Ninul Anatoly Sergeevich - English web-site https://NinulAS.narod.ru/english.html
Large Russian version of this web-site has the inet-address: https://NinulAS.narod.ru
Very compact English version of this web-site has the inet-address: https://Ninul-eng.narod.ru
Information about the author-himself is displayed on his Russian professional web-site.
You may use these active inet-links, in that number, to all my math-phys books, here to the end.
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The author does not have commercial profits from possible advertising here and from the location here of his mathematical-physical works in their digital forms as p-books and e-books in the indicated fields. His interests are only purely scientific! All these books with contents of this site are situated as in the public domain.
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In the case of advertising on the site, you may either wait a some time till its end, or download the web page, for example, as mht file and read it calmly and without interferences (better swiching off Internet or transforming the mht file in pdf), but do not forget that the web-site is also subject to the author's copyright, and, when using its content, links to these site and monograph are required! (The website is periodically improved by the author with updates -- during his lifetime.)
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Discussed in the web-site can be problems devoted to different divisions of mathematics, theoretical physics and mathematical chemistry from the fields indicated below. For this reason of especial interest for the author are opinions, remarks and suggestions concerning the contents of his 4-th published scientific monographs in Russian and English with the given web-site. Namely, they are.
1. The monograph: A. S. Ninul, "Tensor Trigonometry. Theory and applications.", with this new math subject and with studied in it a number of fundamental questions and problems of algebra, geometry and theoretical physics, was issued by the main Russian scientific "Publishing House Mir" (Moscow) in October 2004. Note, that in preliminary Part.I, between many other interest novelties, along the way, the elegant (only in one line) proofs of the Hamilton--Caylay Theorem for the characteristic polynomial of a matrix and the Kronecker--Capelli Theorem for linear algebraic equations are given!
2. To the end of 2020, the author prepared the updated English version of this monograph, as the 2-nd edition, and in January 2021 the Russian scientific "Publishing House Fizmatlit" (Moscow) issued it as the monograph Ninul A. S. "Tensor Trigonometry.", where, in particular, in last Ch.10A, the more complete study of the differential trigonometry of world lines in the Minkowski space-time was carried out with concomitant to them hyperboloids II and I.
3. In 2024, the author prepared the last author's significally updated and expanded version of the monograph Ninul A. S. "Tensor Trigonometry", as its 3-rd edition; and to the end of 2024 the Russian scientific "Publishing House Fizmatkniga" (Moscow) issued it in the paper and electronic identical versions, both with own ISBN and DOI; where, in particular, in Ch.7A, for the Harriot's and Lambert's angular deviations in both non-Euclidean geometries and for the Thomas precession in STR, the most brief and clear cosine formula was revealed; in Ch.3A-7A and 9A, the clear trigonometric explanations of all well-known and new relativistic effects were given with their clear physical essence, -- namely, in the space-time of Minkowski (as real-valued) and of Poincaré (as complex-valued), in that number, in the gravitational field and the Higgs field; and, in last Ch.10A, the complete study of the differential trigonometry of any world lines in the 3D and 4D Minkowski space-time with physical kinematics and dynamics and, in parallel, of regular curves in the 3D and 4D quasi-Euclidean space also with own directed frame axis for the angular rotations and motions were executed and finished.
4. The monograph A. S. Ninul, "Optimization of Objective Functions: Analytics. Numerical methods. Design of experiments." was issued by the Russian scientific "Publishing House Fizmatlit" (Moscow) in May 2009. In it we considered in logical order all functional methods of search and identification of the extremum for evolutional objective functions on the basis of their sequential genesis (up to mathematical programming) with the filling up of "blind spots" having in literature on optimization. This book also completed the investigation of some extreme problems considered earlier in the book "Tensor Trigonometry". So, for instance, Ch.4 contains the derivation of complete (extreme in essence) requirements to the coefficients of a real algebraic equation of degree n for the reality, including positivity, of all its roots, what is the solution to the problem posed by the great Descartes and partially solved by him. In Ch.1, along the way, the problem of the connection between the derivatives of the analytical functions y(x) and x(y), with their any degree n, was solved by us, or, equivalently, it is the problem of the connection between the coefficients of direct and inverse analytical series for the same functions, posed and solved for n=1 and n=2 by the great Newton. Here there is also new useful facts about algebraic and functional polynomials.
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At the end, last two books contain own "Physical-Mathematical Kunstkammers", which includes a number of questions and tasks connected with problems discussed in these monographs (math analysis, algebra, geometry, physics).
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In a paper form, all these books can be looked through in large scientific libraries – Russian and foreign. For instance, the book "Tensor Trigonometry. Theory and Applications." – M.: MIR, 2004 is presented in EU in the most known mathematical library Zentral Universitätsbibliothek Göttingen as the monograph “Tenzornaja Trigonometrija - teorija i prilozenija.” – Ninul A. S. (Moscow, Mir, 2004) with link.
In a digital form, these books, as the author's pdf files, are presented, for instance, in the main Russian digital servers Elibrary.ru and Rusneb.ru, in the electronic library of MSU's Mech-Math Faculty (Geometry and Optimization), in a number of other scientific e-libraries of Russia and CIS, and also in Google books; Internet Archive with Open Library, English E-books Directory (Tensor Analysis), etc..
In my web-site you can find the author's files of these books. If something in their contents or proofs are not understand, then it is better to contact the author by his e-mail, indicated below, before he has finished his earthly journey. Some electronic files of these books may be exhibited with correction of noted misprints and small inaccuracies, inevitable at forming such large works without helpers.
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Given further are some results of the author's investigations discussed in his presented here monographs and the domains of his special interests in exact sciences.
1. Algebra and Theory of Exact Matrices.
- The general inequality for all average values (means) from a given set of positive numbers;
- a real-valued algebraic equation of degree n, necessary and sufficient conditions presented to its coefficients for reality (in particular, positivity) of all roots;
- the limit method and formulae for serial calculation of all n roots of a real-valued algebraic equation of degree n or of the eigen values of an nxn-matrix in the case of their positivity;
- the scalar and matrix characteristic coefficients of an nxn-matrix B (their complete structures, properties and applications in Linear Algebra, Geometry and in construction of Tensor Trigonometry.
- The scalar characteristic coefficients of an nxn-matrix B were discovered and investigated by the supereminent Urbain Le Verrier in the first half of the 19-th century with cosmic applications in his purely mathematical discovery of the planet Neptune. They form also coefficients of a matrix B characteristic algebraic equation of degree n with exactness up to signs.
- The matrix characteristic coefficients of an nxn-matrix B were appeared in the middle of the 20-th century in the famous work of the eminent mathematician Jean-Marie Souriau in his article "Une méthode pour la décomposition spectrale et l'inversion des matrices." in the “Proceedings of the French Academy of Sciences”. Namely, in order to inverse non-singular matrix B only through its scalar and matrix characteristic coefficients, J.-M. Souriau suggested in 1948 the very elegant algorithm with parallel successive calculating all these scalar and matrix characteristic coefficients of order t ≥ 1. Note, that Jean-Marie Souriau was by mathematician, mechanic and physicist with a wide range of his scientific works, and he was one from the pioneers of Symplectic Geometry! But a year later, in 1949, academic AMS published the similar article, however in English and under other name and of another, but own national author, who is otherwise unknown in the Science. And it was this article that become widely cited in the USA, while J.-M. Souriau original article was ignored!!! Of course, we cite only the original article by J.-M. Souriau, as the true Science must be pure and free from any nationalism and plagiarism!
- However the structures of both matrix characteristic coefficients for an nxn-matrix B were not known then. Their structures with additional properties of all coefficients, including 1st and 2nd rock, and, as a consequence, the main three parameters of singularity of a matrix B, was established by the author in 1981, in general, for a non-singular matrix B. In the 1-st stage, in the frame of perfectation of Theory of Exact Matrices, the author applicated these results, by introduction of new concepts of null-prime, null-normal and null-defect matrices, further calculated complete structures of all eigen projectors for a singular matrix B, inferred complete structural formula for the classic real-valued and complex quasi-inverse matrix by Moor-Penrose through elements of the original matrix, and, in addition, obtained its simplest and fair limit formula. At my attempt to publish all these new results in the Soviet mathematical journals of the USSR Academy of Sciences, similar to that, how J.-M. Souriau did this with his results in the same field in the journal of the Academy of Sciences de La France (and even “author” of academic AMS above), all these Soviet journals unitedly and consistently refused to allow the unknown author to publish of articles presenting these new results, and without any comments in their reviews on the correctness of his results. The first Reviewer could not resist even and included my limit formula for the matrix by Moor-Penrose in his nearest textbook on Linear Algebra (where this formula of mine was out of place, but beautiful). In order to camouflage this fact, he kept my article for 2 years! Plagiarism of my non-published results continued further, and again, even on a level of doctors of sciences! (Although originals of these articles that were not accepted for publications at that and this time were and are in the home archive of the author. But then these articles circulated across the desks of leisurely and indifferent reviewers for several years.) As can be seen from my monograph “Tensor Trigonometry”, all these characteristic coefficients of an nxn-matrix B with their structures and properties became an essential generating part for the subsequent correct and consistent development of the new subject, named by the author as “Tensor Trigonometry” (with using renovated by him Theory of Exact Matrices), which was issued finally by the “Mir” Publisher in October 2004 thanks to a bright review of the very eminent and encyclopedically versatile mathematician Michail Michailovich Postnikov -- professor at MSU and MIAN! These are the kind of surprises that await unknown authors who dare in mathematics.
- the fundamental relations for basic parameters of singularity of an nxn-matrix B and the explicit form for its minimal annulling polynomial; they are useful also for theoretical analysis and correct using a matrix B, and also for forming parameters of its Jordan form with some inequalities for them;
- null-prime and null-normal singular nxn-matrices, their definitions and properties;
- affine and orthogonal (spherically and hyperbolically) quasi-inverse matrices and eigen projectors, i.e., 8 for real and 12 for complex nxn-matrices, 4 for real and 6 for complex nxm-matrices; the Table of multiplication for all eigen projectors of a matrix;
- affine and orthogonal (spherically and hyperbolically) eigen reflectors and, obviously, with their eigenvalues -1 and +1;
- the quadratic geometric norms for nxn-, nxm- and nxr-matrices (including hierarchical ones);
- trigonometric nature of commutativity and anticommutativity of prime nxn-matrices;
- the trigonometric spectra of nxn-matrix B; the cosine and sine general inequalities for nxn-matrices B and for nxr-lineors A as generalization of the classic algebraic Inequalities by Cauchy and by Hadamard for vectors and vectors-columns;
- adequate, Hermitian and symbiotic complexifications of mathematical notions.
2. Tensor Trigonometry: Euclidean and anti-Euclidean, quasi-Euclidean and pseudo-Euclidean, accordingly in Euclidean and anti-Euclidean, quasi-Euclidean and pseudo-Euclidean k-dimensional spaces (k>1); two latters are direct orthogonal sums as the binary spaces with dimension k=n+q>1, where q is an index of a binary space.
- The middle reflector of a binary tensor angle between two linear objects in the Euclidean or anti-Euclidean space; it is as a progenitor of the given general reflector tensor and metric tensor with an index “q” for any binary spaces entirely.
- Tensor Trigonometry in the general quasi-Euclidean and pseudo-Euclidean binary spaces with their subdimensions n, q and with the given general reflector tensor and metric tensor of a binary space.
- Tensor Trigonometry in the quasi-Euclidean and pseudo-Euclidean binary spaces in its elementary forms at q=1 or at n=1.
- Simplest variants (k=2) at n=2, q=0 or n=0, q=2 (non-binary) and at n=q=1 (binary) correspond to the scalar and tensor trigonometries on the real-valued or imaginary plane and on the quasiplane or on pseudoplane; including two latters on the same eigen planes in the orthogonal mono-binary representations of trigonometric tensors – see in Chs. 5 and 6.
- Revealed bivalent tensor nature of any angles of projective and motive types between two linear geometric objects (vectors, lineors, straight lines, planes, hyperplanes in linear spaces – metric or affine) and also of trigonometric functions of these angles, reflections and rotations of these objects (spherical, quasi-Euclidean, hyperbolic, pseudo-Euclidean).
- Bivalent projective and motive principal tensor angles (spherical Ф and hyperbolic Г) and their tensor trigonometric functions;
- the motive angle θ of orthospherical rotation Rot θ (elementary rot θ) in the Euclidean and/or anti-Euclidean subspaces of a binary space;
- the simplest unity diagonal reflector tensor, pre-stated for the simplest linear quasi-Euclidean and pseudo-Euclidean binary spaces; where n – a quantity of its eigenvalues +1, q – a quantity of its eigenvalues -1 (as for any reflector);
- the common symmetric reflector tensor, pre-stated for the linear quasi-Euclidean and pseudo-Euclidean binary spaces, bonded with this unity diagonal reflector tensor by spherical rotation and/or orthospherical (Euclidean) rotations (i.e., without changes of the base axes orientations);
- two natural quadratic metrics in linear metric spaces: Euclidean and pseudo-Euclidean, their progenitor is the Pythagorean Theorem – quasi-Euclidean and pseudo-Euclidean;
- correspondences between reflector tensor and metric tensor in two metric binary spaces –- quasi-Euclidean and pseudo-Euclidean;
- a reflector tensor, a metric tensor and the kind of a quadratic metric as the concepts for correct definitions of binary metric spaces; the orientational role of a reflector tensor for any binary spaces, but only in elementary variants – if q=1 or if n=1;
- affine binary spaces with their affine Tensor Trigonometry without any metrics for distances and for angles, but conserving relations of parallelism.
- In the quasi-Euclidean binary spaces, the centralized (in E_1) trigonometric objects are defined as the Special oriented “trigonometric hyperspheroid” with its radius-parameter R=1, and, in particular, if q=1 as oriented in direction of the frame axis y.
- In the pseudo-Euclidean binary spaces, the centralized (in E_1) trigonometric objects are defined as the Special “trigonometric hyperboloids II and I” by Minkowski with their radius-parameter R=1, and, in particular, if q=1 as oriented in direction of the frame axis y.
- Bivalent orthogonal and oblique tensors of reflections (i.e., reflectors) of one linear object into another reflective linear object in the binary spaces, in that number, elementary ones if q=1 in their canonical tensor forms in the universal (trigonometric) base E_1;
- bivalent orthogonal tensors of rotations (i.e., rotors) of one linear object into another rotated linear object in the binary spaces, in that number, elementary ones if q=1 in their canonical tensor forms in the universal (trigonometric) base E_1 (these rotations are isomorphic to continuous angular motions along their Special trigonometric objects above);
- bivalent oblique tensors of deformations of one linear object into another deformed linear object in the binary spaces, in that number, elementary ones if q=1 in their canonical tensor forms in the universal base E_1;
- non-collinear and collinear (at e_α=const) two-steps and polysteps elementary reflections-motions (non-continuous), rotations-motions (continuous) and deformations (continuous); Rules and law of motions summation – Chs.5, 6, 11, 7A, 8A, 10A;
- bivalent orthogonal tensors of admissible quasi-Euclidean or pseudo-Euclidean rotations in their elementary canonical forms if q=1 in E_1, with the motive principal angles (spherical Ф or hyperbolic Г) and with their vector of the directional cosines e_α in the Euclidean subspace, in that number, as isomorphic angular motions at R=1 or with factor-parameter R on the Special oriented hyperspheroid and on the oriented hyperboloids II and I by Minkowski;
- bivalent oblique tensors of admissible quasi-Euclidean or pseudo-Euclidean deformations in their elementary canonical forms if q=1 in E_1, with the motive principal angles (spherical Ф or hyperbolic Г) and with their vector of the directional cosines e_α in the Euclidean subspace;
- as example in STR (see further), the elementary purely hyperbolic rotation roth Г, causing the real relativistic scale of time-arrow ct_m dilation by Minkowski in the base E_m of a moving object, with respect to the universal base E_1 (in STR the base of immobility – see in Ch.1A and 3A), and accordingly decreasing its proper time in E_m;
- as example in STR (see further), the elementary purely hyperbolic deformation defh Г, causing for a moving object with the base E_m its seeming relativistic contractions by Lorentz in the universal base E_1 (in STR the base of immobility – see in Chs.1A and 4A);
- tensors of principal spherical or hyperbolic rotations (motions) in their elementary canonical forms at q=1 in E_1 with their constant unity vector of directional cosines e_α – for descriptions only of the principal spherical or hyperbolic rotations, or as isomorphic principal angular motions with the factor-parameter R on the oriented Hyperspheroid or Hyperboloid II and I by Minkowski;
- tensor of lateral orthospherical rotations in its elementary canonical form at q=1 in E_1 with the constant principal spherical or hyperbolic tensor angle (as inclination in E_1) – for descriptions of screwed or pseudoscrewed rotations-motions, in that number, as real-valued or imaginary orthospherical angular motions with the factor-parameter R on the Special oriented quasi-hypercylinder or pseudo-hypercylinder of the radius R; this Special hypercylinder is tangent to the equator of hyperspheroid and to the equator of hyperboloids II (as imaginary-valued) or I (as real-valued);
- orthogonal tensor of orthospherical rotations Rot θ (as elementary rot θ) for quasi-Euclidean and pseudo-Euclidean binary spaces, acting in the Euclidean subspace of a dimension “n>1” or/and in the anti-Euclidean subspace of a dimension “q>1” under spherical or hyperbolic inclination in E_1 and under action of the given reflector tensor of the binary space with the same n and q;
- independent orthogonal tensor of orthospherical rotations Rot θ (as elementary rot θ in its canonical form if q=1 in E_1), acting only in the Euclidean and anti-Euclidean subspaces of these binary spaces, as independently or as induced in polar decompositions of tensors of general or summary non-collinear rotations (motions); the angle θ or dθ is either the Euclidean rotation-shift of the initial base E_1 (as a passive point of view), or the Euclidean rotation-shift of an object in the initial base E_1 (as an active point of view);
- polar decompositions in the initial base E_1 of tensors of general and of summary non-collinear quasi-Euclidean or pseudo-Euclidean rotations (motions) into the tensor of principal rotation (motion) and the tensor of lateral orthospherical rotation – see in Chs.11, 7A and 8A;
- orthogonal and oblique bivalent tensors of reflections with their projective principal tensor angles (spherical or hyperbolic) in general and elementary forms under acting given reflector tensor of a binary space;
- initial generation of the bivalent orthogonal or oblique tensors of reflections with their projective principal tensor angles (spherical or hyperbolic) through the differences of two complementary eigen orthogonal or oblique projectors for the given generating singular matrix (with its eigen linear subspaces) – all in their canonical forms and in the common trigonometric base E_1 - see in Chs.5 and 6;
- initial generation of the bivalent orthogonal or oblique tensors of rotations and deformations (with their motive principal tensor angles – spherical or hyperbolic) through double action of a pair of the orthogonal or oblique reflection tensor (with their projective principal tensor angles – spherical or hyperbolic) and followed by extraction of the trigonometric square root in the same trigonometric base E_1 – see in Chs.5 and 6.
- The non-commutative Special group of admissible quasi-Euclidean rotations (continuous motions) from spherical and orthospherical rotations, introduced by the author in Chs.5 and 6, and corresponding to the pre-stated reflector tensor of the binary space and to its Euclidean quadratic metric.
- The non-commutative Lorentz group of admissible pseudo-Euclidean rotations (continuous motions) from hyperbolic and orthospherical rotations, introduced by Poincaré, and corresponding to the pre-stated reflector tensor of the binary space and to its pseudo-Euclidean quadratic metric.
Correct definitions of both these groups together with their binary spaces were considered by the author in Ch.6.
- The non-commutative subgroup of admissible orthospherical lateral rotations, as intersection of the same two groups in any universal base (in STR the bases of immobility), but acting in the Euclidean subspace of a dimension “n>1” or/and in the anti-Euclidean subspace of a dimension “q>1” under spherical or hyperbolic inclination in E_1 and under action of the given reflector tensor of the binary spaces with the same n and q.
- The Special Quart cycle, clothed from all tensor rotations and deformations in common and elementary forms of matrices – all they in canonical forms in the universal trigonometric base and corresponding each another either on the abstract spherical-hyperbolic analogy, or on the specific sine-tangent analogy – see in Chs.5 and 12;
- the abstract spherical-hyperbolic analogies: direct ф=+iγ and inverse γ=-iф, correct in any admissible bases;
- the specific spherical-hyperbolic analogies: direct γ(ф) and inverse ф(γ), correct only in universal bases;
- the trigonometric tangent identity, defined the infinity set of specific spherical-hyperbolic analogies γ(ф) and ф(γ), correct only in universal bases
- for development of Tensor Trigonometry and its some numerous applications (for example, for more simple inference of a number of its formulae and theorems with their interpretations and also for constructions of isomorphic projective trigonometric models of non-Euclidean geometries), we present mainly the abstract analogy above and most useful specific spherical-hyperbolic analogies below:
- the basic with inverse to it specific sine-tangent analogy γ(ф) and ф(γ), defined through the identity sinhγ=tanф and only in universal bases – usually in E_1 (in STR the bases of immobility); in literature, it can be named also as Lambertian (but as Gudermannian in the Functional analysis – on the affine plane in any its base);
- the complementary with inverse to it specific sine-cotangent analogy γ(ξ)=γ(π/2-ф) and ξ(γ), defined through the identity sinhγ=cotξ (where obviously equality cotξ=tanф) and only (!) in universal bases – usually in E_1 (in STR the bases of immobility);
- the special with inverse to it specific tangent-tangent analogy γ(ф_r) and ф_r(γ), defined through the identity tanhγ=tanф_r and also only in universal bases – usually in E_1 (in STR the bases of immobility – see in Ch.1A); this analogy is also useful for trigonometric projective presentation of stereographic model of non-Euclidean geometries;
- both specific analogies above are applicable for simple geometric duplication and bisection of a hyperbolic angle γ (Ch.6);
- for complementary hyperbolic angle υ(γ), we have the same analogies: sinhυ=tanξ and sinhυ=cotф (where obviously equality cotф=tanξ);
- the similar formulae – direct υ(γ) inverse γ(υ), bonded complementary hyperbolic angles and additional useful relations for them (of course, more complex, than for spherical scalar angles ф(ξ) and ξ(ф));
- and else more complex formulae, bonded tensor angles U and Г in motive and projective kinds;
- the hyperbolic angle or number or transcendent constant ω=arsh1~0,881 as a hyperbolic analog of the number π/4=arc(tan1)~0,785 (sinhω =tanπ/4=1), and also their representations by infinite numeral power series; geometrically ω corresponds to a hyperbola focus.
- solution of a pseudo-Euclidean right triangle on a pseudoplane and in a pseudo-Euclidean space;
- a pseudo-Euclidean right triangle – interior and exterior: complementary hyperbolic angles γ(υ) and υ(γ), infinite δ and zero combined right angle; functional connection of these complementary hyperbolic angles; application of the specific sine-tangent analogy for simplest inferring in a universal base E_1 of connections between spherical and hyperbolic angles on a quasiplane and a pseudoplane;
- the principal angles – spherical and hyperbolic, scalar and tensor play roles, either as the motive angles of rotations (motions) in the quasi-Euclidean and pseudo-Euclidean binary spaces, in particular, for rotations in them and for motions with a factor “R” on their non-Euclidean hypersurfaces; or as the projective angles between linear objects in the same binary spaces and for tensor reflections between these linear objects;
- thus, in the first case, the motive angles produce the sides-segments of geometric figures with a factor “R” as the radius-parameter of curvilinear non-Euclidean hypersurfaces and with the orthospherical angles θ_k at their vertexes (including orthospherical angular deviations, so, in triangles from a sum π – see in Ch.7A);
- the tensor motive and projective angles have a difference only in their tensor trigonometric structures – see in Chs.5 and 6, but scalar motive and projective angles have not a difference (!).
3. Non-Euclidean Geometries with constant radius-parameter R, as +R, -R or +iR, -iR -- spherical geometry on the Special oriented hyperspheroid, two accompanied -- hyperbolic and hyperbolic-elliptical geometries on the oriented hyperboloids II and I by Minkowski, or these two latter entire as the common non-Euclidean geometry of a hyperboloidal type on the three-shifts hyperboloidal hypersurface.
- From the point of view of Tensor Trigonometry (acting in the enveloping binary space), the spherical non-Euclidean geometry is based on the principal spherical angle of motions in forms Ф and dФ on the hyperspheroid (as always up to the factor R), and also, as an exclusion, dϑ for purely screwed motions (as if on the oriented hypercylinder, tangent to the Special hyperspheroid along its equator); hyperbolic non-Euclidean geometry is based on the principal angle of motions in forms Г and dГ on both hyperboloids, and also, as exclusions, diϑ and dϑ for two purely pseudo-screwed motions (as if on the oriented imaginary and real-valued hypercylinders, tangent to the imaginary and real-valued equators of hyperboloids II and I); besides, generally these principal differential motions dФ on the hyperspheroid and dГ on the hyperboloids are accompanied by the secondary (in that number, induced) orthospherical Euclidean rotations, under inclinations Ф или Г to the Euclidean subspace or to the frame axis;
- antipodal non-connected semi-parts of the hyperboloid II by Minkowski with their antipodal geometries of Lobachevsky-Bolyai with principal angles +Г and -Г;
- inversion of the hyperboloids II and I one to another in result of an rotation of their front projections in common pseudo-Euclidean planes at the right angle П/2 with transformation of a real-valued single equator of the hyperboloid I in an imaginary single equator of the hyperboloid II, and vice versa;
- analogical inversions for the catenoids II and I, and the tractricoids II an I with inversion of their metric forms in their Special binary quasi-Euclidean spaces – see in Chs.5A, 6A and below;
- the illustrative non-connected two-sided flat finite tangent model inside of the trigonometric circle with radius R=1 of geodesic hyperbolic motions on the hyperboloid II by Minkowski and in its hyperbolic geometry of Lobachevsky-Bolyai with radius-parameter R – on the basis of our projective hyperbolic formulae (125A) and (138A) in Ch.7A as the Big Pythagorean Theorem (see it also at Figure 4A); their proportional projective model by Beltrami-Klein inside of the Cayley absolute with radius R on the Euclidean projective hyperplane;
- the illustrative connected two-sided flat infinite cotangent model outside of the trigonometric circle with radius R=1 of geodesic hyperbolic and ellipsoidal motions on the hyperboloid I by Minkowski and in its hyperbolic-elliptical geometry with radius-parameter R – on the basis of our projective hyperbolic formulae (278A) and in Euclidean part of (269A) in Ch.10A as the Big Pythagorean Theorem; their proportional projective model outside of the Cayley absolute with radius R on the Euclidean projective hyperplane;
- the illustrative connected two-sided cylindrical finite tangent model on the lateral surface of the hypercylinder with radius R=1 and with radius as of the trigonometric circle R=1 of geodesic motions on the hyperboloid I by Minkowski and in its hyperbolic-elliptical geometry with radius-parameter R as the same, but cylindrical Big Pythagorean Theorem; this cylinder is bounded at the top and bottom by two-sided tangent disk-model of the hyperboloid II (above);
- the united tangent model in the form of an entire cylinder above with its lateral surface and its upper and lower bases-disks, combined them for trigonometric mapping of the united three-sheets hyperboloidal hypersurface with its hyperboloidal non-Euclidean geometry;
- the illustrative connected two-sided flat finite sine model inside of the trigonometric circle with radius R=1, including it, of geodesic principal spherical motions on the Special hyperspheroid and in its spherical geometry with radius R – on the basis of our projective spherical formulae (190A) and (192A) in Ch.8A as the Big Pythagorean Theorem; their proportional projective Special model inside of the circle with radius R, including it, on the Euclidean projective hyperplane; this flat finite sine model of the Special hyperspheroid with its geometry, as the closed and partial Euclidean region, is indeed the sine-tangent specific analog (see above) of the flat finite tangent model of the hyperboloid II by Minkowski with its geometry, as the opened and full Euclidean region, but with the same Big Pythagorean Theorems for summing motions on the Euclidean plane (n=2) or subspace (n=3) – see at Figure 4A, i.e., they differ principally, as closed and opened models;
- the classic founders of hyperbolic non-Euclidean geometry, for inferring its planimetry, used spherical-hyperbolic analogy – abstract and specific, beginning else from the works of Lambert (1786) and Taurinus (1825), however both its complementary hyperbolic angles may be gotten with specific analogy, what be realized by Nikolai Lobachevski (1829), but, in first, with the angle ξ and then with the using sine-cotangent analogy with the main angle γ, i.e., in contrary order, in according to his original axiomatic method of construction of new non-Euclidean geometry in its complete finished form;
- the spherical angle of parallelism by Lobachevsky (П=ξ=π/2- φ) -- countervariant in the spherical (locally) and hyperbolic geometry, but, in the hyperbolic geometry, it is admitted to applications only in the universal base (usually in E_1={I}); this base is used also for simplest infer in it all metric angular formulae of this geometry, with their following propagation into any admissible bases in it; so, using in the base E_1 specific sine-cotangent analogy sinhγ=cotξ (see above), we obtain П=ξ=arc[cot(sinhγ)];
- the covariant angles of parallelism (ф и γ) in spherical (locally) and hyperbolic geometries, and also they are the angles of motions with Lambert measure in these geometries;
- the hyperbolic and spherical equations of the Minding tractrix in only R-factor in the Especial quasi-Euclidean plane and only with respect to its generating time-like hyperbola in a pseudoplane with the same factor R of similarity, under specific sine-tangent analogy between hyperbolic and spherical angles-arguments of motions;
- a tractrix as the hyperbolic analogue of a spherical curve "cycloid" but with only one cycle;
- the natural hyperbolic--orthospherical equations of a Beltrami pseudosphere in only R-factor of similarity in the Especial quasi-Euclidean space of index q=1 and, with respect to its generating one-sheet hyperboloid of Minkowski, with the common reflector tensor of their pseudo-Euclidean and Especial quasi-Euclidean enveloping spaces and hence with the common orthospherical admissible rotations;
- all tractrices (of Huygens and Minding) and (tractricoids II and I) are similar geometric objects with one factor R of similarity (as circles and spheres, hyperbolae (time-like and space-like) and hyperboloids II and I of Minkowski, catenaries of two types and catenoids II and I, cycloids etc.).
Note, in Chapter 6A of the English-language edition of the tensor trigonometry (January 2021), the author proposed to subdivide surfaces of constant Gaussian curvature into “perfect” and “imperfect” from the principle: whether a complete group of motions is given on the surface or it is absent. But, as can be seen from the context, from the point of view of the enveloping metric spaces, the former are “surfaces of constant radius”, but the latter are surfaces of constant curvature, but not of constant radius. It is at a constant surface radius-parameter that the geometric distances - movements are proportional to the changes in the angle of motion, determined initially by the rotational tensor trigonometry of the enclosing metric space. In the first case, we have a complete group of continuous motions, consisting of principal and orthospherical motions, relatively to the space reflector tensor. In the second case, for surfaces of revolution of constant curvature, we have only a subgroup of orthospherical rotations, but we do not have a complete group of continuous motions at absence of principal angular motions. From here, our hyperspheroid and hyperboloids of Minkowski are perfect surfaces of constant Gaussian curvature, while the Beltrami pseudosphere is not. The sphere is such, but only when a reflector tensor is specified, which determines for it a complete group of quasi-Euclidean permissible motions. But then it turns into our hyperspheroid in the enveloping quasi-Euclidean space.
- One-step isometry of the hyperboloid I of Minkowski and the pseudosphere of Beltrami on the base of one-step specific sine-tangent analogy, although they have constant and equal Gaussian curvature and common topology.
- The Infinitesimal Pythagorean Theorems – quasi-Euclidean and pseudo-Euclidean on hyperboloids II and I.
- Summing geodesic motions (segments) -- two-steps and polysteps in non-Euclidean geometries.
- The Big and Small Euclidean Pythagorean theorem with Euclidean-orthogonal mapping of the sum of two principal motions (segments), as of extended, and separately as with differential increment of motion, -- commutative in Euclidean geometry and non-commutative in non-Euclidean geometries, according to secondary orthospherical shfts + or -θ, or dθ in vector form of summing.
- Polar decomposition of the sum of motions (segments) -- two-steps and polysteps into principal and secondary orthospherical rot θ (the latter as an induced orthospherical rotation of an object or as orthospherical shift of the initial base E_1 with its geometric objects).
- Tensor-trigonometric orthospherical interpretation of three angular effects: (1) the angular excess of Harriot in spherical geometry, (2) the angular defect of Lambert in hyperbolic geometry and (3) the secondary lateral rotation rot θ with formulae for it in the polar decomposition above for two-steps and polysteps principal motions.
- The metric angular differential forms of absolute motions Dф or Dγ with a factor-parameter R on the Special hyperspheroid from initial value Ф or on the hyperboloids II and I by Minkowski from initial value Г with radius-parameter R in a kind of orthogonal decomposition into parallel and normal parts (with respect to direction of motion) as the Absolute and Relative differential Pythagorean Theorems in Euclidean and pseudo-Euclidean forms – see in Ch.6A, 7A, 8A and 10A.
- The secondary induced Euclidean differential orthospherical rotation dθ at non-collinear differential increment of a principal motion, and itself in time as the Thomas precession.
- The simplest common cosine formula for induced orthospherical differentials dθ in generation of the angular deviations of Harriot and Lambert, and in generation of the Thomas precession.
- Metric binary curvilinear spaces -- quasi-Riemannian and pseudo-Riemannian ones as respectively infinitesimally quasi-Euclidean and pseudo-Euclidean;
- a reflector-tensor, a metric tensor and a Riemannian metric as initial concepts in definitions of these metric spaces; orientational role of the reflector tensor only in elementary cases if q=1 or n=1.
4. Theory of Relativity.
- The Mathematical Principle of relativity and physical-mathematical isomorphism.
- 4D binary "affine-Euclidean" metric space-time of Lagrange with the geometry and continuous transformations V_G of Galileo, mapping for objects non-relativistic physical movements with velocity tan v (through “parallel rotations” v) and its rotation with velocity w=dθ/dt (through orthospherical rotations θ), where v and θ -- the angular arguments (see in beginning of Ch.1A);
- the absence of metric connectivity in the space-time of Lagrange between the Euclidean space and the time-arrow;
- the subgroup of “parallel rotations” tensor V and scalar v in the 4D "affine-Euclidean" space-tame of Lagrange along the time-arrow, as intermediate ones between principal spherical Ф, φ and hyperbolic Г, ϒ counter-directed rotations with respect to the frame axis of the binary spaces;
- the common subgroup of orthospherical rotations ϴ and θ, as an intersection of three groups, expressed in the universal base E_1 (the base of immobility in STR): i.e., (1) of the group of quasi-Euclidean motions, introduced by us since 2004 in Chs.5 and 6, (2) of the group of "affine-Euclidean" motions of Galileo, and (3) of the group of pseudo-Euclidean motions of Lorentz -- all three of the index q=1 (Chs.5, 6, 1A);
- the pioneer natural transition by Poincaré in June 1905 from the non-completely metric real-valued 4D space-time of Lagrange to the completely metric homogeneous and isotropic complex-valued 4D space-time of Poincaré with pseudo-Euclidean metric, with the imaginary time-arrow as time axis ict and with the continuous Lorentz group of transformations (as 3D rotations in 4D space-time, identical to 3D motions on its non-Euclidean hypersurfaces – trigonometric hyperboloids II and I of Minkowski of the radius-parameter R=1; or identically to such in the realificated 4D Minkowski space-time with the real-valued time-arrow ct, where "c" is a scale factor for time coordinate and as the light speed in a vacuum;
- 4-velocity by Poincaré as 4-vector c=ci_α, where i_α – 4-vector of the principal tangent to a world line of Minkowski in the 4D space-time of Poincaré or of Minkowski; "c" is also 4-vector-radius of the accompanied hyperboloid of velocities with its radius R=ic in the same space-time;
- non-Euclidean hyperbolic tangent and sine projective vector mapping of the coordinate v and proper v* physical velocities as 3-vectors of orthoprojections of the 4-velocity by Poincaré "c" on the 3D Euclidean projective hyperplane in the 4D space-time of Poincaré or of Minkowski; trigonometric Rules of summing coordinate v and proper v* physical velocities in the initial universal base E_1 (the base of immobility in STR), but expressed on the Euclidean projective hyperplane, and with correction in it at the induced orthospherical rotation θ, in the case of non-collinear velocities, as consequences from the general Law of summing hyperbolic motions (rotations) in its tensor trigonometric form with vector and scalar orthoprojections (Ch.7A); accordingly the tangent projections of "c" are mapped into the disk with radius R=c on the Euclidean projective hyperplane, but the sine projections of "c" are mapped into the disk with infinite radius on the Euclidean projective hyperplane – in both cases, according to the Big and Small Pythagorean Theorems, produced by formulae (138A) and (135A) in Ch.7A on the same Euclidean hyperplane; the sine model has a distortion along e_α, the tangent model has distortions along e_α and e_ν; our tangent model of the hyperbolic geometry is identical to the Beltrami-Klein projective model, however our trigonometric infer of the model is very simple, and we give, in addition, the sine model for summing proper velocities v^*, important for the future relativistic cosmonautic;
- the brief (in one line) simplest and universal trigonometric formula for differential angular orthospherical deviations +-dθ in geometric figures and in motions of non-Euclidean geometries: (1) as the excess of Harriot +dθ, (2) as the defect of Lambert -dθ) and in Theory of Relativity (3) as the Thomas precession in coordinate time -dθ/dt – all caused by the same induced orthospherical rotation +-dθ at summing non-Euclidean noncollinear segments or motions (see in the end of Ch.7A and in Ch.8);
- inner tangential 4-acceleration as a derivative with respect to proper time ꚍ in a direction of 4-vector of the principal tangent i_α to a world line by Minkowski: g_a=c’=c(i_α)’=g_α(j_α), where j_α – 4-vector of the principal pseudonormal to a world line by Minkowski; inner normal 3-acceleration as a derivative with respect to proper time ꚍ in a direction of 4-vector of the normal tangent i_ν to a world line by Minkowski: g_ν=c’= c(i_α)’=g_ν(b_s), where b_s – 3-vector of the sine binormal to a world line by Minkowski; and general inner 4-acceleration is a complete derivative with respect to proper time ꚍ in some given direction of 4-vector of the tangent i_β to a world line by Minkowski: g_β=c’= c(i_α)’=g_β(p_β), where p_β is 4-vector of the general pseudonormal -- all these accelerations are bonded by the 4D Absolute Pythagorean Theorem (in Euclidean variant) in the 4D space-time of Poincaré or of Minkowski (!) as g_β^2=g_α^2+g_ν^2 on the hyperboloid II of accelerations, what is also proportionally to 1-st metric form of any world line in the space-time (Ch.10A and to 1-st metric form of the accompanied trigonometric hyperboloid II; the Relative Big and Small Pythagorean Theorems as orthoprojections of the metric form into 3D Euclidean subspace (Chs.7A, 10A);
- γ is an instant hyperbolic angle of relativistic progressive physical motions of a material object or a particle in tensor, vector and scalar interpretations; in particular, γ is a hyperbolic angle of principal 3D rotations in 4D space-time of Minkowski, with respect to a frame axis, and of identical principal 3D motions on its non-Euclidean hypersurfaces – hyperboloids by Minkowski II and I; iγ is the same, but pseudospherical angle in the complex quasi-Euclidean 4D space-time of Poincaré;
- dγ and diγ are hyperbolic and pseudospherical differentials of 3D principal rotations (motions) in space-time of Poincaré and Minkowski along a world line or on the concomitant to it trigonometric hyperboloids Ch.10A; they are proportional to differentials of potential as dP/c^2= gdl/c^2 – either accelerational, or gravitational ones;
- dα and diα (dθ and diθ) are orthospherical and orthopseudohyperbolic differentials of secondary lateral rotations as “parallel circles” in the binary spaces and in the space-time of Poincaré and Minkowski along a world line or on the concomitant trigonometric hyperboloids, or as principal, but orthospherical differentials of pseudoscrewed motions -- time-like and space-like (Ch.10A);
- the trigonometric tensors from these angles-arguments: an orthogonal tensor of relativistic rotations (motions) and a cosogonal tensor of deformations; the latter acts as one-step and only in the universal base E_1, for example, in the relativistic effect "the spherical Terrel-Penrose rotation of an object at its Lorentz contraction" (Ch.4A);
- tensor-trigonometric interpretations of two very known relativistic effects: Minkowski real dilation of time scale leading to decreasing proper time as a consequence of hyperbolic rotational transformations and Lorentz seeming contraction of extent as a consequence of hyperbolic deformational transformations;
- the accompanied relativistic effects leading to the supervelocities of sine, cosine, cotangent and cosecant types with their invariants.
- Discrete character of the well-known relativistic effect “The influence of velocity v = c tanhγ or v^*=c sinhγ of the moving system E_m relative to the resting system Е_1 on the proper time dilation”; in STR this effect is appeared in result of the Lorentzian transformations, for example, from E_1 into E_m, and expressed through velocity “v” or “v^*”, as follows:
cosh^2γ=(dct/dcτ)^2=1/sech^2γ=1/(1-tanh^2γ)=1/[1-(v/c)^2=1+sinh^2γ=1+(v^*/c)^2;
moreover, the first formula is one-step similar to the tangent cross-projection (Ch.5A), but the second formula is polysteps similar to the sine orthoprojection;
- the action of inner acceleration g_a and of equivalent to it (only mathematically) intensity g_f of gravitational field as the root cause of the proper time dilation on a differential level dct in the base Е_m in comparison with the first differential of coordinate time dct in the base Е_1, both at their separate independent actions, and at their combined action (cinematic and gravitational); the latter, for the free motion, leads to a joint doubling of proper time dilation coshγ=dct/dcτ at g_a=g_f);
- the trigonometric equivalent differential and integral cosine formula for proper time dilation in result of the influence of acceleration g_a and intensity g_f:
dсoshγ=gdx/c^2=dP/c^2 и сoshγ=1+int(0,x)[gdx/c^2]=1+int(0,P)[dP/c^2]=1+ΔP/c^2=1+A/E_0,
where g=c(dγ/dτ) is an inner acceleration, E_0=m_0c^2 – see in Chs.5A and 9A; thus, differentially at tangential g we obtain hyperbolic motion and at normal g we obtain pseudoscrewed motion;
- the indicated trigonometric approach with potentials allows one to estimate time dilation differentially and integrally in a continuous manner, since:
dct/dcτ=coshγ=1+coshγ-1=1+int(0, γ)[sinhγdγ]=1+int(0,τ)[(v^*/c)(g/c)dτ]=1+int(0,x)[gdx/c^2]=1+ΔP/c^2,
moreover, by this universal manner, both for g_a, and for g_f; this approach is based on mathematical (but not physical!) equivalence of g_a and g_f;
- contrary, in the Einsteinian GTR, g_a and g_f are accepted as equivalent mathematically and physically and even at the tensor level, as Christoffel symbols, bending space-time together; below we will show that such curving is an unnecessary operation, which significantly complicates the Theory of Relativity; moreover, we arrive to the hyperbolic angle of motion γ in a universal and continuous way for relativistic motions with both kinematic and gravitational accelerations –- tangential and normal in the frame of the unified Theory of Relativity in the Minkowski space-time;
- the free relativistic motion of an object with equivalent g_a and g_f (induced g_a) under a doubling of its proper time dilation by two these factors: kinematic internal acceleration and gravitational field intensity.
- In particular, this leads to a doubling of the Newtonian gravitational deflection of a light ray going close to the Sun, or as if to an additional equivalent refraction of light in a gravitational field, formally according to the same Snellius Law – on the other mechanism, as at the well-known optical deflection of light of an electromagnetic field nature, which we applied in Chapter 9A for the simplest trigonometric interpretation of this complete relativistic effect without using Einstein's curving space-time; therefore, we additionally obtained the same Snellius Law for the refraction of light in a gravitational field, but in another physical interpretation - without changing the light speed, but with a change in its frequency.
- In particular, this leads to a doubling of the relativistic shift of Mercury's perihelion from each three kinematic cosine time dilations, i.e., ultimately to a sixfold time dilation with the same hyperbolic cosine coshγ and to a sixfold shift of Mercury's perihelion, which in Chapter 9A gives a physically clear and simplest, but relativistic trigonometric interpretation of the Gerber's formula for the shift of Mercury's perihelion and again without using Einstein's curving space-time.
- In particular, this explains the single-fold effect of the red shift of the wavelength of sunlight on Earth, since in this case, due to parallelism of the field intensity vector and the direction of motion of light, the photons do not have a normal acceleration and a tangential acceleration is zero too, thanks to constancy of speed of light, that in fact leads only to the Newtonian gravitational decrease in the energy of light quanta and to the simplest quantum-mechanical interpretation of this effect without using the theory of relativity – all these real influences on proper time are discussed in detail in Chs. 3A, 5A, 9A.
- In particular, this explains by such a way the doubling of the theoretical radius of the Michell "black hole" (on the base of the Newtonian theories of mechanics and gravity) compared to its theoretical Schwarzschild radius (on the base of the Einsteinian GTR).
- The tensor-trigonometric models of relativistic kinematics and dynamics -- relative and absolute;
- the trigonometric models of simplest relativistic motions -- hyperbolic and pseudoscrewed ones;
- interpretation of hyperbolic motion in the pseudo-Euclidean space-time as the motion along catenary and tractrix in the both Special quasi-Euclidean space-time – Chs.5A and 6A;
- the proportional hyperbolically orthogonal tensors of a momentum and of a energy (of energy-momentum), produced from the trigonometric dimensionless tensor of hyperbolic rotations by multiplication it on the invariants mo•c and mo•c2;
- the proportional orthospherically orthogonal tensor of a rotational momentum, produced from the trigonometric dimensionless tensor of orthospherical rotations by multiplication it on the invariant mo•c;
- the proportional pseudo-Euclidean orthogonal tensor of a complete energy-momentum, produced from the trigonometric dimensionless tensor of pseudo-Euclidean rotations by multiplication it on the invariant mo•c2;
- a physical nature of absolute matter's motion at the 4-velocity of Poincare "c" along its world line, namely, with its proper momentum Po=mo•c and proper energy E_o=m_o•c^2 as a flow of its proper time to at a velocity "c" in the same direction of Minkowski space-time; and on the contrary: a flow of its proper time as absolute matter's motion;
- the dynamic full characteristics of relative motion of matter in a specific base: a full momentum P=m•c, a full energy E=m•c^2, -- produced as consequence of hyperbolically orthogonal projection of absolute matter's motion onto the time-arrow ct;
- the relative momentum of matter's motion at a velocity v=c•thγ in this specific base as hyperbolically orthogonal projection of the proper momentum into the space E^3, namely, p=Po•shγ=P•thγ=mv and with the same direction cosines in the space E^3 as for vectors v, sinhγ, tanhγ;
- the pseudo-Euclidean right triangle of 3 momenta: P_o (hypotenuse), P and p (cathetuses) -- similar to the triangle, formed by the arrows of time cꚍ and ct with the hyperbolic angle γ between them;
- the pseudo-Euclidean Pythagorean Theorem for moduli of 3 momenta: (iP_o)^2=(iP)^2+p^2, p^2=P^2-P_o^2, from which easily relativistic function m(v) and Einsteinian formula p^2=(mv)^2=(m^2-m_o^2)•c^2=(E^2-E_o^2)/c^2 are resulted; the same in vector representaations;
- the dynamic (at a 4-pseudovelocity "c") 4D pseudoanalogue of the 3D theory by Frenet-Serret in the Minkowski space-time with interdependent local parameters of a world line's rotation with all of the current differential-geometric parameters of this world line until their order 3 at the maximal order of its embedding "4";
- tensor-trigonometric relativistic interpretations for Doppler effects and for an aberration, and also for the Thomas orthoprecession; the angular velocity of this orthoprecession and its connection with Coriolis acceleration of a material object at its non-collinear motion in the pseudo-Euclidean space-time of Minkowski.
Here one should argue the position of the author, which he stated in the discussion Ch. 9A. In it, the mathematical description of relativistic motions of an object was subdivided into real ones (locally at the place of events) and observable ones (at a remote place of their fixation), with a significant difference in the potential of the gravitational field in these places. In GTR by Albert Einstein, this mathematical description is maid according to the changing values of the metric tensor of space-time formed by the gravitational field and inertia of the object (G-field). But an observable motion is accepted as a real motion in all further conclusions. This is a positivist point of view, coming from the views of Ernst Mach in his work: “Die Mechanik in ihrer Entwickelung historisch-kritisch dargestellt. – 1904”. In GTR, a straight line is the trajectory of a ray of light, since on a cosmic scale there is really nothing to tie it to. This idea with using light rays comes from the creators of non-Euclidean geometry. Carl Gauss even tried to test it with his students on a triangle of 3 mountain peaks. However, in GTR, the gravitational effect on moving photons from massive cosmic objects bends not only their path, but also the physical and mathematical space-time itself of GTR. The main contradictions in this approach are the following.
-The real and remotely fixed descriptions of one motion must inevitably differ, since at the place of observation the gravitational field is completely different, that is, information about the real motion reaches the external observer through the gravitational lens.
-In a curved pseudo-Riemannian space-time of GTR, there is not its complete group of motions, and, therefore, there is no uniqueness of motions mapping in various coordinate bases;
- electrically and magnetically curved rays of light are not considered straight lines and do not bend the space-time. So, a ray of light is bent not only in a gravitational field, but also under the influence of fields created by electric charges and magnetic dipoles. Due to the fact that the absolute electric and magnetic permeabilities are many orders of magnitude higher than the gravitational constant in similar CGS units, then these ray curvatures are observed even in terrestrial conditions in media with a variable refractive index, which, if space were curved, would cause complete decadence in its description.
- More widely, in GTR, there is no answer to the cardinal question: why should space-time be curved in a gravitational field, but not curved in other physical fields?
- From here, as before, there is the incompatibility of GTR and Quantum Mechanics, acting according to Pole Dirac in the Minkowski space-time.
- There is inevitable violation of the fundamental Law of conservation of energy-momentum of matter and field, according to Amalia Noether's Theorem, due to the curvature of space-time with violation of its isotropy and homogeneity. This also gives rise to theoretical temptations for the production of matter and energy from nothing as a modern kind of perpetuum mobile.
- According to GTR, in a gravitational field, spatial material objects (for example, cosmologic ones) should also be bent without the influence of any forces of nature.
- What to do with the many other classical Laws of Nature, formulas and theories formulated in time and space; should they also be curved under acting gravity? And else many-many others.
- Note, in Ch.9A, we inferred that the local velocity of light in the field of gravity does not change and equal to “c”. Hence, according to Tensor Trigonometry, under acting gravitation, a light ray may curve only at the lateral orthospherical angle θ or dθ with its Euclidean nature, as the hyperbolic curving must change velocity “v” (but here tanh γ=1=const). This conclusion was demonstrated by us when evaluating the curvature of a light ray in the region of the Sun using a trigonometric approach. In the direction of the ray, the speed of light did not change, but the frequency of photons did change. (It would seem that the second curvature of the ray formally corresponds to Snell's Law, but for other reasons: in an optical medium, the speed of light and its wavelength change at a constant frequency.) But if the hyperbolic curvature of light in a gravitational field is absent, then even if space-time is curved together with the light beam, the new space-time remains pseudo-Euclidean again, but rotated at θ or dθ! Why is it necessary to curve space-time, although it is enough to curve orthospherically only trajectory of a light ray? Then all the contradictions of GTR noted above disappear, but STR remains without a field and with a field of gravity. We call it Theory of Relativity, as the great Max Planck did this.
- Thus, in Minkowski 4D space-time, orthospherical Euclidean rotations of the world line of photons, bending a ray of light with a degree of freedom of 3 under the doubled action of gravity, do not violate the sacred principle of the theory of relativity as the "Constancy of the speed of light". And only such possible bending a light ray does not affect the space-time in any way. If a ray of light is directed along the field intensity vector, then there is no bending the ray of light, but its frequency changes. On the other hand, world lines of material bodies can be bended both orthospherically and hyperbolically with a summary degree of freedom of 3 – see below.
- Analysis of much sharper images of galaxies obtained since 2024 by the new James Webb deep space telescope increasingly testifies in favor of the complete isotropy and homogeneity of our 4-dimensional space-time of Nature and its seen flat geometric structure. However, of course, any image of the World can be curved: it is enough to look at it through an optical lens or, theoretically, through a gravitational lens. But all this only complicates the description of movements and objects in deep space and, additionally, with increasing distances to them, leads to erroneous conclusions. If all known and new relativistic effects are explained logically in the flat Poincaré and Minkowski space and in full accordance with the Law of Conservation of Energy-Momentum and the Laws of Quantum Mechanics, then why introduce the gravitational curving space-time as a real fact? By the way, this curving space-time does not explain the true physical nature of gravity. This mystery remains, perhaps, until the creation of a full-fledged theory of gravity, for example, within the framework of the classical field concept, which does not contradict the fundamental Laws of Nature.
- A normal part of the full differential of rotation Dф or Dγ differs from the lateral differential of orthospherical rotation dθ in that it is the orthoprojection of the latter onto the Euclidean subspace. In quasi-Euclidean space, this orthoprojection acts under a spherical angle, in pseudo-Euclidean space this orthoprojection acts under a hyperbolic angle. Therefore, in Minkowski space-time, the normal projection of motion of a material object along a world line is not purely Euclidean, but lateral orthospherical rotation dθ is always purely Euclidean. There is an analogy with as if the general theorem of Meusnier.
- About decompositions of the metric form of absolute motions Dф or Dγ in the binary spaces into parallel and normal parts with Absolute and Relative Pythagorean Theorems and with the concomitant trigonometric objects (unity hyperspheroid or unity hyperboloids) see in the last 10A.
5. Formal Complex Analysis.
- Formal analyticity of nonholomorphic functions (complex and real) of complex conjugate arguments (one-dimensional and many-dimensional).
- Formal differentiation and integration.
- Formal power series and completeness of a differential.
6. Differential Analysis and Methods of Optimization for evolutionary Objective Functions.
- Analytic optimization for functions in a scalar variable.
- Analytic unconditional optimization for functions in a vector variable.
- Analytical conditional optimization for the function in a vector variable - either depended on some parameters or restricted by some connections equations.
- Analytical conditional optimization limit methods with big and small parameters, connected between them.
- The exact characteristic (secular) equation for conventional eigenvalues of a Hesse matrix of a function of 2nd order under linear connected its arguments.
- In passing: isoparametric polynomials (including the mirror ones), differential invariants of the order 2 and higher for flat curves y(x), x(y);
- solution of the Newton problem about the relationship between the coefficients of direct and inverse power series, which is also equivalent to the problem of the relationship between derivatives of any order for direct and inverse to it analytic function;
- analytic optimization for real nonholomorphic functions in complex conjugate or mixed variables with the use of operations of the formal complex analysis;
- one dimensional numerical methods of optimization of the orders 0, 1 and 2;
- iterative optimization techniques;
- one dimensional Newton’s method of 2-nd order and its difference analogues;
- multi-dimensional analytical-numerical optimization of the orders 0, 1 and 2;
- coordinate-wise Seidel method of 0th order;
- gradient Cauchy method of 1st order with a given step;
- modification by the speedy descent or ascent method of 1st order;
- quadratic method of incomplete 2nd order with a calculated step;
- modification by the speedy descent or ascent method of incomplete 2nd order;
- multi-dimensional Newton's method of 2-nd order with a calculated step;
- modification by the speedy descent or ascent method of 2nd order;
- independence of the efficiency of both second-order methods on the initially selected scales along the axes or from the dimensions of the parameters;
- multi-dimensional analytical- numerical conditional optimization of the orders 0, 1 and 2;
- normal projection methods of the orders 1 and 2;
- connected methods with big and small parameters (the first of these is the Courant penalty functions method);
- multi-dimensional planned-calculational optimization methods of the 1st, incomplete and complete 2nd orders based on the selected plan of the location of the argument points to calculate the values of the function in them and on the finite difference method to estimate the first and second partial derivatives of the objective function;
- adequacy criterion for planning-difference models;
- multi-dimensional planned-experimental optimization methods of the 1st, incomplete and complete 2nd orders; in general, this is more well-known under name “Design of experiments on the factor space”, when optimization of some objective scalar function of response is realized through searching for optimal values of the influencing significant factors (in the case of a normal distribution of the random error of finding values of this function of response under exact given values of its factors-arguments;
- planned-experimental optimization of some function of response of the orders: 1st (Box method of the speedy descent or ascent), incomplete 2nd (quadratic method) and 2nd (Box-Wilson method); exact formulae and values for all accompanying parameters of the Box-Wilson plan;
- adequacy criterion for planning-experimental models;
- genesis of all these optimization methods and their consecutive interrelation.
7. Mathematical Chemistry.
- Mathematical modelling of chemical reactions, the general kinetic function;
- modelling of a chemical reaction of one mono- and one polyfunctional substances - the last with all initially identical-active functional groups and with reduction of their activity (step by step);
- kinetic curves as plots of functions and as functionals (including the functions of response in planned-experimental optimization of some chemical reaction parameters);
- kinetic isotherms and isochrones, their theory and applications;
- the direct and indirect kinetics;
- the kinetic of 3D curing of a thermosetting plastics or elastomers (composites) in isothermal and isochronous temperature regimes; the characteristic temperature of 3D structuring;
- torsional indirect curing kinetics (by Lewis-Gillham method) due to the change in the shear modulus, according to the periods of free torsional oscillations; and due to fluctuations of the zero point during structuring, calculating the decrement of oscillations through the ratio of successive angular sectors of oscillations clockwise and counterclockwise.
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Large Russian version of this web-site
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The given monograph, as its 1-st edition, with exposition of this new math subject, was translated into English (which is currently international in the scientific community, especially in the field of exact sciences) and It was issued, as its 2-nd edition, in the beginning of 2021 by the Publishing Haus Fizmatlit, and, as its 3-rd edition last from author-himself, in the beginning of 2025 by the Publishing Haus Fizmatkniga (of course, as significantly innovated and expanded). Below they are given in the adopted now forms as p-book and e-book (from 2025 with own ISBN and DOI) in the usual pdf files from the TEX program and in the standard archived pdf/A files:
Comments: Some Novelties in_the_book Tensor Trigonometry with exposition and development of this new subject of Mathematics
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Pdf file of the book Ninul A. S. "Optimization of Objective Functions: Analytics. Numerical Methods. Design of Experiments.” – Moscow: Fizmatlit, 2009, 336p. (Russian p-book, format A5)
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6 math articles of the author on the topics of the books “Tensor Trigonometry” and "Optimization of Objective Functions” non-published before in soviet academic math journals without “affiliation” (in the Soviet years 1982-1989)
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MCCME – Moscow Center for continuous mathematical education
pdf file of the book Ninul A. S. “Tensor Trigonometry. Theory and Applications.” – M.: MIR, 2004 (in Russian). – in E-library of MCCME (since 17.04.2006)
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E-Library of the Moscow State University, Mechanical-Mathematical Faculty
pdf file of the book Ninul A. S. "Optimization of Objective Functions: Analytics. Numerical Methods. Design of Experiments.” – M.: Fizmatlit, 2009. – in this E-library since 26.11.2010.
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Jpg compressed file of Front Cover of book Нинул А. С. "Тензорная тригонометрия" (drag with the mouse from the screen)
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Jpg compressed file of Front Cover of book Ninul A. S. “Tensor Trigonometry” (drag with the mouse from the screen)
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Jpg compressed file of Front Cover of book Нинул А. С. "Оптимизация целевых функций" (drag with the mouse from the screen)
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Small photoalbum
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P.S.: My Russian professional web-site in Chemical fields
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1. Messages exclusively for the author are giving on the E-mail: NinulAS@yandex.ru
2. Problems for discussions with the author when other users of this site can see the information are giving in Invitation book. Premoderation operates. ========================================================================== Copyright extends both to these two books, and to this complete site! ========================================================================== |
