Transformation Theorem Research Articles

Paley–Wiener theorems for the Bessel wavelet transformation

ABSTRACT In this paper, we defined a class of Colombeau-type generalized functions and established the Paley–Wiener theorem for the Bessel wavelet transformation on E μ , σ 2 and Colombeau-type generalized functions.

Establishment of Kifilideen coefficient tables for positive and negative powers of \(n\) and \(-n\) of Kifilideen trinomial theorem and other development based on matrix and standardized methods

The generation of coefficients of terms of positive and negative powers of \(n\) and \(-n\) of Kifilideen trinomial theorem as the terms are progress is stressful and time-consuming which the same problem is identified with coefficients of terms of binomial theorem of positive and negative powers of \(n\) and \(-n\). This slows the process of producing the series of any particular trinomial expansion. This study established Kifilideen coefficient tables for positive and negative powers of \(n\) and \(-n\) of the Kifilideen trinomial theorem and other developments based on matrix and standardized methods. A Kifilideen theorem of matrix transformation of the positive power of \(n\) of trinomial expression in which three variables \(x, y\), and \(z\) are found in parts of the trinomial expression was originated. The development would ease evaluating the trinomial expression's positive power of \(n\). The Kifilideen coefficient tables are handy and effective in generating the coefficients of terms and series of the Kifilideen expansion of trinomial expression of positive and negative powers of \(n\) and \(-n.\)

Generalized convolution and product theorems associated with the free metaplectic transformation and their applications

Convolution theorems for the free metaplectic transformation (FMT) have great significance in the fields of fractional domain non-stationary signal processing theory and method. The existing results have found many applications in high-dimensional non-stationary signal sampling and filtering. However, a widely accepted closed-form expression has not yet been established. In this paper, we introduce the definitions of generalized convolution and product operators in the FMT domain, and use them to formulate the generalized convolution and product theorems for the FMT. The derived results include particular cases the existing convolution and product theorems for the FMT and also the convolution and product theorems for the Fourier transform, the fractional Fourier transform, and the linear canonical transform. We employ the generalized convolution theorem for the FMT in designing an optimal filter in the FMT domain. We also validate the correctness and effectiveness of the theoretical analysis through numerical experiments, demonstrating the superiority of the proposed optimal filtering method over the conventional ones.

No-go theorems for photon state transformations in quantum linear optics

We give a necessary condition for photon state transformations in linear optical setups preserving the total number of photons. From an analysis of the algebra describing the quantum evolution, we find a conserved quantity that appears in all allowed optical transformations. We give some examples and numerical applications, with example code, and give three general no-go results. These include (i) the impossibility of deterministic transformations which redistribute the photons from one to two different modes, (ii) a proof that it is impossible to generate a perfect Bell state with an arbitrary ancilla from the Fock basis and (iii) a restriction for the conversion between different types of entanglement (converting GHZ to W states). These tools and results can help in the design of experiments for optical quantum state generation.

Convolution theorems for the free metaplectic transformation and its application

The free metaplectic transformation (FMT) is widely used in several fields, including filter design, pattern recognition, image processing, and optics. This study investigated two new convolution theorems in the FMT domain to obtain a more concise and intuitive convolution form. First, on the basis of expressing the generalized translation, we derived the convolution theorem (of the first kind) in the FMT domain, which has the elegance and simplicity of the classical Fourier transform (FT) results. Second, we provide the convolution theorem (of the second kind) in the FMT domain to further study the diversity of the convolution theory. The provided solution can be represented using a simple integral that is easy to implement in multiplying filter designs. Finally, based on this simple form of the convolution theorems, we designed a multiplicative filter in the FMT domain, demonstrating the feasibility and efficiency of the filters designed in this study through simulations.

High impedance grounding fault location method for power cables based on reflection coefficient spectrum

When a high impedance grounding fault occurs in the power cables, the current at the fault point is extremely low and the fault characteristics are too weak to be detected by the reflectometry. Interfered by the multiple reflections, the fault is hard to be located in the field. In order to solve the problem of conventional frequency domain reflectometry, a high impedance grounding fault location method for 10 kV distribution cables is proposed in this paper, which is based on the reflection coefficient spectrum. Firstly, the equivalent model of cable distributed parameters is established, and the reflection coefficient spectrum at the cable head is derived. Then, according to the theorem of integral transformation and generalized orthogonality, the diagnostic function is built in the spatial domain. Besides, the Kaiser window is introduced to improve the location sensitivity. Finally, the feasibility of the method is verified by simulated and experimental results. It is shown that this method can effectively eliminate the impact of multiple reflections, and precisely locate the single- and multi-point grounding faults. The fault location error is within 0.2%.

On the geometry in the large of Hadamard manifolds

A Hadamard manifold is a simply connected, complete Riemannian manifold with nonpositive sectional curvature. The theory of Hadamard manifolds is a topic that has been more and more intensively studied for more than forty years. In the present paper, we prove Liouville-type theorems for conformal, isometric and harmonic pointwise transformations of metrics of Hadamard manifolds.

Generalized upper Sugeno integrals: The transformation theorem and related results

In this paper, we deal with the transformation theorem for the generalized upper Sugeno integral I∘. We formulate the sufficient and necessary conditions under which the transformation holds, thus providing a complete solution to this problem. We also provide some representations of the I∘ integral via left-continuous (right-continuous) modifications of level measures. We characterize the class of binary operations for which the I∘ integral can be represented via the quantile function. As a consequence, we get complete solutions to these problems for the smallest semicopula-based universal integral. We thus solve several open problems related to this class of integrals in the literature.

Computability of convergence rates in the ergodic theorem for Martin-Löf random points

In this paper we look at the convergence rates for the ergodic averages in the pointwise ergodic theorem for computable ergodic transformations on the unit interval. While these rates are layerwise computable for Martin-Löf random points and effectively open sets with Lebesgue measure a computable real, they are also layerwise computable for an arbitrary interval. There are however, effectively open sets for which there are \emph{no} effective rates, in particular, not layerwise computable ones. We also show that, when the measure of the effectively open set is any real $\alpha$, the convergence rates are computable in $\alpha$ and the layers relative to $\alpha$.

Inauguration of Kifilideen theorem of matrix transformation of negative power of – n of trinomial expression in which three variables x, y and z are found in parts of the trinomial expression with some other applications

Inauguration of Kifilideen theorem of matrix transformation of negative power of – n of trinomial expression in which three variables x, y and z are found in parts of the trinomial expression with some other applications

Interaction Behaviours between Soliton and Cnoidal Periodic Waves for Nonlocal Complex Modified Korteweg–de Vries Equation

The reverse space-time nonlocal complex modified Kortewewg–de Vries (mKdV) equation is investigated by using the consistent tanh expansion (CTE) method. According to the CTE method, a nonauto-Bäcklund transformation theorem of nonlocal complex mKdV is obtained. The interactions between one kink soliton and other different nonlinear excitations are constructed via the nonauto-Bäcklund transformation theorem. By selecting cnoidal periodic waves, the interaction between one kink soliton and the cnoidal periodic waves is derived. The specific Jacobi function-type solution and graphs of its analysis are provided in this paper.

On the relationship between possibilistic and standard moments of fuzzy numbers

In this paper we introduce a transformation of the center of gravity, variance and higher moments of fuzzy numbers into their possibilistic counterparts. We show that this transformation applied to the standard formulae for the computation of the center of gravity, variance, and higher moments of fuzzy numbers gives the same formulae for the computation of possibilistic moments of fuzzy numbers that were introduced by Carlsson and Fullér (2001) for the possibilistic mean and variance, and also the formulae for the calculation of higher possibilistic moments as presented by Saeidifar and Pasha (2009). We also present an inverse transformation to derive the formulae for standard measures of central tendency, dispersion, and higher moments of fuzzy numbers, from their possibilistic counterparts. This way a two-way transition between the standard and the possibilistic moments of fuzzy numbers is enabled. The transformation theorems are proven for a wide family of fuzzy numbers with continuous, piecewise monotonic membership functions. Fast computation formulae for the first four possibilistic moments of fuzzy numbers are also presented for linear fuzzy numbers, their concentrations and dilations.

Expansion formulas for multiple basic hypergeometric series over root systems

We extend expansion formulas of Liu given in 2013 to the context of multiple series over root systems. Liu and others have shown the usefulness of these formulas in Special Functions and number-theoretic contexts. We extend Wang and Ma's generalizations of Liu's work which they obtained using q-Lagrange inversion. We use the An and Cn Bailey transformation and other summation theorems due to Gustafson, Milne, Milne and Lilly, and others, from the theory of An, Cn and Dn basic hypergeometric series.

Analytic simulation of the synergy of spatial-temporal memory indices with proportional time delay

In the present work, three space-time trace parameters are appended to physical systems to analytically outline their mutual impact and to characterize the dynamic behaviors of these systems; namely the proportional time delay τ∈(0,1) and the Caputo spatial-temporal fractional derivatives α,β∈(0,1). The adopted analytical approach depends on a novel adaptation of the differential transform method in a higher dimensional fractional space in which the initial value problems (IVPs), under consideration, are transformed into a 2-dimensional recurrence relation. Some central differential transformation theorems in 2-dimensional fractional space are provided to illustrate the influence of the aforementioned parameters. The method has been successfully applied to furnish the solution, in the form of a Cauchy product of absolutely convergent series, for a 2-dimensional extension of advection-dispersion, gas dynamics, convection-diffusion, wave, telegraph, and Klein–Gorden equations. The study concluded that the obtained solutions operate as a homotopic mapping between two states, and the Caputo fractional derivatives can be interpreted as memory indices.

Inauguration of Kifilideen theorem of matrix transformation of negative power of – n of trinomial expression in which three variables x, y and z are found in parts of the trinomial expression with some other applications

Inauguration of Kifilideen theorem of matrix transformation of negative power of – n of trinomial expression in which three variables x, y and z are found in parts of the trinomial expression with some other applications

CTE Solvability, Nonlocal Symmetry, and Interaction Solutions of Coupled Integrable Dispersionless System

The consistent tanh expansion (CTE) method is successfully applied to the coupled integrable dispersionless (CID) system. A nonauto‐Bäcklund transformation (BT) theorem includes two fields f and v1 is obtained by using the CTE method. One obtains the consistent condition in the nonauto‐BT theorem by means of the relation between the fields f and v1. The CID system possesses the CTE solvability property by some detailed analysis. Many interactions between one soliton and multiple resonant solitons, and between one soliton and cnoidal waves are generated by using the nonauto‐BT theorem. The types of bright and gray two front waves are shown by some figures. In the meanwhile, the nonlocal symmetry is obtained by the truncated Painlevé method and the Möbious invariant form. The initial value problem and an auto‐BT are constructed by the localization procedure.

The Sugeno integral. A point of view

For any positive decreasing function g on the positive real axis, we define a generalized fixed point x0 of g and call x0 “the midpoint of g”. It can be proved that the Sugeno integral of any positive measurable function f with respect to any motonone measure μ is the midpoint of the canonical decumulative function of f and μ. As an application, we obtain an immediate proof of a well-known transformation theorem concerning the Sugeno integral. Sufficient conditions insuring the validity of a popular reduction procedure (using a smaller set than the whole positive real axis) for the computation of the Sugeno integral are given. The preceeding two topics are applied to positive simple functions. At the end, using the theory of midpoints, we show that the Hausdorff dimension and the Hirsch index actually are Sugeno integrals.

Non-Markovian dynamics of the driven spin-boson model

The non-Markovian dynamics of the driven spin-boson model at zero and finite temperature is investigated theoretically. Based on a unitary transformation and convolution theorem, the analytical expression of population difference in Laplace space is obtained. This method provides a tool to help us to understand the interference between driving and dissipation within the non-Markovian frame. We study the transient and the steady-state non-Markovian dynamics of population difference P(t) in the case of different temperature, driving and spin–bath coupling (SBC) strength. It has been shown that it is necessary to use non-Markovian methods when SBC is strong. At zero temperature, the curve of steady-state amplitude P lt versus Rabi driving amplitude Ω is N-like shaped. While at finite temperature, the curve becomes Λ-like shaped. These curves come from the interference between driving and dissipation. Also, from the interference pattern above, we know that large coherent oscillations can be obtained by optimizing the parameters: Rabi driving amplitude, temperature and SBC strength.

Fuzzy Sawi Decomposition Method for Solving Nonlinear Partial Fuzzy Differential Equations

The main goal of this paper is to propose a new decomposition method for finding solutions to nonlinear partial fuzzy differential equations (NPFDE) through the fuzzy Sawi decomposition method (FSDM). This method is a combination of the fuzzy Sawi transformation and Adomian decomposition method. For this purpose, two new theorems for fuzzy Sawi transformation regarding fuzzy partial gH-derivatives are introduced. The use of convex symmetrical triangular fuzzy numbers creates symmetry between the lower and upper representations of the fuzzy solution. To demonstrate the effectiveness of the method, a numerical example is provided.

Reconstruction of blade tip-timing signals based on the MUSIC algorithm

As a multi-frequency identification method, the MUSIC method has received more and more attention in the field of blade tip-timing (BTT) signal reconstruction. So far, an improved MUSIC method and a subspace dimension-reduced MUSIC (SDR-MUSIC) method were proposed to analyze BTT signals. However, they must rely on the expanded snapshot matrix to overcome the spectrum aliasing. The computational complexity of these methods was analyzed. The computation time of BTT signal reconstruction increased cubically as the size of the expanded matrix increased. This is unfavorable for real-time processing. In this paper, reconstruction conditions based on the MUSIC algorithm were proposed. This makes the traditional MUSIC method overcome spectrum aliasing without expanding the snapshot matrix. The computational complexity is greatly reduced. The feasibility of the reconstruction conditions was verified through simulations and aero-engine experiments. Under the reconstruction conditions, the frequency identification accuracy of the traditional MUSIC can reach that of the improved MUSIC method. Besides, an amplitude identification method was proposed based on the discrete Fourier transformation (DFT) and remainder theorem. It provides a feasible solution for the spectrum methods that cannot directly extract amplitude from the pseudo spectrum, such as the improved MUSIC, SDR-MUSIC, etc. The error of the amplitude identification was analyzed by simulations. And its feasibility was further verified by identifying all blade vibration amplitudes in the aero-engine experiments.