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In standard books on calculus the existence of primitive functions of continuous functions is proved, in one way or another, using Riemann sums. In this note we present a completely different self-contained, however probably folkloristic, proof of this existence. Our proof combines, on the one hand, the so-called Stone Weierstrass theorem on uniform approximation of continuous functions on the unit interval by polynomials, and, on the other hand, a classical result from calculus on the existence of limits of differentiated sequences of functions. The sought for primitive is then constructed as the limit of primitives of the polynomials approximating the original function. [ABSTRACT FROM AUTHOR]

Denote by τ k (n) , ω (n) and μ 2 (n) the number of representations of n as a product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [ t ] be the integral part of real number t. For f = ω , 2 ω , μ 2 , τ k , we prove that ∑ n ≤ x f x n = x ∑ d ≥ 1 f (d) d (d + 1) + O (x f + ) for x → ∞ , where ω = 5 3 1 1 0 , 2 ω = 9 1 9 , μ 2 = 2 5 , τ k = 5 k − 1 1 0 k − 1 and > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordellès. [ABSTRACT FROM AUTHOR]

We show that, in general, the characteristic polynomial of a hypergraph is not determined by its "polynomial deck", the multiset of characteristic polynomials of its vertex-deleted subgraphs, thus settling the "polynomial reconstruction problem" for hypergraphs in the negative. The proof proceeds by showing that a construction due to Kocay of an infinite family of pairs of 3-uniform hypergraphs which are non-isomorphic but share the same hypergraph deck, in fact, have different characteristic polynomials. The question remains unresolved for ordinary graphs. [ABSTRACT FROM AUTHOR]

In the present paper we shall obtain the following result: Let n ≥ 2 and m ≥ 1 be integers. Let p (x 1 , ... , x m) be a polynomial with zero constant term in non-commutative variables over an algebraically closed field K. Suppose that 1 ≤ r ≤ n − 1 , where r is the order of p. Then p (T n (K)) is a dense subset of T n (K) (r − 1) (with respect to the Zariski topology). This is a solution of a problem presented by Gargate and de Mello and a supplement to a result obtained by Panja and Prasad in 2023. [ABSTRACT FROM AUTHOR]

Finding the sharp estimate of max | z | = 1 | p ′ (z) | in terms of max | z | = 1 | p (z) | for the class of polynomials p(z) satisfying p (z) ≡ z n p (1 / z) has been a well-known open problem for a long time and many papers in this direction have appeared. The earliest result is due to Govil, Jain and Labelle [9] who proved that for polynomials p(z) satisfying p (z) ≡ z n p (1 / z) and having all the zeros either in left half or right half-plane, the inequality max | z | = 1 | p ′ (z) | ≤ n 2 max | z | = 1 | p (z) | holds. A question was posed whether this inequality is sharp. In this paper, we answer this question in the negative by obtaining a bound sharper than n 2 . We also conjecture that for such polynomials max | z | = 1 | p ′ (z) | ≤ ( n 2 - 2 - 1 4 (n - 2)) max | z | = 1 | p (z) | and provide evidence in support of this conjecture. [ABSTRACT FROM AUTHOR]

This paper investigates a real-time process generator of wideband signals, which calculates waveforms in a field-programmable gate array (FPGA) using the high-level synthesis (HLS) method. To obtain high-resolution and wide-swath images, the generator must produce multiple modes of large time-bandwidth product (TBP) linear frequency modulation (LFM) signals. However, the conventional storage method is unrealistic as it requires huge storage resources to save pre-computed waveforms. Therefore, this paper proposes a novel processing approach that calculates waveforms in real-time based simply on parameters such as the sampling frequency, bandwidth, and time width. Additionally, this paper implements predistortion through the polynomial curve to approximate phase errors of the system. The parallelizing process in the FPGA is necessary to satisfy the high-speed requirement of a digital-to-analog converter (DAC); however, repeatedly multiplexing real-time calculation consumes extensive logic and DSP resources, potentially exceeding FPGA limitations. To address this, this paper proposes a piecewise linear algorithm to conserve resources, which processes the polynomial only once, acquires the difference in two adjacent values through the register and pipeline, and then adds this increment to facilitate parallel computations. The performance of this proposed generator is validated through simulation and implemented in experiments with an X-band airborne SAR system. [ABSTRACT FROM AUTHOR]

For B ∈ F q [ T ] of degree 2 n ≥ 2 , consider the number of ways of writing B = E 2 + γ F 2 , where γ ∈ F q ⁎ is fixed, and E , F ∈ F q [ T ] with deg E = n and deg F = m < n. We denote this restricted sum-of-squares function by S γ ; m (B). We obtain an exact formula for the variance of S γ ; m (B) over short intervals in F q [ T ] where q is an odd prime power. Interestingly, when m + 1 ≤ n ≤ 2 m , the variance vanishes on some (not all) short intervals, which is in contrast to the standard function that counts representations as a sum of two squares. We use the method of additive characters and Hankel matrices that we previously used for the variance and correlations of the divisor function. In Section 2, we give a short overview of our approach. Section 3 gives new results relating to values of quadratic forms, over F q , that are defined from Hankel matrices. [ABSTRACT FROM AUTHOR]

In this work we consider the regular parabolic manifold X and polynomials on it. It is proved some properties of regular parabolic manifolds and described polynomials on complements of Weierstrass algebroid sets. [ABSTRACT FROM AUTHOR]

The traditional linear discriminant analysis (LDA) is a classical dimensionality reduction method. But there are two problems with LDA. One is the small-sample-size (SSS) problem, and the other is the classification ability of LDA is not very strong. In this paper, by studying several common methods of LDA, an implicit rule of the criterion of LDA is found, and then a general framework of LDA combined with matrix function is presented. Based on this framework, the polynomials are used to reconstruct the LDA criterion, and then a new method called polynomial linear discriminant analysis (PLDA) is proposed. The proposed PLDA method has two contributions: first, it addresses the small-sample-size problem of LDA, and second, it enhances the distance of the between-class sample through polynomial function mapping, improving the classification ability. Experimental results on ORL, FERET, AR, and Coil datasets show the superiority of PLDA over existing variations of LDA. [ABSTRACT FROM AUTHOR]

In this article, we design new distance-based topological indices which are computed by a recent polynomial approach. Also, we present a chemical application on the suitability of these indices with some drugs used for the treatment of COVID-19 via QSPR analysis. Curvilinear regression models are obtained and analysed for the physico-chemical properties of the COVID-19 drugs. Our models and findings could aid in the development of new drugs for the treatment of COVID-19. [ABSTRACT FROM AUTHOR]

In this note it is established that a finite family of positive linear operators acting from an Archimedean vector lattice into an Archimedean -algebra with unit is disjointness preserving if and only if the polynomial presented in the form of the product of powers of these operators is orthogonally additive. A similar statement is established for the sum of polynomials represented as products of powers of positive operators. [ABSTRACT FROM AUTHOR]

In this paper, we sharpen some of the known results of polar derivative of a polynomial by establishing the L q -version of a known inequality on the polar derivative of a polynomial. Our result generalizes as well as improves upon some well-known polynomial inequalities in this direction. [ABSTRACT FROM AUTHOR]

We prove a classification of additive polynomial superfunctors, which allows us to compute some extensions of a superfunctor of the form F ∘ A where F is a classical polynomial functor and A is additive. We get a formula which relates these extensions to the classical ones of F. A possible generalisation is conjectured at the end. [ABSTRACT FROM AUTHOR]

For rational functions , where is a polynomial of degree at the most and , with we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erdős-Lax and Turán to rational functions . In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erdős-Lax and Turán type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of the given polynomial. As a further factor of consideration, during the course of this paper we will demonstrate how some recently obtained results could have been proved without invoking the results of Dubinin [Distortion theorems for polynomials on the circle, Sb. Math. 191(12) (2000) 1797–1807]. [ABSTRACT FROM AUTHOR]

Geodesics are of fundamental interest in mathematics, physics, computer science, and many other subjects. The so-called leapfrog algorithm was proposed in [L. Noakes, J. Aust. Math. Soc., 65 (1998), pp. 37--50] (but not named there as such) to find geodesics joining two given points x0 and x1 on a path-connected complete Riemannian manifold. The basic idea is to choose some junctions between x0 and x1 that can be joined by geodesics locally and then adjust these junctions. It was proved that the sequence of piecewise geodesics\{γk} k≥1 generated by this algorithm converges to a geodesic joining x0 and x1. The present paper investigates leapfrog's convergence rate τi,n of ith junction depending on the manifold M. A relationship is found with the maximal root λn of a polynomial of degree n-3, where n (n >3) is the number of geodesic segments. That is, the minimal τi,n is upper bounded by λn(1+c+), where c+ is a sufficiently small positive constant depending on the curvature of the manifold M. Moreover, we show that λn increases as n increases. These results are illustrated by implementing leapfrog on two Riemannian manifolds: the unit 2-sphere and the manifold of all 2 × 2 symmetric positive definite matrices. [ABSTRACT FROM AUTHOR]

Telehealth and cloud-based communication platforms now hide medical data. Text and multimedia data are hidden using encryption, steganography, and watermarking. Steganography may hide medical data in images, music, video, text, and more. Using optical lights, optical steganography deciphers concealed messages. This study established an effective optical steganography model for telemedical data security. Polynomial Cryptographic LSB Substitution (PCLSB-S) reduces the distortion and increases medical data imperceptibility by using two-level randomised cryptographic technique with polynomial variable estimation. Advanced encryption standard and polynomial variables for video cover increase data security in which the medical text, picture, and audio payload capacity increased with PCLSB-S model. PCLSB-S model boosts payload and resilience, according to experiments. PCLSN-S model worked for optical steganography. [ABSTRACT FROM AUTHOR]

The uniqueness problems of entire function that share a non-zero finite value have been studied and many results on this topic have been obtained. In this paper we prove a uniqueness theorem for an entire function, which shares polynomials with its higher order derivatives. In particular, the result of the paper is an improvement of the corresponding results of H. Zhong [8] and I. Lahiri and G.K. Ghosh [5]. [ABSTRACT FROM AUTHOR]

The interplays between topology and algebraic geometry present a set of interesting properties. In this paper, we comprehensively revisit the Bézout theorem in terms of topology, and we present a topological proof of the theorem considering n-dimensional space. We show the role of topology in understanding the complete and finite intersections of algebraic curves within a topological space. Moreover, we introduce the concept of symmetrically complex translations of roots in a zero-set of a real algebraic curve, which is called a fundamental polynomial, and we show that the resulting complex algebraic curve is additively decomposable into multiple components with varying degrees in a sequence. Interestingly, the symmetrically complex translations of roots in a zero-set of a fundamental polynomial result in the formation of isomorphic topological manifolds if one of the complex translations is kept fixed, and it induces repeated real roots in the fundamental polynomial as a component. A set of numerically simulated examples is included in the paper to illustrate the resulting manifold structures and the associated properties. [ABSTRACT FROM AUTHOR]

Local polynomial trend models are a special class of state-space models that can be used without having the full information about the process under study, since most of their parameters are embodied in the state vector and estimated immediately. This makes them attractive for use in signal processing. The present work considers problems that arise when using a polynomial model with a local quadratic trend for Bayesian fusion of two humidity sensors. The unknown sensor biases make it impossible for the model to satisfy the observability conditions. There is currently no general solution to this problem. To overcome this difficulty, an approach is presented where the humidity measurement result implicitly includes the bias of one of the sensors. The results of the study can be used to fuse quantities other than humidity when two or more sensors are available. [ABSTRACT FROM AUTHOR]

We study the question of the algebraic difference independence of the Euler gamma function Γ and the functions in a certain class F , which contains those Dirichlet series as L-functions in the extended Selberg class S ♯ and some periodic functions. The main results in this paper are the difference analogues of the corresponding results from Lü (J Math Anal Appl 462(2):1195–1204, 2018) that showed that the Euler gamma function Γ and the functions in F can not satisfy a class of algebraic differential equations with meromorphic coefficients ϕ of Nevanlinna's characteristics satisfying T (r , ϕ) = o (r) , as r → ∞. Examples are provided to show that the main results in this paper, in a sense, are best possible. [ABSTRACT FROM AUTHOR]

In this paper we study the polynomiality and co-polynomiality of association schemes. These two notions were introduced by Ito. We show that P -polynomial association schemes are co-polynomial and Q -polynomial association schemes are polynomial. We also determine the polynomiality and co-polynomiality for 3-class association schemes. Further we give examples of association schemes that are polynomial but not co-polynomial, or co-polynomial but not polynomial. [ABSTRACT FROM AUTHOR]

If is a polynomial of degree n having all its zeros in |z| 6 k, k > 1, V.K. Jain [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 59 (2016), 339-347] proved We first obtain a generalization as well as improvement of the above inequality. Further, we extend our first result to a more generalized result which yields improved results of some known inequalities as particular case. [ABSTRACT FROM AUTHOR]

In this paper, the radial basis function finite difference method is used to solve two-dimensional steady incompressible Navier–Stokes equations. First, the radial basis function finite difference method with polynomial is used to discretize the spatial operator. Then, the Oseen iterative scheme is used to deal with the nonlinear term, constructing the discrete scheme for Navier–Stokes equation based on the finite difference method of the radial basis function. This method does not require complete matrix reorganization in each nonlinear iteration, which simplifies the calculation process and obtains high-precision numerical solutions. Finally, several numerical examples are obtained to verify the convergence and effectiveness of the radial basis function finite difference method based on Oseen Iteration. [ABSTRACT FROM AUTHOR]

Combinatorial t-designs have nice applications in coding theory, finite geometries and engineering areas. t-designs can be constructed from image sets of a fixed size of some special polynomials. This paper constructs t-designs from the quadratic polynomial x 4 + x 3 over GF (2 n) and determine their parameters. We yield 2- 2 n , 3 · 2 n - 2 , 3 · 2 n - 2 3 · 2 n - 2 - 1 designs for n even and 3- 2 n , 2 n - 1 , 2 n - 1 2 n - 2 - 1 designs for n odd. [ABSTRACT FROM AUTHOR]

As is well-known, the choice of variable ordering while computing cylindrical algebraic decomposition (CAD) has a great effect on the time and memory use of the computation as well as the number of sample points computed. In this paper, we indicate that typical CAD algorithms, if executed with respect to a special kind of variable orderings (called "the perfect elimination orderings", PEO), naturally preserve chordality, which is well compatible with an important (variable) sparsity pattern called "the correlative sparsity". If the associated graph of the polynomial system in question is chordal (resp. , is nearly chordal), then a PEO of the associated graph (resp. , of a minimal chordal completion of the associated graph) can be a better variable ordering for the CAD computation than other naive variable orderings in the sense that it results in a much smaller full set of projection polynomials and thus more efficient computation. A new (m , d) -property of the full set of CAD projection polynomials obtained via a PEO is given, which indicates that when the corresponding perfect elimination tree has a lower height, the full set of projection polynomials also tends to have a smaller "size". Furthermore, combining the lower-tree-height rule with Brown's heuristics, a new procedure is proposed to choose better PEOs for CAD computation. [ABSTRACT FROM AUTHOR]

The current paper recovers cubic–quartic optical solitons in fiber Bragg gratings having polynomial law of nonlinear refractive index structures. Lie symmetry analysis is carried out, starting with the basic analysis. Then, it is followed through with improved Kudryashov and generalized Arnous schemes. The parameter constraints are also identified for the existence of such solitons. Numerical surface plots support the adopted applied analysis. [ABSTRACT FROM AUTHOR]

Given a polynomial function f : R n → R and an unbounded closed semi-algebraic set S ⊂ R n , we show that the conditions listed below are characterized exactly in terms of the so-called tangency variety of the restriction of f on S: f is bounded from below on S; f attains its infimum on S; The sublevel sets { x ∈ S | f (x) ≤ λ } for λ ∈ R are compact; f is coercive on S. Besides, we also provide some stability criteria for boundedness and coercivity of f on S. [ABSTRACT FROM AUTHOR]

A model of a particle flow forming a copy of some image and the distance between the copy and the image are estimated using a special probability metric. The ability of the flow of balls to cover the surface, when grinding the balls, was investigated using formulas of stochastic geometry. Reconstruction of characteristics of an inhomogeneous Poisson flow by inaccurate observations is analysed using the Poisson flow point colouring theorem. The dependence of the Poisson parameter of the distribution of the number of customers in a queuing system with an infinite number of servers and a deterministic service time on the peak load created by an inhomogeneous input Poisson flow is estimated. All these models consist of an inhomogeneous Poisson flow of points and marks glued to each point of the flow and are characterised by their mass, area, volume, observability (or non-observability), and service time. The presence of an asymptotic power–law relationship between model objective functions and parameters of mark crushing is established. These results may be applied in nanotechnology, powder metallurgy, ecology, and consumer services in the implementation of the "Smart City" program. The proposed approach is phenomenological in nature and is justified by the results of real observations and experiments. [ABSTRACT FROM AUTHOR]

We give a new algorithmic method of detection of atypical values for 2-variables real polynomial functions with emphasis on the effectivity. [ABSTRACT FROM AUTHOR]

This paper contains the finding of some upper bound estimates for the maximal modulus of a lacunary polynomial of degree n on a circle of radius 0 < r ≤ R ≤ k under the assumption that the polynomial has no zero in a disk of radius k; k > 0. Our result extends some known inequalities concerning derivative of a polynomial into integral analogues and it further generalizes as well as sharpens some other results in this direction. [ABSTRACT FROM AUTHOR]

Polynomials are common algebraic structures, which are often used to approximate functions, such as probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of systems rather than to assume polynomials for only approximating known or empirically estimated distributions. Polynomial distributions offer great modeling flexibility and mathematical tractability. However, unlike canonical distributions, polynomial functions may have non-negative values in the intervals of support for some parameter values; their parameter numbers are usually much larger than for canonical distributions, and the interval of support must be finite. Hence, polynomial distributions are defined here assuming three forms of a polynomial function. Transformations and approximations of distributions and histograms by polynomial distributions are also considered. The key properties of the polynomial distributions are derived in closed form. A piecewise polynomial distribution construction is devised to ensure that it is non-negative over the support interval. A goodness-of-fit measure is proposed to determine the best order of the approximating polynomial. Numerical examples include the estimation of parameters of the polynomial distributions and generating polynomially distributed samples. [ABSTRACT FROM AUTHOR]

Logistic regression is one of statistical methods that used to analyze the correlation between categorical response variables and predictor variables which are categorical or continuous. Many studies on logistic regression have been carried out by assuming that the predictor variable and its logit link function have a linear relationship. However, in several cases it was found that the relationship was not always linear, but could be quadratic, cubic, or in the form of other curves, so that the assumption of linearity was incorrect. Therefore, this study will develop a bivariate polynomial ordinal logistic regression (BPOLR) model which is an extension of ordinal logistic regression, with two correlated response variables in which the relationship between the continuous predictor variable and its logit is modeled as a polynomial form. There are commonly two correlated response variables in scientific research. In this study, each response variable used consisted of three categories. This study aims to obtain parameter estimators of the BPOLR model using the maximum likelihood estimation (MLE) method, obtain test statistics of parameters using the maximum likelihood ratio test (MLRT) method, and obtain algorithms of estimating and hypothesis testing for parameters of the BPOLR model. The results of the first partial derivatives are not closed-form, thus, a numerical optimization such as the Berndt–Hall–Hall–Hausman (BHHH) method is needed to obtain the maximum likelihood estimator. The distribution statistically test is followed the Chi-square limit distribution, asymptotically. [ABSTRACT FROM AUTHOR]

We generalize two results in the papers [1] and [2] about sums of subsets of to the more general case in which the sum is replaced by , where is a rather general polynomial. In particular, a lower bound is obtained for the cardinality of the range of , where the variables and belong to a subgroup of the multiplicative group of the field . We also prove that if a subgroup can be represented as the range of a polynomial for and , then the cardinalities of and are close in order to . [ABSTRACT FROM AUTHOR]

The principal objective of the paper is to study the uniqueness of meromorphic functions when two differential polynomials share a nonzero polynomial. A result of Yang and Hua (Ann Acad Sci Fenn Math 22:395–406, 1997) is the starting point of this kind of research. In this direction, Majumder (Math Slovaca 67(1):151–164, 2017) proved two results, which contain a major gap in their proofs. In the paper, we not only present a corrected version of the results of Majumder but also provide some improvement of those. [ABSTRACT FROM AUTHOR]

The paper deals with a certain trinomial with two strictly complex coefficients. The root locus technique is utilized to obtain a distribution of roots with respect to a unit circle in the complex plane. A number of roots inside the unit disk is described in a relation with the two parameter values. Several appropriate graphs illustrate the trinomial roots distribution in particular cases. [ABSTRACT FROM AUTHOR]

We establish primitive recursive (PR) versions of some known facts about computable ordered fields of reals and computable reals, and apply them to prove primitive recursiveness of several important problems in linear algebra and analysis. One of the central results of this paper is a partial PR analogue of Ershov–Madison's theorem about real closures of computable ordered fields. It allows us, in particular, to obtain PR root-finding algorithms in the PR real and algebraic closures of PR fields with a certain property (PR splitting). We also relate the corresponding fields to the PR reals, as well as introduce and study the notion of a PR metric space. It enables us to derive sufficient conditions for PR computing of normal forms of matrices and solution operators of symmetric hyperbolic systems of PDEs. The methods represent a mix of symbolic and approximate algorithms. [ABSTRACT FROM AUTHOR]

In India, research on the development of new composite materials is extensively increased. In the present study, composites are developed with the renewable materials of plant and animal-based natural fibers. The present study will lead to the experimental and statistical investigation of the composite produced with a combination of plant-based natural fibers, i.e., jute and coco peat powder with the animal-based natural fibers, i.e., silk. In the present study, jute and silk were utilized as reinforcement material, cocopeat powder as a filler material (5%), and epoxy resin as matrix material (35%). The composites were prepared with the varying composition of reinforcement material, and also the filler and matrix material were kept constant. Furthermore, mechanical properties such as tensile, flexural, and impact tests were performed on the developed composites. Further, a comparative study was drawn on the mechanical test results of tensile and flexural strength of the composite using power and polynomial regression model. The regression coefficient (R2) was used to study the correlation between the experimental and predicted values. The results showed that the polynomial regression is the best suitable mathematical model than the power regression model for predicting the composites' tensile and flexural test performance. [ABSTRACT FROM AUTHOR]

The Erdős–Lax Theorem states that, if P(z) is a polynomial of degree n having no zeros in | z | < 1 , then max | z | = 1 | P ′ (z) | ≤ n 2 max | z | = 1 | P (z) |. The problem of generalizing the Erdős–Lax Theorem to the class of polynomials having no zeros in | z | < K , K ≤ 1 is still not settled. Motivated by the above open problem, in this paper we prove some inequalities for a class of polynomials having no zeros in | z | < K , K ≤ 1 , and all these zeros lie either on a ray L emanating from the origin or zeros are symmetrically placed along this line L. Several important consequences are also discussed. Our results are the best possible and examples for the equality cases have been presented. [ABSTRACT FROM AUTHOR]

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb {Z}[x]$. We use an explicit version of Mertens' theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb {Z}[x]$ , our result yields a finite list of primes that certifies the number of distinct irreducible factors of f. [ABSTRACT FROM AUTHOR]

Let N p denote the Newton map induced by a complex polynomial p. Schleicher (Ergod Theory Dyn Syst 22:935–945, 2002) showed that there exists a uniform bound A d , such that for every polynomial p of degree d and for every point z 0 in the immediate basin of a root α of p, we have | z 0 - α | ≤ A d | N p (z 0) - z 0 | . Schleicher also proved that A d ≤ f d , where f d = d 2 (d - 1) 2 (2 d - 1) 2 d d ∼ 4 d - 1 d 2 π d . In 2020, the authors showed that A d < 3 d (3.02) d if d ≥ 12. The goal of this paper is to establish a better bound for A d using collaboration between roots of p. We establish that A d < (1.77) d for all sufficiently large d. As a consequence, it gives a better bound on the expected total number of iterations of Newton's method required to reach all roots of every polynomial p within a given precision. [ABSTRACT FROM AUTHOR]

Modelling of sensor signals is a key element in implementing the procedures for estimation and tracking, sensor data fusion, fault detection and diagnosis, etc. In many cases, using traditional approaches to solve this problem is impossible due to lack of prior information about the observed process or particular sensor characteristics. Starting only from the finite rate of change of the signal at the output of real sensors, the Weierstrass theorem assumes the existence of a polynomial that approximates the signal, but does not specify how to find it. The present work attempts to solve this problem by assuming that the approximating polynomial has to retain that information in the signal which would allow its Bayesian estimation. The described approach has been successfully applied to model a humidity sensor signal, but it can be used in other cases without any problem. [ABSTRACT FROM AUTHOR]

With an emerging development in transport technologies, many applications in vehicular environments access roadside unit (RSU) services for tasks offloading. These applications use different types of paid services (e.g., communication, WiFi, and computing nodes) for their execution. The vehicular fog cloud computing (VFCN) is a cooperative computing environment to handle vehicle applications' mobility and resource allocation. However, the aforementioned collaborative VFCN environment still suffers from mobility and security issues in the existing systems. To address the aforementioned issues, in this work we present a cost-efficient and secure VFCN which consists of a mobility-aware multi-scenario offloading phase (MAMSOP) to deal with the mobility and offloading costs. The objective of the work is to execute applications with minimum delays and costs in a secure way. For the security concern, we propose a security scheme based on fully-homomorphism encryption, which encrypts and decrypts data locally and involves computation on encrypted data rather than decrypting it into its original form. As most of the applications have deadline and mobility constraints, so we propose the fully polynomial-time approximation scheme (FPTAS) based search task scheduling (STS) to ensure the execution of all applications within their given constraints. The goal of STS-FPTAS is to determine the delay and cost-aware scheduling into knapsack-aware resources. The knapsack allows all tasks to be allocated to the available length of resources. The results show that the proposed work optimizes the costs by 40% compared to existing systems. [ABSTRACT FROM AUTHOR]

In this paper, a critic learning structure based on the novel utility function is developed to solve the optimal tracking control problem with the discount factor of affine nonlinear systems. The utility function is defined as the quadratic form of the error at the next moment, which can not only avoid solving the stable control input, but also effectively eliminate the tracking error. Next, the theoretical derivation of the method under value iteration is given in detail with convergence and stability analysis. Then, the dual heuristic dynamic programming (DHP) algorithm via a single neural network is introduced to reduce the amount of computation. The polynomial is used to approximate the costate function during the DHP implementation. The weighted residual method is used to update the weight matrix. During simulation, the convergence speed of the given strategy is compared with the heuristic dynamic programming (HDP) algorithm. The experiment results display that the convergence speed of the proposed method is faster than the HDP algorithm. Besides, the proposed method is compared with the traditional tracking control approach to verify its tracking performance. The experiment results show that the proposed method can avoid solving the stable control input, and the tracking error is closer to zero than the traditional strategy. [ABSTRACT FROM AUTHOR]

The growth of solutions of certain second order nonhomogeneous linear differential equation f’’+A (z) f’+B (z) f=H (z) was investigated in this paper.When the coefficients A (z), B (z) are entire functions and satisfy certain conditions, every nontrivial solution of nonhomogeneous linear differential equation f’’+A (z) f’+B (z) f=H (z) is of infinite order. [ABSTRACT FROM AUTHOR]

For a polynomial having all zeros inside a unit disk, Dubinin [Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros, Journal of Mathematical Sciences, 143 (2007), 3069-3076] proved: max | z | = 1 | p ′ (z) | ≥ 1 2 n + | a n | - | a 0 | | a n | + | a 0 | max | z | = 1 | p (z) |. Various refinements and generalizations of this result were proved, we prove some results which include the results of Wali and Shah [Some applications of Dubinin's lemma to rational functions with prescribed poles, J. Math. Anal. Appl., 450, (2017), 769-779] besides the above inequality due to Dubinin as special cases. [ABSTRACT FROM AUTHOR]

A wide range of precision applications requires video measuring systems that achieve a large number of successive measurements and deliver fast results. Their efficiency is essentially given by the technical performances of the used equipment and by the measurement technique on which they operate. In order to enhance the reliability of such a system, the paper presents a new method of measuring the distance with a single video camera intended to assess the distance at which the object of interest to the camera is located. The technique makes use of a least squares-based sharpness cost function and determines the distance between the camera and the object of interest by minimizing the least squares deviation of the current sharpness values from the sharpness values obtained by calibration. It involves the current sharpness calculation phase, the normalization phase, the phase of calculating the deviations of the current sharpness from the dependencies obtained by calibration and the phase of determining the minimum deviation index. [ABSTRACT FROM AUTHOR]

In this paper, two localized meshless schemes using polynomial basis functions have been presented for solving space–time problems. The proposed methods are especially attractive for solving time-dependent problems in 3D spacious domain. The results showed the simplicity and high accuracy of the proposed methods. Six numerical examples are given to illustrate the effectiveness of the proposed methods. We also presented the pros and cons of the proposed methods by comparing to the results of some recent publications using global polynomial basis functions and localized radial basis functions. [ABSTRACT FROM AUTHOR]

The goal of the paper is to give a description of the images of polynomials with zero constant term on 3 × 3 upper triangular matrix algebras over an algebraically closed field. [ABSTRACT FROM AUTHOR]

This paper studies the saddle point problem of polynomials. We give an algorithm for computing saddle points. It is based on solving Lasserre's hierarchy of semidefinite relaxations. Under some genericity assumptions on defining polynomials, we show that: (i) if there exists a saddle point, our algorithm can get one by solving a finite hierarchy of Lasserre-type semidefinite relaxations; (ii) if there is no saddle point, our algorithm can detect its nonexistence. [ABSTRACT FROM AUTHOR]