Complex Numbers

The shape of the optimal rod determined in the work meets the condition of mass conservation in relation to the reference rod. At the same time, this rod shows a significant increase in resistance to axial force. In the examples presented, this increase was <i>80</i>% and <i>117</i>%, respectively, for rods with slenderness of 125 and 175. A practical benefit from the use of compression rods of the proposed shapes is clearly visible. The example presented in this publication shows how great the utility in the structural mechanics can be, resulting from the applications of complex analysis (complex numbers). This approach to many problems can find its solutions, while they are lacking in the real numbers domains. What is more, although these are operations on complex numbers, these solutions have often their real representations, as the numerical example shows. There are too few applications of complex numbers in the technique and science, therefore it is obvious that the use of complex analysis should have an increasing range. One of the first people to use complex numbers was Girolamo Cardano. Cardano, using complex numbers, was solving cubic equations, unsolvable to his times – as the famous Franciscan and professor of mathematics Luca Pacioli put it in his paper <i>Summa de arithmetica, geometria, proportioni et proportionalita</i> (1494). It is worth mentioning that history has given Cardano priority in the use of complex numbers, but most probably they were discovered by another professor of mathematics – Scipione del Ferro (cf. [1]). We can see, that already then, they were definitely important (complex numbers).

Wave-particle duality is one of the basic features of quantum mechanics, giving rise to the use of complex numbers in describing states of quantum systems and their dynamics and interaction. Since the inception of quantum theory, it has been debated whether complex numbers are essential or whether an alternative consistent formulation is possible using real numbers only. Here, we attack this long-standing problem theoretically and experimentally, using the powerful tools of quantum resource theories. We show that, under reasonable assumptions, quantum states are easier to create and manipulate if they only have real elements. This gives an operational meaning to the resource theory of imaginarity. We identify and answer several important questions, which include the state-conversion problem for all qubit states and all pure states of any dimension and the approximate imaginarity distillation for all quantum states. As an application, we show that imaginarity plays a crucial role in state discrimination, that is, there exist real quantum states that can be perfectly distinguished via local operations and classical communication but that cannot be distinguished with any nonzero probability if one of the parties has no access to imaginarity. We confirm this phenomenon experimentally with linear optics, discriminating different two-photon quantum states by local projective measurements. Our results prove that complex numbers are an indispensable part of quantum mechanics.

The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR) type setting is investigated using the mathematical formalism of geometric algebra (GA). The main advantage of this framework is that the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, and hence the quantum game becomes a proper extension of the classical game, avoiding a criticism of other quantum game frameworks. We produce a general solution for two-player games, and as examples, we analyze the games of Prisoners' Dilemma and Stag Hunt in the EPR setting. The use of GA allows a quantum-mechanical analysis without the use of complex numbers or the Dirac Bra-ket notation, and hence is more accessible to the non-physicist.

On the Use of Complex Numbers

Spin-momentum locking of evanescent waves describes the relationship between the propagation constant of an evanescent mode and the polarization of its electromagnetic field, giving rise to applications in light nano-routing and polarimetry among many others. The use of complex numbers in physics is a powerful representation in areas such as quantum mechanics or electromagnetism; it is well known that a lossy waveguide can be modelled with the addition of an imaginary part to the propagation constant. Here we explore how these losses are entangled with the polarization of the associated evanescent tails for the waveguide, revealing a well-defined mapping between waveguide losses and the Poincar\'e sphere of polarizations, in what could be understood as a "polarization-loss locking" of evanescent waves. We analyse the implications for near-field directional coupling of sources to waveguides, as optimized dipoles must take into account the losses for a perfectly unidirectional excitation. We also reveal the potential advantage of calculating the angular spectrum of a source defined in a complex, rather than the traditionally purely real, transverse wavevector space formalism.

Quantum chromodynamics with permanently confined colored quarks and gluons is a remarkably successful theoretical scheme. It is quite possible that the use of such hidden degrees of freedom is — for all practical purposes — unavoidable like, for example, the use of complex numbers in quantum mechanics. Nevertheless, alternative approaches in which the quark degrees of freedom have a more direct physical interpretation should also be pursued.

In this paper we explore the use of complex numbers as means of representing angular statistics for surface normal data. Our aim is to use the representation to construct a statistical model that can be used to describe the variations in fields of surface normals. We focus on the problem of representing facial shape. The fields of surface normals used to train the model are furnished by range images. We compare the complex representation with one based on angles, and demonstrate the advantages of the new method. Once trained, we illustrate how the model can be fitted to brightness images by searching for the set of parameters that both satisfy Lambert’s law and minimize the integrability error.KeywordsZenith AngleAzimuth AngleRange ImageIntegrability ErrorStatistical Shape ModelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In 1892, in search for special algebras, Corrado Segre published a paper in which he introduced an infinite family of algebras whose elements are called bicomplex numbers, tricomplex numbers… , n – complex numbers. In that paper Segre introduced idempotent elements of bicomplex numbers. Idempotent elements play a central role in the theory of bicomplex numbers. This paper introduces the algebraic structure of Tricomplex Numbers exploring some of their fundamental properties. We have identified and characterized sixteen distinct idempotent elements of Tricomplex Numbers. We also discuss their properties and establish the relationships among them.

Discrete fuzzy complex-valued function and complex fuzzy Caputo fractional difference equations

The surface nuclear magnetic resonance (SNMR) method is a geophysical method designed for non-invasive groundwater investigations. Inversion of experimental data provides the spatial distribution of the water content in the subsurface. However, SNMR inversion is ill-posed and admits many solutions because of the imaging equation properties that are compounded by experimental error. SNMR data sets are conveniently presented as complex numbers, thus possessing phase and amplitude components. Subsurface electroconductive formations and fluctuations of the Earth's magnetic field cause non-negligible phase shifts. Consequently, the forward modelling of the SNMR signal generated by 3-D water saturated formations is achieved in the complex domain. Nevertheless, in many cases, phase measurements are less reliable than amplitude measurements and water content rendering cannot be carried out using the complex SNMR signal. This problem is resolved by performing inversion using complex forward modelling whose resulting signal amplitude is used for comparison with the data. Along with water content boundaries ranging from 0 to 1, this property turns the linear initial value inversion problem into a non-linear one. In such a situation, the comprehensive analysis of inversion uncertainties is achieved by performing a solution space exploration based on a Monte Carlo approach. An adapted Metropolis-Hastings algorithm has been used on SNMR 3-D data sets to perform such an exploration. Computing time depends on the problem dimensions. With a standard laptop computer about 10 hr were necessary for the inversion of our field data set. The resulting model collection is used to calculate the probability density functions of the water content. From there, it is possible to estimate the uncertainty of the water content imagery. Using both synthetic and experimental data, we show that our routine provides robust estimates of the spatial distribution of the water content for the SNMR 3-D initial amplitude inversion.

The succession of enlargements from ℤ (the system of the integers) to ℚ (that of the rational numbers) and thence to ℝ (the real numbers) and then ℂ (the complex numbers) is generally seen as one of increasing richness of structure. But gains are often accompanied by losses: for instance the step from ℝ to ℂ leads to the glories of complex analysis but away from ordinal properties once thought essential to any concept of number. As far as group structure is concerned there are gains and losses at each stage of the progression. The following exposition is intended to demonstrate this without assuming more than a minimal knowledge of groups and of the theoretical foundations of analysis. Beyond some elementary arithmetical algebra, and an acquaintance with standard notation and vocabulary for maps in general, essentials taken for granted include the idea of a homomorphism (of groups), the existence and fundamental properties of the exponential function (of a real variable), and the Fundamental Theorem of Algebra (which is actually a theorem in complex analysis).

SummarySerial high-resolution structural magnetic resonance imaging scans of the brain can now be precisely aligned, with six degrees of freedom (three mutually orthogonal translational and three rotational degrees of freedom around three mutually orthogonal axes), using a rigid-body subvoxel registration technique. This is driven by the in-plane point spread function for images acquired in the Fourier domain with data obtained over a bounded region of k-space, namely the sinc interpolation function, where sinc z = (sin z)/z, with z being any complex number (including zero). Computational subtraction of the three-dimensional Cartesian spatial representation matrices of serially acquired scan data allows for the determination of structural cerebral changes with great precision, since voxel signals from unchanged structures are almost completely cancelled. Thus changes readily show up against a background of noise. Furthermore, lateral ventricular changes can now be accurately quantified using a semi-automated method involving contour production, threshold computation, binary image creation and ventricular extraction. These techniques have been applied to the investigation of the effects on cerebral structure of intervention with fatty acids, particularly the long-chain polyunsaturated n-3 fatty acid eicosapentaenoic acid (EPA), in disorders such as schizophrenia, treatment-resistant depression, chronic fatigue syndrome (myalgic encephalomyelitis or ME), and Huntington's disease.

We proposed a superresolved Wave which utilizes fundamental properties of the wave, i.e. diffraction, refraction, reflection, and attenuation. It solves the Helmholtz equation which has a complex wave number including both refractive index and attenuation coefficient, and reconstructs superresolved complex images of the objects. The fundamentals of the wave CT are the following two points: first, the wavelength of the wave used is much longer than the structure of the object to be imaged; second, the phase information of the wave traveled through the object contributes much more to the image reconstruction than the amplitude information. Just as the X-ray CT is the extremum of the short-wavelength imaging, which may be called Particle CT, the wave CT is the extremum of the long-wavelength imaging. Therefore, we named it Wave CT. The mathematical method of the wave CT cannot be independent of itself, and we proposed a combination of FEM and BEM. It is not a perturbation method unlike the so-called...

Magnetic resonance (MR) spectroscopy allows the demonstration of relative tissue metabolite concentrations along a two- or three-dimensional spectrum based on the chemical shift phenomenon. An MR spectrum is a plot of the signal intensity and frequency of a chemical or metabolite within a given voxel. At proton MR spectroscopy, the frequency at which a chemical or compound occurs depends on the configuration of the protons within the structure of that chemical. At in vivo proton MR spectroscopy, the frequency location of water is used as the standard of reference to identify a chemical. The frequency shift or location of chemicals relative to that of water allows generation of qualitative and quantitative information about the chemicals that occur within tissues, forming the basis of tissue characterization by MR spectroscopy. MR spectroscopy also may be used to quantify liver fat by measuring lipid peaks and to diagnose malignancy, usually by measuring the choline peak. Interpretation of MR spectroscopic data requires specialized postprocessing software and is subject to technical limitations including low signal-to-noise ratio, masking of metabolite peaks by dominant water and lipid peaks, partial-volume averaging from other tissue within the voxel, and phase and frequency shifts from motion. MR spectroscopy of the liver is an evolving technology with potential for improving the diagnostic accuracy of tissue characterization when spectra are interpreted in conjunction with MR images.

The effects of elevated blood lead levels on the development of children have been examined only in the context of behavioral and neuropsychological evaluations. No studies have examined the effects of lead on brain metabolism in vivo or on structural and/or functional correlates of brain function in children. In the human brain, magnetic resonance spectroscopy (MRS) provides a noninvasive, risk-free method to monitor the biochemistry of acute and chronic stages of disease. The purpose of this study was to examine in vivo the use of MRS for the evaluation of the neurotoxic effects of lead on the nervous system, by detection of brain metabolism, especially N-acetylaspartate, a metabolite shown to decrease in processes that involve neuronal loss. Two male cousins who live in the same household and share the same socioeconomic background and home environment were studied. The subject, a 10-year-old boy, had elevated blood lead levels. His cousin, a 9-year-old boy, was not exposed to lead. Both underwent a comprehensive neuropsychological evaluation and both were evaluated using the magnetic resonance imaging (MRI) and MRS at the University of Pennsylvania Medical Center. High-resolution MRI and MRS were performed using a 3-in surface coil. Localized proton spectra were obtained from contiguous 6 x 6 x 10-mm voxels using one-dimensional phase encoding, with a 2000-millisecond repetition time and a 31-millisecond echo time. Neuropsychological evaluation demonstrated areas of impairment in the lead-exposed child, including difficulties in academic skills of reading, writing, and arithmetic, as well as deficient linguistic skills and attentional mechanism. By contrast, studies of the cousin, who was not exposed to lead, showed overall neuropsychological functioning within normal limits. Although both children had a normal MRI examination of the brain, studies of the lead-exposed boy showed a significant alteration in brain metabolites, with a reduction in the N-acetylaspartate:creatine ratio for both gray and white matter on the MRS examination, compared with his cousin. The present study is a first attempt to determine in vivo metabolic differences in the brain of a child exposed to lead compared with a healthy control subject. This is a unique case because these children were matched on a number of variables usually regarded as confounders in behavioral lead studies, and therefore can be regarded as matched controls. The present study demonstrates that MRS is a feasible, noninvasive technique for in vivo examination of the brains of children exposed to lead. We were able to obtain high-quality spectra from voxels as small as 0.36 cm at 1.5T. The spatial resolution used in the present study is sufficient to obtain spectra from voxels almost exclusively composed of gray matter. The one-dimensional phase-encoding approach used presents the advantage of obtaining several spectra simultaneously in a relatively short time. The present study has allowed us to examine the spectroscopic patterns of frontal gray and white matter after lead exposure relative to the normal pattern seen in healthy children and adults. The MRS study of the healthy, nonlead-exposed cousin demonstrated spectra entirely consistent with the spectral pattern reported in previous studies of healthy individuals. By contrast, the spectra obtained from the lead-exposed child deviated from the expected pattern in all metabolite ratios analyzed. Because N-acetylaspartate has been shown as a measure of neuronal viability, the level of N-acetylaspartate may enable us to evaluate the degree of neuronal loss in children exposed to lead. The MRI examination indicated no structural abnormalities or cortical thinning, and no abnormal findings in either case. By contrast, MRS indicated a significant change from normal values for the lead-exposed child. This supports the idea that lead neurodevelopmental toxicity may be related to inter