Observer's mathematics applications to number theory, geometry, analysis, classical and quantum mechanics Текст научной статьи по специальности «Математика»

Том 153, кн. 3

УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА

Физико-математические пауки

2011

UDK 530.12

OBSERVER'S MATHEMATICS APPLICATIONS TO NUMBER THEORY, GEOMETRY, ANALYSIS, CLASSICAL AND QUANTUM MECHANICS

B. Khots, D. Khots

Abstract

When we consider and analyze physical events with the purpose of creating corresponding models, we often assume that the mathematical apparatus used in modeling is infallible. In particular. this relates to the use of infinity in various aspects and the use of Newton's definition of a limit in analysis. We believe that is where the main problem lies in contemporary study of nature. This work considers mathematical and physical aspects in a setting of arithmetic, algebra, geometry, and topology provided by Observer's Mathematics, see www.mathrelativity.com.

Key words: Hilbert, solit.on, wave, Sclirodinger, Lorent.z, Schwartz, observer.

Introduction

Today, when we see the classical definition of a limit of a sequence (sequence an approaches a limit b if for any arbitrarily small number e > 0 there is an integer N, such that |an — b| < e for all n > N), we feel somewhat uneasy: what does "arbitrarily small" really mean? Also, what does "sufficiently large" mean? This is because the answer depends on the point of view, depends on an observer, i.e.. has relativistic characteristics.

Consider, for example, geometry. When we speak about lines, planes, or geometrical bodies, we understand that all these objects exist only in our imagination: even if we grind a metal plate we would never get an ideal plane because of instrument and operation. Moreover, we would never reach an ideal plane shape because of the atomic structure of the metal, i.e., we are not able to approach this shape with an arbitrary accuracy. In order to avoid the use of infinity, David Hilbert had created geometrical bases practically without the use of continuity axioms: Archimedes and completeness.

We find similar problems occurring in arithmetic, and in entire mathematics, since it is "arithmetical" in nature.

Physics encounters such problems as well. It is known fact that the dynamics of some systems change when we change the scale (distances, energies) at which we probe it. For example, consider a fluid. At each distance scale, we need a different theory to describe its behavior:

1. At ~ 1 cm - classical continuum mechanics (Navier - Stokes equations);

2. At ~ 10-5 cm - theory of granular structures;

3. At ~ 10-8 cm - theory of atom (nucleus + electronic cloud);

4. At ~ 10-13 cm - nuclear physics (nucleons);

5. At ~ 10-13 — 10-18 cm - quantum chromodynamics (quarks);

6. At ~ 10-33 cm - string theory.

The mathematical apparatus that is applied here for physical data processing and building mathematical models does not contain any barriers, it is universal, omnivorous, and can manipulate with any numbers. This creates a possibility to produce an incorrect

output. Observer's mathematics was created as an attempt to do away with the concept of infinity.

Proof of all theorems stated below can be found in fl 9].

1. Observer's mathematics applications to number theory

1.1. Analogy of Fermat's last problem. This result was presented by authors at the International Congress of Mathematicians in Madrid in 200C.

To begin, we present a few notes. It is obvious that the classical Fermat's Last problem (for any integer m, m > 3, there do not exist positive integers a, b, c, such that am + bm = cm) may be reformulated not just for integers a, b, c, but for any real a, b, c

Note, in observer's mathematics the power operation is not always associative. For illustrative purposes, we give a W2 example. Consider 1.49 e W^^en 1.49 x21.49 = = 2.14 Mid 1.49 x2 2.14 = 3.16. On the other hand, 1.49 x2 3.16 = 4.67 Mid 2.14 x2 2.14 = 4.57, i.e., ((1.49 x2 1.49) x2 1.49) x2 1.49 = (1.49 x2 1.49) x2 (1.49 x2 1.49).

Theorem 1. For any integer n, n > 2, and for any integer m, m > 3, m e Wn there exist positive a, b, c e Wn, such that am +n bm = cm. Here xm means ((... (x xn x) xn ...) xn x)).

v-V-'

m

For example, if n = 2, we can calculate that 13 +2 13 = 1.283.

Note that the main reason of cardinal difference between standard mathematics and observer's mathematics results is the following. The negative solution of classical Fermat's problem requires the Axiom of Choice to be valid. But in observer's mathematics this axiom is invalid.

1.2. Analogy of Mersenne's and Fermat's numbers problems. Mersenne's numbers are defined as Mk = 2k — 1, with k = 1, 2,... The following question is still open: is every Mcrscriric's number sqnarc-frcc?

o f

Fermat's numbers are defined as Fk = 22 +1, k = 0,1, 2,... The following question is still open: is every Fermat's number sqnarc-frcc?

We begin with some comments. It is obvious that if some integer number is sqnarc-frcc in the set of all real integers, then this number is sqnarc-frcc in the set of all real rational numbers.

Theorem 2. There exist integers n, k > 2, Mersenne's numbers Mk, with {k, Mk} e Wn, and posit ive a e Wn, such th at Mk = a2.

Theorem 3. There exist integers n, k > 2, Fermat's numbers Fk, {k,Fk} e Wn, and positive a e Wn, such th at Fk = a2.

1.3. Analogy of Waring's problem. It is known (Lagrange) that the minimum number of squares to express all positive integers is four. What is the minimum number

k

problem in standard arithmetic.

Theorem 4. For any integer k, k > 2, there exist integer n, n > 2, (k e Wn) and some x e Wn, such that any equality of the form x = af + af + ... + ak is not possible for any integer l e Wn and any positive nu mbers ai, a2, ..., a; e Wn.

Note that for n = 2 and for any x e W2, x e [0,1], there do not exist more than four numbers a, b, c, d e W2, such th at x = ((a2 +2 b2) +2 c2) +2 d2.

Fig. 1. Nadezlida effect

1.4. Tenth Hilbert problem in observer's mathematics. Wo provide the following

Theorem 5. For any positive integers m, n, k G Wn, n G Wm, m > log10(1 + + (2 • 102n — 1)k), from the point of view of the Wm —observer, there is an algorithm that takes as input a multivariable polynomial f(x1,...,xk) of degree q in Wn and outputs YES or NO according to whether there exist a1,...,ak G Wn, such that f (a1,..., afc) = 0.

Therefore. Hilbert's tenth problem in observer's mathematics has positive solution. We think that Hilbert expected a positive answer for his tenth problem. Note that the main reason of cardinal difference between standard mathematics and observer's mathematics results is the following. The negative solution of the classical tenth problem requires the Axiom of Choice to be valid. But in observer's mathematics this Axiom is invalid.

2. Observer's Mathematics application to geometry: Nadezhda effect

In this section we consider an open square Q centered at the origin with sides of length 2 located on a plane Wn x Wn. We will calculate the distance D between the origin (0,0) and any point of Q as follows. D = p((0,0), (x, y)) = \]x1 + y2 = = ^Jx xn x +n y xn y, where yfa = b means b xnb = a, x, y G Q. i.e.. |x| < 1. |y| < 1

Fig. 1 below contains an illustration of the fact that for some points on Wn x Wn the concept of distance from the origin does not exist, while for others it does exist. The illustration below is for n =3 (Q c W3 x W3). Points with no distance to the origin are indicated by black, while points where distance from the origin exists are indicated in white.

This means that the distance D does not always exist, i.e., not every segment on

n

''black holes" and ''white cross" as the Nadezhda effect (see Fig. 1). This effect gives us

now possibilities for discovering physical processes and developing their mathematical models.

3. Observer's mathematics application to analysis and physics

In classical physics, it has been realized for centuries that the behavior of idealized vibrating media (such as waves on string, on a water surface, or in air), in the absence of friction or other dissipativo forces, can be modeled by a number of partial differential equations known collectively as dispersive equations. Model examples of such equations include the following:

• The free wave equation utt — c2Au = 0 where u : R x Rd ^ R represents the amplitude u(t, x) of a wave at a point in spacetime with d spatial dimensions,

d 52 §2u

A = V is the spatial Laplacian on Rd. utt is short for tt-t- . and c > 0 is a fixed j=i 5x2 5t2

constant.

h2

• The linear Schrodinger equation ihut + -—Au = Vu where u : R x Rd —► R is the

2m

wave function of a quantum particle, h,m > 0 are physical constants and V : Rd ^ R

x

The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory-arid practice, that once nonlinear effects are taken into account, solitary wave and soliton solutions can be created, which can be stable enough to persist indefinitely.

Wn

they "think" that they "live" in W and deal with W) a real function y of a real variable x, y = y(x), is called differentiable at x = x0 if there is a derivative

// n , • y(x) — y(x0)

y (x0) = hrn -.

x^xo,x=xo x — xo

Wm

m > n

|(y(x) -n y(x0)) -n (y'(x0) xn (x -n xq))| < 0.0_^_01

n

whenever

Iy{x) -n y{xo) \ = 0. 0 .. .0yi yl+1 . ,. yn 1

|(x -n Xq ) | = 0. 0 . . . O.Tfc Xk+l . .. xn k

for 1 < h, l < ^^d xk being non-zero digit. The following theorems have been proven:

Wm

Wn ( m > n)

Theorem 7. From the point of view of a Wm-observe r (with m > n) |y/(x0)| < < Chk, where Clnk e Wn is a constant defined only by n, l, k and not dependent y(x)

Theorem 8. From, the point of view of a Wm -observer, when a Wn -observer (with m > n > 3) calculates the second derivative

y(x3) ~ y{x 1) _ y(x2) - y(x0)

//, n v (x3 — xi) x2 — xo

y (Xa) = lim

xi^xo,xi=xo,x2^xo,x2=xo,x3^xi,x3=xi xi — xo

we get the following '¡inequality:

(|x2 -„ x0| xn |x3 -„ Xi|) xn \x! -„ x0| > 0.0_^01

n

y//(x0) = 0

3.1. Free wave equation. We consider the case when d = 1, i.e., u : Wn x Wn ^ ^ Wn, from Wm -observer point of view, with m > n, where Wn x Wn means Cartesian

Wn

Utt —n ((c xn c) xn Uxx) = 0.

Then we have the following Theorem 9. Let

c = co.ci . .. cf ck+i . .. cn

u x x = ±u 0 *u 1 . . . u x u x + 1 • • • un

with 2k < n, l < n, c0 = c1 = ... = ck = 0, ck+1 = 0, uxx = uxx = ... = ux° = 0 and u < k + l + 2, then utt = 0.

Next, we have the following

Theorem 10. If do > i^J), with 0 < p < n and uqx > with 0 < q < n

p q

and n < p + q, then there is no utt, such that utt = ((c xn c) xn uxx).

3.2. Schrodinger equation. Consider the following:

— (ft xn ft) xn *xx +n ((2 xn m) xn V) xn ^ = i((2 xn m) xn ft)^t,

where ^ = ^(x,t), ft is the Planck's constant, ft = 1.054571628(53) • 10-34 m2kg/s. Then we have the following

Theorem 11. Let 36 < n < 68, m = m0.m1 ... mkmk+1 ... mn, with m e Wn, mo = m1 = ... = mf = 0, mf+1 = 0, k + 35 < n, V = 0, then = ^0.^1......^0 = ...^t = 0, ^t+1,...,are free and

in {0,1,...,9}, where l = n — k — 36, i.e., is a random variable, with e

n

G {(0.0_^_0,A*.. .*)}, where. * G {0,1,..., 9}. l

Corollary 1. Let 36 < n < 68, m = m0.m1... mk mk+1 . ..mn, with m e Wn, m0 = m1 = ... = mk = 0, mk+1 = 0. Also, let V = u0.u1 ...usus+1 . ..un,

with V e Wn, uo = U1 = ... = Us = 0, us+1 = 0, with | +35 ,

1 k + s + 2 > n

iften = ^0.^1......and ^0 = ...^t = 0, ^t+V-,are free

and in {0,1,...,9}, where l = n — k — 36, i.e., is a random variable, with

n

G {(0. where. * G {0,1,..., 9}.

l

3.3. Two-slit interference. Quantum mechanics treats the motion of an electron. neutron or atom by writing down the Schrodinger equation:

h2 o2^ TrT

---r-r + V^ = lb—,

2m ox2 ot

where m is the particle mass and V is the external potential acting on the particle. As these particles pass through the two slits of any of the experiments they are moving

V=0

Now. consider the following:

— (h Xn h) Xn txx +n ((2 Xn m) Xn V) Xn ^ = i((2 Xn m) Xn h)^t,

where ^ = ^(x,t), h is the Planck's constant, h = 1.054571628(53) • 10-34 m2kg/s. Then we have the following

Theorem 12. Let 36 < n < 68, m = m0 .m1... mk mk+1 . ..mn, with m G Wn, m0 = m1 = ... = mk = 0, mk+1 = 0, k + 35 < n, V = 0, then

= ^0.^1... ^t+1... in {0,1,..., 9}, where l = n — k — 36, i.e.

and ^0 = ... ^t = 0,

¡+1

, ^n are free an<i

^t is a random variable, with e

{(0.0_^_0*.. .*)}, where * G {0,1,..., 9}.

i

The wave at the point of combination will be the sum of those from each slit. If is the wave from slit 1 and is the wave from slit 2, then ^ = + . The result gives the predicted interference pattern. Then by Theorem 1, we have

1t = *?t.*1t... *it*i+1... *nt,

where ^

¡1 + 1

*2t = *0t^1t ... *2t*2!1 ...

1t

0,

, ^nt are free and in {0,1,..., 9}, and

2t

2t

¡2 + 1

, ^nt are free anfl in {0,1,..., 9} where l = n — k — 36.

where ^22t

Now we have the following

Theorem 13. 1. // 1 + ^2+1 > 9, then + is not a wave.

2. If 1 + 1 < 9, then + ^2 is a wave.

3. If 1 + 1 = 9, then + may or may not be a, wave.

3.4. Lorentz transform. Let K and K' be two inertial coordinate systems with x-axis and x'-axis permanently coinciding. We consider only events which are localized on the x(x')-axes. Any such event is represented with respect to the coordinate system K by the abscissa x and the time t, and with respect to the system k' by the abscissa x' and the time t' when x and t are given. A light signal, which is proceeding along the positive x—axis, is transmitted according to the equation x = c Xn t or x —n c Xn t = 0 Since the same light signal has to be transmitted relative to k' with the velocity c, the propagation relative to the system k' will be represented by the analogous equation

C n c X n t — 0.

Those space-time points (events) which satisfy the first equation must also satisfy the second equation. Obviously there will be the case when the relation Ai xn(x' —ncxnt') = = Mi xn (x n c xn t) is fulfilled in general, where Ai,Mi G W„, |Ai|, |mi| > 1 are constants; for, according to the last equation, the disappearance of (x —n c xn t) involves the disappearance of (x' —n c xn t').

Note that classical equation x' — ct' = A(x — ct) is not valid since if A < 1, x — ct = = O.th^Ol, then A xn (x —n c x t) = x' —n c xnt' = 0. If we apply quito similar

n — 1

x

obtain the condition A2 xn (x' +n c xn t') = xn (x +n c xn t) with A2,m2 G Wn,

|A2|, M > 1.

3.5. Schwarzian derivative. The Schwarzian derivative S(f (x)) is defined as

f '''(x) 3 (f ''(x)\2

S(f(x)) = '---( ' I Here /(x) is a function in one real variable and

f '(x) 2\ f '(x)7

f'(x), f''(x), f'''(x) are its derivatives. The Schwarzian derivative is ubiquitous and tends to appear in seemingly unrelated fields of mathematics including classical complex analysis, differential equations, and one-dimensional analysis, as well as more recently, Teichrm'illor Theory, integrable systems, and conformal field theory. For example, let's consider the Lorentz plane with the metric g = dxdy and a curve y = f (x). If f '(x) > 0, then its Lorentz curvature can be easily computed via p(x) = f ''(x)(f'(x))-3/2 and the

S (f)

Schwarzian enters the game when one computes p' = . Tims, informally speaking,

vf'

the Schwarzian derivative is enrvatnro.

Consider now the Schwarzian enrvatnro from observer's mathematics point of view. Now we have the following

S(f(x))

• S(f (x)) is a random variable;

• |S(f(x)| < 101-fc+i, where

(2 xn (/'(x) xn /'(x))) = O.O^Oo^oí+i . ..an

i

with ai =0 and

(2 x„ (/"'(x) x„ /'(x))) -„ (3 x„ (/"(x) x„ /"(x))) = ±0.0_^06fc6fc+1 ... bn

k

with bk = 0 and 1 < l, k < n.

Резюме

Б, Хоц, Д. Хоц, Применение математики наблюдателя к теории чисел, геометрии, анализу, классической и квантовой механике.

При рассмотрении и анализе физических событий с целыо создания соответствующих моделей мы часто предполагаем, что математический аппарат, используемый в моделировании. непогрешим. В частности, это касается использования бесконечности в различных аспектах и применения ньютоновского определения предела в анализе. Мы считаем, что имешго в этом заключается основная проблема в современном изучении природы. В па-стоящей работе рассматриваются математические и физические аспекты арифметики, алгебры, геометрии и топологии математики наблюдателя (см. www.mathrelativity.com).

Ключевые слова: Гильберт, солитоп. волна, Шредипгер. Лоренц. Шварц, наблюдатель.

References

1. Khots B., Khots D. Mathematics of Relativity. - 2004. - URL: www.mathrelativity.com.

2. Khots B., Khots D. An Introduction to Mathematics of Relativity // Lecture Notes in Theoretical and Mathematical Physics / Ed. A.V. Aminova. - Kazana: Kazan State University, 2006. - V. 7. - P. 269-306.

3. Khots B., Khots D. Observer's Mathematics - Mathematics of Relativity // Appl. Math. Comput. - 2007. - V. 187, No 1. - P. 228-238.

4. Khots D., Khots B. Quantum Theory and Observer's Mathematics // AIP Conf. Proc. -2007. - V. 962. - P. 261-264.

5. Khots B., Khots D. Analogy of Fermat's last problem in Observer's Mathematics - Mathematics of Relativity // Talk at the International Congress of Mathematicians. - Madrid, Spain, 2006. - URL: http://icm2006.org/v_f/AbsDef/Shorts/abs_0358.pdf.

6. Khots B., Khots D. Analogy of Hilbert's tenth problem in Observer's Mathematics // Talk at the International Congress of Mathematicians. - Hyderabad, India, 2010. - URL: http://www.icm2010.in/wp-content/icmfiles/abstracts/Contributed-Abstracts-5July2010.pdf.

7. Khots B., Khots D. Non-Euclidean Geometry in Observer's Mathematics // Acta Physica Debrecina. - 2008. - T. XLII, - P. 112-119.

8. Khots B., Khots D. Quantum Theory from Observer's Mathematics point of view // AIP Conf. Proc. - 2010. - V. 1232. - P. 294-298.

9. Khots B., Khots D. Solitary Waves and Dispersive Equations from Observer's Mathematics point of view // Geometry "in large", topology and applications. - Kharkov, Ukraine, 2010. - P. 86-95.

Поступила в редакцию 18.12.10

Khots, Boris Doctor of Science, Professor, Compressor Controls Corporation, Des Moines, IA, USA, Application Engineering Leader.

Хоц Борис Соломонович доктор математических паук, начальник отдела проектирования, компания "Compressor Controls Corporation", г. Де-Мойп, шт. Айова, США.

E-mail: bkhotsQuuylobal.com

Khots, Dmitriy PliD, Professor, West Corporation, Omaha, NE, USA, Director, Analytic Insight.

Хоц Дмитрий Борисович кандидат математических паук, директор по аналитике, компания "West. Corporation", г. Омаха, шт. Небраска, США.

E-mail: dkhotsecox.net