Computing some classes of Cauchy type singular integrals with Mathematica software

We constructed an algorithm, [SInt], for computing some classes of Cauchy type singular integrals on the unit circle. The design of [SInt] was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithm. Furthermore, we show how the factorization algorithm described in Conceição et al. (2010) allowed us to construct and implement the [SIntAFact] algorithm for calculating several interesting singular integrals that cannot be computed by [SInt]. All the above techniques were implemented using the symbolic computation capabilities of the computer algebra system Mathematica. The corresponding source code of [SInt] is made available in this paper. Several examples of nontrivial singular integrals computed with both algorithms are presented.     

Symbolic Computation Applied to the Study of the Kernel of a Singular Integral Operator with Non-Carleman Shift and Conjugation

On the Hilbert space \(\widetilde{L}_{2}(\mathbb {T})\) the singular integral operator with non-Carleman shift and conjugation \(K=P_{+}+(aI+AC)P_{-}\) is considered, where \(P_{\pm }\) are the Cauchy projectors, \(A=\sum \nolimits _{j=0}^{m}a_{j}U^{j}\), \(a,a_{j}\), \(j=\overline{1,m}\), are continuous functions on the unit circle \(\mathbb {T}\), U is the shift operator and C is the operator of complex conjugation. We show how the symbolic computation capabilities of the computer algebra system Mathematica can be used to explore the dimension of the kernel of the operator K. The analytical algorithm [ADimKer-NonCarleman] is presented; several nontrivial examples are given.     

Constructive D-Module Theory with Singular

We overview numerous algorithms in computational D-module theory together with the theoretical background as well as the implementation in the computer algebra system Singular. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein–Sato polynomials and also algorithms, recovering any kind of Bernstein–Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein–Sato polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We also address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.     

Problems of Stability of Dynamical Systems and Computer Algebra

Examples of problems of motion stability solved with the help of computer algebra systems (CAS) are presented. The authors have experience in developing and applying problem-oriented systems of symbolic computations and software packages for solving problems of dynamics of multi-body systems. The algorithms under consideration are implemented (completely or partly) with the aid of up-to-date CAS. They are intended for inclusion in the ``Stability' package of symbolic computation. Bibliography: 6 titles.     

Small sample statistical condition estimation for the total least squares problem

In this paper, under the genericity condition, we study the condition estimation of the total least squares (TLS) problem based on small sample condition estimation (SCE), which can be incorporated into the direct solver for the TLS problem via the singular value decomposition (SVD) of the augmented matrix [A, b]. Our proposed condition estimation algorithms are efficient for the small and medium size TLS problem because they utilize the computed SVD of [A, b] during the numerical solution to the TLS problem. Numerical examples illustrate the reliability of the algorithms. Both normwise and componentwise perturbations are considered. Moreover, structured condition estimations are investigated for the structured TLS problem.     

Unification of calculus on Time Scales with mathematica

In 1990 Hilger defined the Time Scale Calculus which is the unification of discrete and continuous analysis in his PhD. In 2005 Yantir and Ufuktepe showed delta derivative with Mathematica[5]. In this study we give many computations of Time Scale Calculus with Mathematica such as the numerical and symbolic computation of forward jump operator and delta derivative for a particular time scale, graphs of functions, and definite integral on a time scale. We also improve and extent the Time Scale package for symbolics computations.     

On the optimality and sharpness of Laguerre’s lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix A ∈ R ^m×m play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on Tr(A ^?1) and Tr(A ^?2) have attracted attention recently, because they can be computed in O(m) operations when A is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed solely from Tr(A ^?1) and Tr(A ^?2) and show that the so called Laguerre’s lower bound is the optimal one in terms of sharpness. Next, we study the gap between the Laguerre bound and the smallest eigenvalue. We characterize the situation in which the gap becomes largest in terms of the eigenvalue distribution of A and show that the gap becomes smallest when {Tr(A ^?1)}²/Tr(A ^?2) approaches 1 or m. These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms.     

Symbolic computation of exact solutions expressible in rational formal hyperbolic and elliptic functions for nonlinear partial differential equations

With the aid of symbolic computation, some algorithms are presented for the rational expansion methods, which lead to closed-form solutions of nonlinear partial differential equations (PDEs). The new algorithms are given to find exact rational formal polynomial solutions of PDEs in terms of Jacobi elliptic functions, solutions of the Riccati equation and solutions of the generalized Riccati equation. They can be implemented in symbolic computation system Maple. As applications of the methods, we choose some nonlinear PDEs to illustrate the methods. As a result, we not only can successfully obtain the solutions found by most existing Jacobi elliptic function methods and Tanh-methods, but also find other new and more general solutions at the same time.     

Computing generalized inverses of a rational matrix and applications

In this paper we investigate symbolic implementation of two modifications of the Leverrier-Faddeev algorithm, which are applicable in computation of the Moore-Penrose and the Drazin inverse of rational matrices. We introduce an algorithm for computation of the Drazin inverse of rational matrices. This algorithm represents an extension of the papers [11] and [14]. and a continuation of the papers [15, 16]. The symbolic implementation of these algorithms in the package mathEmatica is developed. A few matrix equations are solved by means of the Drazin inverse and the Moore-Penrose inverse of rational matrices.     

Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem

A two-point boundary value problem is considered on the interval [0, 1], where the leading term in the differential operator is a Riemann-Liouville fractional derivative of order 2 ? δ with 0 < δ < 1. It is shown that any solution of such a problem can be expressed in terms of solutions to two associated weakly singular Volterra integral equations of the second kind. As a consequence, existence and uniqueness of a solution to the boundary value problem are proved, the structure of this solution is elucidated, and sharp bounds on its derivatives (in terms of the parameter δ) are derived. These results show that in general the first-order derivative of the solution will blow up at x = 0, so accurate numerical solution of this class of problems is not straightforward. The reformulation of the boundary problem in terms of Volterra integral equations enables the construction of two distinct numerical methods for its solution, both based on piecewise-polynomial collocation. Convergence rates for these methods are proved and numerical results are presented to demonstrate their performance.     

Projectile motion with Mathematica

This note describes how to use the computer algebra system (CAS) Mathematica to analyse projectile motion with and without air resistance. For a projectile fired from ground level with an initial velocity ν ft/s at an angle θ degrees from the horizontal (0 < θ < 90°), it is well known that in the absence of air resistance, the projectile follows a parabolic path. However, this is not true if air resistance is taken into account. In the presence of air resistance, the equations of motion become complicated, thus making traditional handcalculation methods quite ineffective, and a powerful CAS such as Mathematica becomes an invaluable tool to better understand projectile motion. The note discusses how Mathematica can be used to create simulated experiments of projectiles with and without air resistance. These experiments result in several conjectures, leading to theorems.     

The FFTRR-based fast direct algorithms for complex inhomogeneous biharmonic problems with applications to incompressible flows

We develop analysis-based fast and accurate direct algorithms for several biharmonic problems in a unit disk derived directly from the Green’s functions of these problems and compare the numerical results with the “decomposition” algorithms (see Ghosh and Daripa, IMA J. Numer. Anal. 36(2), 824–850 [17]) in which the biharmonic problems are first decomposed into lower order problems, most often either into two Poisson problems or into two Poisson problems and a homogeneous biharmonic problem. One of the steps in the “decomposition algorithm” as discussed in Ghosh and Daripa (IMA J. Numer. Anal. 36(2), 824–850 [17]) for solving certain biharmonic problems uses the “direct algorithm” without which the problem can not be solved. Using classical Green’s function approach for these biharmonic problems, solutions of these problems are represented in terms of singular integrals in the complex z?plane (the physical plane) involving explicitly the boundary conditions. Analysis of these singular integrals using FFT and recursive relations (RR) in Fourier space leads to the development of these fast algorithms which are called FFTRR based algorithms. These algorithms do not need to do anything special to overcome coordinate singularity at the origin as often the case when solving these problems using finite difference methods in polar coordinates. These algorithms have some other desirable properties such as the ease of implementation and parallel in nature by construction. Moreover, these algorithms have O(logN) complexity per grid point where N ² is the total number of grid points and have very low constant behind this order estimate of the complexity. Performance of these algorithms is shown on several test problems. These algorithms are applied to solving viscous flow problems at low and moderate Reynolds numbers and numerical results are presented.     

Weighted boundedness of some integral operators on weighted <Emphasis Type="Italic">λ</Emphasis>- central Morrey space

In this paper, the authors prove the weighted boundedness of singular integral and fractional integral with a rough kernel on the weighted λ-central Morrey space. Moreover, the weighted estimate for commutators of singular integral with a rough kernel on the weighted λ-central Morrey space is also given.     

Non-triviality conditions for integer-valued polynomial rings on algebras

Let D be a commutative domain with field of fractions K and let A be a torsion-free D-algebra such that \(A \cap K = D\). The ring of integer-valued polynomials on A with coefficients in K is \( Int _K(A) = \{f \in K[X] \mid f(A) \subseteq A\}\), which generalizes the classic ring \( Int (D) = \{f \in K[X] \mid f(D) \subseteq D\}\) of integer-valued polynomials on D. The condition on \(A \cap K\) implies that \(D[X] \subseteq Int _K(A) \subseteq Int (D)\), and we say that \( Int _K(A)\) is nontrivial if \( Int _K(A) \ne D[X]\). For any integral domain D, we prove that if A is finitely generated as a D-module, then \( Int _K(A)\) is nontrivial if and only if \( Int (D)\) is nontrivial. When A is not necessarily finitely generated but D is Dedekind, we provide necessary and sufficient conditions for \( Int _K(A)\) to be nontrivial. These conditions also allow us to prove that, for D Dedekind, the domain \( Int _K(A)\) has Krull dimension 2.     

Quantitative weighted estimates for rough homogeneous singular integrals

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L ²(w), we obtain a bound that is quadratic in A ₂ constant \({\left[ w \right]_{{A_2}}}\). We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.     

Boundedness of para-product operators on spaces of homogeneous type

We obtain the boundedness of Calderón-Zygmund singular integral operators T of non-convolution type on Hardy spaces H ^p (X) for 1/(1 + ε) < p ? 1, where X is a space of homogeneous type in the sense of Coifman and Weiss (1971), and ε is the regularity exponent of the kernel of the singular integral operator T. Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast to what was used in the literature.     

On a result by geronimus

In 1935, Ya.L. Geronimus found the best integral approximation on the period [?π,π) of the function sin(n + 1)t ? 2q sin nt, q ∈ ?, by the subspace of trigonometric polynomials of degree at most n ? 1. This result is an integral analog of the known theorem by E.I. Zolotarev (1868). At present, there are several methods of proving this fact. We propose one more variant of the proof. In the case |q| ≥ 1, we apply the (2π/n)-periodization and the fact that the function | sin nt| is orthogonal to the harmonic cos t on the period. In the case |q| < 1, we use the duality relations for Chebyshev’s theorem (1859) on a rational function least deviating from zero on a closed interval with respect to the uniform metric.     

A characterization of BMO in terms of endpoint bounds for commutators of singular integrals

We provide a characterization of BMO in terms of endpoint boundedness of commutators of singular integrals. In particular, in one dimension, we show that ∥b∥_BMO ? B, where B is the best constant in the endpoint L log L modular estimate for the commutator [H, b]. We provide a similar characterization of the space BMO in terms of endpoint boundedness of higher order commutators of the Hilbert transform. In higher dimension we give the corresponding characterization of BMO in terms of the first order commutators of the Riesz transforms. We also show that these characterizations can be given in terms of commutators of more general singular integral operators of convolution type.     

An exact method for minimizing the total treatment time in intensity-modulated radiotherapy

In this paper, we solve a combinatorial optimization problem that arises from the treatment planning of a type of radiotherapy where intensity is modulated by multileaf collimators (MLC) in a step-and-shoot manner. In Ernst et al [INFORMS Journal on Computing 21 (4) (2009): 562–574], we proposed an exact method for minimizing the number of MLC apertures needed for a treatment. Our method outperformed the fastest algorithms available at the time. We refer to our method as the CPI method. We now attempt to minimize the total treatment time by modifying our CPI method. This modification involves non-trivial work, as some of the search space elimination schemes used in the CPI method cannot be applied in here. In our numerical experiments, we again compare our new method with the fastest algorithms currently available. There has been significant recent research in this area; hence we compare our method with those published in Wake et al [Computers and Operations Research 36 (2009): 795–810], Ta?kin et al [Operations Research 58 (3) (2010): 674–690] and Cambazard et al [CPAIRO (2010): 1–16]. The numerical comparisons indicate that our method generally outperformed the first two, while being competitive with the third.