Extracto
2
Für Aaron und alle unsere Lieben.
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Table of Contents
PART A: Mathematics ... 4
I. Basic Structure ... 4
II. Number Systems ... 5
III. Functions ... 20
IV. Functionals ... 21
V. Cardinal Arithmetic ... 23
VI. Measure Theory ... 25
VII. Functional Analysis ... 28
VIII. The Structure of Functions and Functionals ... 32
IX. Stochastic Analysis ... 35
X. Atiyah-Singer Index Theorem ... 40
XI. Net Theory ... 45
PART B: Physics ... 56
XII. Definition of Elementary Physical Structures ... 56
XIII. Derivation of Composed Physical Structures ... 60
XIV. Physical Measurements and Conservation Laws ... 71
XV. Special Relativity and Quantum Mechanics ... 77
XVI. Complementary Coordinates ... 82
XVII. The Structure of Information and Causality Spaces ... 86
XVIII. Inner and Outer Structures of Vector Spaces ... 91
XIX. Standard Model of Particle Physics and Cosmology ... 105
XX. Standard Model of Physics ... 128
XXI. Observations and Experiments... 168
Bibliography ... 169
4
PART A: Mathematics
I. Basic Structure
0 Sets
"Under a set we shall understand an aggregation of distinguishable elements
which are merged to a whole. (Georg Cantor, 1895)."
Hence, we have:
= {
,
}
The elements of are distinguishable but not ordered.
Two sets are equal if they have the same elements [1].
The building of a set proceeds in two steps. First, we build a multeity
,
, an
accumulation, a range. Then, this multeity is merged to a unity, to a set [1].
Following definitions are useful.
:
= {}
The empty set does not comprise any element.
: ( ) = { }
The power set is the set of all subsets of . If the set has elements, then the
power set has
2 elements.
The set shall be called infinite, if a true subset
exists which can be
mapped in a bijective manner to ; hence, if there is a
with | | = | | [1].
A set shall be called finite, if it is not infinite [1].
Two sets
are of equal cardinality | | = | | if they can be mapped in a
bijective manner to each other. The power set of a set is always of bigger cardinality
that the set itself [1].
5
II. Number Systems
1 The Natural Numbers: The Set
The natural numbers are formed from the set . The idea is that all elements of the
set shall have an ordering, giving each element of a "name"/a "number". The
natural numbers define a set
in which one element 0 is distinguished (the first
element); and a self-mapping
: (
) exists, such that the
following axioms apply [2]:
( 1)
( 2)0 ()
( 3)
0
-
, =
The set
starts with the point 0 and proceeds with the successor, with its successor,
and so forth. We can therefore characterize the natural numbers with an index:
:
:
:
= ( + 1) - ( )
This index exists for each natural number, since the difference between a successor
and its predecessor is always
1. But this index does not exist for the elements of the
set , since said elements are not ordered, such that no subtraction can be carried
out. Hence, this index is a characteristic of the natural numbers.
The successor function can be applied in an infinite manner to the natural numbers.
The limit number of
which cannot be reached by a finite number of steps shall be
called . The cardinality of
shall be called
[1].
The successor function defines one direction, namely from an element towards its
successor. Therefore, the natural numbers define a structure with a single direction,
hence an open structure.
6
2 The Integers: The Set
The natural numbers allow an addition in every case, but do not allow the inverse
operation, the subtraction, in any case. In order to remove this disadvantage, the
natural numbers are extended to the integers. Every integer can be defined as the
difference
( - ) of two natural numbers , [2]. The integer ( - ) shall be written
as a pair (
, ) [2]. We consider the following relation on × : ( , )~( , ) +
= + ( ( - = - ) [2].
The integers are defined as equivalence classes of the relation
~ [2]. The integers
set up a commutative group with respect to the addition [2]. The integers set up an
integrity ring (commutative, zero division free ring with one-element) [2]. Every
integer
( , ) possesses an inverse integer ( , ) [2]. The integers define a set [2].
We can therefore characterize the integers with an index:
:
:
:
= ( - ) - ( - )
,
This index exists for each integer
( - ), since we can always build an inverse
integer
( - ). But this index does not exist for the elements of the set , since said
elements do not allow negative numbers, such that the index cannot be always
computed. Hence, this index is a characteristic of the integers.
The cardinality of
is the same like the cardinality of [1]. Hence, has also the
cardinality
. The integers possess, like the natural numbers, one direction given by
the addition operation of the successor function. The integers possess, contrary to
the natural numbers, also the opposite direction given by the subtraction operation
which can be always computed in
but not in . Hence, the integers define a
structure with two opposite directions, hence a closed structure.
The integers (and also the natural numbers) possess the following ordering type.
:
/
:
, <
,
( , )
, <.
( , )
, < :
= sup( )
= inf( )
< .
is then the successor of in
, <, written = + 1. is then the predecessor of
in
, <, written = - 1. Figure 1 shows the structure of a step/jump.
Figure 1: Step/Jump of a Linear Ordering
7
3 The Rational Numbers: The Set
The integers allow an addition, a subtraction, and a multiplication in every case, but
do not allow a division in any case. In order to remove this disadvantage, the
integers are extended to the rational numbers. Every rational number can be
described as a quotient of two integers
, [2]. The rational number shall be
written as a pair (
, ) [2]. We consider the following relation on
× \{0}: ( , )~( , ) = ( = ) [2].
The rational numbers are defined as equivalence classes of the relation
~ [2]. By
using the relation
: , () the set is mapped in an isomorph
manner to the subring
() [2]. The field is the smallest field containing as a
subring [2]. Every rational number
( , ) possesses an inverse rational number ( , )
[2]. We can therefore characterize the rational numbers with an index:
:
:
:
= -
,
This index exists for each rational number , since we can always build an inverse
rational number . But this index does not exist for the elements of the set
, since
said elements do not allow floating point numbers, such that the index cannot be
always computed. Hence, this index is a characteristic of the rational numbers.
The cardinality of
is the same like the cardinality of [1]. Hence, has also the
cardinality
.
The rational numbers possess the following ordering type [1].
:
:
, <
:
( ) , <
,
( ) , <
,
( ) , <
.
Gaps of a linear ordering: Shall
, <define a linear ordering, and shall ( , ) define
a cut in
, <. ( , ) is named a gap in , < if sup( ) does not exist. Figure 2
shows the structure of a gap.
Figure 2: Gap of a Linear Ordering
8
4 The Real Numbers: The Set
The rational numbers allow an addition, a subtraction, a multiplication, a division, and
an exponentiation in every case, but do not allow the root extraction and the
extraction of logarithms in any case. Also transcendental numbers like or cannot
be defined by rational numbers. In order to remove these disadvantages, the rational
numbers are extended to the real numbers. This extension is performed by Dedekind
cuts [2].
A Dedekind cut is [1]: Shall
, < define a linear ordering. A Dedekind cut in , <
is a pair
,
, with the following properties:
( ) , , = , =
( )
,
<
( )
( ) Sup( )
The supremum
( ) exists for each rational number. The supremum
( ) does
not exist for irrational numbers. Here, we have a gap which is closed by the
Dedekind cut. Otherwise stated, the Dedekind cuts close all gaps of
and lead to
the set
which is complete [2]. We can define the set in the following manner [1]:
Every non-empty upper bound subset of
possesses a supremum (smallest
upper bound); there is hence a
:
(1)
(2)
,
is called the supremum of . We have therefore
=
( ). We can therefore
characterize the real numbers with an index:
:
:
:
=
( )
,
-
,
(
)
This index exists for each real number, since we can always define a supremum with
a Dedekind cut. But this index does not exist for the elements of the set
, since said
elements do not always possess a supremum, such that the index cannot be always
computed. Hence, this index is a characteristic of the real numbers.
The cardinality of
is greater than the cardinality of [1]. As proved in [1], the
cardinality of
is the power set of the cardinality of ; hence the cardinality of is
= 2
. Hence,
has the cardinality .
The real numbers possess the following ordering type [1].
9
:
:
, <
.
:
( ) , <
,
( ) , <
,
( ) , <
.
Separable means: Between two real numbers there is always a rational number.
Hence, the real numbers possess a dense subset. For a dense subset we have:
Shall
. is named dense in , if: for all , with < there is a
with
< < . Figure 3 shows the structure of a continuity.
Figure 3: Continuity of a Linear Ordering
10
5 The Complex Numbers: The Set
The real numbers allow an addition, a subtraction, a multiplication, a division, an
exponentiation, a root extraction and an extraction of logarithms in any case, but do
not always lead to a solution of algebraic equations. Indeed, when said solution
demands the extraction of a square root from a negative real number, then the real
numbers do not allow a solution. In order to remove this disadvantage, the real
numbers are extended to the complex numbers. This extension is performed in the
following manner [2].
The set
× of all ordered real number pairs ( , ) defines by the natural
addition (
, ) + ( , ) (
+
,
+
) an Abel group. We define a
multiplication in
× in the following manner [1]:
( , ) ( , ) (
-
,
+
)
(1,0) is the one-element. (0,1) is the imaginary unit with = -1 [2]. We
have then:
(
+
) ( + ) ((
-
) + (
+
))
=
is called the essential real unit.
=
is called the essential complex unit.
Hence, we can write any complex number as
=
+
,
The set
is not ordered. We can define [2] a conjugation × ,
with
= +
and with
= - as an automorphism which maps in itself, and which
maps into the second, in principal equal, zero point
- of
+ 1 = 0. We can
therefore characterize the complex numbers with an index:
:
:
:
= ( ) = -
This index exists for each complex number including the essential complex unit,
since we can always build a complex conjugate for any such complex number. But
this index does not exist for the elements of the set
, since said elements do not
possess an imaginary unit, such that the index cannot be computed. Hence, this
index is a characteristic of the complex numbers.
The cardinality of
is the same like the cardinality of [1]. Hence, has also the
cardinality
.
11
6 The Quaternions: The Set
The complex numbers allow the solution of any algebraic equation by introducing
one imaginary unit. The question arises, if one can generalise the complex numbers
by introducing further imaginary units. The first generalisation leads to the
quaternions [2]. We shall define the algebra
of quaternions as: In the four
dimensional
vector space the standard basis is defined by the ordered real
quadruples:
= (1,0,0,0);
= (0,1,0,0);
= (0,0,1,0);
= (0,0,0,1). is the one-
element. The nine products
, = 1,2,3 are defined by the following
multiplication table of Figure 4.
Figure 4: Multiplication Table of Quaternions
We have:
=
=
=
= -1. We also have
=
.
is called the essential quaternion unit. Hence, we can write any quaternion as
=
+
+
+
, , ,
12
The Hamiltonian algebra
is an associative division algebra [2]. We have
=
- [2]. We can therefore characterize the quaternions with an index:
:
:
:
= ( , ) = -
,
This index exists and is not zero for each quaternion including at least two complex
units, since said quaternions do not commute. But this index is always zero for the
elements of the set
, since said elements always commute. Hence, this index is a
characteristic of the quaternions.
The cardinality of
is the same like the cardinality of [1]. Hence, has also the
cardinality
.
13
7 The Octonions: The Set
The quaternions are the first generalization of the complex numbers. The second
generalization leads to the octonions [2]. For elements of
× we define a product
[2]:
, ( , ) ( , )
-
,
+
,
The set defines an alternative division algebra [2]. The 49 products
, = 1 7 are defined by the following multiplication table of Figure 5.
is called the essential octonion unit. Hence, we can write any octonion as
=
+
+
+
+
+
+
+
, , , , , , ,
We have 7 quaternionic groups in the octonions:
( , , ), ( , , ), ( , , ), ( , , ), ( , , ), ( , , ), ( , , ).
Figure 5: Multiplication Table of Octonions
14
We have:
=
( )
= -1. The alternative algebra
implies [2]:
: ( ) = ( )
: ( ) = ( )
We have
( ) = -( ) [2]. We can therefore characterize the
octonions with an index:
: :
:
= ( , , ) = ( ) - ( )
, ,
This index exists and is not zero for each octonion including units from different
quaternionic groups, since said octonions do not associate. But this index is always
zero for the elements of the set
, since said elements always associate. Hence, this
index is a characteristic of the octonions.
The four algebras
, , ,
are the only division algebras; hence algebras
without zero divisors [2].
The cardinality of is the same like the cardinality of
[1]. Hence, has also the
cardinality
.
15
8 The Sedenions: The Set
The generalization of the octonions leads to the sedenions. The set of the
sedenions is built by the product of
× by [3]:
, ( , ) ( , )
-
,
+
,
The 225 products
, = 1 15 are defined by the following multiplication
table depicted in Figure 6 [3].
is called the essential sedenion unit. Hence, we can write any sedenion as
=
+
+
+
+
+
+
+ +
+
+
+
+
+
+
+
, , , , , , , , , , , , , , ,
We have octonionic groups in the sedenions.
Figure 6: Multiplication Table of Sedenions
The multiplication of sedenions is not commutative, not associative, and not
alternative [2]. Said multiplication is only power associative,
= ( ) ( ) , and
flexible, (
, , ) = ( ) - ( ) = 0 [2]. The multiplication of sedenions
possesses zero divisors depicted in Figure 7 [3].
16
Figure 7: The 84 Zero Divisors of the Sedenions
Sedenions set up a non-commutative Jordan algebra [2]. It is proven that the space
of pairs of norm-one sedenions that multiply to zero is homeomorphic to the compact
17
form of the exceptional Lie group [3]. We can therefore characterize the
sedenions with an index:
: :
: = ( , , ) = ( ) - ( )
,
This index exists and is not zero for each sedenion including units from different
octonionic groups, since said sedenions are not alternative. But this index is always
zero for the elements of the set , since said elements are always alternative.
Hence, this index is a characteristic of the sedenions.
Remark 1: If
= 0 such that ( ) = 0, for , , hence if is a zero
divisor, we have
0, since is never a zero divisor according to Figure 7. Then,
we also have ( )
0, since ( ) is also never a zero divisor according to
Figure 7. Hence, the
: is always existing and not zero for every
sedenion including units from different octonionic groups, even if
is a zero
divisor.
The cardinality of is the same like the cardinality of
[1]. Hence, has also the
cardinality
.
If we further generalize the sedenions to higher dimensional complex numbers, no
new algebraic structures occur [2]. We can finally define an index
:
which is zero for all numbers:
:
= ( , , ) = ( ) - ( ) = 0
, , , , , , , ,
18
9 Summary of the Structures
, , , , , ,
The structures of the sets
, , , , , ,
are summarized in Figure 8 below.
Figure 8: The structures of the sets
, , , , , ,
The form, ordering type, and operator define three independent features, in the
sense that they can be combined independently from one another. Hence, we can
use sedenions with rational coefficients which are always positive.
The elements of a single feature cannot be, however, combined. Indeed, we cannot
combine an open with a closed form, since they are excluding each other. Once we
have selected the form, its index (counter
, orientator
) is decided upon and
cannot be changed.
Also, we cannot combine a discrete, a dense, and a continuous ordering type with
one another, since they are also excluding each other. Once we have selected the
ordering type, its index (orientator
, interleaver
, complementator
) is decided
upon and cannot be changed.
Finally, once we have selected how many dimensions a number has, the operators
leading to its index (complementator
, conjugator
, commutator
, associator
, alternator ) are decided upon and cannot be changed.
Hence, we can derive 30 different number types from
, , , , , ,
, see
Table 1.
19
Number Description
Natural Number
Integer
Positive Rational Number
Rational Number
Positive Real Number
Real Number
Complex Number with Natural
Coefficients
Complex Number with Integer
Coefficients
Complex Number with Positive Rational
Coefficients
Complex Number with Rational
Coefficients
Complex Number with Positive
Coefficients
Complex Number
Quaternion with Natural Coefficients
Quaternion with Integer Coefficients
Quaternion with Positive Rational
Coefficients
Quaternion with Rational Coefficients
Quaternion with Positive Coefficients
Quaternion
Octonion with Natural Coefficients
Octonion with Integer Coefficients
Octonion with Positive Rational
Coefficients
Octonion with Rational Coefficients
Octonion with Positive Coefficients
Octonion
Sedenion with Natural Coefficients
Sedenion with Integer Coefficients
Sedenion with Positive Rational
Coefficients
Sedenion with Rational Coefficients
Sedenion with Positive Coefficients
Sedenion
Table 1: 30 Different Number Types
20
III. Functions
10 The Functions and the Set
We define a function in the following manner [1]:
.
,
, ,
: ( , )
( , )
:
=
(
)
, ,
, , , , , , , ,
.
Functions are therefore single valued relations. A relation is hence a function
exactly when for each
( ) there is exactly one with ( , ) (
( )).
We write then: ( )
=
( , ) .
Let and define sets. The cardinality of is smaller or equal than the cardinality of
, | |
| |, if [1]:
:
( )
:
( )
:
We further have:
| | = | |:
:
: : :
:
;
:
;
:
A real function is a
: . The set of all functions shall be named . We have [1]:
|| < | |
|| = | ()|
| | = | ()|
Hence, the cardinality of is greater than the cardinality of
[1]. As proved in [1],
the cardinality of is the power set of the cardinality of
; hence the cardinality of
is
= 2
. Hence, has the cardinality
.
21
IV. Functionals
11 The Functionals and the Set
A continuous linear mapping between normed spaces
, is named continuous
operator [4]. If the range of the mapping is the scalar field, we call the mapping a
functional instead of operator.
/
: :
The spaces
, are function spaces ; hence, their elements are functions. Hence,
the continuous operators/functionals are functions operating on functions. The space
( , )
= = of the continuous linear functionals of a normed space
is called the dual space of , is forming the set of all functionals, and is called .
We have [4]:
|| =
|| = | ()| = = 2
| | = | ()| = = 2
| | = | ( )| = = 2
Hence, the cardinality of is greater than the cardinality of [1]. As proved in [1],
the cardinality of is the power set of the cardinality of ; hence the cardinality of
is
= 2
. Hence, has the cardinality
.
A functional can be written as [4]:
( ) =
( ) ( )
If we further generalize the mapping process, such that we define
operators/functionals operating between functionals of the spaces , then we still
have the principle of using operators/functionals operating on functionals defined by
a combination of functions; and this does not add something significantly different to
the principle of functionals.
Otherwise stated, we begin with defining functions by operating on numbers, and
end with defining operators/functionals by operating on functions. Hence, the sets
and complement the sets of the number systems
, , , , , , , . Now we
therefore have the relevant set sequence:
, , , , , , , , , .
This set sequence has four cardinalities, see Table 2.
22
Set
Cardinality
Table 2: The four cardinalities of the numbers, functions, and functionals.
By the mentioned generalization of the mapping process, we can easily achieve a
higher cardinality
with each generalization step , although said generalization
does not add something significantly different to the principle of functionals. Hence,
we can define a limit cardinality
which is never reached by applying said
generalization steps in a finite manner. The set comprising all said generalization
steps including the limit cardinality
shall be called
. Hence, we generalize Table
2 including
to Table 3.
Set
Cardinality
Table 3: The five cardinalities of the numbers, functions, functionals, and the limit
cardinality
. This gives then the set sequence:
, , , , , , , , , ,
.
23
V. Cardinal Arithmetic
12 Cofinalities, Covering Numbers, and the Index
Until now, we have operated with numbers, functions, functionals/operators, and
have defined a limit cardinality
. Now, we shall define some structures of the
infinite cardinalities.
First, we define the cofinality, , of a linear ordering [1].
, <
,
.
, <,
.
, <,
( , <)
:
( , <) =
{| || ,
, <}.
The cofinality of a linear ordering is hence the minimal number of steps which we
need to reach the end of the linear ordering [1].
Next we define filters and ultrafilters [1].
-
,
( ).
,
,
, :
( ) ,
( )
, ,
( ) , , .
,
:
( ) - .
Now, we can use the cofinality principle to derive covering numbers for infinite
cardinalities [16].
( , , , )
:
,
< ,
&| | < (
)
&| | < &
.
We always assume
, , > 1, > 1, [
= &
< ]
so that
( , , , ) is well defined [16].
The covering number can then be used to define a pseudo power,
, of infinite
cardinalities [16]:
( ) =
( , , , 2)
>
( ) =
< & ( ) =
( ) =
In this way we know the structure of power operations with infinite sets.
24
Knowing the structure of power operations, we can now proceed to
mentioned in
IV.11 as an infinite step generalization of functionals.
, =
()
()
,
() <
()
( ) <
() | ()|
This formula is proved in [16]. This formula is further refined in [16] to:
()
()
<
() | ()|
Setting
= () + (), this formula above leads to Shelah's famous formula [16]:
( ) < (
)
Hence, even in the case where the functionals are generalized to
and power
operations on
are considered, there is a limit cardinality covering ( ), namely
. The set comprising the set
including the covering of
by
shall be called
. Hence, we generalize Table 3 including
to Table 4.
Set
Cardinality
Table 4: The six cardinalities of the numbers, functions, functionals, the limit
cardinality
, and its covering
. This gives then the set sequence:
, , , , , , , , , ,
,
.
The power formula can be refined in the following manner according to [16].
<
,
( ) <
Hence, if, for example,
< & ( ) =
( ) < . This special case is of
great importance, since the cardinalities
, , ,
are the cardinalities of
the natural numbers, the real numbers, the functions, and the functionals,
respectively, which play a crucial role for finite and infinite structures at the same
time. Thus we have proved the following Theorem 1.
Theorem 1: The cardinality
defines an index characterizing the finite structures
of the natural numbers, the real numbers, the functions, and the functionals,
including their infinite extensions with power operations.
25
VI. Measure Theory
13 Algebras and Measures
Until now, we have defined sets with different mathematical properties and with
different cardinalities. Now, we shall refine these properties by introducing measures
on the sets. First we define a algebra [17]:
:
,
:
(i)
( )
( )
( )
:
Next, we define a pre-measure [17].
-
:
,
[0, +].
-
() = 0
( )
:
=
( )
(
) =
( ) is called content.
Every pre-measure defined on a algebra of a set
is named measure on
[17].
The pre-measure , defined on the ring
of -dimensional figures in is called
Lebesgue-pre-measure in
and is written [17].
The elements of the algebra generated by the system
of half-open intervals in
are called Borel-sets of the space [17].
Every countable subset of
is a Lebesgue-Borel-null-set [17].
A real function defined on
is named -elementary function (or not negative step
function) if said function is not negative, is -measureable, and takes only a finite
number of values [17].
We define an integral in the following manner [17].
26
-
:
.
=
1
.
=
( )
.
-
(
)
A numerical function on
is called -integrable if said function is -measurable
and if the integrals
and
define real numbers [17]. We have [17]:
=
-
Shall define a property, such that for every point
it is defined if possesses
said property or not. We shall say " -almost points
possess the property " or
" is valid -almost everywhere on
", if there is a -null set such that all points
N possess the property [17].
A null set is defined as [17]:
-
-
:
andN
and(N) = 0
Shall (
, ) define a measure space, and shall and define measures on . We
write
if is -continuous [17]. Hence, if the second measure is continuous
with respect to the first measure , we write
.
We call singular with respect to (or -singular) and write
if a null set
exists with ( )
= 0 and ( ) = 0 [17]. Hence, if the first measure is zero with
respect to , and the second measure is zero with respect to
, we write .
We immediately see that the relation
is symmetric in and . The sets and
interchange their roles during the transition . Therefore, the measures and
on with
are called singular with respect to each other, or mutual singular
[17].
We have in general [17]:
( ) = ( ) + ( )
: ( ) = ( )
( ) = 0
In this case, the measure is carried by a -null set.
27
The unit measure or Dirac-measure is defined as [17]:
( ) = 1,
0,
The terms -continuity and -singularity are opposite to each other. For the
Lebesgue-Borel-measure and for every Dirac-measure on
we have
( ) [17].
28
VII. Functional Analysis
14 Spaces and Dual Spaces
Until now, we have introduced measures on sets. Now, we shall define measures on
spaces on which functions and functionals are operating.
We will consider vector spaces of the fields
= and = . We will call said
vector spaces vector spaces. We first define a semi-norm and a norm [4].
-
.
: [0, )
-
:
( ) ( , ) = | | ( ) , ,
( ) ( + ) ( ) + ( ) ,
. ( )
:
( ) ( ) = 0 = 0,
( )
.
Norms will also be depicted by
. . The pair ( , ) is called a semi-normed or
eventually a normed space. A normed space (
, . ) possesses a natural metric [4]:
( , ) = - ,
A sequence ( ) of elements of a (semi-) normed space is called a Cauchy-
sequence if [4]
> 0 ( ) ,
( )
-
<
A metric space in which every Cauchy-sequence converges is called complete. A
complete normed space is called Banach-space [4].
The following three norms are of big importance:
=
| |,
=
| | ,
/
,
= max
,...,
| |.
The complete semi-normed vector space
( ) can be defined by using the
Lebesgue integral defined in the previous chapter 13. The interval
can be
open, semi-open, or closed, limited or unlimited. is the Lebesgue measure. For
1 we define [4]:
29
( ) =
: :
,
| |
< ,
=
| |
/
( )
The space
( ) is, as already said, a complete semi-normed vector space. The
limits in this space are not uniquely determined; they are determined only modulo the
elements of the core of the semi-norm
.
, namely
= { : = 0
} [4]. It is therefore obvious to identify functions
which coincide almost everywhere. The appropriate mathematical procedure is
hence to replace functions with their equivalence classes [ ], in the corresponding
quotient vector space
( ) = ( )/
We have then the following Lemma [4]:
( , .
)
-
.
( ) = { :
= 0}
-
.
( )[ ] =
.
( )
,
.
Normed spaces have the following properties [4]:
:
( ) dim < ,
( )
= { : 1}
;
( )
.
Hence, normed spaces have a finite dimensionality!
A metric (or topological) space is called separable, if said space possesses a
countable dense subset [4]. is called dense in when
= . Hence, the set is
dense when each nonempty open set contains a point of . Every point in is limit
of a sequence of [4].
In [4] it is proved that:
:
[ , ]
[ , ], ,
( [ , ], . )
As already stated in IV.11 a continuous linear mapping between normed spaces
,
is named continuous operator [4]. If the range of the mapping is the scalar field, we
call the mapping a functional instead of operator.
30
/
: :
A continuous operator
: fulfills hence one of the equivalent conditions [4]:
( ) lim
= , lim
=
( )
> 0
> 0
-
-
( )
,
( ) = { :
}
Next we define an operator norm [4].
( ) =
( , ),
.
( )
,
( , )
,
.
-
,
-
,
( , );
= .
:
= .
In the next step dual spaces are defined [4].
( , )
Hence, if the space is defined by vectors in
, then the dual space has the
cardinality
. If the space is defined by vectors containing functions, then the dual
space has the cardinality
.
Hilbert spaces are Banach spaces with a scalar product as an additional structure
[4].
.
. , . : ×
(
) :
( ) + , = , + , ,
( ) , = , , ,
( ) , = , ,
( ) , 0
( ) , = 0 = 0
In Hilbert spaces we have the Cauchy-Schwartz-inequality [4]:
-
-
:
. , . .
:| , | , , , .
.
For a Hilbert space with infinite dimensions we have [4]:
31
:
( )
( )
( )
A resolvent set and a spectrum are [4]:
:
( ) ( ) = { :( - )
( )}
( )
: ( ) ( ),
: =
( ): = ( - )
( )
:
( ) =
( )
The following structure of infinite dimensional spaces is useful [4]:
( ). :
( )
,
0 ( ).
( ) (
)
( ) {0}
.
( )
( ) {0}
,
ker( - )
.
,
= ( ) ( )
( ) ( ),
( ) ( )
( )
ker( - )
( - )
( ) ( ).
( ) ( )
0
32
VIII. The Structure of Functions and Functionals
15 Quarkonial Representation
Until now, the structure of spaces has been described. Now, we shall refine this
description by structuring the function spaces. We ask for quarkonial representations
and Taylor series of functions, distributions/functionals/operators, and function
spaces. (Distributions are a class of linear functionals that map a set of test functions
into the set of real numbers.) Then, in 16, we shall ask for spectral representations
by Fourier and Laplace transformations.
For holomorphic functions we have the Taylor series of a function
( ) [19]:
( ) =
-
( )
is a non-negative
function in
with compact support such that
{ ( - ):
} is a resolution of unity in . Let
( ) =
( ) where
and
. This representation can be refined to a quarkonial representation [19]:
( ) =
2
-
This resembles, at least formally, the Weierstrassian approach to holomorphic
functions (in the complex plane), combined with the wavelet philosophy: translations
- where and dyadic dilations 2 where [19].
Hence, the idea is to describe a function or distribution/functional/operator locally as
a sum of simple functions or distributions/functionals/operators which include
translations and dilations.
Let
( ) be a complex valued function in a domain of the complex plane . There
are two versions of conditions under which
( ) is called holomorphic [19]:
(i) The Riemannian (or qualitative) approach:
( ) is holomorphic in if it is complex
differentiable at any point
[19].
(ii) The Weierstrassian (or quantitative) approach:
( ) is holomorphic in if it can
be expanded locally in a Taylor series at any point
[19].
Function spaces are usually defined in the Riemannian spirit. In [19] this standard is
complemented by a Weierstrassian approach.
Let (
) be the Schwartz space of all complex valued, rapidly decreasing, infinitely
differentiable function on
. By ( ) we denote its topological dual, the space of
all tempered distributions on
[19].
33
In [19] it is shown that the coefficients
can be represented as:
= 2
2
| |
,
,
,
( ) ( )
, ,
Hence, the coefficients
can be represented as pairings in (
) and ( ) [19].
Otherwise stated, the space of functions (
)of cardinality , and the space of
distributions/functionals/operators
( ) of cardinality jointly define the
coefficients
of the quarkonial decomposition.
34
16 Spectral Representation
According to [14] we can approximate a function
( ) by:
( ) =
( )
,
with
( ) =
( )
,
The function ( ) is called the Fourier transform of the function ( ). The
coordinates and are inverse to one another, in the sense that if
= defines a
portion of space, then
= defines the spatial frequency [14]. The function ( )
defines the spectrum of the function ( ) [14].
The broader the width of ( ) is, the narrower the width of ( ) is, and vice versa
[14].
Hence, the idea is to describe a function globally as an integral of spectral functions,
considering the entire spectrum from
- to .
The Fourier transform has the peculiarity that the exponent in the integral is pure
imaginary (
). This is very useful when examining the spectrum of a function,
since it shows which frequencies are present. But a pure imaginary exponent does
not allow a damping factor which is, however, important in physical processes. In
order to consider damping, the Fourier transform can be generalized to the Laplace
transform [34].
According to [34] we can approximate a function
( ) by:
1
2
( )
=
( )
0
0
< 0
>
> 0,
with
( ) =
( )
The function ( ) is called the Laplace transform of the function ( ). ( ) exists if
| ( )|
>
> 0,
> 0
> 0 and if | ( )|
<
> 0 [34]. Since possesses a real and an imaginary part, real damping
factors can be jointly considered with imaginary spectral factors.
35
IX. Stochastic Analysis
17 Stochastic and Deterministic Processes
Until now, we have focused on spaces and described their structures. Now, we shall
concentrate on changes in spaces, said changes describing deterministic and
stochastic evolutions.
Basis for our investigation will be the probability space [18]
(; ; )
a sample space which is the set of all possible outcomes; a set of events, where
each event is a set containing zero or more outcomes; and the probability measure
function. A measurable space
( ; )
is defined by the non empty set and its -algebra [18].
Stochastic processes describe the evolution (in time) of a phenomenon under an
additional probabilistic dependence [18]. They are hence functions of two different
variables which strongly differ from one another with respect to their character: a
deterministic parameter of a set ; and a probabilistic parameter of a probability
space
(; ; ) [18]. A stochastic process can be formally defined as [18]:
:
(; ; )
;
( ; )
,
-
.
( ; )
(; ; )
= ( )
(
:
)
:(; ; ) ( ; ); .
:(; ; )
,
( ) ( ) =
( )
,
( )
.
We observe that the path mapping
is itself
/ -measurable. Hence, this path
mapping defines a probabilistic variable from which one can recover the entire
process by
=
, [18]. All stochastic information of the process is hence
gained from the distribution
of the path mapping [18].
Final del extracto de 170 páginas
- Citar trabajo
- Dr. Alexander Mircescu (Autor), 2018, General Index Theory: Its Mathematical and Physical Structures, Múnich, GRIN Verlag, https://www.grin.com/document/423592
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