Ninul Anatoly Sergeevich – the personal web-site http://Ninul-eng.narod.ru/
Its large Russian version: http://NinulAS.narod.ru/
Its large English version: http://NinulAS.narod.ru/English.html
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The author welcomes visitors to this web-site with good intentions and hopes that you will receive interesting and useful information with its further correct use in own scientific works in similar and adjacent fields.
The author has no commercial purposes or profit from his electronic resources here, other than scientific goals.
So, at the end of this Main page, you will see active internet links to the author's electronic files of his mathematical monographs in Russian and English. All these books with contents of the site are here in the public domain. These books files are exhibited, if necessary, with correction of noticed misprints and small inaccuracies, inevitable at forming such very large works. (If something in their contents or infers is not clear to some-body, then it is better to contact with their author before he has finished own earthly journey!) You may leave your review, opinion or suggestions in the moderated Invitation book or write to me in any e-mails at the end of this page. General information about the author is displayed on the related web-sites in Russian.
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 and remarks concerning of his three scientific monographs. Namely, they are.
1. In Russian: Ninul A. S. "Tensor Trigonometry. Theory and applications.” – Moscow: MIR, 2004, 336 p., 8 ill., 1 table, issued in October of 2004 (ISBN-10 : 5-03-003717-9 and ISBN-13 : 978-5-03-003717-2).
2. In English: Ninul A. S. "Tensor Trigonometry.” – Moscow: Fizmatlit, 2021, 320 p., 8 ill., 1 table, issued in January of 2021 (ISBN-13 : 978-5-94052-278-2). This book is the author's English updated version of its 1st 2004-edition.
These books are devoted to a new subject of mathematics called by the author "Tensor Trigonometry", with preliminary coverage of a number of necessary questions in Theory of Exact Matrices, Linear Algebra, and in directions of Complexification. Besides, they contain numerous its applications to some fundamental problems concerning Algebra, Geometry and Theoretical Physics. Both these books contain the large Appendix "Trigonometric models of motions in STR and non-Euclidean Geometries.”. If necessary, one can easily compare both editions of the Tensor Trigonometry by numbered 750 formulae and 8 figures, even without their textual parts, for example, to state strictly the priority of results.
3. In Russian: Ninul A. S. "Optimization of Objective Functions: Analytics. Numerical Methods. Design of Experiments.” – Moscow: Fizmatlit, 2009, 336 p., 18 illustrations and diagrams, 5 tables, issued in May of 2009 (ISBN-13 : 978-5-94052-175-4).
This book has as a main goal to consider in logical order all basic methods of search and identification of extremes for various types of objective functions generating here on the basis of their sequential genetic interconnections (up to math programming), with parallel filling up of some "blind spots” encountered in math literature. So, in particular, Chapter 1 gives the full solution to the Newtonian Problem about a relationship between coefficients of direct and inverse power series, which is equivalent to the Problem of a relationship between derivatives of any orders for direct and inverse analytic functions. Chapter 4 contains the validation of complete and extremal in essence requirements to coefficients of a real algebraic equation of the degree "n" for the reality (or positivity, or negativity) of all its roots. In fact, this is the full solution to the Problem posed by the great Descartes and partially solved by him, etc.
At the end, last two books contain small parts named "Physical-Mathematical Kunstkammer”, which include a number of especial questions and tasks connected with some problems discussed in these monographs.
In a paper form, these books can be looked in well-known state and scientific libraries – Russian, CIS and other foreign. For instance, the first book is presented in the most known European math library "Zentral Universitätsbibliothek Göttingen" with its link.
In a digital form, these books are presented, for instance, in the largest Russia Scientific resources Elibrary.ru and RusNEB.ru, in the E-library of MSU's Mech-Math Faculty, and also in Google books, Internet Archive with Open Library, English E-books Directory, etc..
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Given further are domains of the author's especial interests and some results of his investigations gotten in these monographs.
1. Some results in Algebra, Linear Algebra, Theory of Exact Matrices.
– the general inequality for all averages (means) of positive number values with its important applications:
Special sign-alternating reduced form of a real algebraic equation of extent "n" (!) for its descriptive analysis and solution,
Limit method and formula for serial calculation of a real algebraic equation roots or of an nxn-matrix eigenvalues in the case of their positivity,
More strict necessary condition to coefficients of a real algebraic equation for all its roots reality (positivity, negativity), than Descartes' Rule,
Necessary and sufficient conditions to coefficients of a real algebraic equation for all its roots reality (positivity, negativity),
Complete set of hierarchical quadratic norms of nxn and nxm matrices and matrix objects including Frobenius and general norms;
– the scalar and matrix characteristic coefficients of a square matrix B: their structures and properties with numerous applications in Linear Algebra, for example:
One-line inferring of the Hamilton-Cayley Theorem and of the Kronecker--Capelli Theorem,
Minimal annulling polynomial of a square matrix B in its explicit form with powers s0= r"– r'+1 (!),
Connections and inequalities for all singularity parameters of a square matrix B in its Jordanian form,
All eigen quasi-inverse matrices in algebraic and explicit forms including the Moor-Penrose matrix: their nature, properties, structures,
All orthogonal and oblique paired eigen projectors in algebraic and explicit forms (spherical and hyperbolic): their nature, properties, structures, namely, 8 for real-valued and 12 for complex singular nxn matrices B, 4 for real-valued and 6 for complex nxm matrix A;
Table of Multiplications for all eigen projectors (!) with a goal of simplest inferring formulae of Tensor Trigonometry in its projective and reflective versions,
Limit formulae for all eigen projectors and quasi-inverse matrices,
All orthogonal and oblique paired eigen reflectors (spherical and hyperbolic), tied one-to-one with mutual paired eigen projectors,
Complete explicit spectral presentation of a square matrix B in its decomposition into prime and nilpotent matrices of the type B = P + O;
– the null-prime nxn matrices: definition, features, properties and applications, their cosine trigonometric spectrum with general cosine relation and inequality;
– the nxr lineors A as matrices and geometric objects;
– for a pair of equirank lineors A1 and A2, their cosine trigonometric spectrum with general cosine relation and inequality including a particular cases at r=1 as the cosine Cauchy Inequality for a pair of vectors,
– for a pair of arbitrary lineors, their sine trigonometric spectrum with general sine relation and inequality including a particular cases at r=1 as the sine Hadamard algebraic and trigonometric Inequality for a pair of vectors;
– the null-normal nxn-matrices: definition, features, properties and applications (in that number in Theory of Optimization);
– the Principle of binarity for simplest inferring formulae of Tensor Trigonometry in all its three versions;
– the main variants of complexifications of introduced notions for the development of the complex kinds of Tensor Trigonometry.
2. Tensor Trigonometry as the new math subject with own notions, formulae and theorems.
– the binary tensor angles between two subspaces: projective ones in accordance with their eigen projectors or eigen reflectors and motive ones in accordance with their eigen rotations, – in affine or in homogeneous and isotropic spaces,
– their projective and motive mutual tensor trigonometric functions: sine - cosine, tangent - secant, etc.,
(all they may be affine or metric notions without or with spaces' quadratic metrics)
– the connections of these types of tensor angles and functions in their trigonometric bases;
– the Euclidean Tensor Trigonometry in n-dimensional Euclidean space with the specific middle reflector tensor (!) of each or some binary spherical tensor angle;
– the quasi-Euclidean binary space of an index q (!) and its quasi-Euclidean Tensor Trigonometry,
– the pseudo-Euclidean binary space of an index q discovered by Poincaré (rediscovered and developed later by Minkowski) and its pseudo-Euclidean Tensor Trigonometry,
– the fundamental reflector tensor (!) for definitions of both quasi-Euclidean and pseudo-Euclidean spaces of dimension (n+q) – more or equal to 2;
– the complete solutions of internal and external pseudo-Euclidean right triangles in the pseudo-Euclidean space (in particular, on the pseudoplane ) with functional connection of their complementary hyperbolic angles (!);
– the abstract spherical-hyperbolic analogy for formal introducing scalar and tensor hyperbolic angles from spherical ones in any own bases E;
– the specific covariant spherical-hyperbolic analogies for functional connections of hyperbolic (g) and spherical (ф) angles only in any universal base E1 including the most important Lambert's vector sine-tangent analogy (in the Euclidean n-subspace) with additional scalar cosine-secant analogy (in the time-like ordinate axis) for principal hyperbolic and spherical angles in pseudo-Euclidean and quasi-Euclidean (n+1)-spaces. (In this Lambert's analogy, there is formally the Lambertian functional connection g(ф): sine ф = tanh g / tan ф = sinh g (rediscovered and developed later by Gudermann for goals of functional math analysis);
– the especial hyperbolic angle and number ω = arsinh1 ≈ 0,881 (as the hyperbolic analogue of the spherical angle and number π/4 =Arctan1 ≈ 0,785 in the universal base E1, with their parallel representations by numeral power series) and its further wide applications in Tensor Trigonometry, non-Euclidean Geometry, and STR;
– the sine-cosine and cosine-sine quadratic invariants of spherical scalar and tensor principal angles of projective and motive types in any own bases E;
– the sine-cosine and cotangent-cosecant (!) quadratic invariants of hyperbolic scalar and tensor principal angles of projective and motive types in any own bases E;
– the spherical (quasi-Euclidean) and hyperbolic (pseudo-Euclidean) orthogonal tensors of principal rotations or motions (!) in these binary spaces and in non-Euclidean Geometries;
– the orthospherical orthogonal tensor of secondary rotations there. (So, it causes the Thomas precession in STR and the angular deviations in non-Euclidean figures – see further.);
– the Lorentzian non-commutative, in general, group of pseudo-Euclidean motions in P(n+q);
– the Special non-commutative, in general, group (!) of quasi-Euclidean motions in Q(n+q);
– the non-commutative, in general, subgroup of orthospherical rotations (as an intersection of last two groups in the universal base E1);
– the polar decomposition of two-step and multi-step principal motions or mixed homogeneous motions in both these binary spaces into principal and secondary orthospherical ones;
– the spherical and hyperbolic non-orthogonal tensors of trigonometric deformations in both these binary spaces;
– the orthogonal and oblique tensor reflectors in both these binary spaces with spherical and hyperbolic projective angles, the middle tensor reflector of a tensor angle;
– the Special Quart Cycle generating all these specific tensors of rotations and deformations of spherical and hyperbolic types;
– the trigonometric nature of commutativity and anticommutativity for prime matrices P1 and P2;
– the elementary tensors of all these types rotations and deformations, and also the elementary tensor of orthospherical rotations in the binary space (at q = 1) with the oriented frame axis for counting tensor angles (!);
– summing two-step and multi-step principal motions in quasi- and pseudo-Euclidean geometries with revealing of a secondary orthospherical rotation.
3. Tensor Trigonometry and non-Euclidean Geometries.
– the Lorentzian group of elementary pseudo-Euclidean rotations (q = 1) realized as non-Euclidean motions from conjugate points M1 and M2 on the accompanied hyperboloids I (one-sheet) and II (two-sheets) of Minkowski oriented in E1 of P(n+1) with the frame axis as two objects with a constant radius-parameter R;
– the trigonometric projective models of two associated hyperbolic non-Euclidean geometries (of Lobachevsky-Bolyai and of Minding-Beltrami) on these paired oriented hyperboloids of Minkowski as tangent-cotangent projections into the projective plane or cylinder;
– the two antipodal parts of the complete Lobachevsky-Bolyai geometry with positive and negative hyperbolic angles of motions;
– the complete three-sheets hyperbolic-orthospherical geometry (!) with its Lorentzian group of homogeneous motions;
– the Special group of elementary quasi-Euclidean rotations (q = 1) realized as non-Euclidean motions from a point M on the hyperspheroid (!) oriented in E1 of Q(n+1) with the frame axis as an object with a constant radius-parameter R;
– the trigonometric projective model of the spherical non-Euclidean geometry on the oriented hyperspheroid as a sine projection into the projective plane;
– the infinitesimal Pythagorean theorems: Euclidean on a two-sheet hyperboloid II and pseudo-Euclidean on an one-sheet hyperboloid I, and also Euclidean on a hyperspheroid;
– the subgroup of elementary orthospherical rotations as an intersection of these Lorentzian and Special groups in the universal base E1 of P(n+1) and Q(n+1);
– summing two or many motions (segments) in both these non-Euclidean geometries by polar decomposition of a result into principal and orthospherical ones with their explicit formulae;
– Identity of the rotational orthospherical scalar angle and the scalar angular defect or excess of triangles or other closed figures in non-Euclidean geometries (!);
– the non-commutative Big and Small Pythagorean Theorems (!) for summing two segments in non-Euclidean geometries (but commutative in the Euclidean geometry);
– the contravariant Lobachevskian parallelism spherical angle (П), connected with the hyperbolic angle of motion (g) in hyperbolic geometries as g=arcoth(sin П) under the contravariant sine-cotangent analogy in the universal base E1, and connected with the spherical angle of motion (ф) in spherical non-Eucliden geometry as ф = п/2 – П in any own quasi-Cartesian base E;
– the covariant angles of parallelism – spherical (ф) or hyperbolic (g) as also the angles of principal motions in both these types of non-Euclidean geometries in any own bases E. (In the universal base E1, both these parallelism angles are connected by the Lambert's sine-tangent analogy between them: g(ф) or ф(g).),
– the natural hyperbolic equations of a tractrix (in only R-factor) in the Especial quasi-Euclidean plane, namely, with respect to its generating time-like hyperbola (with the same radius R) in the pseudoplane, under the specific scalar sine-tangent (cosine-secant) analogy between their spherical and hyperbolic angles-arguments;
– the natural hyperbolic--orthospherical equations of a Beltrami pseudosphere (in only R-factor) in the Especial quasi-Euclidean (2+1)-space, namely, with respect to its generating one-sheet hyperboloid I of Minkowski (with the same radius R) in the pseudo-Euclidean space, under the same specific analogy. (Both these enclosing spaces have the common reflector tensor and hence the common orthospherical rotations.),
– Isomorphism of these hyperboloid I and pseudosphere, both with one factor R, in their two types' metric and formed in the universal base E1 of both these enclosing spaces (!),
– all these tractrices and pseudospheres are similar geometric objects, proportional only to factor R, (as circles and spheres, hyperbolae and hyperboloids, catenaries and catenoids, etc.);
– "perfect" and "imperfect surfaces of constant Gaussian curvature from the principle: whether a complete group of motions on them (here with respect to their enclosing spaces) is given or it is absent; the perfect surfaces have a constant radius-parameter, the imperfect surfaces have not a constant it (in some directions).
4. Tensor Trigonometry and Theory of Relativity.
– "parallel rotations" in the affine-Euclidean 4D space-time of Lagrange L(3+1) as intermediate between spherical and hyperbolic rotations;
– the mathematical principle of relativity and physical-mathematical isomorphism;
– dimensionless tensors: a hyperbolically orthogonal tensor of relativistic motions-rotations and a spherically quasi-orthogonal tensor of relativistic deformations in the universal base E1;
– hyperbolically orthogonal physical tensors of an impulse and an energy, proportional to the trigonometric dimensionless tensor of motions, by multiplication on the invariants mоc and mоc2;
– the Einsteinian dilation of time as consequence of the time-arrow hyperbolic rotation;
– the Lorentzian contraction of extent as consequence of the Euclidean subspace hyperbolic deformation (the latter is quasi-analogous to this subspace principal spherical rotation in the base E1!),
– a pair of additional relativistic effects associated with these main effects of STR;
– relativistic progressive physical motions of a material object or a particle, expressed in scalar, vector and tensor interpretations with an instant tensor hyperbolic angle Г and its 3 directional cosines, but with induced orthospherical rotations;
– the general laws of summing two or many velocities in STR with revealing the induced orthospherical rotation (!);
– the orthospherical angular shift and, in time, the Thomas precession, these orthospherical relativistic effects also as a consequence of an induced Coriolis acceleration (!) in P(3+1);
– the trigonometric descriptions of hyperbolic motions by time-like hyperbola, catenary and tractrix with radius-parameter R and current hyperbolic angle;
– the trigonometric description of pseudoscrewed motions by time-like pseudoscrew with radius-parameter R at constant hyperbolic angle;
– the tensor-trigonometric kinematic and dynamic in P(3+1);
– a nature of absolute matter’s motion at 4 pseudovelocity "c" along its world line with its proper momentum Po=moc and proper energy Eo=moc2, as a flow of its proper time at the same velocity in the Minkowskian space-time;
– the pseudo-Euclidean right triangle of three momenta: Po=moc (hypotenuse), P=mc and p=mv (cathetuses) with the pseudo-Euclidean Pythagorean theorem for moduli of them: (iРо)2=(iP)2+p2 ↔ p2=P2–Ро2;
– the tensor trigonometric interpretations for relativistic Doppler effects;
– the general trigonometric formulae for Star's light ray aberration;
– the complete differential geometry of world lines, accompanied by the unit y Minkowskian hyperboloids I and II in P(3+1):
4D pseudoanalog of 3D Frenet – Serret theory with rotated tetrahedron along 4D world lines,
2D relative Euclidean, pseudo-Euclidean and 3D absolute non-Euclidean (Rimannian and pseudo-Riemannian) two-step metric forms along 3D world lines,
4D absolute non-Euclidean three-step metric form for 4D world lines with its absolute local curvature.
5. Formal Complex Analysis.
– formal analyticity of nonholomorphic functions in complex conjugate arguments (one- and many-dimensional);
– formal differentiation and integration;
– formal power series and completeness of a differential.
6. Differential Analysis and Optimization Methods for 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 connected by some equations;
– analysis, procedure and interrelation of limit methods of conditional optimization with big and small parameters;
– the characteristic (secular) equation for conventional eigenvalues of a Hesse matrix;
– isoparametric polynomials (including the mirror ones), differential invariants of the order 2 and higher namely for flat curves;
– analytic optimization for real nonholomorphic functions in complex conjugate or mixed variables with the use of operations of the formal complex analysis;
– numerical optimization of the orders 0, 1 and 2;
– planned-calculational optimization of the 1st, incomplete and complete 2nd orders;
– planned-experimental optimization of the incomplete and complete 2nd order, or design of experiments on the factor space when searching for optimal conditions (in the case of a normal distribution of the random error of finding the function of response and at exact values of factors-variables), probabilistic estimating of a portion of the systematic error, the parameters of optimal plans;
– genesis of the optimization methods and their consecutive interrelation.
7. Mathematical Chemistry.
– mathematical modelling of chemical reactions, the general kinetic function;
– modelling of a reaction of one mono- and one polyfunctional monomer (with initially identical activity);
– kinetic curves as plots of functions and as functionals (including the ones of response); kinetic isotherms and isochrones, their theory and applications.
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Invitation book:
http://ninulas.narod.ru/gb/
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Pdf file of the book "Tensor Trigonometry. Theory and Applications.” – Moscow: MIR, 2004, 336p. (Russian book, format A5):
http://ninul-eng.narod.ru/NinulAS_Tensor_Trigonometry_M_Mir_2004_Rus-ebook.pdf
Pdf file of the book "Tensor Trigonometry.” – Moscow: Fizmatlit, 2021, 320p. (English ebook, format A4):
http://ninul-eng.narod.ru/NinulAS_Tensor_Trigonometry_M_FM_2021_Eng-ebook.pdf
Pdf file of the book "Tensor Trigonometry.” – Moscow: Fizmatlit, 2021, 320p. (English pbook, format B5):
http://ninul-eng.narod.ru/NinulAS_Tensor_Trigonometry_M_FM_2021_Eng-pbook.pdf
Pdf file of the book "Optimization of Objective Functions: Analysis. Numerical Methods. Design of Experiments.” – Moscow: Fizmatlit, 2004, 336p. (Russian book, format A5):
http://ninul-eng.narod.ru/NinulAS_Optimization_of_Functions_M_FM_2009-ebook.pdf
In e-library of MCCME – Moscow Center for continuous mathematical education:
The book Ninul A. S. "Tensor Trigonometry. Theory and Applications." – M.: MIR, 2004, 336p. (in Russian)
https://mccme.ru/free-books/init/ninul.pdf (since 17.04.2006)
In e-library of MSU’s Mech-Math Faculty in the same sequence of these books:
http://lib.mexmat.ru/books/24506 (since 28.08.2007)
http://lib.mexmat.ru/books/199397 (since 20.08.2021)
http://lib.mexmat.ru/books/77483 (since 26.11.2010)
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Small photoalbum:
http://ninulas.narod.ru/photoalbum.html
1. Messages for the author are reported in e-mails: NinulAS@yandex.ru / NinulAS2004@gmail.com
2. Discussions with the author when other readers can see such information are giving in the Invitation book with moderation.
All copyrights for the site and the books belong only to the author!
