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This paper develops the analysis of discrete-time periodically time-varying linear systems over finite fields. It is shown that the conditions for the existence of Floquet Transform for periodic linear systems over reals (or complex) do not carry forward for this case over finite fields. The existence of Floquet Transform is shown to be equivalent to the existence of an Nth root of the monodromy matrix for the class of non-singular periodic linear systems. As the verification of existence and computation of the Nth root is a computationally hard problem, an independent analysis of the solutions of such systems is carried without the use of Floquet Transform. It is proved that all initial conditions of such systems lie either in a periodic orbit or a chain leading to a periodic orbit. A subspace of the state space is also identified, containing all initial conditions lying on a periodic orbit. Further, whenever the Floquet Transform exists, more concrete results on the orbit lengths are established depending on the coprimeness of the orbit length with the system period.

We propose a new method and hardware architecture to compute the functions expressed as $X^{Y}$ ( $X$ and $Y$ are arbitrary floating-point numbers), which can support arbitrary Nth root, exponential and power operations. Because of the complexity of direct computation, we usually convert it to logarithm, multiplication, and antilogarithm operations. Traditional approaches suffer from long latency, large area and high power consumption. To solve this problem, we propose a symmetric-mapping lookup table (SM-LUT) to be capable of computing $\log _{2}x$ ( $x\in [{1, 2}]$ ) and $2^{x}$ ( $x\in [{0, 1}]$ ) simultaneously. It lays the foundation for computing $X^{Y}$ . To further improve hardware performance of our architecture, we propose a multi-region address searcher to speed up the calculation of SM-LUT. In addition, we use an optimized Vedic multiplier to shorten the critical path and improve the efficiency of multiplication, which is included in computing $X^{Y}$ . Under the TSMC 40nm CMOS technology, we design and synthesize a reference circuit to compute $X^{Y}$ with a maximum relative error of 10−3. The report shows that the reference circuit achieves the area of $14338.50~\mu \text {m}^{2}$ and the power consumption of 4.59 mW at the frequency of 1 GHz. In comparison with the state-of-the-art work under the same input range and similar precision, it saves 78.57% area and 80.42% power consumption for $\sqrt [N]{R}$ computation and 82.89% area and 81.89% power consumption for $R^{N}$ computation averagely. On top of that, our architecture reduces the computation latency by 62.77% averagely and has one more order of magnitude of energy efficiency than others.

In recent studies, it has been shown that the theory of fourth-root metrics plays a very important role in physics, theory of space-time structures, gravitation, and general relativity. The class of weakly Berwald metrics contains the class of Berwald metrics as a special case. We establish necessary and sufficient conditions under which the fourth-root Finsler space with an (α, β)-metric is a weakly Berwald space.

As the existing complex number Nth root computation methods are relatively discrete, we propose a general method and architecture based on coordinate rotation digital computer (CORDIC) to compute arbitrary complex number Nth root for the first time. Our method performs the tasks of computing complex modulus, complex phase angle, real Nth root, sine function and cosine function, which can be implemented by circular CORDIC, linear CORDIC and hyperbolic CORDIC. Based on these CORDICs, our proposed architecture can not only improve the hardware efficiency just through shift-add operations, but also flexibly adjust the precision and the input range of complex number Nth root. To prove its feasibility, we conduct a software simulation and implement an example circuit in hardware. Under the TSMC 28nm CMOS technology, we synthesize it and get the report that it has the area of 6561μm2 and the power of 3.95mW at the frequency of 1.5GHz.

Decision-making problems are more often tainted with uncertainty. Fuzzy numbers play utmost important role to band uncertainty, more especially intuitionistic fuzzy number (IFN) which is the extension of fuzzy number (FN). Different types of IFNs such as normal and generalized trapezoidal, triangular and symmetric hexagonal IFNs are explored. However, based on nature of the data, semi-elliptic type of IFN may exists in real world decision-making problems. In this paper, a maiden attempt has been made to study normal semi elliptic intuitionistic fuzzy number (NSEIFN). Our emphasis has been on arithmetic operations of NSEIFNs and comparing with the other existing IFNs. Also rank of NSEIFNs has been proposed based on mean and value. Apart from that inverse, exponential, logarithm, square root and nth root of NSEIFNs are derived. Finally, the proposed ranking method is applied to the decision making problem where criteria and rating of alternatives are represented in terms of NSEIFN. It is observed that the proposed model produces better results and overcome the drawbacks of existing approaches.

We recently showed how nonhomogeneous second-order difference equations that appear in describing the ABJM quantum spectral curve can be solved using a Mellin space technique. In particular, we provided explicit results for anomalous dimensions of twist-1 operators in the sl(2) sector at arbitrary spin values up to the four-loop order. We showed that the obtained results can be expressed in terms of harmonic sums with additional factors in the form of a fourth root of unity, and the maximum transcendentality principle therefore holds. Here, we show that the same result can also be obtained by directly solving the mentioned difference equations in the space of the spectral parameter u. The solution involves new highly nontrivial identities between hypergeometric functions, which can have various applications. We expect that this method can be generalized both to higher loop orders and to other theories, such as the N=4 supersymmetric Yang–Mills theory.

The theory of 4-th root metrics plays an important role in Biology, General Relativity and Seismic Ray Theory. Using the Shen's sy-separating trick, we prove that every conformally flat weakly Einstein 4-th root ( α , β ) -metric on a manifold M of dimension n ≥ 3 is either a Riemannian metric or a locally Minkowski metric. Then, we show that every conformally flat 4-th root ( α , β ) -metric of almost vanishing Ξ-curvature on a manifold M of dimension n ≥ 3 reduces to a Riemannian metric or a locally Minkowski metric.

This paper presents a COordinate Rotation Digital Computer (CORDIC)-based architecture for the computation of Nth root and proves its feasibility by hardware implementation. The proposed architecture performs the task of Nth root simply by shift-add operations and enables easy tradeoff between the speed (or precision) and the area. Technically, we divide the Nth root computation into three different subtasks, and map them onto three different classes of the CORDIC accordingly. To overcome the drawback of narrow convergence range of the CORDIC algorithm, we adopt several innovative methods to yield a much improved convergence range. Subsequently, in terms of convergence range and precision, a flexible architecture is developed. The architecture is validated using MATLAB with extensive vector matching. Finally, using a pipelined structure with fixed-point input data, we implement the example circuits of the proposed architecture with radicand ranging from zero to one million, and achieve an average mean of approximately 10−7 for the relative error. The design is modeled using Verilog HDL and synthesized under the TSMC 40-nm CMOS technology. The report shows a maximum frequency of 2.083 GHz with $197421.00~{\mu }\text{m}^{2}$ area. The area decreases to $169689.98~{\mu }\text{m}^{2}$ when the frequency lowers to 1.00 GHz.

Exceptional points (EPs) could potentially enhance the sensitivity of an optical sensing system by orders of magnitude. Higher-order EP systems, having more complex physics, can further boost this parameter. In this paper, we investigate the response order of high-order non-Hermitian systems and provide a guideline for designing a sensor with high response order. Based on this design rule, we propose and demonstrate an optical sensor with a fourth-order response, and analyze its associated properties. The four resonant wavelengths of our optical sensor simultaneously collapse at a high-order exceptional point in the parameter space, providing a fourth root relation between the amount of wavelength splitting and the amplitude of the perturbation. A large sensitivity enhancement factor over 100, is observed when the wavelength splitting is compared with traditional single resonator-based sensors under small perturbation conditions.

This paper contains a proposition of a new method of determining the fatigue life of elements subjected to non-stationary loads. The model was based on the determination of the time-variable equivalent weighted amplitude. This amplitude is derived on a continuous basis as the arithmetic mean of nth root of the amplitudes that occur up to a given time. The analysis of the proposed model was carried out on the basis of selected literature insights based on specimens made of P91 and P92 steels. The experimental tests were performed under block loading with a zero mean value in the conditions of tension-compression with a small number of cycles.

There are two types of elliptic curves, ordinary elliptic curves and supersingular elliptic curves. In 2012, Sutherland proposed an efficient and almost deterministic algorithm for determining whether a given curve is ordinary or supersingular. Sutherland’s algorithm is based on sequences of isogenies started from the input curve, and computation of each isogeny requires square root computations, which is the dominant cost of the algorithm. In this paper, we reduce this dominant cost of Sutherland’s algorithm to approximately a half of the original. In contrast to Sutherland’s algorithm using j-invariants and modular polynomials, our proposed algorithm is based on Legendre form of elliptic curves, which simplifies the expression of each isogeny. Moreover, by carefully selecting the type of isogenies to be computed, we succeeded in gathering square root computations at two consecutive steps of Sutherland’s algorithm into just a single fourth root computation (with experimentally almost the same cost as a single square root computation). The results of our experiments using Magma are supporting our argument; for cases of characteristic p of 768-bit to 1024-bit lengths, our algorithm runs 43.6% to 55.7% faster than Sutherland’s algorithm.

We studied the dynamics of the order parameter and the winding numbers $W$ formation of a quenched normal-to-superconductor state phase transition in a finite size holographic superconducting ring. There is a critical circumference $\tilde{C}$ below it no winding number will be formed, then $\tilde{C}$ can be treated as the Kibble-Zurek mechanism (KZM) correlation length $\xi$ which is proportional to the fourth root of its quench rate $\tau_Q$, which is also the average size of independent pieces formed after a quench. When the circumference $C \geq 10 \xi$, the key KZM scaling between the average value of absolute winding number and the quench rate $\langle|W|\rangle \propto \tau_Q^{-1/8}$ is observed. At smaller sizes, the universal scaling will be modified, there are two regions. The middle size $5\xi, Comment: 8 pages, 6 figures

An important problem of numerical flow simulation in the frame of an oil field development is the need to improve the quality and quantity of the input data, in particular, the data on cross-well formation properties. Well test is the most informative means of a reservoir borehole-to-borehole survey. When studying the cross-well formation properties based on the interpretation of continuous curves of pressure variations measured by telemetry sensors on down-hole pumps, the impact of adjacent wells and high noise contamination of the obtained data shall be taken into account. In an attempt to solve this issue this article studies multi-well deconvolution as a means to study all components of the pressure variations curve. The multi-well deconvolution can identify the relevant response to a change in the operating mode of a certain well and can process this response in traditional ways. The multi-well deconvolution method makes it possible to assess and account for noise impacts on the pressure curve. This approach also simplifies the curve processing, as the diagnosis of an interpreted formation model is done easier. This work proposes a new approach to building self-influence and influence functions: we propose to set them down as a sum of elementary time functions representing individual filtration modes of the formation. The well bore impact is represented exponentially; the bilinear flow – as a fourth root; the linear flow – as a second root; the radial flow – as a logarithm; the boundary effect – as a linear function. This way the self-influence and influence functions coefficients are set down linearly and Newton method can be used to define them. This approach was tested on two synthetic curves of the bottom-hole pressure obtained by modelling. The simulation and deconvolution curves of the bottom-hole pressure converged closely and this proved that the formation parameters selected for the modelling and retrieved during the self-influence and influence curves processing are close and this, consequently, proves high efficiency of the proposed approach. #COMESYSO1120.

In this chapter, we will introduce exponentiation with integer exponents of real numbers. We will then derive general properties of exponentiation including the negative operator. We will also present root of order n of real numbers. We will then derive general properties of the nth root.

In this chapter, we will present exponentiation in integers. We will then derive general properties of exponentiation including the negative operator. We will also present root of order n in integers and discuss the extension of the nth root to negative numbers.

The limit superior of the nth root of |x(n)|, where x is any nonzero solution of a scalar homogeneous linear difference equation of finite order, is identified in an elementary manner with ...

The set of functions {xq∣q∈R} is linearly independent over R (with respect to any open subinterval of (0, ∞)). The titular result is a corollary for any integer n ≥ 2 (and the domain [0, ∞)). Some more accessible proofs of that result are also given. Let F be a finite field of characteristic p and cardinality pk. Then the pth-root function F → F is a polynomial function of degree at most pk − 2 if pk ≠ 2 (resp., the identity function if pk = 2). Also, for any integer n ≥ 2, every element of F has an nth root in F if and only if, for each prime number q dividing n, q is not a factor of pk − 1. Various parts of this note could find classroom use in courses at various levels, on precalculus, calculus or abstract algebra. A final section addresses educational benefits of such coverage and offers some recommendations to practitioners.

In this work, we present four results for the Laplace inverse transform of functions that involve the nth root of a product of linear factors. In order to find the Laplace inverse transform, we considered a branch cut for the nth root and a region of suitable integration, to avoid the branching points. Due to that, the solution is in terms of integrals, we easily approach this solution for some specific parameters.

The dependence of the Nusselt number on the Rayleigh number at the stage preceding the turbulent regime of convection is substantiated and obtained in analytic form. At this stage, the Nusselt number can be described by the power dependence on the Rayleigh number, which contains five constants. One of these constants is the same for all liquids and is the critical Rayleigh number Racr = 1700 ± 51. The remaining four constants are strictly individual for each type of liquid. For liquid helium, the critical Rayleigh number Racr = 35000 ± 4000 has the threshold value (Ra1mix)th = 37 085.75. This value is determined by the small ratio of the diameter of the vessel to its height, as well as the formation of a cylindrical convection cell at the bottom of the vessel with a mode number of 2 and a radial number determined by the fourth root of the firstorder Bessel function of the first kind.

Cannabis is the most frequently used illicit drug in Belgium, where it is mainly cultivated indoor. To improve the fight against this drug, cannabis-profiling methods are required. Cannabis is a natural product and its chemical composition depends on many factors, which cause a high heterogeneity and variability in the secondary metabolites, and make this study challenging. The aim of this study is to combine cannabis profiling with statistical methodology to evaluate the intra (within)- and inter (between)–plantation variabilities with the goal to define a suitable approach linking seized marijuana to given plantations. The data set used contains 46 samples from 9 locations. The chemical profiles, consisting of data from eight cannabinoids, are obtained by gas chromatography - mass spectrometry. The raw data (peak areas) is pretreated with different preprocessing methods. The Pearson correlation coefficients between intra-location profiles were calculated after each pre-treatment, and the 95 and 99 % confidence limits determined. All preprocessed data were then compared with the internal standard normalization reference method with the aim to minimize the overlap between intra- and inter-location results, i.e. to reduce the number of false positives, and to obtain the best discrimination. Furthermore, cross-validation was used to evaluate the model originating from the most efficient data pre-treatment technique. The best results were obtained, when the peak areas were normalized to the internal standard with subsequent calculation of the fourth root. It results in a reduction of false positives for both confidence limits to 11 % and 14 % compared to 21 % and 27 % for the reference method. Cross-validation reveals similar false positive results as for the calibration set. In conclusion, when preprocessing the data, an improved model is obtained resulting in a significant decrease in the number of false positives. After studying the predictive performance of the model, it appears to be representative for the entire plantation information.

Ground-penetrating radar (GPR) is an effective tool for subsurface detection. Due to the influence of the environment and equipment, the echoes of GPR contain significant noise. In order to suppress noise for GPR data, a method based on singular value decomposition (SVD) of a window-length-optimized Hankel matrix is proposed in this paper. First, SVD is applied to decompose the Hankel matrix of the original data, and the fourth root of the fourth central moment of singular values is used to optimize the window length of the Hankel matrix. Then, the difference spectrum of singular values is used to construct a threshold, which is used to distinguish between components of effective signals and components of noise. Finally, the Hankel matrix is reconstructed with singular values corresponding to effective signals to suppress noise, and the denoised data are recovered from the reconstructed Hankel matrix. The effectiveness of the proposed method is verified with both synthetic and field measurements. The experimental results show that the proposed method can effectively improve noise removal performance under different detection scenarios.

As part of our ongoing work on the enumeration of symmetry classes of lozenge tilings of hexagons with certain four-lobed structures removed from their center, we consider the case of the tilings which are both vertically and horizontally symmetric. In order to handle this, we need an extension of Kuo's graphical condensation method, which works in the presence of free boundary. Our results allow us to compute exactly the correlation in a sea of dimers of a macroscopic dent in a 90 degree wedge with mixed boundary conditions. We use previous results to compute the correlation of the corresponding symmetrized system with no boundary, and show that its fourth root has the same log-asymptotics as the correlation of the dent in the 90 degree wedge. This is the first result of this kind involving a macroscopic defect. It suggests that the connections between dimer systems with gaps and 2D electrostatics may be deeper that previously thought., 17 pages

Parabolic equation techniques are based on rational approximations of the square root and other functions. The split-step Pade solution is based on an exponential of the square root [1], which takes into account the range numerics and allows large range steps. An initial condition may be obtained using the self-starter [2], which is based on a far-field approximation of a Hankel function of the square root. Accurate solutions to range-dependent problems may be obtained with the energy-conserving parabolic equation [3], which is based on the fourth root. Three-dimensional parabolic equations are derived by factoring the wave equation without making the uncoupled azimuth approximation [4–6]. When horizontal variations in the environment are sufficiently gradual so that energy coupling between modes may be neglected, three-dimensional calculations can be avoided by solving horizontal wave equations for the mode coefficients [7, 8].

The qubit state is represented by a coherent superposition of its orthonormal basis states (|0〉 and |1〉). The probability distribution of a qubit is a discrete probability distribution of the qubits probability amplitudes. Setting a custom probability amplitudes allows a qubit to evolve to any desired state. In this paper a controlled n-root of X gate is simulated on a real quantum processor using IBMs Quantum Experience platform. A custom quantum gate could be constructed where the only inputs are the control and target qubits and the desired probability distribution for the target qubit.The qubit state is represented by a coherent superposition of its orthonormal basis states (|0〉 and |1〉). The probability distribution of a qubit is a discrete probability distribution of the qubits probability amplitudes. Setting a custom probability amplitudes allows a qubit to evolve to any desired state. In this paper a controlled n-root of X gate is simulated on a real quantum processor using IBMs Quantum Experience platform. A custom quantum gate could be constructed where the only inputs are the control and target qubits and the desired probability distribution for the target qubit.

This chapter proposes a novel approach to resolve the nth root of NOT problem for quantum computers using (−1, 0, 1) permutation matrices. Only logic NOT and exchange operations are required. This result provides a complete solution to design and implement the nth root of NOT operators of quantum computers.

In this paper we prove an nth root test for series as well as a Cauchy–Hadamard type formula and Abel's' theorem for power series on universally complete Archimedean complex vector lattices. These results are aimed at developing an alternative approach to the classical theory of complex series and power series using the notion of order convergence.

For a fixed integer $n$, we study the question whether at least one of the numbers $\Re X\omega^k$, $1\leq k\leq n$, is $\varepsilon$-close to an integer, for any possible value of $X\in\mathbb{C}$, where $\omega$ is a primitive $n$th root of unity. It turns out that there is always a $X$ for which the above numbers are concentrated around $1/2\bmod1$. The distance from $1/2$ depends only on the local properties of $n$, rather than its magnitude. This is directly related the so-called "pyjama" problem which was solved recently., Comment: 7 pages

Orthogonal Frequency Division Multiplexing (OFDM) is an efficient modulation technique for high data rate communication systems such as Digital audio broadcasting, DSL internet access, 4G etc. Peak to Average Power Ratio (PAPR) is one of major problems in OFDM system. In this paper, a combined Nth root technique with other conventionally established methods such as Clipping, SLM and PTS is proposed. An elaborate analysis is carried out by plotting CCDF curve in MATLAB demonstrating a picturesque comparison between our proposed technique and other methods individually. The obtained results show that the proposed system is better than its competitors when considering the three performance criteria together. For depicting PAPR reduction CCDF curve will be plotted by MATLAB.

This paper presents a new algorithm to approximate the nth root of a given real number z. The proposed algorithm is a hybrid algorithm between the Bisection method and the combination of the inverse of sine series and Newton’s method. The proposed algorithm will be tested to find the numerical results on Matlab programming. The results showed that if z > 1, the proposed method converges with the initial interval [1, z] and if 0 < z < 1, it converges with initial interval [0,4]. The comparison of the obtained result with Newton’s method and AS-Newton’s method shown that the efficiency of the proposed algorithm may better than both of them concerning the initial value.

Let $q$ be an $n^{th}$ root of unity for $n > 2$ and let $T_n(q)$ be the Taft (Hopf) algebra of dimension $n^2$. In 2001, Susan Montgomery and Hans-J��rgen Schneider classified all non-trivial $T_n(q)$-module algebra structures on an $n$-dimensional associative algebra $A$. They further showed that each such module structure extends uniquely to make $A$ a module algebra over the Drinfel'd double of $T_n(q)$. We explore what it is about the Taft algebras that leads to this uniqueness, by examining actions of (the Drinfel'd double of) Hopf algebras $H$ "close" to the Taft algebras on finite-dimensional algebras analogous to $A$ above. Such Hopf algebras $H$ include the Sweedler (Hopf) algebra of dimension 4, bosonizations of quantum linear spaces, and the Frobenius-Lusztig kernel $u_q(\mathfrak{sl}_2)$., 28 pages, 1 table

Given a knot $K$ we may construct a group $G_n(K)$ from the fundamental group of $K$ by adjoining an $n$th root of the meridian that commutes with the corresponding longitude. For $n\geq2$ these "generalised knot groups" determine $K$ up to reflection (Nelson and Neumann, 2008; arXiv:0804.0807). The second author has shown that for $n\geq2$, the generalised knot groups of the square knot $SK$ and the granny knot $GK$ can be distinguished by counting homomorphisms into a suitably chosen finite group (arXiv:0706.1807). We extend this result to certain generalised knot groups of square and granny knot analogues $SK_{a,b}=T_{a,b}\# T_{-a,b}$, $GK_{a,b}=T_{a,b}\# T_{a,b}$, constructed as connect sums of $(a,b)$-torus knots of opposite or identical chiralities. More precisely, for coprime $a,b\geq2$ and $n$ satisfying a certain coprimality condition with $a$ and $b$, we construct an explicit finite group $G$ (depending on $a$, $b$ and $n$) such that $G_n(SK_{a,b})$ and $G_n(GK_{a},b)$ can be distinguished by counting homomorphisms into $G$. The coprimality condition includes all $n\geq2$ coprime to $ab$. The result shows that the difference between these two groups can be detected using a finite group., Comment: 18 pages, 3 figures

In this note we consider the sequence whose n th term is the number of M -sequences of length n . We show that the n th term of this sequence is bounded above by the n th Fibonacci number and bounded below by the number of integer partitions of n into distinct parts.

Summary In the context of surface microseismic processing, a nonlinear stack method, the phase weighted nth root stack, was evaluated and benchmarked against linear stack. From a synthetic analysis, the parameter choice was evaluated to improve the detectability of small amplitude microseismic events. The choice and impact of small exponent values for the nonlinear stack method were confirmed on one stage of a multiwell, multistage hydraulic fracturing in the Marcellus shale formation. Compared to the linear stack, up to 30 % more events were detected and located for this data example.

When considering rings acting on groups, one may specialize to subrings of \(\mathbb {Q}.\) A group action by a subring of \(\mathbb {Q}\) may be interpreted as root extraction in the said group, which is the underlying subject of Chap. 5. In Sect. 5.1, we discuss the theory of isolators and isolated subgroups. This topic is related to the study of root extraction in nilpotent groups, which is the theme of Sect. 5.2. We discuss some of the theory of root extraction in groups, emphasizing the main results for nilpotent groups. Residual properties of nilpotent groups are also discussed in Sect. 5.2, as well as embeddings of nilpotent groups into nilpotent groups with roots. Section 5.3 contains an introduction to the theory of localization of nilpotent groups. A group is said to be P-local, for a set of primes P, if every group element has a unique nth root whenever n is relatively prime to the elements of P. The main result in this section is: Given a nilpotent group G, there exists a nilpotent group G P of class at most the class of G which is the best approximation to G among all P-local nilpotent groups.

New methods for finding the nth root of a positive number m, to any degree of accuracy, are discussed. These methods are based on finding eigen values and eigen vectors of a special matrix. For even order matrices, the method is founded on the well-known power method. The desired root and its higher powers can also be obtained from the same matrix.

In this paper we propose two original iterated maps to numerically approximate the nth root of a real number. Comparisons between the new maps and the famous Newton-Raphson method are carried out, including fixed point determination, stability analysis and measure of the mean convergence time, which is confirmed by our analytical convergence time model. Stability of solutions is confirmed by measuring the Lyapunov exponent over the parameter space of each map. A generalization of the second map is proposed, giving rise to a family of new maps to address the same problem. This work is developed within the language of discrete dynamical systems.

In this paper new method is introduced for complex numbers; this method does not include imaginary number “i” but produces the same results that occur in Addition, Subtraction, Multiplication & Division of complex numbers, also proof of Eluer Formula eiθ and De Moivre theorem without using imaginary number “i”. Furthermore placing the light on the square root of a negative number, the square root of a negative number is equal to the square root of the same positive real number but with an angle of 90 degree to real line. The intention is that there’s nothing mystical about imaginary number “i”. The square root of minus one is just as real as any other number. Complex numbers exists without imaginary number “i”. This paper is limited to the results which are already established.

We construct a sequence of rational functions, which will be called disc wavelets, parametrized by points in the unit disc λk,n(γ)=(1−γ2n)(ωn)k,n∈N, 0≤k