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This paper explores the approximate analytical solution of non-linear Klein-Gordon equations (NKGE) by using multistep modified reduced differential transform method (MMRDTM). Through this proposed strategy, the non-linear term is substituted by associating Adomian polynomials obtained by utilization of a multistep approach. The NKGE solutions can be obtained with a reduced number of computed terms. In addition, the approximate solutions converge rapidly in a wide time region. Three examples are provided to illustrate the effectiveness of the proposed method to obtain solutions for the NKGE. Graphical results are shown to represent the behavior of the solution so as to demonstrate the validity and accuracy of the MMRDTM. [ABSTRACT FROM AUTHOR]

In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied. [ABSTRACT FROM AUTHOR]

We study the transformation of "predicate introduction" in non-monotonic logics. By this, we mean the act of replacing a complex formula by a newly defined predicate. From a knowledge representation perspective, such transformations can be used to eliminate redundancy or to simplify a theory. From a more practical point of view, they can also be used to transform a theory into a normal form imposed by certain inference programs or theorems. In this paper, we study predicate introduction in the algebraic framework of "approximation theory"; this is a fixpoint theory for nonmonotone operators that generalizes all main semantics of various non-monotonic logics, including logic programming, default logic and autoepistemic logic. We prove an abstract, algebraic equivalence result in this framework. This can then be used to show that, in logic programming, certain transformations are equivalence preserving under, among others, both the stable and well-founded semantics. Based on this result, we develop a general method of eliminating universal quantifiers in the bodies of rules. Our work is, however, also applicable beyond logic programming. In a companion paper, we demonstrate this, by using the same algebraic results to derive a transformation which reduces the nesting depth of the modal operator K in autoepistemic logic. [ABSTRACT FROM AUTHOR]

Let M be a 2 × 2 matrix of Laurent polynomials with real coefficients and symmetry. In this paper, we obtain a necessary and sufficient condition for the existence of four Laurent polynomials (or finite-impulse-response filters) u1, u2, v1, v2 with real coefficients and symmetry such that [This symbol cannot be presented in ASCII format] and [Su1](z)[Sv2](z) = [Su2](z)[Sv1](z), where [Sp](z) = p(z)/p(1/z) for a nonzero Laurent polynomial p. Our criterion can be easily checked and a step-by-step algorithm will be given to construct the symmetric filters u1, u2, v1, v2. As an application of this result to symmetric framelet filter banks, we present a necessary and sufficient condition for the construction of a symmetric multiresolution analysis tight wavelet frame with two compactly supported generators derived from a given symmetric refinable function. Once such a necessary and sufficient condition is satisfied, an algorithm will be used to construct a symmetric framelet filter bank with two high-pass filters which is of interest in applications such as signal denoising and image processing. As an illustration of our results and algorithms in this paper, we give several examples of symmetric framelet filter banks with two highpass filters which have good vanishing moments and are derived from various symmetric low-pass filters including some B-spline filters. [ABSTRACT FROM AUTHOR]

Polynomial root-finding is a classical and highly developed problem of mathematics and computational mathematics. There is a vast literature in approximating the roots of a polynomial and their convergence rate is quite high, but the matrix methods are among the most popular practical choices for approximating all the roots of a polynomial simultaneously and they are still considered competitive. This is the reason why in this paper we propose a new method for finding the zeros of a polynomial by computing the eigenvalues of the corresponding companion matrix. The method proposed finds the multiple roots of a polynomial by constructing a new companion matrix which has only simple eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper we propose a Fully Polynomial Time Approximation Scheme for a class of optimization problems where the feasible region is a polyhedral one and the objective function is the sum or product of ratios of linear functions. The class includes the well known ones of Linear (Sum-of-Ratios) Fractional Programming and Multiplicative Programming. [ABSTRACT FROM AUTHOR]

This paper presents an efficient technique for synthesis and optimization of the polynomials over GF(2m), where m is a nonzero positive integer. The technique is based on a graph-based decomposition and factorization of the polynomials, followed by efficient network factorization and optimization. A technique for efficiently computing the coefficients of the polynomials over GF(pm), where p is a prime number, is first presented. The coefficients are stored as polynomial graphs over GF(pm). The synthesis and optimization is initiated from this graph-based representation. The technique has been applied to minimize multipliers over the fields GF(2κ ), where k = 2,..., 8, generated with all the 51 primitive polynomials in the 0.18-/~m CMOS technology with the help of the Synopsys design compiler. It has also been applied to minimize combinational exponentiation circuits, parallel integer adders and multipliers, and other multivariate bit- as well as word-level polynomials. The experimental results suggest that the proposed technique can reduce area, delay, and power by significant amounts. We also observed that the technique is capable of producing 100% testable circuits for stuck-at faults. [ABSTRACT FROM AUTHOR]

This paper considers the problem of approximating an integrable function by piecewise linear functions that keep the integral and positivity of the original function. [ABSTRACT FROM AUTHOR]

This paper offers a sparse, multiscale representation of objects. It captures the object appearance by selection from a very large dictionary of Gaussian differential basis functions. The learning procedure results from the matching pursuit algorithm, while the recognition is based on polynomial approximation to the bases, turning image matching into a problem of polynomial evaluation. The method is suited for coarse recognition between objects and, by adding more bases, also for fine recognition of the object pose. The advantages over the common representation using PCA include storing sampled points for recognition is not required, adding new objects to an existing data set is trivial because retraining other object models is not needed, and significantly in the important case where one has to scan an image over multiple locations in search for an object, the new representation is readily available as opposed to PCA projection at each location. The experimental result on the COIL-100 data set demonstrates high recognition accuracy with real-time performance. [ABSTRACT FROM AUTHOR]

The top part of Figure 1.1 shows some classes from the (truth-table) bounded-query and boolean hierarchies. It is well known that if either of these hierarchies collapses at a given level, then all higher levels of that hierarchy collapse to that same level. This is a standard "upward translation of equality" that has been known for over a decade. The issue of whether these hierarchies can translate equality downwards has proven vastly more challenging. In particular, with regard to Figure 1.1, consider the following claim: \[ \psigkmtt = \psigkmponett \implies \diffmsigk = \codiffmsigk = \bh(\sigmak). (*) \] This claim, if true, says that equality translates downwards between levels of the bounded-query hierarchy and the boolean hierarchy levels that (before the fact) are immediately below them. Until recently, it was not known whether (*)ever held, except for the degenerate cases m = 0 and k = 0. Then Hemaspaandra, Hemaspaandra, and Hempel [SIAM J. Comput., 28 (1999), pp. 383--393] proved that (*) holds for all m, for k > 2. Buhrman and Fortnow [J. Comput. System Sci., 59 (1999), pp. 182--199] then showed that, when k = 2, (*) holds for the case m = 1. In this paper, we prove that for the case k = 2, (*) holds for all values of m. Since there is an oracle relative to which "for k = 1, (*) holds for all m" fails (see Buhrman and Fortnow), our achievement of the k = 2 case cannot be strengthened to k = 1 by any relativizable proof technique. The new downward translation we obtain also tightens the collapse in the polynomial hierarchy implied by a collapse in the bounded-query hierarchy of the second level of the polynomial hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism s ? Aut( R ), we use the notion of a s-compatible module M R to obtain results on the couniform dimension of the polynomial modules M [ x ], M [ x -1 ], and M [ x , x -1 ] over suitable skew extension rings. [ABSTRACT FROM AUTHOR]

In this paper we show that a generalization of a popular motion planning puzzle called Lunar Lockout is computationally intractable. In particular, we show that the problem is PSPACE-complete. We begin with a review of NP-completeness and polynomial-time reductions, introduce the class PSPACE, and motivate the significance of PSPACE-complete problems. Afterwards, we prove that determining whether a given instance of a generalized Lunar Lockout puzzle is solvable is PSPACE-complete. [ABSTRACT FROM AUTHOR]

The paper establishes lower bounds on the provability of D = NP and the MRDP theorem in weak fragments of arithmetic. The theory I[sup 5] E[sub l] is shown to be unable to prove D = NP. This non-provability result is used to show that I[sup 5] E[sub 1] cannot prove the MRDP theorem. On the other hand it is shown that I¹ E[sub1] proves D contains all predicates of the form (∀i ≤ |b|) P(i, x) ο Q(i, x) where ο is =, <, or ≤, and I[sup 0] E[sub1] proves D contains all predicates of the form (∀i ≤ b) P(i, x) = Q(i, x). Here P and Q are polynomials. A conjecture is made that D contains NLOGTIME. However, it is shown that this conjecture would not be sufficient to imply D = NP. Weak reductions to equality are then considered as a way of showing D = NP. It is shown that the bit-wise less than predicate, ≤2, and quality are both co-NLOGTIME complete under FDLOGTIME reductions. This is used to show that if the FDLOGTIME functions are definable in D then D = NP. [ABSTRACT FROM AUTHOR]

This paper examines the application of a bootstrap test error estimation of radial basis functions, specifically thin-plate spline fitting, in surface smoothing. The presence of noisy data is a common issue of the point set model that is generated from 3D scanning devices, and hence, point set denoising is one of the main concerns in point set modelling. Bootstrap test error estimation, which is applied when searching for the smoothing parameters of radial basis functions, is revisited. The main contribution of this paper is a smoothing algorithm that relies on a bootstrap-based radial basis function. The proposed method incorporates a k-nearest neighbour search and then projects the point set to the approximated thin-plate spline surface. Therefore, the denoising process is achieved, and the features are well preserved. A comparison of the proposed method with other smoothing methods is also carried out in this study. [ABSTRACT FROM AUTHOR]

This study is concerned with finding a level ideal (LI) of a partially ordered set (poset). Given a finite poset P, the level of each element p ∈ P is defined as the number of ideals that do not include p, then the problem is to find the ith LI-the ideal consisting of elements whose levels are less than a given integer i. The concept of a level ideal is naturally derived from the generalized median stable matchings, introduced by Teo and Sethuraman [Teo, C. P., J. Sethuraman. 1998. The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23(4) 874-891] in the context of "fairness" of matchings in a stable marriage problem. Cheng [Cheng, C. T. 2010. Understanding the generalized median stable matchings. Algorithmica 58(1) 34-51] showed that finding the ith LI is #P-hard when i = θ (N), where N is the total number of ideals of P. This paper shows that finding the ith LI is #P-hard even if i = θ (N 1/c), where c is an arbitrary constant at least one. Meanwhile, we present a polynomial time exact algorithm when i = O(logN)c', where c' is an arbitrary positive constant. We also devise two randomized approximation schemes for the ideals of a poset, by using an oracle of an almost-uniform sampler. [ABSTRACT FROM AUTHOR]

Continuous linear programs have attracted considerable interest due to their potential for modeling manufacturing, scheduling, and routing problems. While efficient simplex-type algorithms have been developed for separated continuous linear programs, crude time discretization remains the method of choice for solving general (nonseparated) problenl instances. In this paper we propose a more generic approximation scheme for nonseparated continuous linear programs, where we approximate the functional decision variables (policies) by polynomial and piecewise polynomial decision rules. This restriction results in an upper bound on the original problem, which can be computed efficiently by solving a tractable semidefinite program. To estimate the approximation error, we also compute a lower bound by solving a dual continuous lineal program in (piecewise) polynomial decision rules. We establish the convergence of the primal and dual approximations under Stater-type constraint qualifications. We also highlight the potential of our method for optimizing large-scale multiclass queueing systems and dynamic Leontief models. [ABSTRACT FROM AUTHOR]

This paper surveys some results on acute and nonobtuse simplices and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplices called path-simplices, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute sad nonobtuse simplices in other areas of mathematics. [ABSTRACT FROM AUTHOR]

We investigate the following existence problem for rational functions: for a given collection Π of partitions of a number n to define whether there exists a rational function f of degree n for which Π is the branch datum. An important particular case when the answer is known is the one when the collection Π contains a partition consisting of a single element (in this case, the corresponding rational function is equivalent to a polynomial). In this paper, we provide a solution in the case when Π contains a partition consisting of two elements. [ABSTRACT FROM AUTHOR]

Copyright of Petroleum - Gas University of Ploiesti Bulletin, Mathematics - Informatics - Physics Series is the property of Petroleum - Gas University of Ploiesti 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 this paper we are interested in the polynomial Krylov approximations for the computation of φ(A)v, where A is a square matrix, v represents a given vector, and φ is a suitable function which can be employed in modern integrators for differential problems. Our aim consists of proposing and analyzing innovative a posteriori error estimates which allow a good control of the approximation procedure. The effectiveness of the results we provide is tested on some numerical examples of interest. [ABSTRACT FROM AUTHOR]

This paper brings together two fundamental topics: polyhedral projection and parametric linear programming. First, it is shown that, given a parametric linear program (PLP), a polyhedron exists whose projection provides the solution to the PLP. Second, the converse is tackled and it is shown how to formulate a PLP whose solution is the projection of an appropriately defined polyhedron described as the intersection of a finite number of halfspaces. The input to one operation can be converted to an input of the other operation and the resulting output can be converted back to the desired form in polynomial time-this implies that algorithms for computing projections or methods for solving parametric linear programs can be applied to either problem class. [ABSTRACT FROM AUTHOR]

In this paper we study the optimal control problem described by a hemivariational parabolic inclusion. We derive the existence of optimal solutions. Then we prove the convergence of optimal values for approximated control problems to the one for the original problem. Finally, we give a simple example. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide an error analysis for the multicategory support vector machine (MSVM) classificaton problems. We establish the uniform convergency approach for MSVMs and estimate the misclassification error. The main difficulty we overcome here is to bound the offset vector. As a result, we confirm that the MSVM classification algorithm with polynomial kernels is always efficient when the degree of the kernel polynomial is large enough. Finally the rate of convergence and examples are given to demonstrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the relation between the rate of convergence for the derivatives of the combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated. We give some direct and inverse results on pointwise simultaneous approximation by the combinations of Baskakov operators. We also give a new equivalent result on pointwise approximation by these operators. [ABSTRACT FROM AUTHOR]

The study of Tchebycheff spaces (generalizing the space of algebraic polynomials) and extremal problems related to them began one and a half centuries ago. Recently, many facts of approximation theory have been understood and reinterpreted from the point of view of general principles of the theory of extremum and convex duality. This approach not only allowed one to prove the previously known results for algebraic polynomials and generalized polynomials in a unified way, but also enabled one to obtain new results. In this paper, we work out this direction with special attention to the optimal recovery problems. [ABSTRACT FROM AUTHOR]

The detection of errors in arithmetic operations is an important issue. This paper discusses the detection of multiple-bit errors due to faults in bit-serial and bit-parallel polynomial basis (PB) multipliers over binary extension fields. Our approach is based on multiple parity bits. Experimental results presented here show that due to an increase hi the number of parity bits, the area overhead tends to increase linearly, but the probability of error detection approaches unity fairly quickly, e.g., for eight parity bits. In bit-serial implementation of a GF(2163) PB multiplier using eight parity bits, the area overhead and the probability of error detection are 10.29% and 0.996, respectively. This is achieved without any increase in the computation time of the GF(2163) PB multiplier. [ABSTRACT FROM AUTHOR]

In this paper, large commutative subalgebras of quantum algebras (in particular, quantum polynomial, matrix, and Weyl algebras) are studied. The notions of the transcendence degree and the Krull dimension are discussed. [ABSTRACT FROM AUTHOR]

The paper deals with approximation of a continuous function, on a finite interval, which changes convexity finitely many times, by algebraic polynomials which are coconvex with it. We give final answers to open questions concerning the validity of Jackson type estimates involving the weighted Ditzian-Totik moduli of smoothness. [ABSTRACT FROM AUTHOR]

In this paper, we solve the problem of the exact order of decrease of uniform moduli of smoothness for the classes of 2π-periodic functions of several variables with a given majorant of the sequence of total best approximations in the metric of L p, 1 ≤ p < ∞. [ABSTRACT FROM AUTHOR]

This paper presents novel coding algorithms based on tree-structured segmentation, which achieve the correct asymptotic rate-distortion (R-D) behavior for a simple class of signals, known as piecewise polynomials, by using an R-D based prune and join scheme. For the one-dimensional case, our scheme is based on binary-tree segmentation of the signal. This scheme approximates the signal segments using polynomial models and utilizes an R-D optimal bit allocation strategy among the different signal segments. The scheme further encodes similar neighbors jointly to achieve the correct exponentially decaying R-D behavior (D(R) ∼ c02-c1R), thus improving over classic wavelet schemes. We also prove that the computational complexity of the scheme is of O(N log N). We then show the extension of this scheme to the two-dimensional case using a quad tree. This quadtree-coding scheme also achieves an exponentially decaying R-D behavior, for the polygonal image model composed of a white polygon-shaped object against a uniform black background, with low computational cost of O(N log N). Again, the key is an R-D optimized prune and join strategy. Finally, we conclude with numerical results, which show that the proposed quadtree-coding scheme outperforms JPEG2000 by about 1 dB for real images, like cameraman, at low rates of around 0.15 bpp. [ABSTRACT FROM AUTHOR]

The paper computes the noncommutative A-ideal of the figure-eight knot, a noncommutative generalization of the A-polynomial. It is shown that if a knot has the same A-ideal as the figure-eight knot, then all colored Kauffman brackets are the same as those of the figure eight knot. [ABSTRACT FROM AUTHOR]

The computability of reals was introduced by Alan Turing [20] by means of decimal representations. But the equivalent notion can also be introduced accordingly if the binary expansion, Dedekind cut or Cauchy sequence representations are considered instead. In other words, the computability of reals is independent of their representations. However, as it is shown by Specker [19] and Ko [9], the primitive recursiveness and polynomial time computability of the reals do depend on the representation. In this paper, we explore how the weak computability of reals depends on the representation. To this end, we introduce three notions of weak computability in a way similar to the Ershov's hierarchy of Δ02-sets of natural numbers based on the binary expansion, Dedekind cut and Cauchy sequence, respectively. This leads to a series of classes of reals with different levels of computability. We investigate systematically questions as on which level these notions are equivalent. We also compare them with other known classes of reals like c. e. and d-c. e. reals. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Let α(E) be the continuous analytic capacity of a compact set E ⊂ C. In this paper we obtain a characterization of α in terms of curvature of measures with zero linear density, and we deduce that α is countably semiadditive. This result has important consequences for the theory of uniform rational approximation on compact sets. In particular, it implies the so-called inner boundary conjecture. [ABSTRACT FROM AUTHOR]

This paper is motivated by S. Park [10] in which the injective cover of left R[x]-module M[x[sup -1]] of inverse polynomials over a left R-module M was discussed. The author considers the Ω-covers of modules and shows that if η: P → M is an Ω-cover of M, then [η[sup S,≤]] : [P[sup S,≤]] → [M[sup S,≤]] is an [Ω[sup S,≤]]-cover of left [[R[sup S,≤]]]-module [M[sup S,≤]], where Ω is a class of left R-modules and [M[sup S,≤]] is the left [[R[sup S,≤]]]-module of generalized inverse polynomials over a left R-module M. Also some properties of the injective cover of left [[R[sup S,≤]]]-module [M[sup S,≤]] are discussed. [ABSTRACT FROM AUTHOR]

A (u1, u2, . . . )-parking function of length n is a sequence (x1, x2, . . . , xn) whose order statistics (the sequence (x(1), x(2), . . . , x(n)) obtained by rearranging the original sequence in non-decreasing order) satisfy x(i)≤ u(i). The Gončarov polynomials gn (x; a0, a1, . . . , an-1) are polynomials biorthogonal to the linear functionals ε(ai) Di, where ε(a) is evaluation at a and D is differentiation. In this paper, we give explicit formulas for the first and second moments of sums of u-parking functions using Gončarov polynomials by solving a linear recursion based on a decomposition of the set of sequences of positive integers. We also give a combinatorial proof of one of the formulas for the expected sum. We specialize these formulas to the classical case when ui=a+ (i-1)b and obtain, by transformations with Abel identities, different but equivalent formulas for expected sums. These formulas are used to verify the classical case of the conjecture that the expected sums are increasing functions of the gaps ui+1 - ui. Finally, we give analogues of our results for real-valued parking functions. [ABSTRACT FROM AUTHOR]

Cameron–Praeger designs with parameters $t-(v,k,\lambda)$ are studied. Cameron and Praeger showed that in such designs, $t=2$ or 3. In 1989, Delandtsheer and Doyen proved that if $t=2$ then \[ v \leqslant \left(\left({k \atop 2}\right)-1\right)^2. \] In 2000, Mann and Tuan improved this equality and showed that if $t=3$ then \[ v \leqslant \left({k \atop 2}\right)+1. \] Three infinite families of Cameron–Praeger 3-designs for which this bound is met have been constructed by Mann and Tuan and by Sebille. The paper constructs infinitely many infinite families of such Cameron–Praeger 3-designs via a study of a divisibility problem for polynomials. Further, the construction generalizes the previous constructions. [ABSTRACT FROM AUTHOR]

The class of indexed languages properly includes the class of context-f ret languages and is properly included in the class of context-dependent languages [1]. The emptiness problem (the problem of determining whether or not the given language is empty) is polynomial-time complete for the class of context-free languages and is undecidable for the class of context-dependent languages. The recognition problem (the problem, given a language L and word W, of determining whether or not W belongs to L) is polynomial-time complete for the class of context-free languages and is polynomial-space complete for the class of context-dependent languages. This paper shows that both the emptiness and recognition problems are exponential-time complete for the class of indexed languages. It is known in the pebble game [2] that the problem of determining whether or not the first player baa the winning strategy is exponential-time complete. This paper reduces the problem to the emptiness problem for the class of indexed languages in the logarithmic space, indicating the exponential-time difficulty of the emptiness problem for the indexed language. Since Also has shown that the problem can be answered in exponential time, the exponential-time completeness is shown. The exponential-tine difficulty is also directly indicated from the fact that the emptiness problem is exponential-time complete. Consequently, the recognition probe tern is also exponential-time complete. [ABSTRACT FROM AUTHOR]

Reliability functions have the property that if they intersect the diagonal y = x it is done in an S-shaped form (Moore and Shannon). In this paper a wider class of polynomials is presented, the members of which form a dense grid on the unit square. The S-shapedness intersection property holds whenever a reliability function intersects a polynomial in this set. The members of this set are all reliability polynomials. The diagonal y = x is also a member of this set, thus a generalization of Moore-Shannon result is achieved. [ABSTRACT FROM AUTHOR]

Geometric programming is a mathematical programming technique that is designed to determine the constrained minimum value of a generalized polynomial objective function. To date, most applications of the technique have been restricted to certain classes of engineering problems. This paper presents a brief summary of geometric programming and then illustrates its application to managerial problems by applying it to three well-known inventory models. [ABSTRACT FROM AUTHOR]

In this paper, the order of simultaneous approximation and Voronovskaja-type results with quantitative estimate for complex q-Kantorovich polynomials (q > 0) attached to analytic functions on compact disks are obtained. In particular, it is proved that for functions analytic in {z ∊ C : ∣z∣ < R}, R > q, the rate of approximation by the q- Durrmeyer - Stancu operators (q > 1) is of order q-n versus 1/n for the classical q-Durrmeyer - Stancu operators. [ABSTRACT FROM AUTHOR]

Earlier, D. Yu. Grigoriev and N. N. Vorob'yov obtained an upper bound $d^\{O(\{n+d\; \backslash choose\; n\})\}$ for the number of vectors of multiplicities for solutions of systems of the form $g\_1=\; \backslash cdots\; =\; g\_n=0$ (assuming that the system has a finite number of solutions). In the system above, $g\_1,\; \backslash dots\; ,\; g\_n\; \backslash in\; F[X\_1,\; \backslash dots\; ,\; X\_n]$ are polynomials of degrees $\backslash deg\; (g\_i)\; \backslash leq\; d$. In the present paper, we show that the order of growth of this bound is close to the exact one. In particular, in the case $d=n$, the construction provides a doubly-exponential (in n) number of vectors of multiplicities. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

We consider a homogeneous degenerate center of order 2m + 1 and perturb it by a homogeneous polynomial of order 2m. We study the Lyapunov constants around the origin to estimate the number of limit cycles. To do it, we classify the parameters and study their effect on the number of limit cycles. Finally, we find that the perturbed degenerate center without any condition has at least two limit cycles, and the number of the bifurcated limit cycles could reach 2m + 3. [ABSTRACT FROM AUTHOR]

Copyright of Rad HAZU: Matematicke Znanosti is the property of Croatian Academy of Sciences & Arts (HAZU) 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.)

Copyright of Rad HAZU: Matematicke Znanosti is the property of Croatian Academy of Sciences & Arts (HAZU) 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.)

We show that many spaces of multivariate splines possess additional smoothness (supersmoothness) at certain faces where polynomial pieces join together. This phenomenon affects the dimension and interpolating properties of splines spaces. The supersmoothness is caused by the geometry of the underlying partition. [ABSTRACT FROM AUTHOR]

Abstract: In this research paper using the Chebyshev expansion, we explicitly determine the best uniform polynomial approximation out of (the space of polynomials of degree at most ) to a class of rational functions of the form on , where is the first kind of Chebyshev polynomial of degree and . In this way we give some new theorems about the best approximation of this class of rational functions. Furthermore we obtain the alternating set of this class of functions. [Copyright &y& Elsevier]