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The concept of unity roots plays a central role in the theory of field extensions and polynomials roots' computing. The objective of this paper is to find the algebraic formula for computing the symbolic 2-plithogenic and 3-plithogenic complex roots of unity, where a general formula will be provided with many related examples up to the exponent 3. [ABSTRACT FROM AUTHOR]

Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z) = nP(z) + (α − z)P ′ (z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we prove some Lr inequalities for the polar derivative of a polynomial have all zeros in |z| ≤ 1. Our theorem generalizes a result of Dewan and Mir [K. K. Dewan, A. Mir, Inequalities for the polar derivative of a polynomial, J. Interd. Math. 10 (2007), no. 4, 525–531] and includes as special cases several interesting many known results [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n. The polar derivative of p(z) with respect to a complex number a is defined by Da p(z) = np(z)+(a -z)p'(z). If p(z) has all its zeros in |z| ≥ k, k ≤ 1, then for |a| ≥ k, Aziz and Rather [Math. Inequal. Appl., 1, (1998), 231-238] proved. In this paper, we first improve as well as generalize the above inequality. Besides, we are able to prove an improvement of a result due to Govil and Mctume [Acta Math. Hungar., 104, (2004), 115-126] and also prove an inequality for a subclass of polynomials having all its zeros in |z| ≥ k, k ≤ 1. [ABSTRACT FROM AUTHOR]

The purpose of this article is to generalize the ring of q-Appell polynomials to the complex case. The formulas for q-Appell polynomials thus appear again, with similar names, in a purely symmetric way. Since these complex q-Appell polynomials are also q-complex analytic functions, we are able to give a first example of the q-Cauchy-Riemann equations. Similarly, in the spirit of Kim and Ryoo, we can define q-complex Bernoulli and Euler polynomials. Previously, in order to obtain the q-Appell polynomial, we would make a q-addition of the corresponding q-Appell number with x. This is now replaced by a q-addition of the corresponding q-Appell number with two infinite function sequences C;q(x; y) and S;q(x; y) for the real and imaginary part of a new so-called q-complex number appearing in the generating function. Finally, we can prove q-analogues of the Cauchy-Riemann equations. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n. The polar derivative of p(z) with respect to a complex number α is defined by D α p (z) = n p (z) + (α - z) p ′ (z). If p (z) = z s ∑ j = 0 n - s c j z j , 0 ≤ s ≤ n , has all its zeros in | z | ≤ k , k ≥ 1 , then for | α | ≥ k , Kumar and Dhankhar [Bull, Math. Soc. Sci. Math., 63(4), 359-367 (2020)] proved max | z | = 1 | D α p (z) | ≥ n (| α | - k) 1 + k n - s 1 + (| c n - s | k n - | c 0 | k s) (k - 1) 2 (| c n - s | k n + | c 0 | k s + 1) max | z | = 1 | p (z) |. In this paper, we first improve the above inequality. Besides, we are able to prove an improvement of a result due to Govil and Mctume [Acta. Math. Hungar., 104, 115-126 (2004)] and also prove an inequality for a subclass of polynomials having no zero in | z | < k , k ≤ 1 . [ABSTRACT FROM AUTHOR]

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if it has an ideal periodic auto-correlation function. This paper proposes a method for constructing sparse perfect Gaussian integer sequences (SPGISs) in which most of the sequence elements are zero. The proposed SPGISs are obtained by linearly combining four base sequences or their cyclic-shift equivalents using nonzero Gaussian integer coefficients of equal magnitudes. Each base sequence contains four nonzero elements belonging to the set \\pm 1, \pm j\. The number of nonzero elements of the constructed SPGISs depends on the choice of complex coefficients and cyclic shifts. However, each SPGIS has at most 16 nonzero elements, irrespective of the sequence length. A systematic investigation is performed into the properties of the SPGISs and their Fourier dual equivalents. Finally, a general expression is derived for a perfect Gaussian integer sequence (PGIS) of length $4n$, where $n$ is any positive integer and most of the sequence elements are nonzero. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniqueness of meromorphic functions with q-shift difference-differential polynomials F = [P(f)L(z, f)s](k) and G = [P(g)L(z, g)s](k), where P(z) is a non-constant polynomial with degree n sharing a finite value. The results of this paper are an extension of the previous theorems given by Harina P. Waghamore and Rajeshwari S [19]. [ABSTRACT FROM AUTHOR]

Let p (z) be a polynomial of degree n having no zero in | z | < k , k ≤ 1 , then Govil [Proc. Nat. Acad. Sci., 50(1980), 50-52] proved max | z | = 1 | p ′ (z) | ≤ n 1 + k n max | z | = 1 | p (z) | , provided | p ′ (z) | and | q ′ (z) | attain their maxima at the same point on the circle | z | = 1 , where q (z) = z n p (1 z ‾) ‾. In this paper, we present integral mean inequalities of Turán- and Erdös-Lax-type for the polar derivative of a polynomial by involving some coefficients of the polynomial, which refine some previously proved results and one of our results improves the above Govil inequality as a special case. These results incorporate the placement of the zeros and some coefficients of the underlying polynomial. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to some previously established results. [ABSTRACT FROM AUTHOR]

A dynamic visualization of Julia and Mandelbrot fractals involves creating animated representations of these fractals that change over time or in response to user interaction which allows users to gain deeper insights into the intricate structures and properties of these fractals. This paper explored the dynamic visualization of fractals within Julia and Mandelbrot sets, focusing on a generalized rational type complex polynomial of the form Sc(z) = azn + b/zm + c, where a; b; c ∈ C with |a| > 1 and n;m ∈ N with n > 1. By applying viscosity approximation-type iteration processes extended with s-convexity, we unveiled the intricate dynamics inherent in these fractals. Novel escape criteria was derived to facilitate the generation of Julia and Mandelbrot sets via the proposed iteration process. We also presented graphical illustrations of Mandelbrot and Julia fractals, highlighting the change in the structure of the generated sets with respect to the variations in parameters. [ABSTRACT FROM AUTHOR]

In the current paper, we introduce new subclasses of analytic and bi-univalent functions involving Caputo-type fractional derivatives subordinating to the Lucas polynomial. Furthermore, we find non-sharp estimates on the first two Taylor–Maclaurin coefficients a 2 and a 3 for functions in these subclasses. Using the values of a 2 and a 3 , we determined Fekete–Szegő inequality for functions in these subclasses. Moreover, we pointed out some more subclasses by fixing the parameters involved in Lucas polynomial and stated the estimates and Fekete–Szegő inequality results without proof. [ABSTRACT FROM AUTHOR]

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Let $\mathscr{F}$ be a family of solutions of Laplace equations in a domain D and for each $f\in\mathscr{F}$, f has only zeros of multiplicity at least k. Let n be a positive integer and such that $n\geq\frac{1+\sqrt{1+4k(k+1)^{2}}}{2k}$. Let a be a complex number such that $a\neq0$. If for each pair of functions f and g in $\mathscr{F}$, $f^{n}f^{(k)}$ and $g^{n}g^{(k)}$ share a value in D, then $\mathscr{F}$ is normal in D. [ABSTRACT FROM AUTHOR]

The polar derivative of a polynomial P(z) of degree n with respect to a complex number γ is a polynomial nP(z) + (γ-z)P′(z) of degree at most n-1 and is denoted by DγP(z). We consider the class of polynomials P(z) = a0 + Σv=μN avzv, μ ≥ 1, of degree n such that P(z) ≠0 in |z| < k, k ≥ 1 and establish some upper bound estimates for the maximum modulus of DγP(z) on the unit disk by involving some of the coefficients of P(z). The obtained results refine and generalize some well known polynomial inequalities. [ABSTRACT FROM AUTHOR]

The well-known Erdös-Lax and Turán-type inequalities, which relate the uniform norm of a univariate complex coefficient polynomial to its derivative on the unit circle in the plane, are discussed in this paper. We create some new inequalities here when there is a restriction on its zeros. The obtained results strengthen some recently proved ErdösLax and Turán-type inequalities for constrained polynomials and also produce various inequalities that are sharper than the previous ones known in a very rich literature on this subject. [ABSTRACT FROM AUTHOR]

In this paper, we introduced two novel subclasses of bi-univalent functions, MΣ(α, B(x, ξ)) and HΣ(α, µ, B(x, ξ)), utilizing Lucas-Balancing polynomials. Within these function classes, we established bounds for the Taylor-Maclaurin coefficients |a2| and |a3|, addressing the Fekete-Szegö functional problems specific to functions within these new subclasses. Moreover, we illustrated how our primary findings could lead to various new outcomes through parameter specialization. [ABSTRACT FROM AUTHOR]

In this paper, we establish some upper bound estimates for the maximum modulus of the higher order derivative of a polynomial with restricted zeros on a circle of |z| = R, R ≥ 1, in terms of the maximum modulus of the same polynomial on the unit circle |z| = 1. These results generalize and sharpen some known results in this direction. [ABSTRACT FROM AUTHOR]

This study focused on "Extension of synthetic division and its applications". The study was designed to show synthetic division and its extension and to point out the applications of synthetic division and its extension. The study found out that the concepts of polynomial and rational expressions in single variables are basic concepts to deal extension of synthetic division and its applications. Using the preliminary concepts, the concept of synthetic division is extended in this study. Also, the study found out that an extension of synthetic division is used for finding the oblique asymptote of the graph of a rational function, evaluating the integration of some rational functions, representing polynomial expression by factorial function in numerical analysis, and so on. [ABSTRACT FROM AUTHOR]

Let ... be a polynomial of degree n having all its zeros in ..., then Aziz [Proc. Am. Math. Soc., 89, (1983) 259--266] proved ... . In this paper, we prove a polar derivative extension which sharpens the above inequality. As a consequence, we also derive a result on Bernstein type inequality for the class of polynomials having all its zeros in ... . [ABSTRACT FROM AUTHOR]

In this article, we investigate the problem of sharing values between entire functions f(z) and f1(z) = b-1 + ... bi f(ki)(z+iη) share two distinct values a with counted and b with ignoring multiplicities, where b-1 and bi(i = 0,1· · ·, n) are small meromorphic functions of f(z), ki ≥ 0 (i = 0,1, · · ·, n) are integers. In relation to previous research, we obtain results that improve and generalise the findings conducted by Yang and Qi [CMFT, 20.1 (2020): 159-178]. [ABSTRACT FROM AUTHOR]

In the present study, we use several identities from the q-calculus to define the concept of q-Hermite polynomials with three variables and present their associated formalism. Many properties and new results of q-Hermite polynomials of three variables are established, including their generation function, series description, summation equations, recurrence relationships, q-differential formula and operational rules. [ABSTRACT FROM AUTHOR]

Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form Q C (p) = a p n + m p + c , where n ≥ 2 . It establishes escape criteria that play a vital role in generating these sets and provides escape time results using different iterative schemes. In addition, the study includes the visualization of graphical images of Julia and Mandelbrot sets, revealing distinct patterns. Furthermore, the study also explores the impact of parameters on the deviation of dynamics, color, and appearance of fractals. [ABSTRACT FROM AUTHOR]

This paper investigates various aspects of the distribution of roots and critical points of a complex polynomial, including their variance and the relationships between their moduli using an inequality due to de Bruijn. Making use of two other inequalities-again due to de Bruijn-we derive two probabilistic results concerning upper bounds for the average moduli of the imaginary parts of zeros and those of critical points, assuming uniform distribution of the zeros over a unit disc and employing the Markov inequality. The paper also provides an explicit formula for the variance of the roots of a complex polynomial for the case when all the zeros are real. In addition, for polynomials with uniform distribution of roots over the unit disc, the expected variance of the zeros is computed. Furthermore, a bound on the variance of the critical points in terms of the variance of the zeros of a general polynomial is derived, whereby it is established that the variance of the critical points of a polynomial cannot exceed the variance of its roots. Finally, we conjecture a relation between the real parts of the zeros and the critical points of a polynomial. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n with zero of multiplicity s at the origin and the remaining zeros be 0. In this paper, we investigate the relative growth of a polynomial p(z) with respect to two circles z=r and z = R and obtain inequalities about the dependence of p(rz) on p(Rz), where z = 1, for 0 while taking into account the placement of the zeros of the underlying polynomial. Our results improve as well as generalize certain well-known polynomial inequalities. Some numerical examples are also given in order to illustrate and compare graphically the obtained inequalities with some recent results. [ABSTRACT FROM AUTHOR]

This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations $af(z^{r})=f(z)+R(z)$, where $a$ is a nonzero complex number, $r$ an integer greater than 1, and $R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained. [ABSTRACT FROM AUTHOR]

Let P(z) be a polynomial of degree n having no zeros in |z| < k where k ≥ 1. Then it is known that for every real or complex number α with |α| ≥ 1, max |z|=1 |DαP(z)| ≤ n (|α| + k/1 + k) max |z|=1 |P(z)|, where DαP(z) = nP(z) + (α - z)P' (z) denotes the polar derivative of the polynomial P(z) of degree n with respect to a point α ∈ ℂ. In this paper, by a simple method, a refinement of the above inequality and other related results are obtained. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 be a fixed integer, and denote by M n the algebra of all n × n complex matrices. Fix also an integer k such that 1 ≤ k < n. We prove that if φ : M n → M n is a linear map which preserves the polynomial numerical radius of order k , there exist then a unitary n × n complex matrix U and a complex number ξ of modulus one such that either φ (T) = ξ U ⁎ T U for all T ∈ M n , or φ (T) = ξ U ⁎ T t U for all T ∈ M n. [ABSTRACT FROM AUTHOR]

We introduce a new homomorphic encryption scheme that is natively capable of computing with complex numbers. This is done by generalizing recent work of Chen, Laine, Player and Xia, who modified the Fan–Vercauteren scheme by replacing the integral plaintext modulus t by a linear polynomial X − b. Our generalization studies plaintext moduli of the form Xm + b. Our construction significantly reduces the noise growth in comparison to the original FV scheme, so much deeper arithmetic circuits can be homomorphically executed. [ABSTRACT FROM AUTHOR]

In this paper, we have proved an Lp inequality for polar derivative of the polynomial having all its zeros in the disk |z| ≤ k, k ≤ 1. Our result generalizes and improves the earlier known results. [ABSTRACT FROM AUTHOR]

Let Pm be a class of all polynomials of degree at most m and let Rm;n = Rm;n(d1; :::; dn) = fp(z)=w(z); p 2 Pm;w(z) = Qn j=1(z dj) where jdj j > 1; j = 1; :::; n and m ng denote the class of rational functions. It is proved that if the rational function r(z) having all its zeros in jzj 1, then for jzj = 1The main purpose of this paper is to improve the above inequality for rational functions r(z) having all its zeros in jzj k 1 with t-fold zeros at the origin and some other related inequalities. The obtained results sharpen some well-known estimates for the derivative and polar derivative of polynomials. [ABSTRACT FROM AUTHOR]

In this paper, a new parametric class of optimal fourth-order iterative methods to estimate the solutions of nonlinear equations is presented. After the convergence analysis, a study of the stability of this class is made using the tools of complex discrete dynamics, allowing those elements of the class with lower dependence on initial estimations to be selected in order to find a very stable subfamily. Numerical tests indicate that the stable members perform better on quadratic polynomials than the unstable ones when applied to other non-polynomial functions. Moreover, the performance of the best elements of the family are compared with known methods, showing robust and stable behaviour. [ABSTRACT FROM AUTHOR]

The article investigates a class of polynomials known as convolved Pell polynomials. This class generalizes the standard class of Pell polynomials. New formulas related to convolved Pell polynomials are established. These formulas may be useful in different applications, in particular in numerical analysis. New expressions are derived for the high-order derivatives of these polynomials, both in terms of their original polynomials and in terms of various well-known polynomials. As special cases, connection formulas linking the convolved Pell polynomials with some other polynomials can be deduced. The new moments formula of the convolved Pell polynomials that involves a terminating hypergeometric function of the unit argument is given. Then, some reduced specific moment formulas are deduced based on the reduction formulas of some hypergeometric functions. Some applications, including new specific definite and weighted definite integrals, are deduced based on some of the developed formulas. Finally, a matrix approach for this kind of polynomial is presented. [ABSTRACT FROM AUTHOR]

In this research article, we have studied the results of P. Sahoo and H. Karmakar, intending to determine, in any manner, whether it is possible to relax the nature of sharing by replacing the shift of non-constant transcendental entire functions of finite order with the product of shift. In this direction, we have investigated the uniqueness of shift difference polynomials of two entire functions when they share a non-zero polynomial with a finite weight and one being the Mobius transformation of the other satisfying n = 2d-s +3, and also when they share a small function with a finite weight satisfying n = m + s + 5. We also investigate the same situation when the original functions f and g share the value zero counting multiplicities (CM) satisfying n > 2(G1 + d) - s. Our results in this paper extend and generalizes the previous results of P. Sahoo and H. Karmakar [Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 2017, 52 (2), pp. 102-110]. [ABSTRACT FROM AUTHOR]

We investigate the uniqueness problem of entire functions that share two polynomials with their kth derivatives and obtain some results which improve and generalize the recent result due to Lü and Yi (2011). Also, we exhibit some examples to show that the conditions of our results are the best possible. [ABSTRACT FROM AUTHOR]

In this paper, on the basis of zero set of reverse indexed polynomial, first introduced by us, we have investigated the uniqueness of meromorphic functions under weighted sharing in wider sense criteria [4], which in turn extend some earlier results in different directions. We have succeeded to identify a subclass of meromorphic functions for which uniqueness property exists for higher reverse indexed polynomial in literature. In the last section, we have presented the application of our results in case of derivatives of the concerned functions accompanied by series of examples. [ABSTRACT FROM AUTHOR]

Cryptography is the process of transforming data so that only the recipient of the message can read it. It uses an algorithm and a key to convert an input into an encrypted output. In this study, we introduce a novel method for protecting readable messages through the utilization of a polynomial root-finding technique in conjunction with the decimated output of Zaslavsky equations. The innovative approach we have developed is a block encryption algorithm that offers both protection for sensitive information as well as adaptability to various block sizes, including dynamically changing block sizes. Precise security analysis is provided for the proposed algorithm using key space analysis and strong statistical tests. The presented results show that the novel block encryption protection provides a high level of security. [ABSTRACT FROM AUTHOR]

For a given polynomial ... of degree n with complex coefficients, the Cauchy radius r0 is a unique positive root of the equation... It refers to a radius of the circular region z ≤ r0 in which all the zeros of p(z) lie. The basic aim has been to determine the smallest radius, thereby, minimizing the area of the circular region. In this present paper, we have obtained a result which gives an improvement of the Cauchy radius. Also, we produce an annular region whose center is different from the origin in which the zeros of p(z) lie. Moreover, in many cases, our results give better approximations for estimating the region of polynomial zeros than that obtained from many other well-known results. [ABSTRACT FROM AUTHOR]

The fractional q-calculus has attracted the interest of a large number of academics over the last four decades or so, due mainly to a wide range of applications that cover natural sciences to social sciences. Many fractional q-calculus operators, particularly those involving various q-special functions, have been deeply studied and widely applied. In this paper, we aim to establish certain image formulas of Saigo fractional q-integral operators involving the product of generalized q-hypergeometric series and a general class of q-polynomials that are primarily expressed in terms of generalized q-hypergeometric series in a systematic manner. We demonstrate their use by studying q-Konhouser biorthogonal polynomials and q-Jacobi polynomials. Additionally, some fascinating special cases of our main findings are taken into consideration, and pertinent connections between some of the findings presented here and those from earlier studies are also made. [ABSTRACT FROM AUTHOR]

It is remarkable that, in recent years, intensive studies have been done for degenerate versions of many special polynomials and numbers and have yielded many interesting results. The aim of this paper is to study the generalized degenerate (r, s)-Stirling numbers of the second and their natural extensions to polynomials, namely the generalized degenerate (r, s)-Bell polynomials, arising from certain 'degenerate boson normal ordering.'Wederive some properties, explicit expressions and generating functions for those numbers and polynomials. The generalized degenerate (r, s)-Stirling numbers of the second and the degenerate boson normal ordering are respectively degenerate versions of the generalized (r, s)-Stirling numbers of the second and the boson normal ordering studied earlier by Blasiak-Person-Solomon. [ABSTRACT FROM AUTHOR]

This present work provides the initial co-efficient bounds of the function f(z) which is defined in open unit disk D . We introduced bi-univalent class C Σ (λ , t , ν) . Making use of Horadam polynomials h n (ν) and the generating function Π (ν , z) , we estimated the bounds of a 2 and a 3 for the given function to be in the defined class. Moreover Fekete-Szego inequalities are calculated. In addition to all the results obtained mathematically, we provide application for Horadam polynomial in computer vision. [ABSTRACT FROM AUTHOR]

The present paper addresses linear parameter-varying systems with high-order time-varying parameter dependency known as polynomial LPV systems and their controller design. Throughout this work, a procedure ensuring a state-feedback controller from a parameterized linear matrix inequality (PLMI) solution is presented. As the main contribution of this paper, the controller is designed by considering the time-varying parameter rate as a tuning parameter with a continuous control gain in such a way that the closed-loop eigenvalues lie in a complex plane subset, with high-order time-varying parameters defining the system dynamics. Simulation results are presented, aiming to show the effectiveness of the proposed controller design. [ABSTRACT FROM AUTHOR]

We suggest an ansatz for representation of affine Yangian Y gl ̂ 1 by differential operators in the triangular set of time-variables Pa,i with 1 ⩽ i ⩽ a, which saturates the MacMahon formula for the number of 3d Young diagrams/plane partitions. In this approach the 3-Schur polynomials are defined as the common eigenfunctions of an infinite set of commuting "cut-and-join" generators ψn of the Yangian. We manage to push this tedious program through to the 3-Schur polynomials of level 5, and this provides a rather big sample set, which can be now investigated by other methods. [ABSTRACT FROM AUTHOR]

In this article, we construct a new type of degenerate q-sigmoid (DQS) polynomial for sigmoid functions containing quantum numbers and find several difference equations related to it. We check how each point moves by iteratively synthesizing a quartic degenerate q-sigmoid (DQS) polynomial that appears differently depending on q in the space of a complex structure. We also construct Julia sets associated with quartic DQS polynomials and find their features. Based on this, we make some conjectures. [ABSTRACT FROM AUTHOR]

This article, published in the Turkish Journal of Mathematics, explores the topic of interpolation polynomials associated with linear recurrences. The authors investigate the use of polynomial interpolation using basic points given by (t, Gt) for a linear recurrence (Gn). They develop formulas related to finite differences and provide an explicit formula for the interpolation polynomials. The article also includes introductory information on finite differences and properties of recursive sequences. The authors provide examples and demonstrate the application of their approach in obtaining interpolation polynomials. The research was supported by multiple grants. [Extracted from the article]

In this paper, we consider a class of rational functions r(s(z)) of degree at most mn, where s(z) is a polynomial of degree m and obtain a certain sharp compact generalization of well-known inequalities for rational functions. [ABSTRACT FROM AUTHOR]

Abstract We consider the problem of scheduling complex-valued demands over a discretized time horizon. Given a set of users, each user is associated with a set of demands representing different power consumption preferences. A demand is represented by a complex number, a time interval, and a utility value obtained if it is satisfied. At each time slot, the magnitude of the total selected demands should not exceed a given generation capacity. This naturally captures the supply constraints in alternating current (AC) electric systems. In this paper, we consider maximizing the aggregate user utility subject to power supply limits over a time horizon. We present approximation algorithms characterized by the maximum angle ϕ between any two complex-valued demands. More precisely, a PTAS is presented for the case ϕ ∈ [ 0 , π 2 ] , a bi-criteria FPTAS for ϕ ∈ [ 0 , π - ε ] for any polynomially small ε , assuming the number of time slots in the discretized time horizon is a constant. Furthermore, if the number of time slots is part of the input, we present a reduction to the real-valued unsplittable flow problem on a path with only a constant approximation ratio. Finally, we present a practical greedy algorithm for the single time slot case with an approximation ratio of 1 2 cos ⁡ ϕ 2 and a running time complexity of only O (N log ⁡ N) , N standing for the aggregate number of user demands, which can be implemented efficiently in practice. [ABSTRACT FROM AUTHOR]

In this paper, we establish some inequalities involving the modulus of the derivative of rational functions with prescribed poles and restricted zeros. The obtained results generalize some known inequalities for rational functions. Moreover, our results also contain certain known polynomial inequalities. [ABSTRACT FROM AUTHOR]

The principal objective of the paper, to study the uniqueness problem of Q-shift differencedifferential polynomials of a meromorphic function sharing finite value. A result Chung-Chun Yang and Xinhou. Hua (In Annales-Academiae Scientiarum Fennicae Series A 1 Mathematica 22:395-406,1997) is the starting point of this kind of research. In addition to this, one of our result improve and extend that of Harina R Waghamore and Rajeshwari (International Journal of Pure and Applied Mathematics 107(4):971-981, 2016). Some examples have been exhibited to validate our certain claims in the main result. [ABSTRACT FROM AUTHOR]

Stenger conjectures are claims about the location of the eigenvalues of matrices whose elements are certain integrals involving basic Lagrange interpolating polynomials supported on the zeros of orthogonal polynomials. In this paper, we show the validity of the extended Stenger conjecture for families of classical orthogonal polynomials. We also show the validity of the restricted Strenger conjecture for a family of Jacobi and generalized Laguerre orthogonal polynomials. A connection with the A-stability of the collocation Runge-Kutta methods is investigated. [ABSTRACT FROM AUTHOR]