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Three- and Four-Dimensional Generalized Pythagorean Numbers

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Advances in Pure Mathematics 2024年第1期14卷 1-15页

作者： alfred Wünsche Institute of Physics Humboldt University Berlin Germany

The Pythagorean triples (a, b | c) of planar geometry which satisfy the equation a2+b2=c2 with integers (a, b, c) are generalized to 3D-Pythagorean quadruples (a, b, c | d) of spatial geometry which satisfy the equation a2+b2+c2=d2 with integers (a, b, c, d). Rules for a parametrization of the numbers (a, b, c, d) are derived and a list of all possible nonequivalent cases without common divisors up to d2 is established. The 3D-Pythagorean quadruples are then generalized to 4D-Pythagorean quintuples (a, b, c, d | e) which satisfy the equation a2+b2+c2+d2=e2 and a parametrization is derived. Relations to the 4-square identity are discussed which leads also to the N-dimensional case. The initial 3D- and 4D-Pythagorean numbers are explicitly calculated up to d2, respectively, e2.

关键词： Number Theory Pythagorean Triples Tesseract 4-Square Identity Diophantine Equation

Nutritional Steps to Maintain Health and Reduce Cancer Risk

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Journal of Cancer Therapy 2019年第10期10卷 829-834页

作者： alfred Ordman Beloit College Beloit USA

At the 2017 Linus Pauling meeting, it was announced that based on metabolomic analysis, vitamin K at levels far above the daily value activates a variety of enzymes that are beneficial. The enzymes have been shown to remove calcium where it harms our health, deposit it where it benefits bone, may reduce hip fractures, and decrease calcification of the circulatory system, reducing by 50% the risk of heart disease and high blood pressure. Linus Pauling demonstrated that vitamin C can kill cancer cells. One form of vitamin K regenerates vitamin C so that it is even more effective at killing cancer cells. I previously reported a trial that determined an oral dosage of vitamin C that produces a concentration in the bladder able to kill recurring non-invasive muscle carcinoma cells. Nature in 1999 published an article on foods that reduce the risk of cancer. Since then there is additional evidence that nutrients can extend the human healthspan. Other nutrients are effective at reducing the risk of cancer, such as the active ingredient in cruciferous vegetables. These nutrients that reduce the risk of cancer and improve health are GRAS. They are safe and likely to substantially reduce the risk of cancer, heart disease, hip fractures, and other undesirable conditions. While research on treating cancer dominates media attention, pharmaceutical development, and Federal funding, it is likely that people can prevent cancer by consuming nutrients that have been shown to kill human cancer cells.

关键词： Vitamin K Vitamin C Healthspan Cancer Prevention Calcium Bladder Cancer

Rapid correction of hyperglycemia:A necessity but at what price?A brief report of a patient living with type 1 diabetes

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World Journal of Diabetes 2023年第11期14卷 1710-1716页

作者： Priscille Huret Philippe Lopes Randa Dardari alfred Penfornis Claire Thomas Dured Dardari LBEPS University EvryIRBAUniversitéParis SaclayEvry 91000France Diabetology Al FourkanAleppo 90000Syria Diabetology Centre Hospitalier Sud FrancilienCorbeil Essonne 91100France

BACKGROUND The correction and control of chronic hyperglycemia are the management goals of patients living with *** hyperglycemia is the main factor inducing diabetes-related ***,in certain situations,the rapid and intense correction of chronic hyperglycemia can paradoxically favor the onset of microvascular *** SUMMARY In this case report,we describe the case of a 25-year-old woman living with type 1 diabetes since the age of 9 *** diabetes was chronic and unstable but without *** an unplanned pregnancy,her diabetes was intensely managed with the rapid correction of her ***,over the following 2 years,she developed numerous degenerative microvascular complications:Charcot neuroarthropathy with multiple joint involvement,severe proliferative diabetic retinopathy,gastroparesis,bladder voiding disorders,and end-stage renal failure requiring *** In the literature to date,the occurrence of multiple microvascular complications following the rapid correction of chronic hyperglycemia has been rarely described in the same individual.

关键词： Unstable diabetes Chronic hyperglycemia Microvascular complication Type 1 diabetes Case report

MEASUREMENT OF AMPLITUDE AND DELAY OF STIMULUS FREQUENCY OTOACOUSTIC EMISSIONS

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Journal of Otology 2013年第1期8卷 57-62页

作者： Tianying Ren Jiefu Zheng Wenxuan He alfred L. Nuttall Oregon Hearing Research Center Department of Otolaryngology and Head & Neck SurgeryOregon Health and Science University3181 SW Sam Jackson Park RoadPortlandOR 97239 Department of Radiology Nuclear Medicine University of Iowa Hospitals and Clinics200 Hawkins DriveIowa CityIA 52242

Although stimulus frequency otoacoustic emissions (SFOAEs) have been used as a non-invasive measure of cochlear mechanics, clinical and experimental application of SFOAEs has been limited by difficulties in accurately deriving quantitative information from sound pressure measured in the ear canal. In this study, a novel signal processing method for multicomponent analysis (MCA) was used to measure the amplitude and delay of the SFOAE. This report shows the delay-frequency distribution of the SFOAE measured from the human ear. A low level acoustical suppressor near the probe tone significantly suppressed the SFOAE, strongly indicating that the SFOAE was generated at characteristic frequency locations. Information derived from this method may reveal more details of cochlear mechanics in the human ear.

关键词： MEASUREMENT OF AMPLITUDE AND DELAY OF STIMULUS FREQUENCY OTOACOUSTIC EMISSIONS MCA SDC LDC Figure

Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials

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Applied Mathematics 2015年第12期6卷 2142-2168页

作者： alfred Wunsche Institut für Physik Nichtklassische StrahlungHumboldt-UniversitatBerlinGermany

The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of operator disentanglement some operator identities are derived in an appendix. They allow calculating convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions.

关键词： Laguerre and Hermite Polynomials Laguerre 2D Polynomials Jacobi Polynomials Mehler Formula SU(1,1)Operator Disentanglement Gaussian Convolutions

Confucius and American Democracy

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China Today 2024年第6期73卷 48-50页

作者： alfred HORNUNG The Obama Institute for Transnational American Studies Johannes Gutenberg UniversityMainzGermany

UPON the establishment of European universities in the Middle Ages,the educational system in Western countries became characterized by an essentially Eurocentric *** was reflected in the study of Latin and Greek in high school,and in basing public life and political systems on philosophies of classical *** work of the influential Chinese philosopher Confucius(551-479 B.C.),a near-contemporary of the major Greek philosophers Socrates(469-399 B.C.),Plato(428-348 B.C.)and Aristotle(384-322 B.C.),whose ethical and political ideas entered into the constitutions of European states,remained unknown.

关键词： Latin Middle Confucius

Optic Axes and Elliptic Cone Equation in Coordinate-Invariant Treatment

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Journal of Modern Physics 2022年第6期13卷 1001-1043页

作者： alfred Wünsche Institut für Physik Humboldt-Universit&auml t Newtonstr Berlin Germany

We derive for crystal optics in coordinate-invariant way the cone approximation of refraction vectors in the neighborhood of optic axes and determine its invariants and eigenvectors. It proved to describe an elliptic cone. The second invariant of the operator of the wave equation with respect to similarity transformations determines the special cases of degeneration including the optic axes where the polarization of the waves due to self-intersection of the dispersion surface is not uniquely determined. This second invariant is included in all investigations and it is taken into account in the illustrations. It is biquadratic in the refraction vectors and the corresponding forth-order surface in three-dimensional space splits in two separate shells and a non-rational product decomposition describing this is found. We give also a more general classification of all possible solutions of an equation with an arbitrary three-dimensional operator.

关键词： Permittivity Tensor Principal Permittivities Three-Dimensional Operator of Wave Equation Operator Invariants Refraction Vector Ray Vector Cone Ap-proximation in Neighborhood of Optic Axis Conical Refraction

Mathematical Aspects of SU (2) and SO(3,R) Derived from Two-Mode Realization in Coordinate-Invariant Form

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Journal of Modern Physics 2023年第3期14卷 361-413页

作者： alfred Wünsche Institut für Physik Humboldt-Universit&#228t Newtonstr. 15 Berlin Germany

Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.

关键词： Boson Operators Lie Algebra Root Diagram Invariant Integration Hamilton-Cayley Identity Cayley-Gibbs-Fyodorov Parametrization Composition Law Quaternion Stereographic Projection Fractional Linear Transformation

Two Concepts in Optics of Anisotropic Dispersive Media and Polariton Case in Coordinate-Invariant Way

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Journal of Modern Physics 2022年第4期13卷 574-619页

作者： alfred Wünsche Institut für Physik Humboldt-Universit&auml t Berlin Germany

Two concepts of phenomenological optics of homogeneous, anisotropic and dispersive media are compared, the younger and more general concept of media with spatial dispersion and the older concept of (bi)-anisotropic media with material tensors for electric and magnetic induction which only depend on the frequency. The general algebraic form of the polarization vectors for the electric field and their one-dimensional projection operators is discussed without the degenerate cases of optic axis for which they become two-dimensional projection operators. Group velocity and diffraction coefficients in an approximate equation for the slowly varying amplitudes of beam solutions are calculated. As special case a polariton permittivity for isotropic media with frequency dispersion but without losses is discussed for the usual passive case and for the active case (occupation inversion of two energy levels that goes in direction of laser theory) and the group velocity is calculated. For this active case, regions of frequency and wave vector with group velocities greater than that of light in vacuum were found. This is not fully understood and due to large diffraction is likely only to realize in guided resonator form. The notion of “negative refraction” is shortly discussed but we did not find agreement with its assessment in the original paper.

关键词： Spatial and Frequency Dispersion Bi-Anisotropic Media Uniaxial Media Passive and Active Media Negative Refraction Operator Invariants Complementary Operator Group Velocity