Special Summation Research Articles

Copper ratio obtained by generalizing the Fibonacci sequence

In this study, we define a new generalization of the Fibonacci sequence that gives the copper ratio, and we will call it the copper Fibonacci sequence. In addition, inspired by the copper Fibonacci definition, we also define copper Lucas sequences, and then we give the relationships between the terms of these sequences. We present some properties, such as the Binet formulas, special summation formulas, special generating functions, etc. We find the relationships between the roots of the characteristic equation of these sequences and the general terms of these sequences. What is interesting here is that the relationships obtained from that between the roots of the characteristic equation of these new sequences and the terms of the sequences are satisfied in both roots. In addition, we examine the relationships between these sequences with the classic Fibonacci and Lucas sequences. Moreover, we calculate some identities of these sequences, such as Cassini and Catalan. Then Catalan transformation is applied to these sequences, and their terms are found. Furthermore, we apply Hankel transform to the Catalan transform of these sequences. Besides, we associate the terms of the Hankel transformation of the Catalan copper Fibonacci sequence with the classical Fibonacci numbers and the terms of the Hankel transformation of the Catalan copper Lucas sequence with the terms of the copper Lucas sequence. We present the application of copper Fibonacci and copper Lucas sequences to hyperbolic quaternions. Finally, the terms of the copper Fibonacci and copper Lucas sequences are associated with their hyperbolic quaternion values.

Calculation of Rayleigh type sums for zeros of the equation arising in spectral problem

For zeros of the equation (arising in the oblique derivative problem)μJn′(μ)cosα+inJn(μ)sinα=0,μ∈ℂ,with parameters n ∈ ℤ, α ∈ [−π/2, π/2] and the Bessel function Jn(μ) special summation relationships are proved. The obtained results are consistent with the theory of well-known Rayleigh sums calculating by zeros of the Bessel function.

The role played by evanescent fields in the process of formation of radiation of combined radiating systems

The problem of active controlling of the structure of fields of combined radiating systems within their near zone is studied. The characteristic size of this zone is on the order of the wavelength and is characterized by the presence of evanescent (nonpropagating) fields, which are formed, among other things, due to the interference interaction of radiators of the system. Using multipole expansions for fields and special summation formulas for such expansions allows one to obtain concise expressions convenient in carrying out numerical calculations. The results of calculations confirm that the evanescent fields’ structure plays a significant part in the process of the formation of the radiation field.

Control of the magnitude and spatial distribution of interference energy flows in near fields of systems of identical radiators

The problem of active control for both the magnitude and spatial distribution of individual components of the interference component of the Poynting vector within the near zone of a system of radiators is studied. The characteristic size of this zone is on the order of the wavelength and is characterized by the presence of evanescent (nonpropagating) fields, which are formed due to the interference interaction of radiators. Using multipole expansions for fields and special summation formulas for such expansions allows one to obtain concise expressions convenient in carrying out numerical calculations. The results of calculations confirm the feasibility of the above-mentioned control in principle in solving problems of medium and object sensing.

BISC

The performance of a depth-first frequent itemset (FI) miming algorithm is closely related to the total number of recursions. In previous approaches this is mainly decided by the total number of FIs, which results in poor performance when a large number of FIs are involved. To solve this problem, a three-strategy adaptive algorithm, bitmap itemset support counting (BISC), is presented. The core strategy, BISC1, is used in the innermost steps of the recursion. For a database D with only s frequent items, a depth-first approach need up to s levels of recursions to detect all the FIs (up to 2 s ). BISC1 completely replaces these recursions with a special summation that directly calculates the supports of all the possible 2 s candidate itemsets. With BISC1 the run-time is entirely independent of the database after one database scan, and the per-candidate cost is only s . To offset the exponential growth of cost (both time and space) with BISC1 as s increases, a second strategy, BISC2, is introduced to effectively double the acceptable range of s . BISC2 divides an itemset into prefix and suffix and improves the performance by pruning all the itemsets with infrequent prefixes. If the total number of frequent items in D is high, the classic database projection strategy is used. In this case for the first s items a single run of BISC (1 or 2) is applied. For each of the remaining items, a projected database is created and the mining process proceeds recursively. To achieve optimal performance, BISC adaptively decides which strategy to use based on the dataset and minimum support. Experiments show that BISC outperforms previous approaches in all the datasets tested. Even though this does not guarantee that BISC will always perform the best, the result is impressive given the fact that most existing algorithms are only efficient in some types of datasets. The memory usage of BISC is also comparable to those of other algorithms.

Winding numbers and summation processes

Answering a question asked by I.M. Gelfand, H. Brezis gave a simple formula for the winding number of a mapping of the circle into itself in terms of the absolute values of its Fourier coefficients. This formula applies only to good classes of mappings and may fail otherwise. It has the form of a series and this series may need a convenient process of summation. The article contains optimal results about classes of functions and a special summation process. This summation process is not classical and the last part of the article explains its relation with classical summation processes.

A note on true desingularisation of boundary integral methods for three-dimensional potential problems

A desingularized boundary element formulation for the three-dimensional potential problem will be presented. It is based on integral identities for the fundamental solution. The shown approach has the advantage that the singular terms on both influence matrices can be directly calculated by replacing it with a special summation of the other off-diagonal elements. It is an extension of the so-called 4 π rule in which the strongest singularity is removed by replacing the terms of one of the influence matrices by 4 π minus the sum of the off-diagonal terms of the same row. It is shown here that a similar method can also be applied to the weakest singularity, thereby completely desingularizing the method. Both integral equations and their corresponding matrix–vector notation will be presented.

A SUMMATION FORMULA OF6F5(1)

The authors aim at obtaining an interesting result for a special summation formula for (1), by comparing two generalized Watson`s theorems on the sum of a obtained earlier by Mitra and Lavoie et. al.

The Properties of Functions and Approximation by Summation Rational Operators on the Real Axis

In the space \(C(\mathbb{R})\), we construct special summation operators of Jackson type with rational kernels and under a suitable choice of poles we obtain order estimates for deviations of these operators from functions belonging to certain classes.

Generalized Pascal matrix and recurrence sequences

This paper gives a product formula of the generalized Pascal matrix φ n [ x, y], from this, getting the striking simplicity of the powers of φ n [ x, y]. Furthermore, this paper defines the IEGP-matrix Ω n[x,y] . It is shown that not only can Ω n[x,y] be factorized by special summation matrices, but also has the closely relation with generalized Fibonacci sequences W n ( x, y), an explicit diagonalization of Ω n[x,y] is given.

An extension of the generalized pascal matrix and its algebraic properties

The extended generalized Pascal matrix can be represented in two different ways: as a lower triangular matrix Φ n[ x, y] or as a symmetric Ψ n[ x, y]. These matrices generalize P n [ x], Q n [ x], and R n[ x], which are defined by Zhang and by Call and Velleman. A product formula for Φ n[ x, y] has been found which generalizes the result of Call and Velleman. It is shown that not only can Φ n[ x, y] be factorized by special summation, but also Ψ n[ x, y] as Q n [ xy]Φ s T[ y,1/ x] or Φ n [ x, y] P n T [ y/x]. Finally, the inverse of Ψ n [ x, y] and the values of det Φ n [ x, y], det Φ n −1[ x, y], det Ψ n [ x, y], and det Ψ n −1[ x, y] are given.

An extension of the generalized pascal matrix and its algebraic properties

The extended generalized Pascal matrix can be represented in two different ways: as a lower triangular matrix Φ n [ x, y ] or as a symmetric Ψ n [ x, y ]. These matrices generalize P n [ x ], Q n [ x ], and R n [ x ], which are defined by Zhang and by Call and Velleman. A product formula for Φ n [ x, y ] has been found which generalizes the result of Call and Velleman. It is shown that not only can Φ n [ x, y ] be factorized by special summation, but also Ψ n [ x, y ] as Q n [ xy ]Φ s T [ y ,1/ x ] or Φ n [ x, y ] P n T [ y/x ]. Finally, the inverse of Ψ n [ x, y ] and the values of det Φ n [ x, y ], det Φ n −1 [ x, y ], det Ψ n [ x, y ], and det Ψ n −1 [ x, y ] are given.

The linear algebra of the pascal matrix

Pascal's triangle can be represented as a square matrix in two basically different ways: as a lower triangular matrix P n or as a full, symmetric matrix Q n. It has been found that the P n P n T is the Cholesky factorization of Q n . P n can be factorized by special summation matrices. It can be shown that the inverses of these matrices are the operators which perform the Gaussian elimination steps for calculating Cholesky's factorization. By applying linear algebra we produce combinatorial identities and an existence theorem for diophantine equation systems. Finally, an explicit formula for the sum of the kth powers is given.

The linear algebra of the Pascal matrix

Pascal's triangle can be represented as a square matrix in two basically different ways: as a lower triangular matrix P n or as a full, symmetric matrix Q n . It has been found that the P n P n T is the Cholesky factorization of Q n . P n can be factorized by special summation matrices. It can be shown that the inverses of these matrices are the operators which perform the Gaussian elimination steps for calculating Cholesky's factorization. By applying linear algebra we produce combinatorial identities and an existence theorem for diophantine equation systems. Finally, an explicit formula for the sum of the k th powers is given.

Pointwise vanishing of two-loop contributions to the one, two, three-point functions in the Neveu-Schwarz-Ramond formalism

i. It is often argued that in a properly defined superstring theory the one, two, three-point functions for massless particles must, together with the partition functions, vanish in all perturbation orders (in flat spacetime) [i]. However, explicit realization of such a theory outside the single-loop approximation is as yet unknown. It is widely believed that a correct theory will arise from a fermion string after a special summation over the spinor structures, this being analogous to the Gliozzi-Scherk-Olive projection. Unfortunately, the situation here is not so simple. The main problem is that the theory of a fermionic string depends on the choice of the odd moduli [2,3]. If they are chosen arbitrarily, nothing like a superstring is obtained. Moreover, even if the arbitrariness can be eliminated (it actually reduces to arbitrariness in the choice of the odd moduli on the boundary of the space of the ordinary moduli, where the ideas of physical factorization are valid) integration over the moduli space in an arbitrary coordinate system need not necessarily lead to simple expressions (see [3] in connection with a general analysis of this kind). Fortunately, there is as yet no reason not to believe that the formalism corresponding to a superstring can be simple if one uses a special coordinate system on the supermoduli space of the fermionic string, i.e., in this coordinate system the moduli space for the superstring and for the fermionic string have the simplest connection. Moreover, it was shown in [4] that the partition function at the two-loop level is determined pointwise by the zero measure on the moduli space if one of the odd moduli is placed at the branch point. There has since been a successful generalization of this coordinate system [5]: The odd moduli must be placed at the zeros of the holomorphic l-differential W, which must then be made to tend to ~ (similar arguments are also given in [6]). It is readily verified that for such a choice of the coordinates the expressions for the one, two, three-point functions in two loops also vanish pointwise on the moduli space. This prescription also gives promising results for higher genuses. However, the experience gained from the calculation of the two-loop partition functions gives a hint that the prescription could be too restrictive -for the partition functions it was sufficient to fix only one of the odd moduli at the branch point. We shall show below that for pointwise vanishing of the one, two, three-point functions the position of the second modulus must be positioned at the same point, in accordance with the general prescription of [5]. In addition, several of the expressions used in this paper may be helpful for the further development of the technique of calculations on Riemann surfaces of genus 2. We shall also give a more accurate formulation and proof of the important identities in the Appendix to [2], which synthesize Riemann's identities and Riemann's zero theorem.

Estimation and removal of power line interference in the electrocardiogram: A comparison of digital approaches

Global filtering of AC interference in the digitized ECG is introduced as a new concept. Two different filters embodying a global approach are presented. One is based on a least-squares error fit, the other uses a special summation method. Both methods are compared with a local predictive filter by applying each filter to artificial signals and to real ECGs. A critical evaluation of the results is given.

A Fourier series kernel based on Chebyshev polynomials

In most cases evaluation of Fourier series requires that special summation methods be applied or that the coefficients of the series be suitably modified to suppress strong oscillations at discontinuties of the approximated function. All these methods may be described as a substitution of the Dirichlet kernel by other kernels. In this paper eight of these kernels are briefly reviewed and compared with a ninth kernel which is based on Chebyshev polynominals. A closed-form representation has been derived for the Fourier coefficients of this kernel as well as a recursive relation for their practical computation. Furthermore, an error criterion is given which allows the determination of an upper bound of the difference between Fourier series and approximated function provided upper limits on both the variation and the second derivative of the latter are known.

On the field representations in the ELF range

Spectral propagation characteristics of ELF atmospherics or slow-tail wave forms are usually determined from mode theory or a residue series representation [Jean et al., 1961; Jones, 1970; Taylor and Sao, 1970; Hughes and Theisen, 1970]. The azimuthal or distance variation of the fields is characterized by Legendre functions of complex order where special simplified forms exist near the source, in the asymptotic range, or near the antipode. The successive Schumann resonances (or circumferential resonances of the earth to ionosphere cavity, which are excited by the worldwide distribution of thunderstorm activity) can be described by Legendre polynomials of increasing integer order n [Wait, 1962; Large and Wait, 1968]. The Legendre polynomials or zonal harmonics can be also used at higher frequencies; special summation techniques are required to assure a more rapid convergence of the series [Johler and Berry, 1962], and the representation of zonal harmonics yields fields consistent with other representations [Galejs, 1969, Figure 4].

Electromagnetic radiation from a cylindrically capped bi-wedge

The cylindricalLy capped bi-wedge is an arrangement of two perfectly conducting wedges, apex to apex and exactly opposing; it is truncated by a cylinder whose axis is the apex of the biwedge. A line source at the apex excites this geometry. To find the radiation properties of this configuration the corresponding boundary-value problem is solved. Truncating the bi-wedge by a cylinder defines two regions-the interior and the exterior. The fields in the interior and exterior region are given by appropriate series expansions which are solutions to Maxwell's equations. A relationship between the coefficients of the two fields is obtained when thoe fields are matched across the aperture. A set of infinite simultaneous equations is generated. These can be solved for the coefficients to which special summation techniques have been applied to make a truncation of the infinite matrix valid. The radiated field of a bi-wedge is then expressed as a series with unknown coefficients. These can now be calculated by solving a set of finite simultaneous equations. A bi-wedge in the resonance region was chosen to illustrate this procedure. The computations for ka = \pi/2 were performed and the far-field radiation patterns plotted for various wedge angles. The limiting cases of the Rayleigh region, the thin bi-wedge, and the bi-wedge when the angle \theta_{0} approaches \pi/2 are then analyzed. It is found that for the thin and the very thick bi-wedge the matrix decouples, and the field can be given exactly for these two limiting cases. Furthermore, it was found that the simple formulas of the two limiting cases could be extended to give usable results for other angles, thereby avoiding a solution of the infinite matrix.