Trigonometric Set Research Articles

Do Dogs Know Calculus With Early Transcendentals?

Summary A standard applications problem in differential calculus involves minimizing the travel time needed for a dog on a beach to reach a toy floating in the water, taking into account the different velocities the dog can achieve when running versus swimming. The solution usually presented to such problems are formulated in terms of a Cartesian coordinate. Here, a solution that is based in trigonometry is presented that, while quite natural, appears to be relatively unknown. The solution to the optimization problem in the trigonometric setting has a clear interpretation which is lacking in the Cartesian solution.

Frobenius algebras and root systems: the trigonometric case

We construct Frobenius structures on the \({\mathbb {C}}^{\times }\)-bundle of the complement of a toric arrangement associated with a root system, by making use of a one-parameter family of torsion free and flat connections on it. This gives rise to a class of Frobenius manifolds parameterizing Frobenius algebras in terms of root systems in a trigonometric setting. We then determine their potential functions, which amounts to giving the trigonometric solutions of WDVV equations.

Online harmonic elimination pulse width modulation technique for modular multilevel cascade converter

This paper presents a novel online technique, which simultaneously exploits the global search ability of differential evolution (DE) and rapid convergence of Newton Raphson (NR) methods (named as DE-NR) to solve intricate simultaneous transcendental trigonometric set of harmonic elimination pulse width modulation equations for modular multilevel cascade converters based power systems. Major contribution of this paper is a harmonically efficient online algorithm with rapid convergence. Switching angles for an extensive range of modulation index (M,0.86⩽M≤9.97) are computed online to demonstrate the harmonically efficient working of proposed method. Total harmonic distortion values of the filtered line-to-line output voltages during online operation are restricted to 0.35% of the fundamental component, which depict successful removal of unwanted harmonics and are significantly better than the values allowed by IEEE standard 519-2014. Comparison between DE-NR, differential evolution and Newton Raphson methods demonstrates the rapid convergence behavior of DE-NR as it requires only (on average) 4 iterations compared to 490 and 182 iterations respectively. It has also shown superior harmonic control than the recently devised online Middle-Level Selective Harmonic Elimination Pulse-Amplitude Modulation method. Successful simulation and experimental validations of DE-NR method have been done by developing three-phase modular multilevel cascade converter in MATLAB-Simulink and single-phase eleven-level modular multilevel cascade converter hardware prototype.

On the computation of moving mass/beam interaction utilizing a semi-analytical method

This article proposes a new orthogonal basis for the spatial discretization tracking the dynamic response of shear deformable beams. The corresponding initial boundary value differential equations of motion are dealt with focusing on Timoshenko and Reddy–Bickford beam theories. The presented technique takes advantage of a compatible trigonometric set of functions defining the rotation field in combination with orthogonal splines for the transverse deformation. A broad survey is conducted gaining the beam natural frequencies in free vibration; the forced vibration of the beam is attacked considering a traveling inertial body interacting with the base beam. For different beam end-fixity cases, the orthogonal basis could be straightforwardly constructed representing a rapid convergence. In this regard, a remedy is achieved for the inconvenient procedure of creating the shape functions in other competing methodologies.

Trigonometric generalized T-splines

The paper’s main aim consists of extending the T-spline approach to trigonometric generalized B-splines, a particularly relevant case of non-polynomial splines. Such goal can be achieved by a careful revision of some results concerning the basic properties of the trigonometric generalized B-splines and by a formalization of the concept of T-splines in the trigonometric setting. Moreover, fundamental for the use of this new tool is the study of the noteworthy case with constant frequencies and of the linear independence of the corresponding blending functions, which can be proved to be strongly linked to the linear independence of the polynomial blending functions associated to the same T-mesh.

An Orthonormally Weighted Standardized Time Series for the Error Estimation of Local Tallies in Monte Carlo Criticality Calculation

An orthonormally weighted standardized time series (OWSTS) was investigated for the statistical error estimation of local tallies in Monte Carlo criticality calculation. Unlike the original implementation of a standardized time series, the computation of standard deviation via OWSTS can be made free of the grouping of iteration cycles into batches. The characteristic aspect of OWSTS is the application of an arbitrary number of weighting functions to a standardized series of tallies such that asymptotically independent and unbiased estimates are produced based on the statistics of Brownian bridge. In the present work, a trigonometric set of weighting functions is extended and applied to local power tallies in the three-dimensional model of a pressurized water reactor core. Numerical results demonstrate that the OWSTS error estimation is unbiased for a sufficiently large number of iteration cycles.

Bessel functions for root systems via the trigonometric setting

By taking an appropriate limit, we obtain the Bessel functions related to root systems as limit of Heckman-Opdam hypergeometric functions . A more general class of Bessel functions is also investigated, which we shall call the \Theta-Bessel functions. Explicit formuals for the \Theta-Bessel functions are obtained when the multiplicity functions are even and positive integer-valued. This class encloses the Bessel functions on the tangent space at the origin of non-compact causal symmetric spaces, were an integral representation for these special functions is shown.

Positivity of Dunkl's Intertwining Operator via the Trigonometric Setting

AbstractIn this note, a new proof for the positivity of Dunkl’s intertwining operator in thecrystallographic case is given. It is based on an asymptotic relationship between theOpdam-Cherednik kernel and the Dunkl kernel as recently observed by M. de Jeu, andon positivity results of S. Sahi for the Heckman-Opdam polynomials and their non-symmetric counterparts. 2000 AMS Subject Classification: 33C52, 33C67. 1 Introduction In [R], it was proven that Dunkl’s intertwining operator between the rational Dunkl operatorsfor a fixed finite reflection group and nonnegative multiplicity function is positive. As aconsequence we obtained an abstract Harish-Chandra type integral representation for theDunkl kernel, the image of the usual exponential kernel under the intertwiner. The proofwas based on methods from the theory of operator semigroups and a rank-one reduction.In the present note, we give a new, completely different proof of these results under theonly additional assumption that the underlying reflection group has to be crystallographic.In contrast to the proof of [R], where precise information on the supports of the representingmeasures could only be obtained by going back to estimates of the kernel from [J1], thisinformation is now directly obtained. Our new approach relies first on an asymptotic rela-tionship between the Opdam-Cherednik kernel and the Dunkl kernel as recently observed byde Jeu [J2], and second on positivity results of Sahi [S] for the Heckman-Opdam polynomialsand their non-symmetric counterparts.

A HIERARCHICAL FUNCTIONS SET FOR PREDICTING VERY HIGH ORDER PLATE BENDING MODES WITH ANY BOUNDARY CONDITIONS

In this paper a new hierarchical functions set is proposed to predict flexural motion of plate-like structures in the medium frequency range. This functions set is built from trigonometric functions instead of polynomials as classically encountered in the literature. It is shown that such a trigonometric set presents all the advantages of a classical hierarchical polynomials set and additional ones which are of interest if very high order functions are intended to be used. It is stated that this new trigonometric set can be used at very high orders, up to 2048 without taking care of computer round-off errors, while the polynomials set fail, at order 46 because of the limited numerical dynamics of computers. This trigonometric set can be easily implemented on a computer. It does not require quadruple precision pre-computed arrays. Only a very low number (which does not depend on the function order) of basic operations is needed when calling such functions. Moreover, it is shown that this trigonometric set presents a better convergence rate than polynomials when predicting high order natural flexural modes of rectangular plates with any boundary conditions.

A new trigonometric functions set for predicting plate bending, using the hierarchical finite element method

This communication proposes a new hierarchical functions set to predict flexural motion of platelike structures in the medium-frequency range. This functions set is built from trigonometric functions instead of polynomials as classically encountered in the literature. It is shown that such a trigonometric set presents all the advantages of a classical hierarchical polynomials set and additional ones which are of interest if very high-order functions are desired to be used. It is stated that this new trigonometric set can be used at very high orders, up to 2048, without taking care of computer roundoff errors, while the polynomials set fails at order 46 because of the limited numerical dynamic of computers. This trigonometric set can be easily implemented on a computer. It does not require quadruple precision precomputed arrays. Only a very low number (which do not depend on the function order) of basic operations is needed when calling such functions. Moreover, it is shown that this trigonometric set presents a better convergence rate than polynomials when predicting high-order natural flexural modes of rectangular plates with any boundary conditions.

On the search for weighted norm inequalities for the Fourier transform

| (see e.g. [2] forthe trigonometric case), although our sufficient condition is somehow arestatement of a generalized Hausdorf-Young inequality (see e.g. [4, page200]). In this connection we must point out the work of Hardy, Littlewoodand Paley (see e.g. [5, Chapter VII, §8 and Chapter XII, §§3, 5 and 6]). Bythe way, both of our conditions may be easily translated to the trigono-metric setting, where relationships between / and its Fourier coefficientsare considered, / appearing either on the left or right hand side of theinequality. For a similar inequality to that of our necessary condition see,for instance, [5, Chapter XVI, Example 8, page 298] where, however, theexponents are less than 2.In §2 we give some introductory ideas which give some feeling forthe subject and use these in §3 to show a simple necessary condition when