi = c, and provide a criterion to identify it. We consider the global property termed absolute concentration robustness (ACR), meaning that all steady states are contained in a hyperplane of the form xi = c. We clarify and formalize the relation between the two approaches. In particular, we show that ACR implies zero sensitivity and identify when the two properties do not agree, via an intermediate property we term local ACR. For families of systems arising from modeling biochemical reaction networks, we obtain the first practical and automated criterion to decide upon (local) ACR. [ABSTRACT FROM AUTHOR]

Let q = pr be the power of a prime p and (β1, . . . , βr) be an ordered basis of Fq over Fq. For ξ = ∑j=1r xj βj ∈ Fq with digits xj ∈ Fq, we define the Rudin-Shapiro function R on Fq by R(ξ) = ∑j=1r−1 xixi+1, ξ ∈ Fq. For a non-constant polynomial f(X) ∈ Fq[X] and c ∈ Fq we study the number of solutions ξ ∈ Fq of R(ƒ(ξ)) = c. If the degree d of ƒ(X) is fixed, r ≥ 6 and p → ∞, the number of solutions is asymptotically pr−1 for any c. The proof is based on the Hooley-Katz Theorem. [ABSTRACT FROM AUTHOR]

In this article we compute the radii of the largest disks for which the functions in the class S of normalized, analytic and univalent functions belong to certain subclasses of it. [ABSTRACT FROM AUTHOR]

An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the leading monomials of polynomials in the ideal and the Specht polynomials contained in the ideal. This provides applications in several contexts. Most notably, this connection gives information about the solutions of the corresponding set of equations. From another perspective, it restricts the isotypic decomposition of the ideal viewed as a representation of the symmetric group. [ABSTRACT FROM AUTHOR]

We examine quadratic surfaces in 3-space that are tangent to nine given figures. These figures can be points, lines, planes or quadrics. The numbers of tangent quadrics were determined by Hermann Schubert in 1879. We study the associated systems of polynomial equations, also in the space of complete quadrics, and we solve them using certified numerical methods. Our aim is to show that Schubert's problems are fully real. [ABSTRACT FROM AUTHOR]

We investigate the complexity of solving systems of polynomial equations over finite groups. In 1999 Goldmann and Russell showed NP -completeness of this problem for non-Abelian groups. We show that the problem can become tractable for some non-Abelian groups if we fix the number of equations. Recently, Földvári and Horváth showed that a single equation over groups which are semidirect products of a p-group with an Abelian group can be solved in polynomial time. We generalize this result and show that the same is true for systems with a fixed number of equations. This shows that for all groups for which the complexity of solving one equation has been proved to be in P so far, solving a fixed number of equations is also in P . Using the collecting procedure presented by Horváth and Szabó in 2006, we furthermore present a faster algorithm to solve systems of equations over groups of order pq. [ABSTRACT FROM AUTHOR]

Homotopy function is used in conjunction with homotopy continuation methods (HCM) and a classical numerical method to approximate the roots of nonlinear algebraic equations, particularly in solving polynomial equations. In this paper, we develop a new function, known as quadratic parameter homotopy function, and apply it to solve several non- application equations and several application equations. This homotopy function is used in two homotopy methods i.e., Newton-HCM and Ostrowski-HCM. The results obtained indicate that the proposed homotopy function performs better than the existing homotopy functions. [ABSTRACT FROM AUTHOR]

This paper primarily discusses and proves the Hyers-Ulam stability of three types of polynomial equations: xn+a1x+a0 = 0, anxn+...+a1x+a0 = 0, and the infinite series equation: ... aixi = 0, in dislocated quasi-metric spaces under certain conditions by constructing contraction mappings and using fixed-point methods. We present an example to illustrate that the Hyers-Ulam stability of polynomial equations in dislocated quasi-metric spaces do not work when the constant term is not equal to zero. [ABSTRACT FROM AUTHOR]

In this paper, we present a complete method for finding the roots of all polynomials of the form ϕ (z) = c n z n + c n − 1 z n − 1 + ⋯ + c 1 z + c 0 over a given octonion division algebra. When ϕ (z) is monic, we also consider the companion matrix and its left and right eigenvalues and study their relations to the roots of ϕ (z) , showing that the right eigenvalues form the conjugacy classes of the roots of ϕ (z) and the left eigenvalues form a larger set than the roots of ϕ (z). [ABSTRACT FROM AUTHOR]

Identifiability of statistical models is a fundamental regularity condition that is required for valid statistical inference. Investigation of model identifiability is mathematically challenging for complex models such as latent class models. Jones et al. used Goodman's technique to investigate the identifiability of latent class models with applications to diagnostic tests in the absence of a gold standard test. The tool they used was based on examining the singularity of the Jacobian or the Fisher information matrix, in order to obtain insights into local identifiability (ie, there exists a neighborhood of a parameter such that no other parameter in the neighborhood leads to the same probability distribution as the parameter). In this paper, we investigate a stronger condition: global identifiability (ie, no two parameters in the parameter space give rise to the same probability distribution), by introducing a powerful mathematical tool from computational algebra: the Gröbner basis. With several existing well‐known examples, we argue that the Gröbner basis method is easy to implement and powerful to study global identifiability of latent class models, and is an attractive alternative to the information matrix analysis by Rothenberg and the Jacobian analysis by Goodman and Jones et al. [ABSTRACT FROM AUTHOR]

Two multi‐parametric iterative algorithms are developed to solve the forward discrete periodic Lyapunov matrix equation associated with discrete‐time linear periodic systems. An important feature of one of the proposed algorithms is that the information in the current and the last steps is used to update the iterative sequence. Necessary and sufficient conditions for these two algorithms to be convergent are provided in terms of the roots of a set of polynomial equations. Based on these conditions, a two‐dimensional section method is established to search suboptimal tuning parameters for these two algorithms. In addition, convergence properties are also analysed for some special cases of the obtained algorithms. Finally, numerical examples are provided to illustrate the effectiveness of the proposed algorithms. It can be found that the presented algorithms have better convergence performance than some existing iterative algorithms by choosing proper tuning parameters. [ABSTRACT FROM AUTHOR]

Newton's method is a widely applicable and empirically efficient method for finding the solution to a set of equations. The recently developed algebraic geometry analyses provide information-theoretic bounds on the sampling rate to ensure the existence of a unique completion. A remained open question from these works is to retrieve the sampled data when the sampling rate is very close to the mentioned information-theoretic bounds. This paper is concerned with proposing algorithms to retrieve the sampled data when the sampling rate is too small and close to the mentioned information-theoretic bounds. Hence, we propose a new approach for recovering a partially sampled low-rank matrix or tensor when the number of samples is only slightly more than the dimension of the corresponding manifold, by solving a set of polynomial equations using Newton's method. In particular, we consider low-rank matrix completion, matrix sensing, and tensor completion. Each observed entry contributes one polynomial equation in terms of the factors in the rank factorization of the data. By exploiting the specific structures of the resulting set of polynomial equations, we analytically characterize the convergence regions of the Newton's method for matrix completion and matrix sensing. Through extensive numerical results, we show that the proposed approach outperforms the well-known methods such as nuclear norm minimization and alternating minimization in terms of the success rate of data recovery (noiseless case) and peak signal-to-noise ratio (noisy case), especially when the sampling rate is very low. [ABSTRACT FROM AUTHOR]

In 2003, Moreno and Castro proved that the covering radius of a class of primitive cyclic codes over the finite field F2 having minimum distance 5 (resp. 7) is 3 (resp. 5). We here give a generalization of this result as follows: the covering radius of a class of primitive cyclic codes over F2 with minimum distance greater than or equal to r+2 is r, where r is any odd integer. Moreover, we prove that the primitive binary e-error correcting BCH codes of length 2f-1 have covering radii 2e-1 for an improved lower bound of f. [ABSTRACT FROM AUTHOR]

Information about the current mode of the electric power system (EPS) is received by the dispatch control centers in the form of telemetry and tele-signals by the SCADA complexes, and from phasor measurement units (PMU) devices. Tele-measurements include information about the mode parameters, and the state of the switching equipment. Since the system of equation of state is nonlinear, the problem of state estimation is traditionally solved using iterative methods. This article presents the results of solving the state estimation problem using a method based on the Kipnis-Shamir re-linearization method, which allows solving it by non-iterative method. The results of the solution are given on the example of a power transmission line with 500 kV voltage.

We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over C. Applying recent results of Baker and DeMarco (2011) and of Ghioca, Krieger, Nguyen and Ye (2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang (2015). As a by-product, we also slightly improve a result of Silverman (2008) and recover a result of Akbary and Ghioca (2009) as special extreme cases of our estimates. [ABSTRACT FROM AUTHOR]

This study aimed to formulate suitable nanovesicles (NVs) for transdermal delivery of Ondansetron. It also illustrated a practical example for the importance of Box-Cox transformation. A 2 3 full factorial design was used to enable testing transfersomes, ethosomes, and transethosomes of Ondansetron simultaneously. The independent variables (IVs) studied were sodium taurocholate amount, ethanol volume in hydration medium and sonication time. The studied dependent variables (DVs) were: particle size (PS), zeta potential (ZP) and entrapment efficiency (EE). Polynomial equations were used to study the influence of IVs on each DV. Numerical multiple response optimization was applied to select an optimized formula (OF) with the goals of minimizing PS and maximizing ZP absolute value and EE. Box-Cox transformation was adopted to enable modeling PS raised to the power of 1.2 with an excellent prediction R 2 of 1.000. ZP and EE were adequately represented directly with prediction R 2 of 0.9549 and 0.9892 respectively. Response surface plots helped in explaining the influence of IVs on each DV. Two-sided 95% prediction interval test and percent deviation of actual values from predicted ones proved the validity of the elucidated models. The OF was a transfersomal formula with desirability of 0.866 and showed promising results in ex-vivo permeation study. [ABSTRACT FROM AUTHOR]

Particle filtering (PF) is a popular nonlinear estimation technique and has been widely used in a variety of applications such as target tracking. Within the PF framework, one critical design choice is the selection of the proposal distribution from which particles are drawn. In this paper, we advocate using as proposal distribution a Gaussian-mixture-based approximation of the posterior probability density function (pdf) after taking into account the most recent measurement. The novelty of our approach is that the parameters of each Gaussian used in the mixture are determined analytically to match the modes of the underlying unknown posterior pdf. As a result, particles are sampled along the most probable regions of the state space, hence reducing the probability of particle depletion. Based on the analytically determined proposal distribution, we introduce a novel PF, termed analytically guided sampling-based PF, which is validated in range-only and bearing-only target tracking. [ABSTRACT FROM AUTHOR]

Given a central division algebra D of degree d over a field F, we associate to any standard polynomial $$\phi (z)=z^n+c_{n-1} z^{n-1}+\cdots +c_0$$ over D a 'companion polynomial' $$\Phi (z)$$ of degree nd with coefficients in F. The roots of $$\Phi (z)$$ in D are exactly the set of conjugacy classes of the roots of $$\phi (z)$$ . When D is a quaternion algebra, we explain how all the roots of $$\phi (z)$$ can be recovered from the roots of $$\Phi (z)$$ . Along the way, we are able to generalize a few known facts from $$\mathbb {H}$$ to any division algebra. The first is the connection between the right eigenvalues of a matrix and the roots of its characteristic polynomial. The second is the connection between the roots of a standard polynomial and left eigenvalues of the companion matrix. [ABSTRACT FROM AUTHOR]

In this paper the problem of existence of solution of polynomial equations over the field of real numbers is considered. In particular, the explicit necessary and sufficient conditions are established for the equation A(z)X + B(z)Y = C(z) in polynomial matrices to have a solution for X and Y over the field of real numbers, with X being non-singular, for every polynomial matrix C(z) from a given class. [ABSTRACT FROM AUTHOR]

A method is proposed for calculating the cutting characteristics in the end milling of stainless steel workpieces. The calculation is based on analysis of the parameters for the chip-formation zone and takes account of softening of the material under the action of the cutting temperature. The calculated tool life and the torque are approximated by polynomial equations, which may be used to calculate the constraints on milling and to optimize the machining conditions. [ABSTRACT FROM AUTHOR]

We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the polynomial equations. Our algorithm works in time polynomial in the number of equations and the logarithm of the size of the field, whenever the degree of the polynomial equations is constant. As a consequence we design polynomial time quantum algorithms for two algebraic hidden structure problems: for the hidden subgroup problem in certain semidirect product p -groups of constant nilpotency class, and for the multi-dimensional univariate hidden polynomial graph problem when the degree of the polynomials is constant. 1 [ABSTRACT FROM AUTHOR]

We give a full description of the dynamics of the Abel equation \(z'=z^3+f(t)\) for some special complex valued \(f\). We also prove the existence of at least three periodic solutions for equations of the form \(z'=z^n+f(t)\) for odd \(n \geq 5\).

We give a few sufficient conditions for the existence of periodic solutions of the equation \(\dot{z}=\sum_{j=0}^n a_j(t)z^j-\sum_{k=1}^r c_k(t)\overline{z}^k\) where \(n \gt r\) and \(a_j\)'s, \(c_k\)'s are complex valued. We prove the existence of one up to two periodic solutions.

In this paper, the concept of canonical representation is proposed to find fuzzy roots of fuzzy polynomial equations. We transform fuzzy polynomial equations to system of crisp polynomial equations, this transformation is perform by using canonical representation based on three parameters Value, Ambiguity and Fuzziness.

The double wishbone (DWB) and the MacPherson strut (MPS) suspension systems are commonly used independent suspensions in passenger cars. Their kinematics are complicated, and have not been analysed comprehensively in existing literature. This paper presents an analysis of the position kinematics of the complete spatial model of these suspension systems. The presented solution is built upon two key elements: the use of Rodrigue's parameters to develop an algebraic set of equations representing the kinematics of the mechanisms, and the computation of Gröbner basis as a method of solving the resulting set of equations. It is found that the final univariate equation representing all the kinematic solutions for a given pair of steering and road-profile inputs, in the general case, is of 64 degree, for both the suspension mechanisms. It is also shown that in certain special cases, both the suspensions generate 28 solutions, instead of 64. Numerical accuracy of the solutions obtained is established by computing the residuals of the original set of kinematic constraint equations. The configurations of the mechanisms for the real solutions are depicted graphically. Finally, the responses of the suspensions to continuously varying steering and road-profile inputs are computed using a branch-tracking technique. [ABSTRACT FROM AUTHOR]

Homotopy continuation methods (HCMs) can be used to find the solutions of polynomial equations. The advantages of HCMs over classical methods such as the Newton and bisection methods are that HCMs are able to resolve divergence and start- ing value problems. In this paper, we develop Super Ostrowski-HCM as a technique to overcome the starting value problem. We compare the performance of this proposed method with Ostrowski-HCM. The results provide evidence of the superiority of Super Ostrowski-HCM. [ABSTRACT FROM AUTHOR]

The main objective of this article is to make a formal description of the polynomial, polynomial equations with definitions and their properties. Besides studying some of its uses in real life situations, we shall discuss polynomial approximation using higher order derivatives. [ABSTRACT FROM AUTHOR]

We analyze Tartaglia's account, in 1546, of the circumstances leading to his breakthrough regarding the solution of cubic equations. He claims that he solved x 3 + r x 2 = q in 1530, well before he could handle, in 1535, equations with a linear term px (and no quadratic term). This claim is at variance with Cardano's narrative as well as with later treatments of the problem, in which the solution of equations of the latter type provides the basis for the solution of all the other types of cubic equations. We show that Tartaglia's claim is supported in his text by the use of the theory of continued proportions, that occurs as a Leitmotiv. We show that relations on continued proportions stressed by Pacioli as basic “keys” provide a simple derivation of the results given by Tartaglia, that is consistent with their chronological order. Thus, his narrative contains not only priority claims, but also proposes an account of the mathematical steps that led him to his results. [ABSTRACT FROM AUTHOR]

In this paper, we study the number of different interference alignment (IA) solutions in a $K$ -user multiple-input multiple-output (MIMO) interference channel, when the alignment is performed via beamforming and no symbol extensions are allowed. We focus on the case where the number of IA equations matches the number of variables. In this situation, the number of IA solutions is finite and constant for any channel realization out of a zero-measure set and, as we prove in this paper, it is given by an integral formula that can be numerically approximated using Monte Carlo integration methods. More precisely, the number of alignment solutions is the scaled average of the determinant of a certain Hermitian matrix related to the geometry of the problem. Interestingly, while the value of this determinant at an arbitrary point can be used to check the feasibility of the IA problem, its average (properly scaled) gives the number of solutions. For single-beam systems, the asymptotic growth rate of the number of solutions is analyzed and some connections with classical combinatorial problems are presented. Nonetheless, our results can be applied to arbitrary interference MIMO networks, with any number of users, antennas, and streams per user. [ABSTRACT FROM PUBLISHER]

In this paper, the design of science orbits for the observation of a celestial body has been carried out using polynomial equations. The effects related to the main zonal harmonics of the celestial body and the perturbation deriving from the presence of a third celestial body have been taken into account. The third body describes a circular and equatorial orbit with respect to the primary body and, for its disturbing potential, an expansion in Legendre polynomials up to the second order has been considered. These polynomial equations allow the determination of science orbits around Jupiter's satellite Europa, where the third body gravitational attraction represents one of the main forces influencing the motion of an orbiting probe. Thus, the retrieved relationships have been applied to this moon and periodic sun-synchronous and multi-sun-synchronous orbits have been determined. Finally, numerical simulations have been carried out to validate the analytical results. [ABSTRACT FROM AUTHOR]