Imaginary Numbers Research Articles

Cardan Polynomials, Chebyshev Exponents, Ultra-Radicals and Generalized Imaginary Units

The concept of generalized imaginary numbers and the theory of special polynomials as well are pivoting elements in pure and applied Mathematics. Special polynomials can be viewed as a realization of the algebraic abstract notion of generalized imaginary unit. We unify different concepts, emerged in the past or in more recent times, like Cardan polynomials, Chebyshev exponents and Ultra-Radicals, within a common framework yielding new tools to deal with the theory of algebraic equations.

Free Vibration Analysis of a Periodic Viaduct Supported by Pile Foundations

In this study, a numerical model is developed for the analysis of the free vibration of an open-type periodic structure, namely, a periodic viaduct supported by pile foundations. The viaduct is supposed to be composed of infinite spans, with each span consisting of a pile foundation, a pier, two longitudinal beams (left and right beams), and three linking springs. The boundary-element method (BEM) is used to solve the pile-soil interaction problem first. Based on the developed BEM model for the pile-soil system, the compliances of the pile foundation are obtained. By using the compliances of the pile foundation, the joint conditions at the beam-beam-pier (BBP) junction and the transfer matrix method, the eigenvalue equation for the periodic viaduct is established. Numerical results for the energy bands of the viaduct show that there are three kinds of characteristic waves propagating in the periodic viaduct. The first and second kinds of waves are highly decaying. The third characteristic wave can propagate in pseudopassbands with a small attenuation. Also, it is found that, unlike a closed periodic structure, because of the energy transmission from the viaduct to the half-space soil, all energy bands for the open viaduct considered in this study are complex bands with nonzero imaginary wave numbers.

Ideal Elements in Hilbert's Geometry

Hilbert first mentioned ideal elements in his 1898-99 lectures on geometry. He described them as important, fruitful, and of frequent occurrence in mathematics, pointing to the examples of negative, irrational, imaginary, ideal and transfinite numbers. In geometry, he had in mind the examples of points, lines, and planes at infinity, whose introduction gives geometry a certain completeness, by making theorems such as those of Pappus and Desargues universally valid. In this article I will discuss how Hilbert transformed our view of the Pappus and Desargues theorems by showing that they express the underlying algebraic structure of projective geometry. I will compare this result with another of Hilbert's great contributions, his calculus of ends. By studying the ideal elements of the hyperbolic plane, Hilbert similarly extracted algebraic structure from the axioms of hyperbolic geometry. Hilbert's treatments of projective and hyperbolic geometry have another important common element: construction of real numbers. To achieve this, Hilbert has to add an axiom of continuity to the geometry axioms, but he evidently wants to show that the real numbers can be put on a geometric foundation.

수학적 지식의 발달에서 연속성 원리의 역할

When imaginary numbers were first encountered in the 16th century, mathematicians were able to calculate the imaginary numbers the same as they are today. However, it required 200 years to mathematically acknowledge the existence of imaginary numbers. The new mathematical situation that arose with a development in mathematics required a harmony of real numbers and imaginary numbers. As a result, the concept of complex number became clear. A history behind the development of complex numbers involved a process of determining a comprehensive perspective that ties real numbers and imaginary numbers in a single category, complex numbers. This came after a resolution of conflict between real numbers and imaginary numbers. This study identified the new perspective and way of mathematical thinking emerging from resolving the conflicts. Also educational implications of the analysis were discussed.

Creating Majorana fermions in topological insulators

Having not been verified in experiments yet, the concept of Majorana fermions was originally hypothesized in 1937 to describe the spin-1/2 particles that are their own antiparticles, in contrast to Dirac fermions [1], which differ from their antiparticles and covermost of wellknown fermions including quarks, leptons, electrons and their antiparticles. The different properties of these two classes of fermions are the mathematical consequence of modification that Majorana did to Dirac’s equation in depicting the spin-1/2 particles. After the modification, the imaginary numbers indispensable to understand antiparticles inDirac’s equation canbe abandoned and thus only real numbers are involved in Majorana’s equation. Recently, Majorana fermions have attracted a great deal of interest in condensedmatter physics, mainly because of their non-Abelian statistics that can be utilized for fault-tolerant quantum computing [2,3]. Theoretical investigations suggest that some exotic fermionic quasiparticles in a variety of solid state matters, such as the fractional quantum Hall state at filling factor of 5/2, ultra-cold atoms and chiral p-wave superconductors, should be identical to their antiquasiparticles and thus be Majorana fermions. In an Abrikosov vortex core of a superconductor, for example, the electron and hole excitations within the superconducting gap , i.e. the Andreev bound states, play the role of particle and antiparticle. The Majorana fermion comes out of the equal superposition of an electron (filled state at energy E above the Fermi level, EF) and a hole (empty state at energy –E below EF) excited exactly at the Fermi level, i.e. the Andreev bound state atE= 0. In a usual s-wave superconductor, where Cooper pairs have orbital angular momentum 0 and the electrons obey a Schrodinger-like, nonrelativistic equation, such a zero energy excitation is strictly forbidden and the lowest bound state is at E ∼ /2EF. In contrast, the zero mode Majorana particle is predicted to occur if the Cooper pairs have orbital angular momentum 1 (px ± ipy-wave). Majorana fermions in condensedmatters occur always in pair, whose superposition forms a conventional fermion, such as an electron or a hole. What scientists are chasing to discover are Majorana fermions that are spatially separated and thus prevented from overlapping. Such a highly delocalized pair of Majorana fermions forma fermionic state that can be protected from most types of decoherence, since a local perturbation may affect only one Majorana fermion of the pair. This topologically protected fermionic state can be used to store quantum information as a qubit and manipulated by physical exchange of the Majorana fermions basedon their non-Abelian statistics. Actually, the object showing non-Abelian statistics, i.e. an Ising anyon referred in the literature on topological quantum computation, is not the Majorana fermion by itself, but a topological defect hosting a Majorana fermion, such as an Abrikosov vortex of a px ± ipy-wave superconductor. Unfortunately, px ± ipy-wave superconductors in nature are not easily available, so the boom in searching for Majorana fermions in condensed matters had not come until 2008, when Fu and Kane at University of Pennsylvania theoretically proposed realistic proximity effect induced superconductivity in a strong topological insulator by combining it with an ordinary s-wave superconductor [4]. In this engineered superconductor, Cooper pairs are induced to the topological surface state via proximity effect, so that zero energy Andreev bound states at the vortex cores can occur. This is because the electrons in the normal state obey a Dirac-like equation and have their spin and momentum locked, so that a closed orbit produces a phase shift of π from the 360o rotation of the spin. This additional Berry phase converts destructive interference at E = 0 into constructive interference and thus results in a Majorana fermion at the Abrikosov vortex core. Here, the necessary underlying physical ingredient is the strong spin–orbit coupling that gives rise to an odd number of bands with momentumdependent spin crossing EF. Inspired by this idea, other proximity-induced superconductors were proposed using semiconductor nanowires and/or quantum wells with a strong spin–orbit coupling, in which Kramer’s degeneracy is lifted by Zeeman splitting due to an external magnetic field or amagnetic insulator in proximity [5–7]. Following the rapid theoretical developments, a number of experimental groups have taken up the challenge to create Majorana fermions by constructing the artificial p-wave-like superconductors. Last year, owing to welldeveloped nanoscale fabrication technology, Kouwenhoven’s group at Delft University of Technology succeeded in

Humanitarianism, Displacement, and the Politics of Nothing in Postwar Georgia

After the 2008 war with Russia, many internally displaced people (IDPs) in the Republic of Georgia complained that they had nothing, despite the fact that international donors gave more than $450 million in humanitarian aid. What was nothing? How was it related to forced migration? Why did humanitarianism continually focus the IDPs' attention on what they had lost rather than the help they had been given? In this article, I use the work of existentialist philosopher Alain Badiou to argue that humanitarianism creates four forms of absence: anti-artifacts, black holes, imaginary numbers, and absolute zero. These forms of nothingness force displaced people into having nothing, doing nothing, and being nothing, which in turn prevents them from reassembling the fragments of their previous lives into meaningful forms of existence in the present.

동영상 저작권보호를 위한 FFT 기반 정보 은닉 기법

FFT based fingerprinting to conceal more information has developed for video copyright protection. More complex information of video is necessary to prove an ownership and legal distributions in invisible form. This paper describes a method to insert more information and to detect them. 3x3 points structure is used to present information. The possible ways to show are 8bit, 28 = 256 where one point of 9 is always turn on. The points are marked in frequency domain that both real and imaginary party numbers are modified. The five successive frames of same scenes are used to mark because the same scene has very similar shape in FFT result. However, the detail values of coefficients are totally different each other to recognize the marked points. This paper also describes a method to detect the marked points by averaging and correlation algorithm. The PSNRs of marked images by our method had 51.138[dB] to 51.143[dB]. And we could get the correlation values from 0.79 to 0.87

A complementary set theory for quaternary code designs

Quaternary code (QC) designs form an attractive class of nonregular factorial fractions. We develop a complementary set theory for characterizing optimal QC designs that are highly fractionated in the sense of accommodating a large number of factors. This is in contrast to existing theoretical results which work only for a relatively small number of factors. While the use of imaginary numbers to represent the Gray map associated with QC designs facilitates the derivation, establishing a link with foldovers of regular fractions helps in presenting our results in a neat form.

The Harmonic Neutron Hypothesis: Derivation of the Mass Equivalents of the Quarks from the Frequency Equivalents of the Ionization Energy of Hydrogen and the Annihilation Energy of the Neutron

ABSTR ACT: The harmonic neutron hypothesis states that all of the fundamental constants, when expressed as frequency equivalent coupling constant ratios, are related to the annihilation frequency of the neutron, vn 2.271859 × 1023 Hz. The constants are interrelated by classic standing wave, harmonic number properties, and have classic quantum spectral characteristics. If vns is raised by the exponents of a consecutive integer quantum fraction (qf ) series ([n ± 1]/n) for the principal quantum numbers, then n = 1 to ∞, representing many of the degenerate exponent values of the fundamental constants, including the quarks. The qf values have been shown to be symmetrically split by sign when plotted on an ln ln plane, similar to imaginary numbers. The hypothesis also states that, when plotted, the associated entities should align in linear patterns, and their principal quantum numbers should simultaneously be products of smaller harmonic numbers for higher hierarchical particle generations. The slopes and intercepts of the lines are derived solely from the properties of hydrogen, and the neutron have been published and used to derive other constants. It has been previously predicted that the quarks are associated with the slope, defined by the line connecting the points for Planck’s constant (−1, 0) and (−1/3, known exponent of the ionization energy of hydrogen −2/3), which is referred to as the electromagnetic line, or em line, whose principal quantum number is 3. Assuming that the quarks fall on integer slope multiples of the line connect (1, 0), the inverse slope of the em line, the quark values can be derived and compared to the known values. The quarks, sign, principal quantum numbers, quantum fractions, and known and predicted eV values include: up (−10, 9/10, 1.7–3.1 MeV, 2.3 MeV); down (−11, 10/11, 4.1–5.7 MeV, 4.7 MeV); strange (−30, 29/30, 100; +30 or −20 MeV, 106.6 MeV); charm (109, 110/109, 1,290; +50 or –110 MeV, 1,280 MeV); bottom (32, 33/32, 4190 +180 or −60 MeV, 4,214 MeV); and top (10, 11/10, 172.9 ± 1.5 GeV, and 172.2 GeV). The three harmonic numbers associated with the quarks are 3, 10, and 11. The strange quark is related to the product of 3 and 10, the bottom quark 3 and 11, and the charm quark 10 and 11. The paper explains the origin and rationale for the mass equivalents of the quarks and makes predictions of more accurate values than are known.

Mixed Transforms Generated by Tensor Product and Applied in Data Processing

Finding orthogonal matrices in different sizes is very complex and important because it can be used in different applications like image processing and communications (e.g. CDMA and OFDM). In this paper we introduce a new method to find orthogonal matrices by using tensor products between two or more orthogonal matrices of real and imaginary numbers with applying it in images and communication signals processing. The output matrices will be orthogonal matrices too and the processing by our new method is very easy compared to other classical methods those use basic proofs .The results are normal and acceptable in communication signals and images but it needs more research works. General Terms Mixed Transforms, Tensor Operation, Orthogonal Matrices, Images Compression, Communication system.

Man-induced regime shifts in small estuaries—I: theory

This is Part I of two papers on man-induced regime shifts in small, narrow, and converging estuaries, with focus on the interaction between effective hydraulic drag, fine sediment import, and tidal amplification, induced by river engineering works, e.g., narrowing and deepening. In this part, a simple linear analytical model is derived, solving the linearized shallow water equations in exponentially converging tidal rivers. Distinguishing reflecting and non-reflecting conditions, a non-dimensional dispersion equation is derived which yields the real and imaginary wave numbers as a function of the estuarine convergence number and effective hydraulic drag. The estuarine convergence number describes the major geometrical features of a tidal river, e.g., intertidal area, convergence length, and water depth. This model is used in Part II analyzing the historical development of the tide in four rivers. Part I also presents a conceptual model on the response of tidal rivers to narrowing and deepening. It is argued that, upon the loss of intertidal area, flood-dominant conditions prevail, upon which fine sediments are pumped into the river, reducing its effective hydraulic drag. Then a snowball effect may be initiated, bringing the river into a hyper-turbid state. This state is self-maintaining because of entrainment processes, and favorable from an energetic point of view, and therefore highly stable. We may refer to an alternative steady state.

Imaginary numbers for combining linear equation models via Dempsterʼs rule

This paper proposes the concept of imaginary extreme numbers, which are like traditional imaginary number a+bi with i=−1 being replaced by e=1/0, along with the usual operations on these numbers including addition, subtraction, and division. It then applies the concept to representing linear equations in knowledge-based systems. It proves that the combination of linear equations via Dempsterʼs rule is equivalent to solving a system of simultaneous equations or finding a least-squares estimate when they are overdetermined.

How to conciliate Popper with Cartesius: Comment on: “Consciousness in the universe. A review of the ‘Orch OR’ theory” by S. Hameroff and R. Penrose

Hameroff and Penrose review evidence that coherent quantum processes evolve in defined biological structures terminating according to a specific scheme of objective reduction of the quantum state [1]. Both the specific scheme of objective reduction of the quantum state and the biological implications of these processes were proposed since several years by Hameroff and Penrose [2], and under the name of Diosi–Penrose theory [3]. How progress in physics, cosmology, quantum biology and neurosciences fits with these proposals? From the point of view of the history and philosophy of science this vision is almost as far-reaching as the theory-ofeverything, with the added relevance of putting us human beings in the centre of the Humean scene at a cosmological level, loading us with an intrinsically central role. Reading this review leaves the Observer between two foci. The first is: “I” must absolutely contribute to the solidification of this major discovery. The other is best summarized by Faraday’s aphorism: “it is always better to have an obscure and distorted vision of the facts than to have none” [4]. It is reassuring that Hameroff and Penrose dub their work as Theory, not as Interpretation. The interpretation of a phenomenon based on a slippery, not completely solidified ground is like an equation whose solution is reached through imaginary numbers. The word imaginary might seem particularly appropriate in a context dealing with consciousness. In the systematization through quantum physics of the collective phenomena embedded in the word consciousness a positive facet is that each of the two terms of the problem may help to solve the other, as in an open equation. Does quantum physics provide a sufficiently solidified ground? As far as consciousness is concerned one is tempted to leave the question partially open, bringing into the discussion the apparently unrelated comment that goes under the label of Weinberg’s nightmare.

수학사와 지적 흥미를 고려한 복소수의 두 가지 제시 방법

It has been proposed since modern times that we need to consult the history of mathematics in teaching mathematics, and some modifications of this principle were made recently by Lakatos, Freudenthal, and Brousseau. It may be necessary to have a direction which we consult when modifying the history of mathematics for students. In this article, we analyse the elements of the cognitive interest in Hamilton's discovery of the quaternions and in the history of discovery of imaginary numbers, and we investigate the effects of these elements on attention of the students of nowadays. These works may give a direction to the historic-genetic principle in teaching mathematics.

Magneto-optical effects of transparent magnetic diffraction gratings

Magneto-optical effects of transparent magnetic diffraction gratings composed of bismuth-substituted yttrium-iron-garnet (Bi2YFe5O12) were investigated. The off-diagonal components of the dielectric tensor of the grating are imaginary and real numbers for the light with 450nm and 532nm wavelengths, respectively. From precise observation of the first-order transmitted diffraction beam from the grating, the intensity imbalance and relative phase shift caused by the off-diagonal components are clearly observed to be 0.021 and 0.012rad, respectively.

Графика мнимых чисел

Графика мнимых чисел

Complex roots of polynomials and their computation with the help of Scientific Calculators

Real numbers are something which are associated with the practical life. This number system is one dimensional. Situations arise when the real numbers fail to provide a solution. Perhaps the Italian mathematician Gerolamo Cardano is the first known mathematician who pointed out the necessity of imaginary and complex numbers. Complex numbers are now a vital part of sciences and are used in various branches of engineering, technology, electromagnetism, quantum theory, chaos theory etc. A complex number constitutes a real number along with an imaginary number that lies on the quadrature axis and gives an additional dimension to the number system. Therefore any computation based on complex numbers, is usually complex because both the real and imaginary parts of the number are to be simultaneously dealt with. Modern scientific calculators are capable of performing on a wide range of functions on complex numbers in their COMP and CMPLX modes with an equal ease as with the real numbers. In this work, the use of scientific calculators (Casio brand) for efficient determination of complex roots of various types of equations is discussed. DOI: http://dx.doi.org/10.3126/bibechana.v9i0.7148 BIBECHANA 9 (2013) 18-27

An Improved CS Algorithm Based on the Curved Trajectory in Geosynchronous SAR

In Geosynchronous (GEO) Synthetic Aperture Radar (SAR), the linear trajectory model fits the actual slant range history in a poor approximation. For example, when the satellite passes the apogee, the velocity and squint angle based on the linear trajectory model will be imaginary numbers; when the satellite passes the perigee and equator, the linear trajectory model can cause large phase error such as hundreds of radians. Therefore, the classical chirp scaling (CS) algorithm under the linear trajectory model is not useable in GEO SAR. In this paper, an improved CS algorithm is proposed to solve the imaging in GEO SAR. Firstly, the true slant histories (curved trajectory) are accurately modeled based on vector expression. Then, an accurate Two-Dimension spectrum in GEO SAR is obtained for the first time based on the high order Taylor expansion and series reversion method. The new CS factor, range migration factor, chirp frequency rate, range and azimuth compression functions are all analytically derived over again in sequence. Finally, the simulation results show that the improved CS algorithm can realize the focusing of large scene and long synthetic aperture time (hundreds of seconds) in GEO SAR.

Measurement of evanescent wave properties of a bulk acoustic wave resonator [Letters

Acoustic wave fields in a thin-film bulk acoustic wave resonator are studied using a heterodyne laser interferometer. The measurement area is extended outside the active electrode region of the resonator, so that wave fields in both the active and surrounding regions can be characterized. At frequencies at which the region surrounding the resonator does not support laterally propagating acoustic waves, the analysis of the measurement data shows exponentially decaying amplitude fields outside the active resonator area, as suggested by theory. The magnitude of the imaginary wave vectors is determined by fitting an exponential function to the measured amplitude data, and thereby the experimentally determined dispersion diagram is extended into the region of imaginary wave numbers.

Redefining the Wilhelmy and Young equations to imaginary number space and implications for wettability measurements

AbstractAlthough Wilhelmy balance measurements have been reported to yield undefined values of the type cos θ > 1, this phenomenon often goes unnoticed because commercial instruments fail to report this error, listing a contact angle of zero instead. On rough superhydrophilic surfaces such “undefined” values appear much more frequent, but a mathematical framework for evaluation and quantification is lacking. A solution to the problem of cos θ > 1 was found by implementing the imaginary number i. It will be shown that both the classical and novel contact angles can be described by numbers in an imaginary space hitherto not accessible to the Wilhelmy and Young equation system. It will be exemplified that Wilhelmy balance data classed as undefined because of cos θ > 1, can easily be converted to imaginary numbers allowing the extrapolation of a novel imaginary advancing θai,AH2O = 0.36i rad and receding contact angle θai,RH2O = 0.37i rad at zero immersion depth as in classical tensiometry. The two imaginary angles compare to classical angles of ∣20°∣–∣25°∣ . The postulated core wettability range for superhydrophilicity in the special case of the “inverse lotus effect” is suggested to extend from the classical angle of cos (10°) to the imaginary angle of cos (0.37i rad). Knowledge obtained from such analyses should be of use in constructing novel artificial surfaces of extreme wettability, e. g. superhydrophilicity, not only in the medical field of implantology but also in chemistry, physics and engineering.