Introduction to Dynamical SystemsIntroduction to Dynamical Systems Herbert Wiklicky...
Transcript of Introduction to Dynamical SystemsIntroduction to Dynamical Systems Herbert Wiklicky...
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Introduction toDynamical Systems
Herbert [email protected]
Imperial College London
Bertinoro, June 2013
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The Land of Oz
The Land of Oz is blessed with many things, but not by goodweather. They never have two nice days in a row. If they have anice day, the chance of rain or snow the next day are the same.If there is rain or snow the chances are even that the weatherstays the same for the next day. If there is a change from snowor rain, only half of the time is this a change to a nice day.
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The Land of Oz
From this we obtain the transition probabilities between nice(N), rainy (R) and snowy (S) days:
N
12
12
R1
2 8814
44
14
HH
S 12ff
14tt
14
VV
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The Land of Oz
We can then define the following transition matrix:
P =
12
14
14
12 0 1
214
14
12
From Grinstead & Snell: Introduction to Probability, p406;available as GNU book on http://www.dartmouth.edu/∼chance
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The Land of Oz
We can then define the following transition matrix:
P =
12
14
14
12 0 1
214
14
12
From Grinstead & Snell: Introduction to Probability, p406;available as GNU book on http://www.dartmouth.edu/∼chance
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Discrete Time Markov Chain
Given a finite set of states S = s1, . . . , sr.
A discrete time Markov chain (DTMC) on S is defined via astochastic matrix P as a above, i.e. an r × r (square) matrixwith entries 0 ≤ pij ≤ 1 and such that all row sums are equal toone, i.e. ∑
j
pij = 1.
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Discrete Time Markov Processes
Let P be the transition matrix of a DTMC. The entry in p(n)ij in
the n-th matrix power Pn gives the probability that the Markovchain, starting in state si , will be in state sj after exactly n steps.
At any time step we can describe the probabilities of being in acertain state si by a probability ui . These probabilities define aprobability distribution, i.e. a row vector
u = (u1,u2, · · · ,ur )
such that 0 ≤ ui ≤ 1 and∑
i
ui = 1.
For any stochastic matrix P and probability distribution u themultiplication uP is again a probability distribution.
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Discrete Time Markov Processes
Let P be the transition matrix of a DTMC. The entry in p(n)ij in
the n-th matrix power Pn gives the probability that the Markovchain, starting in state si , will be in state sj after exactly n steps.
At any time step we can describe the probabilities of being in acertain state si by a probability ui . These probabilities define aprobability distribution, i.e. a row vector
u = (u1,u2, · · · ,ur )
such that 0 ≤ ui ≤ 1 and∑
i
ui = 1.
For any stochastic matrix P and probability distribution u themultiplication uP is again a probability distribution.
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Discrete Time Markov Processes
Let P be the transition matrix of a DTMC. The entry in p(n)ij in
the n-th matrix power Pn gives the probability that the Markovchain, starting in state si , will be in state sj after exactly n steps.
At any time step we can describe the probabilities of being in acertain state si by a probability ui . These probabilities define aprobability distribution, i.e. a row vector
u = (u1,u2, · · · ,ur )
such that 0 ≤ ui ≤ 1 and∑
i
ui = 1.
For any stochastic matrix P and probability distribution u themultiplication uP is again a probability distribution.
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The Land of Oz
Consider the initial probability distributions u = (0,1,0) andv = (1
3 ,13 ,
13) in the Oz Example. The vector u describes a
situation where we are certain that we start with a nice day (N),while v corresponds to one where we assume the samechances of having a rainy (R), nice (N) or snowy (S) day.
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The Land of Oz
Consider the initial probability distributions u = (0,1,0) andv = (1
3 ,13 ,
13) in the Oz Example.
uP =
(12,0,
12
)uP2 =
(38,14,38
)
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The Land of Oz
Consider the initial probability distributions u = (0,1,0) andv = (1
3 ,13 ,
13) in the Oz Example.
vP =
0.416670.166670.41667
T
vP2 =
0.395830.208330.39583
T
vP3 =
0.401040.197920.40104
T
vP4 =
0.399740.200520.39974
T
· · · vP100 =
0.400000.200000.40000
T
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Convention
Note that in the theory of Markov chains one usually isconcerned with probability distributions as row vectors.Therefore, probability vectors are post-multiplied by thestochastic matrix P defining a Markov chain.
The usual pre-multiplication could be realised via:
Pu = (uT PT )T
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Dynamical Systems
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Dynamical Systems (Birkhoff 1927)
Introductory remarks. In dynamics we deal with physicalsystems whoes state at time t is completely specified by thevalues of n real variables
x1, x2, . . . , xn.
Accordingly the system is such that the rates of change ofthese variables, namely
dx1/dt ,dx2/dt , . . . ,dxn/dt ,
merely depend upon the values of the variables themselves, sothat the laws of motion can be expresses by means of ndifferential equations of the first order
dxi/dt = Xi(x1, x2, . . . , xn) (i = 1, . . . ,n).
George D. Birkhoff. Dynamical Systems, volume 9 ofColloquium Publications. AMS, 1927.
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Dynamical Systems (Bhatia/Szegö 1970)
. . . the symbol X denotes a metric space [. . . ] and R stands forthe set of real numbers.
1.1 Definition. A dynamical system on X is a triplet (X ,R, π),where π is a map from the product space X × R into the spaceX satisfying the following axioms:
1.1.1 π(x ,0) = x for every x ∈ X (identity axiom),1.1.2 π(π(x , t1), t2) = π(x , t1 + t2) for every x ∈ X and
t1, t2 ∈ R (group axiom),1.1.3 π is continuous (continuity axiom).
Nam Parshad Bhatia and Giorgio P. Szegö. Stability Theory ofDynamical Systems, volume 161 of Grundlehren dermathematischen Wissenschaften.
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Dynamical Systems
DefinitionA general dynamical system is a triple (G, π,X ) with (G, ·) agroup, X any set and π : G × X → X with:Identity Axiom
π(e, x) = x
for all x ∈ X and e ∈ G unit.Homomorphism Axiom
π(g, π(h, x)) = π(gh, x)
for all x ∈ X and g,h ∈ G.
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Elements of a General Dynamical System
A general dynamical systems is made up of three ingredients:
Phase Space: a set X where “things happen”. This can haveadditional structure (topology, norm, etc.)
Phase Group: the group G which allows us to “combine” thepartial dynamics to obtain a global picture.
Group Action: the way in which the dynamics of the group G isimplement on the phase space X .
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Elements of a General Dynamical System
A general dynamical systems is made up of three ingredients:
Phase Space: a set X where “things happen”. This can haveadditional structure (topology, norm, etc.)
Phase Group: the group G which allows us to “combine” thepartial dynamics to obtain a global picture.
Group Action: the way in which the dynamics of the group G isimplement on the phase space X .
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Elements of a General Dynamical System
A general dynamical systems is made up of three ingredients:
Phase Space: a set X where “things happen”. This can haveadditional structure (topology, norm, etc.)
Phase Group: the group G which allows us to “combine” thepartial dynamics to obtain a global picture.
Group Action: the way in which the dynamics of the group G isimplement on the phase space X .
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Variations of Dynamical System
Typical phase groups are Z (integers) or R (reals) for so calleddiscrete time or continuous time models.
To investigate, for example, symmetries of the phase space it isalso often the case that one considers so-called Lie Groups astransformation groups.
Typically we will request that the group action preserves thestructure of the phase space, i.e. π(g, .) is a structurepreserving morphism on X for all g ∈ G.
An option is to drop invertability to get one-sided dynamicalsystems by taking G to be a semi-group (e.g. the naturals N).
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Variations of Dynamical System
Typical phase groups are Z (integers) or R (reals) for so calleddiscrete time or continuous time models.
To investigate, for example, symmetries of the phase space it isalso often the case that one considers so-called Lie Groups astransformation groups.
Typically we will request that the group action preserves thestructure of the phase space, i.e. π(g, .) is a structurepreserving morphism on X for all g ∈ G.
An option is to drop invertability to get one-sided dynamicalsystems by taking G to be a semi-group (e.g. the naturals N).
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Variations of Dynamical System
Typical phase groups are Z (integers) or R (reals) for so calleddiscrete time or continuous time models.
To investigate, for example, symmetries of the phase space it isalso often the case that one considers so-called Lie Groups astransformation groups.
Typically we will request that the group action preserves thestructure of the phase space, i.e. π(g, .) is a structurepreserving morphism on X for all g ∈ G.
An option is to drop invertability to get one-sided dynamicalsystems by taking G to be a semi-group (e.g. the naturals N).
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Variations of Dynamical System
Typical phase groups are Z (integers) or R (reals) for so calleddiscrete time or continuous time models.
To investigate, for example, symmetries of the phase space it isalso often the case that one considers so-called Lie Groups astransformation groups.
Typically we will request that the group action preserves thestructure of the phase space, i.e. π(g, .) is a structurepreserving morphism on X for all g ∈ G.
An option is to drop invertability to get one-sided dynamicalsystems by taking G to be a semi-group (e.g. the naturals N).
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(Phase) Groups
DefinitionA group G is a set with two maps (product and inverse)
. · . : G ×G→ G and .−1 : G→ G
fulfilling:
(i) (xy)z = x(yz) for all x , y , z ∈ G associativityaxiom.
(ii) ∃e ∈ G such that ex = xe = x f or all x ∈ Gidentity axiom.
(iii) x−1x = xx−1 = e for all x ∈ G inverse axiom.
Here the group is presented multiplicatively, some groups arerepresented additively, e.g. (Z,+) and (R,+).
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(Phase) Groups
DefinitionA group G is a set with two maps (product and inverse)
. · . : G ×G→ G and .−1 : G→ G
fulfilling:
(i) (xy)z = x(yz) for all x , y , z ∈ G associativityaxiom.
(ii) ∃e ∈ G such that ex = xe = x f or all x ∈ Gidentity axiom.
(iii) x−1x = xx−1 = e for all x ∈ G inverse axiom.
Here the group is presented multiplicatively, some groups arerepresented additively, e.g. (Z,+) and (R,+).
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G-Spaces
DefinitionLet G be a group. A G-Space is a set S and a mapτ : G × S → S so that
τ(e, s) = s all s ∈ S
andτ(g, τ(h, s)) = τ(gh, s)
for all g,h ∈ G and s ∈ S. τ is also called an action of G on S.
We write τg(s) = τ(g, s) so τg : S → S and we have τgτh = τghas well as τgτg−1 = τg−1τg = τe = id, see e.g. [3, I.2]
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Phase Spaces
There are many choices for the phase space of a dynamicalsystem, among them we could mention:
Topological Spaces and require that π(g, .) a homeomorphism.Measurable Spaces with π(g, .) to be measure preserving.Vector Spaces like Rn with π a linear map or operator.Strings of Symbols in an alphabet Σ as in Symbolic Dynamics.Differentiable Manifolds as, e.g., in Classical Mechanics.Topological Vector Spaces, Toplogical Groups, Lie Groups, . . .
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Phase Spaces
There are many choices for the phase space of a dynamicalsystem, among them we could mention:
Topological Spaces and require that π(g, .) a homeomorphism.Measurable Spaces with π(g, .) to be measure preserving.Vector Spaces like Rn with π a linear map or operator.Strings of Symbols in an alphabet Σ as in Symbolic Dynamics.Differentiable Manifolds as, e.g., in Classical Mechanics.Topological Vector Spaces, Toplogical Groups, Lie Groups, . . .
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Phase Spaces
There are many choices for the phase space of a dynamicalsystem, among them we could mention:
Topological Spaces and require that π(g, .) a homeomorphism.Measurable Spaces with π(g, .) to be measure preserving.Vector Spaces like Rn with π a linear map or operator.Strings of Symbols in an alphabet Σ as in Symbolic Dynamics.Differentiable Manifolds as, e.g., in Classical Mechanics.Topological Vector Spaces, Toplogical Groups, Lie Groups, . . .
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Phase Spaces
There are many choices for the phase space of a dynamicalsystem, among them we could mention:
Topological Spaces and require that π(g, .) a homeomorphism.Measurable Spaces with π(g, .) to be measure preserving.Vector Spaces like Rn with π a linear map or operator.Strings of Symbols in an alphabet Σ as in Symbolic Dynamics.Differentiable Manifolds as, e.g., in Classical Mechanics.Topological Vector Spaces, Toplogical Groups, Lie Groups, . . .
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Phase Spaces
There are many choices for the phase space of a dynamicalsystem, among them we could mention:
Topological Spaces and require that π(g, .) a homeomorphism.Measurable Spaces with π(g, .) to be measure preserving.Vector Spaces like Rn with π a linear map or operator.Strings of Symbols in an alphabet Σ as in Symbolic Dynamics.Differentiable Manifolds as, e.g., in Classical Mechanics.Topological Vector Spaces, Toplogical Groups, Lie Groups, . . .
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Phase Spaces
There are many choices for the phase space of a dynamicalsystem, among them we could mention:
Topological Spaces and require that π(g, .) a homeomorphism.Measurable Spaces with π(g, .) to be measure preserving.Vector Spaces like Rn with π a linear map or operator.Strings of Symbols in an alphabet Σ as in Symbolic Dynamics.Differentiable Manifolds as, e.g., in Classical Mechanics.Topological Vector Spaces, Toplogical Groups, Lie Groups, . . .
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Phase Spaces
There are many choices for the phase space of a dynamicalsystem, among them we could mention:
Topological Spaces and require that π(g, .) a homeomorphism.Measurable Spaces with π(g, .) to be measure preserving.Vector Spaces like Rn with π a linear map or operator.Strings of Symbols in an alphabet Σ as in Symbolic Dynamics.Differentiable Manifolds as, e.g., in Classical Mechanics.Topological Vector Spaces, Toplogical Groups, Lie Groups, . . .
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Group Action
DefinitionLet (G, π,X ) be a dynamical system. The orbit of a point x ∈ Xis given by
OG(x) = π(g, x) | g ∈ G.
DefinitionLet (G, π,X ) be a dynamical system. The group action π is
transitive iff∀x , x ′ ∈ X : OG(x) = OG(x ′).
faithful iffg 7→ π(g, x) is injective.
free iff∀x ∈ X : g 7→ π(g, x) is injective.
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Group Action
DefinitionLet (G, π,X ) be a dynamical system. The orbit of a point x ∈ Xis given by
OG(x) = π(g, x) | g ∈ G.
DefinitionLet (G, π,X ) be a dynamical system. The group action π is
transitive iff∀x , x ′ ∈ X : OG(x) = OG(x ′).
faithful iffg 7→ π(g, x) is injective.
free iff∀x ∈ X : g 7→ π(g, x) is injective.
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Group Action
DefinitionLet (G, π,X ) be a dynamical system. The orbit of a point x ∈ Xis given by
OG(x) = π(g, x) | g ∈ G.
DefinitionLet (G, π,X ) be a dynamical system. The group action π is
transitive iff∀x , x ′ ∈ X : OG(x) = OG(x ′).
faithful iffg 7→ π(g, x) is injective.
free iff∀x ∈ X : g 7→ π(g, x) is injective.
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Group Action
DefinitionLet (G, π,X ) be a dynamical system. The orbit of a point x ∈ Xis given by
OG(x) = π(g, x) | g ∈ G.
DefinitionLet (G, π,X ) be a dynamical system. The group action π is
transitive iff∀x , x ′ ∈ X : OG(x) = OG(x ′).
faithful iffg 7→ π(g, x) is injective.
free iff∀x ∈ X : g 7→ π(g, x) is injective.
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Group Action
DefinitionLet (G, π,X ) be a dynamical system. The orbit of a point x ∈ Xis given by
OG(x) = π(g, x) | g ∈ G.
DefinitionLet (G, π,X ) be a dynamical system. The group action π is
transitive iff∀x , x ′ ∈ X : OG(x) = OG(x ′).
faithful iffg 7→ π(g, x) is injective.
free iff∀x ∈ X : g 7→ π(g, x) is injective.
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Elements of Ergodic Theory
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Topological Dynamical System
DefinitionA topological dynamical system is a dynamical system (G, π,X )with the elements:
G is a topological group, i.e. . · . is continuous,X is a topological space,
and π fulfills theContinuity Axiom:
π : G × X → X is continuous.
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Toplological Spaces
DefinitionA topological space is a set X together with a family of sub-setsτ ⊆ P(X ), the topology (of open sets), iff
1 ∅ ∈ τ and X ∈ τ .
2
n⋂i=0
Oi ∈ τ for Oi ∈ τ (finite).
3⋃i∈I
Oi ∈ τ for Oi ∈ τ (arbritrary).
The sets O ∈ τ are called open sets. The complementsA = X \O of open sets are closed sets.
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Metric Spaces
DefinitionA metric space is a set X and a real-valued function d(., .), ametric, on X × X which satisfies:
1 d(x , y) ≥ 02 d(x , y) = 0 ⇐⇒ x = y3 d(x , y) = d(y , x)
4 d(x , z) ≤ d(x , y) + d(y , z)
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Complete Metric Spaces
In a metric space we can define a basis for the topology opensets via open balls, i.e. sets B(x , ε) = x ′ | d(x , x ′) < ε, i.e.open sets are those which are unions of open balls.
Given a sequence (xi)i∈N of points in a topological space. Wesay that it converges if there exists x = lim xi such that for allneighbourhoods U(x) of x there ∃N s.t. for n > N : xn ∈ U(x).
A sequence of elements (xi)i∈N in a metric space (X ,d) iscalled a Cauchy sequence if
∀ε > 0 ∃N : n,m ≥ N ⇒ d(xn, xm) < ε.
A metric space (X ,d) in which all Cauchy sequences convergeis called complete (metric) space.
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Complete Metric Spaces
In a metric space we can define a basis for the topology opensets via open balls, i.e. sets B(x , ε) = x ′ | d(x , x ′) < ε, i.e.open sets are those which are unions of open balls.
Given a sequence (xi)i∈N of points in a topological space. Wesay that it converges if there exists x = lim xi such that for allneighbourhoods U(x) of x there ∃N s.t. for n > N : xn ∈ U(x).
A sequence of elements (xi)i∈N in a metric space (X ,d) iscalled a Cauchy sequence if
∀ε > 0 ∃N : n,m ≥ N ⇒ d(xn, xm) < ε.
A metric space (X ,d) in which all Cauchy sequences convergeis called complete (metric) space.
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Complete Metric Spaces
In a metric space we can define a basis for the topology opensets via open balls, i.e. sets B(x , ε) = x ′ | d(x , x ′) < ε, i.e.open sets are those which are unions of open balls.
Given a sequence (xi)i∈N of points in a topological space. Wesay that it converges if there exists x = lim xi such that for allneighbourhoods U(x) of x there ∃N s.t. for n > N : xn ∈ U(x).
A sequence of elements (xi)i∈N in a metric space (X ,d) iscalled a Cauchy sequence if
∀ε > 0 ∃N : n,m ≥ N ⇒ d(xn, xm) < ε.
A metric space (X ,d) in which all Cauchy sequences convergeis called complete (metric) space.
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Complete Metric Spaces
In a metric space we can define a basis for the topology opensets via open balls, i.e. sets B(x , ε) = x ′ | d(x , x ′) < ε, i.e.open sets are those which are unions of open balls.
Given a sequence (xi)i∈N of points in a topological space. Wesay that it converges if there exists x = lim xi such that for allneighbourhoods U(x) of x there ∃N s.t. for n > N : xn ∈ U(x).
A sequence of elements (xi)i∈N in a metric space (X ,d) iscalled a Cauchy sequence if
∀ε > 0 ∃N : n,m ≥ N ⇒ d(xn, xm) < ε.
A metric space (X ,d) in which all Cauchy sequences convergeis called complete (metric) space.
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Continuous Functions
DefinitionA function T : X → X ′ between two topological spaces (X , τ)and (X ′, τ ′) is calledcontinuous iff
∀O ∈ τ ′ : T−1(O) ∈ τ.
homeomorph iff
T is a bijection, and T and T−1 are continuous.
Continuous functions preserve limits, i.e. lim T(xi) = T(lim(xi)).
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Continuous Functions
DefinitionA function T : X → X ′ between two topological spaces (X , τ)and (X ′, τ ′) is calledcontinuous iff
∀O ∈ τ ′ : T−1(O) ∈ τ.
homeomorph iff
T is a bijection, and T and T−1 are continuous.
Continuous functions preserve limits, i.e. lim T(xi) = T(lim(xi)).
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Continuous Functions
DefinitionA function T : X → X ′ between two topological spaces (X , τ)and (X ′, τ ′) is calledcontinuous iff
∀O ∈ τ ′ : T−1(O) ∈ τ.
homeomorph iff
T is a bijection, and T and T−1 are continuous.
Continuous functions preserve limits, i.e. lim T(xi) = T(lim(xi)).
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Continuous Functions
DefinitionA function T : X → X ′ between two topological spaces (X , τ)and (X ′, τ ′) is calledcontinuous iff
∀O ∈ τ ′ : T−1(O) ∈ τ.
homeomorph iff
T is a bijection, and T and T−1 are continuous.
Continuous functions preserve limits, i.e. lim T(xi) = T(lim(xi)).
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Measure Theoretical Dynamical System
DefinitionA measure theoretic dynamical system is a dynamical system(G, π,X ) with
G is a measur(abl)e space,X is a measur(abl)e space,
and π fulfills theMeasurability Axiom:
π : G × X → X is measurable.π(g, .) : X → X is measure preserving ∀g ∈ G.
One can define a product measure on G × X in order to makesense of the first condition.
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Measure Theoretical Dynamical System
DefinitionA measure theoretic dynamical system is a dynamical system(G, π,X ) with
G is a measur(abl)e space,X is a measur(abl)e space,
and π fulfills theMeasurability Axiom:
π : G × X → X is measurable.π(g, .) : X → X is measure preserving ∀g ∈ G.
One can define a product measure on G × X in order to makesense of the first condition.
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Measureable Spaces
DefinitionGiven any set X . A family σ of sub-sets σ ⊆ P(X ) is called aσ-algebra iff
1 ∅ ∈ σ and X ∈ σ.
2
∞⋂i=0
Si ∈ σ for Si ∈ σ (countable).
3 X \ S ∈ σ for S ∈ σ.We say that (X , σ) is a measurable space, and S ∈ σ aremeasurable sets.
By de Morgan we have also:∞⋃
i=0
Si ∈ σ for Si ∈ σ (countable).
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Measureable Spaces
DefinitionGiven any set X . A family σ of sub-sets σ ⊆ P(X ) is called aσ-algebra iff
1 ∅ ∈ σ and X ∈ σ.
2
∞⋂i=0
Si ∈ σ for Si ∈ σ (countable).
3 X \ S ∈ σ for S ∈ σ.We say that (X , σ) is a measurable space, and S ∈ σ aremeasurable sets.
By de Morgan we have also:∞⋃
i=0
Si ∈ σ for Si ∈ σ (countable).
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Measures and Measurable Functions
DefinitionGiven a measurable space (X , σ) then µ : σ → R+ is a (finite)measure if
1 µ(∅) = 0 (for µ(X ) = 1 we have a probability measure).
2 µ(∞⋃
i=0
Si) =∞∑
i=0
µ(Si) for Si ∈ σ with Si ∩ Sj = ∅ for i 6= j .
DefinitionA function T : X → X ′ between two measure spaces spaces(X , σ, µ) and (X ′, τ ′, µ′) is calledmeasurable iff
∀S ∈ σ′ : T−1(S) ∈ σ.
measure preserving iff ∀S ∈ σ′ also µ′(S′) = µ(T−1(S)).
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Topological Mixing Notions
DefinitionGiven a topological dynamical system (G, π,X ).We say that (G, π,X ) is topologically transitive if
∃x ∈ X : OG(x) is dense in X .
We say that (G, π,X ) is (topologically) minimal if
∀x ∈ X : OG(x) is dense in X .
DefinitionA discrete topological dynamical system (T,X ) is calledtopologically (strong) mixing if
∀U,V ⊆ X open and non-empty ∃N : ∀n > N : Tn(U) ∩ V 6= ∅.
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Topological Mixing Notions
DefinitionGiven a topological dynamical system (G, π,X ).We say that (G, π,X ) is topologically transitive if
∃x ∈ X : OG(x) is dense in X .
We say that (G, π,X ) is (topologically) minimal if
∀x ∈ X : OG(x) is dense in X .
DefinitionA discrete topological dynamical system (T,X ) is calledtopologically (strong) mixing if
∀U,V ⊆ X open and non-empty ∃N : ∀n > N : Tn(U) ∩ V 6= ∅.
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Topologically Transitive
TheoremGiven a discrete topological dynamical system (T,X ) on acompact metric space X then the following conditions areequivalent:
1 ∀x ∈ X : OT(x) is dense in X (topologically transitive).2 ∀C ⊆ X closed with T(C) = C ⇒ C = X or C = ∅.3 ∀O ⊆ X open with T(O) = O ⇒ O = X or O = ∅.
4 ∀O ⊆ X open and non-empty, then∞⋃
n=−∞Tn(O) = X.
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Measure Theoretic Mixing Notions
DefinitionGiven a discrete measure theoretic dynamical system (T,X ).We say (T,X ) is measure theoretic transitive or ergodic if
∀S ⊆ X measurable with T(S) = S ⇒ µ(S) = 0 or µ(S) = 1.
DefinitionA discrete measure theoretic dynamical system (T,X ) is calledstrong mixing if
∀S1,S2 ⊆ X measurable limn→∞
µ(T−n(S1) ∩ S2) = µ(S1)µ(S2).
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Measure Theoretic Mixing Notions
DefinitionGiven a discrete measure theoretic dynamical system (T,X ).We say (T,X ) is measure theoretic transitive or ergodic if
∀S ⊆ X measurable with T(S) = S ⇒ µ(S) = 0 or µ(S) = 1.
DefinitionA discrete measure theoretic dynamical system (T,X ) is calledstrong mixing if
∀S1,S2 ⊆ X measurable limn→∞
µ(T−n(S1) ∩ S2) = µ(S1)µ(S2).
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Ergodicity
TheoremGiven a discrete measure theoretic dynamical system (T,X )with T measure preserving. Then the following conditions areequivalent (with ergodic):
1 ∀S ⊆ X measurable T(S) = S ⇒ µ(S) = 0 or µ(S) = 1.2 ∀S ⊆ X measurable and µ(T−1(S)
aS) = 0
⇒ µ(S) = 0 or µ(S) = 1.
3 ∀S ⊆ X measurable and µ(S) > 0⇒ µ(∞⋃
n=−∞T−n(S)) = 1.
4 ∀S1,S2 ⊆ X measurable and µ(S1) > 0 < µ(S2)⇒ ∃n ∈ N such that µ(T−n(S1) ∩ S2) = 0.
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Ergodic Theorem
Given a discrete measure theoretic dynamical system (T,X )and a function (i.e. a random variable) f : X → R.
The phase average of f is defined as µ(f ) =∫
X f (x)dx . The
time average of f is defined as f ∗(x) = limT→∞
1T
∫ T
0f (Tt (x)))dt .
Theorem (Birkhoff)
Given a discrete measure theoretic dynamical system (T,X ),with T measure preserving, and a function f : X → R withf ∈ L1(X , µ) then the following holds:
(T,X ) is ergodic ⇔ µ(f ) = f ∗(x) µ-almost everywhere.
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Ergodic Theorem
Given a discrete measure theoretic dynamical system (T,X )and a function (i.e. a random variable) f : X → R.
The phase average of f is defined as µ(f ) =∫
X f (x)dx . The
time average of f is defined as f ∗(x) = limT→∞
1T
∫ T
0f (Tt (x)))dt .
Theorem (Birkhoff)
Given a discrete measure theoretic dynamical system (T,X ),with T measure preserving, and a function f : X → R withf ∈ L1(X , µ) then the following holds:
(T,X ) is ergodic ⇔ µ(f ) = f ∗(x) µ-almost everywhere.
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Elements of Linear DynamicalSystems
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Linear Dynamical System
DefinitionA linear dynamical system is a dynamical system (G, π,X ) with
G is a group (typically G = Z),X is a vector space
and π fulfils theLinearity Axiom:
π(g, .) : X → X is linear ∀g ∈ G.
Many versions of linear dynamical systems play an importantrole in control theory investigating e.g. feed back loops etc.
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Linear Dynamical System
DefinitionA linear dynamical system is a dynamical system (G, π,X ) with
G is a group (typically G = Z),X is a vector space
and π fulfils theLinearity Axiom:
π(g, .) : X → X is linear ∀g ∈ G.
Many versions of linear dynamical systems play an importantrole in control theory investigating e.g. feed back loops etc.
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Abstract Vector Spaces
DefinitionA Vector Space (over a field K, e.g. R or C) is a set V togetherwith two operations:
Scalar Multiplication . ·. : K× V 7→ VVector Addition .+. : V × V 7→ V
such that (∀x , y , z ∈ V and α, β ∈ K):
1 x + (y + z) = (x + y) + z2 x + y = y + x3 ∃o : x + o = x4 ∃−x : x + (−x) = o
1 α(x + y) = αx + αy2 (α + β)x = αx + βx3 (αβ)x = α(βx)
4 1x = x (1 ∈ K)
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Tuple Spaces
TheoremAll finite dimensional vector spaces are isomorphic to the (finite)Cartesian product of the underlying field Kn (i.e. Rn or Cm).
Finite dimensional vectors can always be represented via theircoordinates with respect to a given base, e.g.
x = (x1, x2, x3, . . . , xn)
y = (y1, y2, y3, . . . , yn)
Algebraic Structure
αx = (αx1, αx2, αx3, . . . , αxn)
x + y = (x1 + y1, x2 + y2, x3 + y3, . . . , xn + yn)
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Linear Operators
DefinitionA map T : V → W between two vector spaces V andW iscalled a linear map iff
1 T(x + y) = T(x) + T(y) and2 T(αx) = αT(x)
for all x , y ∈ V and all α ∈ K (e.g. K = C or R).
The set of all linear maps between V andW is denotedL(V,W). For V =W we talk about a linear operator on V.
On normed vector spaces the continuous or equivalentlybounded linear operators are of particular interest, i.e.
B(V) = T | ‖T‖ = supx∈V
‖T(x)‖‖x‖
<∞ ⊆ L(V) = L(V,V).
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Linear Operators
DefinitionA map T : V → W between two vector spaces V andW iscalled a linear map iff
1 T(x + y) = T(x) + T(y) and2 T(αx) = αT(x)
for all x , y ∈ V and all α ∈ K (e.g. K = C or R).
The set of all linear maps between V andW is denotedL(V,W). For V =W we talk about a linear operator on V.
On normed vector spaces the continuous or equivalentlybounded linear operators are of particular interest, i.e.
B(V) = T | ‖T‖ = supx∈V
‖T(x)‖‖x‖
<∞ ⊆ L(V) = L(V,V).
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Linear Operators
DefinitionA map T : V → W between two vector spaces V andW iscalled a linear map iff
1 T(x + y) = T(x) + T(y) and2 T(αx) = αT(x)
for all x , y ∈ V and all α ∈ K (e.g. K = C or R).
The set of all linear maps between V andW is denotedL(V,W). For V =W we talk about a linear operator on V.
On normed vector spaces the continuous or equivalentlybounded linear operators are of particular interest, i.e.
B(V) = T | ‖T‖ = supx∈V
‖T(x)‖‖x‖
<∞ ⊆ L(V) = L(V,V).
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Normed Spaces
DefinitionA complex vector space V is called a normed (vector) space ifthere is a real valued function ‖.‖ on V that satisfies (∀x , y ∈ Vand ∀α ∈ C):
1 ‖x‖ ≥ 02 ‖x‖ = 0 ⇐⇒ x = o3 ‖αx‖ = |α| ‖x‖4 ‖x + y‖ ≤ ‖x‖+ ‖y‖
The function ‖.‖ is called a norm on V.
We have a Banach space if the topology induced by d(x , y)= ‖x − y‖ is complete – always for finite dimensional spaces.
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Normed Spaces
DefinitionA complex vector space V is called a normed (vector) space ifthere is a real valued function ‖.‖ on V that satisfies (∀x , y ∈ Vand ∀α ∈ C):
1 ‖x‖ ≥ 02 ‖x‖ = 0 ⇐⇒ x = o3 ‖αx‖ = |α| ‖x‖4 ‖x + y‖ ≤ ‖x‖+ ‖y‖
The function ‖.‖ is called a norm on V.
We have a Banach space if the topology induced by d(x , y)= ‖x − y‖ is complete – always for finite dimensional spaces.
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Hilbert Spaces
DefinitionA complex vector space H is called an inner product space (or(pre-)Hilbert space) if there is a complex valued function 〈., .〉on H×H that satisfies (∀x , y , z ∈ H and ∀α ∈ C):
1 〈x , x〉 ≥ 02 〈x , x〉 = 0 ⇐⇒ x = o3 〈αx , y〉 = α 〈x , y〉4 〈x , y + z〉 = 〈x , y〉+ 〈x , z〉5 〈x , y〉 = 〈y , x〉
The function 〈., .〉 is called an inner product on H.
If the topology induced by ‖x‖ =√〈x , x〉 is complete then we
have a Hilbert space – always for finite dimensional spaces.
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Basis Vectors
A set of vectors xi is said to be linearly independent iff
λixi =∑
λixi = 0 implies that ∀ i : λi = 0
Two vectors in a Hilbert space are orthogonal iff 〈x , y〉 = 0An orthonormal system (base if it generates all H) in a Hilbertspace is a set of linearly independent vectors bii with:⟨
bi ,bj⟩
= δij =
1 iff i = j0 iff i 6= j
TheoremFor a Hilbert space there exists an orthonormal basis bi. Therepresentation of each vector is unique:
x =∑
i
xibi =∑
i
〈x ,bi〉bi
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Basis Vectors
A set of vectors xi is said to be linearly independent iff
λixi =∑
λixi = 0 implies that ∀ i : λi = 0
Two vectors in a Hilbert space are orthogonal iff 〈x , y〉 = 0An orthonormal system (base if it generates all H) in a Hilbertspace is a set of linearly independent vectors bii with:⟨
bi ,bj⟩
= δij =
1 iff i = j0 iff i 6= j
TheoremFor a Hilbert space there exists an orthonormal basis bi. Therepresentation of each vector is unique:
x =∑
i
xibi =∑
i
〈x ,bi〉bi
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Basis Vectors
A set of vectors xi is said to be linearly independent iff
λixi =∑
λixi = 0 implies that ∀ i : λi = 0
Two vectors in a Hilbert space are orthogonal iff 〈x , y〉 = 0An orthonormal system (base if it generates all H) in a Hilbertspace is a set of linearly independent vectors bii with:⟨
bi ,bj⟩
= δij =
1 iff i = j0 iff i 6= j
TheoremFor a Hilbert space there exists an orthonormal basis bi. Therepresentation of each vector is unique:
x =∑
i
xibi =∑
i
〈x ,bi〉bi
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Basis Vectors
A set of vectors xi is said to be linearly independent iff
λixi =∑
λixi = 0 implies that ∀ i : λi = 0
Two vectors in a Hilbert space are orthogonal iff 〈x , y〉 = 0An orthonormal system (base if it generates all H) in a Hilbertspace is a set of linearly independent vectors bii with:⟨
bi ,bj⟩
= δij =
1 iff i = j0 iff i 6= j
TheoremFor a Hilbert space there exists an orthonormal basis bi. Therepresentation of each vector is unique:
x =∑
i
xibi =∑
i
〈x ,bi〉bi
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Dual Spaces
A linear functional on a vector space V is a map f : V → K suchthat f (x + y) = f (x) + f (y) and f (αx) = αf (x) for allx , y ∈ V, α ∈ K.
Theorem (Riesz Representation Theorem)Every (bounded) linear functional on a Hilbert space H can berepresented by a vector in the Hilbert space H, such that
f (x) = 〈yf |x〉 = fy (x)
The dual Hilbert space H∗ is isomorphic to the original Hilbertspace H, e.g. for the universal Hilbert space `2(N)∗ = `2(N).
`p(X) =
(xi)i∈X |
(∑i∈X|xi |2
) 1p
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Dual Spaces
A linear functional on a vector space V is a map f : V → K suchthat f (x + y) = f (x) + f (y) and f (αx) = αf (x) for allx , y ∈ V, α ∈ K.
Theorem (Riesz Representation Theorem)Every (bounded) linear functional on a Hilbert space H can berepresented by a vector in the Hilbert space H, such that
f (x) = 〈yf |x〉 = fy (x)
The dual Hilbert space H∗ is isomorphic to the original Hilbertspace H, e.g. for the universal Hilbert space `2(N)∗ = `2(N).
`p(X) =
(xi)i∈X |
(∑i∈X|xi |2
) 1p
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Dual Spaces
A linear functional on a vector space V is a map f : V → K suchthat f (x + y) = f (x) + f (y) and f (αx) = αf (x) for allx , y ∈ V, α ∈ K.
Theorem (Riesz Representation Theorem)Every (bounded) linear functional on a Hilbert space H can berepresented by a vector in the Hilbert space H, such that
f (x) = 〈yf |x〉 = fy (x)
The dual Hilbert space H∗ is isomorphic to the original Hilbertspace H, e.g. for the universal Hilbert space `2(N)∗ = `2(N).
`p(X) =
(xi)i∈X |
(∑i∈X|xi |2
) 1p
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Finite-Dimensional Hilbert Spaces
We represent vectors and their transpose using coordinates:
~x =
x1...
xn
, ~y =
y1...
yn
T
= (y1, . . . , yn)
The adjoint of ~x = (x1, . . . , xn), with .∗ = . denoting complexconjugate in C), is given by
~x† = ~x∗ = (x∗1 , . . . , x∗n )T
The inner product is:⟨~y , ~x
⟩=∑
i
y∗i xi = ~y†~x
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Finite-Dimensional Hilbert Spaces
We represent vectors and their transpose using coordinates:
~x =
x1...
xn
, ~y =
y1...
yn
T
= (y1, . . . , yn)
The adjoint of ~x = (x1, . . . , xn), with .∗ = . denoting complexconjugate in C), is given by
~x† = ~x∗ = (x∗1 , . . . , x∗n )T
The inner product is:⟨~y , ~x
⟩=∑
i
y∗i xi = ~y†~x
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Finite-Dimensional Hilbert Spaces
We represent vectors and their transpose using coordinates:
~x =
x1...
xn
, ~y =
y1...
yn
T
= (y1, . . . , yn)
The adjoint of ~x = (x1, . . . , xn), with .∗ = . denoting complexconjugate in C), is given by
~x† = ~x∗ = (x∗1 , . . . , x∗n )T
The inner product is:⟨~y , ~x
⟩=∑
i
y∗i xi = ~y†~x
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Differential Equations
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Discrete (Time) Dynamical Systems: Collatz
The Colltaz problem is a (one-sided) discrete time dynamicalsystem (C,Z), which we can describe by the followingtransformation:
C : Z→ Z
with
C(n) =
n/2 if n is even
3× n + 1 otherwise.
The unsolved question is:
Does ∃m ∈ N such that Cm(n) = 1 for all n ∈ N?
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Discrete (Time) Dynamical Systems: Collatz
The Colltaz problem is a (one-sided) discrete time dynamicalsystem (C,Z), which we can describe by the followingtransformation:
C : Z→ Z
with
C(n) =
n/2 if n is even
3× n + 1 otherwise.
The unsolved question is:
Does ∃m ∈ N such that Cm(n) = 1 for all n ∈ N?
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Continuous Dynamical Systems
A popular way to specify continuous time dynamical systems isvia (ordinary) differential equations, e.g. Morris W. Hirsch,Stephen Smale, and Robert L. Devaney. Differential Equations,Dynamical Systems and An Introduction to Chaos. Elsevier,2004.
The group action is interpreted as time t ∈ R.
Ordinary Differential Equations
x ′1 = dx1dt = f1(t , x1, x2, . . . , xn)
x ′2 = dx2dt = f2(t , x1, x2, . . . , xn)
. . . . . . . . .
x ′n = dxndt = fn(t , x1, x2, . . . , xn)
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Continuous Dynamical Systems
A popular way to specify continuous time dynamical systems isvia (ordinary) differential equations, e.g. Morris W. Hirsch,Stephen Smale, and Robert L. Devaney. Differential Equations,Dynamical Systems and An Introduction to Chaos. Elsevier,2004.
The group action is interpreted as time t ∈ R.
Ordinary Differential Equations
x ′1 = dx1dt = f1(t , x1, x2, . . . , xn)
x ′2 = dx2dt = f2(t , x1, x2, . . . , xn)
. . . . . . . . .
x ′n = dxndt = fn(t , x1, x2, . . . , xn)
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Differential
Given a function f : R→ R. We say it is differentiable at a pointt ∈ R if there is a linear map Df (t) : R→ R which approximatesf at t . That is, ∀ε > 0 there is a neigborhood U of t such that:
‖f (t ′)− f (t)− Df (t)(t − t ′)‖ < ε‖t − t ′‖ ∀t ′ ∈ U
We also write for the differential (quotient) Df = dfdt .
We also approximate a function f : Rn → Rm by a linear mapDf (x) : Rn → Rm represented by the matrix of partial derivates:
(Df )ij =∂fi∂tj
i = 1, . . . ,m, i = 1, . . . ,n.
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Differential
Given a function f : R→ R. We say it is differentiable at a pointt ∈ R if there is a linear map Df (t) : R→ R which approximatesf at t . That is, ∀ε > 0 there is a neigborhood U of t such that:
‖f (t ′)− f (t)− Df (t)(t − t ′)‖ < ε‖t − t ′‖ ∀t ′ ∈ U
We also write for the differential (quotient) Df = dfdt .
We also approximate a function f : Rn → Rm by a linear mapDf (x) : Rn → Rm represented by the matrix of partial derivates:
(Df )ij =∂fi∂tj
i = 1, . . . ,m, i = 1, . . . ,n.
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Euler’s Number
What is Euler’s e? Metafont users know, it is e = 2.7183 . . .It is the unique number such that for f (t) = et we have
dfdt
(t) = f (t)
The exponential function is the fixed-point or eigen-function ofthe differential operator d
dt . One could show this via the Taylorexpansion of exp(x) =
∑∞n=0
xn
n! as ddt
tn
n! = ntn−1
n(n−1)! = tn−1
(n−1)! .
The simplest differential equation one can think of is perhaps:
x ′(t) =dxdt
(t) = ax(t)
The solution is x(t) = keat = k exp(at) for some constant k(can be determined via an initial/boundary condition, e.g. x(0)).
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Euler’s Number
What is Euler’s e? Metafont users know, it is e = 2.7183 . . .It is the unique number such that for f (t) = et we have
dfdt
(t) = f (t)
The exponential function is the fixed-point or eigen-function ofthe differential operator d
dt . One could show this via the Taylorexpansion of exp(x) =
∑∞n=0
xn
n! as ddt
tn
n! = ntn−1
n(n−1)! = tn−1
(n−1)! .
The simplest differential equation one can think of is perhaps:
x ′(t) =dxdt
(t) = ax(t)
The solution is x(t) = keat = k exp(at) for some constant k(can be determined via an initial/boundary condition, e.g. x(0)).
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Euler’s Number
What is Euler’s e? Metafont users know, it is e = 2.7183 . . .It is the unique number such that for f (t) = et we have
dfdt
(t) = f (t)
The exponential function is the fixed-point or eigen-function ofthe differential operator d
dt . One could show this via the Taylorexpansion of exp(x) =
∑∞n=0
xn
n! as ddt
tn
n! = ntn−1
n(n−1)! = tn−1
(n−1)! .
The simplest differential equation one can think of is perhaps:
x ′(t) =dxdt
(t) = ax(t)
The solution is x(t) = keat = k exp(at) for some constant k(can be determined via an initial/boundary condition, e.g. x(0)).
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Euler’s Number
What is Euler’s e? Metafont users know, it is e = 2.7183 . . .It is the unique number such that for f (t) = et we have
dfdt
(t) = f (t)
The exponential function is the fixed-point or eigen-function ofthe differential operator d
dt . One could show this via the Taylorexpansion of exp(x) =
∑∞n=0
xn
n! as ddt
tn
n! = ntn−1
n(n−1)! = tn−1
(n−1)! .
The simplest differential equation one can think of is perhaps:
x ′(t) =dxdt
(t) = ax(t)
The solution is x(t) = keat = k exp(at) for some constant k(can be determined via an initial/boundary condition, e.g. x(0)).
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Ordinary Linear Differential Equations [1, p129]
Solution to ordinary differential equations via exponentation.
x ′1 = dx1dt = a11x1 + a12x2 + . . .+ a1nxn
x ′2 = dx2dt = a21x1 + a22x2 + . . .+ a2nxn
. . . . . . . . .
x ′n = dx2dt = an1x1 + an2x2 + . . .+ annxn
TheoremLet A be an n × n matrix. Then the unique solution to the initialvalue problem x′ = Ax with x(0) = x0 is given by
x(t) = exp(tA)x0.
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Ordinary Linear Differential Equations [1, p129]
Solution to ordinary differential equations via exponentation.
x ′1 = dx1dt = a11x1 + a12x2 + . . .+ a1nxn
x ′2 = dx2dt = a21x1 + a22x2 + . . .+ a2nxn
. . . . . . . . .
x ′n = dx2dt = an1x1 + an2x2 + . . .+ annxn
TheoremLet A be an n × n matrix. Then the unique solution to the initialvalue problem x′ = Ax with x(0) = x0 is given by
x(t) = exp(tA)x0.
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Computing Solutions
The exponential of a matrix A can be computed as:
exp(A) =∞∑
k=0
Ak
k !
However this anything but an efficient way to compute it.
We can represent matrices e.g. in Jordan normal form:A = D + N where D is a diagonal matrix and N is an upperdiagonal matrix which is nilpotent, i.e. ∃m s.t. Nm vanishes.This boils down to finding the eigenvalues of A (via SVD).
We then have exp(A) = exp(D + N) = exp(D) exp(N) withexp(diag(d1, . . . ,dn)) = diag(exp(d1), . . . ,exp(dn)) and Nk 6= 0only for finitely many terms.
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Computing Solutions
The exponential of a matrix A can be computed as:
exp(A) =∞∑
k=0
Ak
k !
However this anything but an efficient way to compute it.
We can represent matrices e.g. in Jordan normal form:A = D + N where D is a diagonal matrix and N is an upperdiagonal matrix which is nilpotent, i.e. ∃m s.t. Nm vanishes.This boils down to finding the eigenvalues of A (via SVD).
We then have exp(A) = exp(D + N) = exp(D) exp(N) withexp(diag(d1, . . . ,dn)) = diag(exp(d1), . . . ,exp(dn)) and Nk 6= 0only for finitely many terms.
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Computing Solutions
The exponential of a matrix A can be computed as:
exp(A) =∞∑
k=0
Ak
k !
However this anything but an efficient way to compute it.
We can represent matrices e.g. in Jordan normal form:A = D + N where D is a diagonal matrix and N is an upperdiagonal matrix which is nilpotent, i.e. ∃m s.t. Nm vanishes.This boils down to finding the eigenvalues of A (via SVD).
We then have exp(A) = exp(D + N) = exp(D) exp(N) withexp(diag(d1, . . . ,dn)) = diag(exp(d1), . . . ,exp(dn)) and Nk 6= 0only for finitely many terms.
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Computing Solutions
The exponential of a matrix A can be computed as:
exp(A) =∞∑
k=0
Ak
k !
However this anything but an efficient way to compute it.
We can represent matrices e.g. in Jordan normal form:A = D + N where D is a diagonal matrix and N is an upperdiagonal matrix which is nilpotent, i.e. ∃m s.t. Nm vanishes.This boils down to finding the eigenvalues of A (via SVD).
We then have exp(A) = exp(D + N) = exp(D) exp(N) withexp(diag(d1, . . . ,dn)) = diag(exp(d1), . . . ,exp(dn)) and Nk 6= 0only for finitely many terms.
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Smooth Functions
We say a function f is differentiable on R or U ∈ R if it isdifferentiable at every point t ∈ R or t ∈ U. We then writef ∈ C1(R) = C1 or f ∈ C(U).
We can see Df (x) itself as a function Rn → L(Rn,Rm) = Rnm.
As such we can ask if this is itself differentiable. We denote theset of p-times differentiable maps by Cp and by C∞ the set ofinfinitely differentiable or smooth functions.
Note: Differentiation is primarily a real number notion. We needto introduce the notion of a differentiable manifold as a spacewhich looks like Rm locally (with respect to diff. operations).
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Smooth Functions
We say a function f is differentiable on R or U ∈ R if it isdifferentiable at every point t ∈ R or t ∈ U. We then writef ∈ C1(R) = C1 or f ∈ C(U).
We can see Df (x) itself as a function Rn → L(Rn,Rm) = Rnm.
As such we can ask if this is itself differentiable. We denote theset of p-times differentiable maps by Cp and by C∞ the set ofinfinitely differentiable or smooth functions.
Note: Differentiation is primarily a real number notion. We needto introduce the notion of a differentiable manifold as a spacewhich looks like Rm locally (with respect to diff. operations).
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Smooth Functions
We say a function f is differentiable on R or U ∈ R if it isdifferentiable at every point t ∈ R or t ∈ U. We then writef ∈ C1(R) = C1 or f ∈ C(U).
We can see Df (x) itself as a function Rn → L(Rn,Rm) = Rnm.
As such we can ask if this is itself differentiable. We denote theset of p-times differentiable maps by Cp and by C∞ the set ofinfinitely differentiable or smooth functions.
Note: Differentiation is primarily a real number notion. We needto introduce the notion of a differentiable manifold as a spacewhich looks like Rm locally (with respect to diff. operations).
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Differentiable Manifolds
DefinitionLet M be a topological space.
A chart (V ,Φ) is a homeomorphism Φ of an open set V of Minto an open set of Rm.
Two charts (V1,Φ1) and (V2,Φ2) are said to be compatible incase V1 ∩ V2 = ∅ or the restricted maps Φ1 Φ−1
2 and Φ2 Φ−11
are in C∞(Rm).
A atlas is a set of compatible charts that cover all of M. Twoatlases are compatible if all their charts are.
A differentiable manifold is a separable, metrizisable space withan set of compatible (equivalent) atlases.
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Differentiable Manifolds
DefinitionLet M be a topological space.
A chart (V ,Φ) is a homeomorphism Φ of an open set V of Minto an open set of Rm.
Two charts (V1,Φ1) and (V2,Φ2) are said to be compatible incase V1 ∩ V2 = ∅ or the restricted maps Φ1 Φ−1
2 and Φ2 Φ−11
are in C∞(Rm).
A atlas is a set of compatible charts that cover all of M. Twoatlases are compatible if all their charts are.
A differentiable manifold is a separable, metrizisable space withan set of compatible (equivalent) atlases.
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Differentiable Manifolds
DefinitionLet M be a topological space.
A chart (V ,Φ) is a homeomorphism Φ of an open set V of Minto an open set of Rm.
Two charts (V1,Φ1) and (V2,Φ2) are said to be compatible incase V1 ∩ V2 = ∅ or the restricted maps Φ1 Φ−1
2 and Φ2 Φ−11
are in C∞(Rm).
A atlas is a set of compatible charts that cover all of M. Twoatlases are compatible if all their charts are.
A differentiable manifold is a separable, metrizisable space withan set of compatible (equivalent) atlases.
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Differentiable Manifolds
DefinitionLet M be a topological space.
A chart (V ,Φ) is a homeomorphism Φ of an open set V of Minto an open set of Rm.
Two charts (V1,Φ1) and (V2,Φ2) are said to be compatible incase V1 ∩ V2 = ∅ or the restricted maps Φ1 Φ−1
2 and Φ2 Φ−11
are in C∞(Rm).
A atlas is a set of compatible charts that cover all of M. Twoatlases are compatible if all their charts are.
A differentiable manifold is a separable, metrizisable space withan set of compatible (equivalent) atlases.
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Tangents
DefinitionLet f ,g ∈ L(V,W) then we say that f is tangent to g at t iff
limt ′→t
‖f (t ′)− g(t ′)‖‖t ′ − t‖
= 0
Definition
Let M be a manifold and m ∈ M. A curve at m is a C1 mapc : I → M with an open interval in R containing 0 s.t. c(0) = m.
We say that two curves c1 and c2 are tangent if Φ c1 andΦ c2 are tangent at 0.
The tangent space Tm(M) of M at m is the set of (tangent)equivalent classes of curves.
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Tangents
DefinitionLet f ,g ∈ L(V,W) then we say that f is tangent to g at t iff
limt ′→t
‖f (t ′)− g(t ′)‖‖t ′ − t‖
= 0
Definition
Let M be a manifold and m ∈ M. A curve at m is a C1 mapc : I → M with an open interval in R containing 0 s.t. c(0) = m.
We say that two curves c1 and c2 are tangent if Φ c1 andΦ c2 are tangent at 0.
The tangent space Tm(M) of M at m is the set of (tangent)equivalent classes of curves.
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Tangents
DefinitionLet f ,g ∈ L(V,W) then we say that f is tangent to g at t iff
limt ′→t
‖f (t ′)− g(t ′)‖‖t ′ − t‖
= 0
Definition
Let M be a manifold and m ∈ M. A curve at m is a C1 mapc : I → M with an open interval in R containing 0 s.t. c(0) = m.
We say that two curves c1 and c2 are tangent if Φ c1 andΦ c2 are tangent at 0.
The tangent space Tm(M) of M at m is the set of (tangent)equivalent classes of curves.
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Lie Groups
DefinitionA Lie group over a field K is a group G equipped with thestructure of a differentiable manifold over K sich that
. · . : G ×G→ G is differentable.
Using the implicit function theorem, one can also show thatg 7→ g−1 is differentiable (a diffeomorphism).
The fields we are typically interested are K = R or K = C.
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Lie Groups
DefinitionA Lie group over a field K is a group G equipped with thestructure of a differentiable manifold over K sich that
. · . : G ×G→ G is differentable.
Using the implicit function theorem, one can also show thatg 7→ g−1 is differentiable (a diffeomorphism).
The fields we are typically interested are K = R or K = C.
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Lie Groups
DefinitionA Lie group over a field K is a group G equipped with thestructure of a differentiable manifold over K sich that
. · . : G ×G→ G is differentable.
Using the implicit function theorem, one can also show thatg 7→ g−1 is differentiable (a diffeomorphism).
The fields we are typically interested are K = R or K = C.
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Fields
DefinitionA field is a set K together with two operations:
Addition .+. : K×K 7→ KMultiplication . ·. : K×K 7→ K
such that
1 ∀x , y , z ∈ K : x + (y + z) = (x + y) + z2 ∃ o ∈ K, ∀x ∈ K : o + x = x3 ∀ x ∈ K, ∃ −x ∈ K : x + (−x) = o4 ∀x , y ∈ K : x + y = x + y5 ∀x , y , z ∈ K : x · (y · z) = (x · y) · z6 ∃ o 6= e ∈ K, ∀x ∈ K : e · x = x7 ∀ o 6= x ∈ K, ∃x−1 ∈ K : x · x−1 = e8 ∀x , y ∈ K : x · y = y · x9 x · (y + z) = x · y + x · z, ∀x , y , z ∈ K
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Examples of Lie Groups
Examples of Lie groups we can mentions here:
The additive group of the field K = K+.The multiplicative group of the field K×.The “circle” T = z ∈ C | |z| = 1 or eiφ | φ ∈ [0,2π).GLn(K) of invertible matrices of order n over K.SLn(K) of matrices of order n over K with det = 1.On(K) orthogonal matrices over K of order n.Un unitary matrices over C of order n.SOn(K) = On(K) ∩ SLn(K).SUn = Un ∩ SLn(C).
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Lie Algebras
DefinitionA Lie algebra is a vector space g over some field K togetherwith a binary operation [·, ·] : g× g→ g called the Lie bracket,which satisfies the following:
Bilinearity: ∀α, β ∈ K and ∀x , y , z ∈ g
[αx + βy , z] = α[x , z] + β[y , z]
[z, αx + βy ] = α[z, x ] + β[z, y ]
Alternating on g: ∀x ∈ g[x , x ] = 0
Jacobi identity: ∀x , y , z ∈ g
[x , [y , z]] + [z, [x , y ]] + [y , [z, x ]] = 0
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Examples of Lie Algebras
It follows easily ∀x , y ∈ g that [x , y ] = −[y , x ]. One could alsodefine a associative product on an algebra g and then introducethe Lie bracket as [x , y ] = xy − yx .
TheoremGiven a Lie group G then the tangent space at the unit g = TeGis a Lie algebra.
Let g(t) and h(t) be differentiable paths or C1 curves on G.Assume, g(0) = h(0) = e as well as dg
dt (0) = ξ and dhdt (0) = η
then we define a Lie bracket on the tangent space Te(G) via
[ξ, η] =∂2
∂t∂s[g(t),h(s)]|s=t=0
where [g,h] = ghg−1h−1 is the group commutator.
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Examples of Lie Algebras
It follows easily ∀x , y ∈ g that [x , y ] = −[y , x ]. One could alsodefine a associative product on an algebra g and then introducethe Lie bracket as [x , y ] = xy − yx .
TheoremGiven a Lie group G then the tangent space at the unit g = TeGis a Lie algebra.
Let g(t) and h(t) be differentiable paths or C1 curves on G.Assume, g(0) = h(0) = e as well as dg
dt (0) = ξ and dhdt (0) = η
then we define a Lie bracket on the tangent space Te(G) via
[ξ, η] =∂2
∂t∂s[g(t),h(s)]|s=t=0
where [g,h] = ghg−1h−1 is the group commutator.
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Examples of Lie Algebras
It follows easily ∀x , y ∈ g that [x , y ] = −[y , x ]. One could alsodefine a associative product on an algebra g and then introducethe Lie bracket as [x , y ] = xy − yx .
TheoremGiven a Lie group G then the tangent space at the unit g = TeGis a Lie algebra.
Let g(t) and h(t) be differentiable paths or C1 curves on G.Assume, g(0) = h(0) = e as well as dg
dt (0) = ξ and dhdt (0) = η
then we define a Lie bracket on the tangent space Te(G) via
[ξ, η] =∂2
∂t∂s[g(t),h(s)]|s=t=0
where [g,h] = ghg−1h−1 is the group commutator.
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Prescribed Velocities
DefinitionA path or curve g(t) in a Lie group G with t ∈ R is called aone-parameter subgroup if
g(t + s) = g(t)g(s).
We denote by gξ(s) the one-parameter sub-group withg′ = dg
dt (s) = ξ(s) – i.e. with prescribed “velocity” ξ(s).
DefinitionFor a Lie group G and ξ ∈ g, i.e. its Lie algebra, we define:
exp(ξ) = gξ(1)
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Prescribed Velocities
DefinitionA path or curve g(t) in a Lie group G with t ∈ R is called aone-parameter subgroup if
g(t + s) = g(t)g(s).
We denote by gξ(s) the one-parameter sub-group withg′ = dg
dt (s) = ξ(s) – i.e. with prescribed “velocity” ξ(s).
DefinitionFor a Lie group G and ξ ∈ g, i.e. its Lie algebra, we define:
exp(ξ) = gξ(1)
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Prescribed Velocities
DefinitionA path or curve g(t) in a Lie group G with t ∈ R is called aone-parameter subgroup if
g(t + s) = g(t)g(s).
We denote by gξ(s) the one-parameter sub-group withg′ = dg
dt (s) = ξ(s) – i.e. with prescribed “velocity” ξ(s).
DefinitionFor a Lie group G and ξ ∈ g, i.e. its Lie algebra, we define:
exp(ξ) = gξ(1)
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Exponentation
TheoremThe exponential map exp : g→ G maps a neighbourhood ofzero in the tangent algebra g = Te(G) diffeomorphically onto aneighbourhood of the identity in G.
TheoremLet g = a1 ⊕ . . .⊕ ak be a decomposition of a Lie algebra asdirect sum, then ξ1 + . . .+ ξk 7→ exp(ξ1) . . . exp(ξk ) maps aneighbourhood of zero in g diffeomorphically onto aneighbourhood of the identity in G.
If G is the group of invertible elements in an associative algebra(e.g. of non-singular matrices), then
exp(ξ) =∞∑
n=0
ξn
n!.
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Exponentation
TheoremThe exponential map exp : g→ G maps a neighbourhood ofzero in the tangent algebra g = Te(G) diffeomorphically onto aneighbourhood of the identity in G.
TheoremLet g = a1 ⊕ . . .⊕ ak be a decomposition of a Lie algebra asdirect sum, then ξ1 + . . .+ ξk 7→ exp(ξ1) . . . exp(ξk ) maps aneighbourhood of zero in g diffeomorphically onto aneighbourhood of the identity in G.
If G is the group of invertible elements in an associative algebra(e.g. of non-singular matrices), then
exp(ξ) =∞∑
n=0
ξn
n!.
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Exponentation
TheoremThe exponential map exp : g→ G maps a neighbourhood ofzero in the tangent algebra g = Te(G) diffeomorphically onto aneighbourhood of the identity in G.
TheoremLet g = a1 ⊕ . . .⊕ ak be a decomposition of a Lie algebra asdirect sum, then ξ1 + . . .+ ξk 7→ exp(ξ1) . . . exp(ξk ) maps aneighbourhood of zero in g diffeomorphically onto aneighbourhood of the identity in G.
If G is the group of invertible elements in an associative algebra(e.g. of non-singular matrices), then
exp(ξ) =∞∑
n=0
ξn
n!.
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Stochastic Dynamics
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Discrete Time Markov Chains (DTCM)
DefinitionA discrete time Markov chain (DTMC) on S is defined via astochastic matrix P, i.e. an r × r (square) matrix with entries0 ≤ pij ≤ 1 and such that all row sums are equal to one, i.e.∑
j
pij = 1.
This defines a discrete linear dynamical system:
Phase group: Z or N,Phase space: Rr ,Group action: π(n, x) = x · Pn.
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Discrete Time Markov Chains (DTCM)
DefinitionA discrete time Markov chain (DTMC) on S is defined via astochastic matrix P, i.e. an r × r (square) matrix with entries0 ≤ pij ≤ 1 and such that all row sums are equal to one, i.e.∑
j
pij = 1.
This defines a discrete linear dynamical system:
Phase group: Z or N,Phase space: Rr ,Group action: π(n, x) = x · Pn.
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Memoryless Property of DTMC
Let I be a finite (or maybe countable) set. Each i ∈ I is called astate or index. Given a probability space (Ω, σ,P) a randomvariable is a map X : Ω→ I.
A sequence of random variables Xn is a Markov Chain if
P(Xn+1 = i + 1 | X0 = i0,X1 = i1, . . . ,Xn = in) =
= P(Xn+1 = i + 1 | Xn = in) =
= pin,in+1P(Xn = in)
The probability P(i →n j) of reaching state (actually index) jfrom i in exactly n steps is given by p(n)
ij i.e. the entry in row iand column j of Pn.
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Memoryless Property of DTMC
Let I be a finite (or maybe countable) set. Each i ∈ I is called astate or index. Given a probability space (Ω, σ,P) a randomvariable is a map X : Ω→ I.
A sequence of random variables Xn is a Markov Chain if
P(Xn+1 = i + 1 | X0 = i0,X1 = i1, . . . ,Xn = in) =
= P(Xn+1 = i + 1 | Xn = in) =
= pin,in+1P(Xn = in)
The probability P(i →n j) of reaching state (actually index) jfrom i in exactly n steps is given by p(n)
ij i.e. the entry in row iand column j of Pn.
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Memoryless Property of DTMC
Let I be a finite (or maybe countable) set. Each i ∈ I is called astate or index. Given a probability space (Ω, σ,P) a randomvariable is a map X : Ω→ I.
A sequence of random variables Xn is a Markov Chain if
P(Xn+1 = i + 1 | X0 = i0,X1 = i1, . . . ,Xn = in) =
= P(Xn+1 = i + 1 | Xn = in) =
= pin,in+1P(Xn = in)
The probability P(i →n j) of reaching state (actually index) jfrom i in exactly n steps is given by p(n)
ij i.e. the entry in row iand column j of Pn.
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Properties
DefinitionGiven a DTMC with transition matrix P. A state i is said to be
recurrent if P(i →n i for infinitely many n = 1transient if P(i →n i for infinitely many n = 0
DefinitionA DTMC with transition matrix P is called
ergodic or irreducible: if ∀i , j ∃n such that Pnij > 0.
regular: if ∃n such that ∀i , j we have Pnij > 0.
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Properties
DefinitionGiven a DTMC with transition matrix P. A state i is said to be
recurrent if P(i →n i for infinitely many n = 1transient if P(i →n i for infinitely many n = 0
DefinitionA DTMC with transition matrix P is called
ergodic or irreducible: if ∀i , j ∃n such that Pnij > 0.
regular: if ∃n such that ∀i , j we have Pnij > 0.
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Long Run Behaviour
TheoremGiven a DTMC with transition matrix P. If it is regular and v anarbitrary probability vector. Then
limn→∞
vPn = w
where w is the unique probability vector for P.
TheoremGiven a DTMC with transition matrix P. Assume P is ergodic.Let An be the matrix defined by:
An =I + P + . . .+ Pn
n + 1
then An →W where W is a matrix all of whose rows are equalto the unique vector w for P.
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Long Run Behaviour
TheoremGiven a DTMC with transition matrix P. If it is regular and v anarbitrary probability vector. Then
limn→∞
vPn = w
where w is the unique probability vector for P.
TheoremGiven a DTMC with transition matrix P. Assume P is ergodic.Let An be the matrix defined by:
An =I + P + . . .+ Pn
n + 1
then An →W where W is a matrix all of whose rows are equalto the unique vector w for P.
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Continuous Time Markov Chains (CTMC)
DefinitionA continuous time Markov chain (CTMC) on S = s1, . . . , sr isdefined via an r × r (square) generator or Q-matrix Q = (qij)specifying the rates going from an index or state i to an index orstate j and which fullfills:
1 0 ≤ −qii <∞ for all i2 qij ≥ 0 for all i 6= j
3∑
j
qij = 0 for all i
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Continuous Time Markov Chains (CTMC)
DefinitionA continuous time Markov chain (CTMC) on S = s1, . . . , sr isdefined via an r × r (square) generator or Q-matrix Q = (qij)specifying the rates going from an index or state i to an index orstate j and which fullfills:
1 0 ≤ −qii <∞ for all i2 qij ≥ 0 for all i 6= j
3∑
j
qij = 0 for all i
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Computing Transition Probabilities
Again we use exponentation to get the transition probabilities.
P(t) = exp(tQ) =∞∑
k=0
(tQ)k
k !
This gives the unique solutions to the forward equations
ddt
P(t) = P(t)Q with P(0) = I
and the backward equation
ddt
P(t) = QP(t) with P(0) = I
and fulfills the (semi-)group property:
P(s + t) = P(s)P(t).
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Computing Transition Probabilities
Again we use exponentation to get the transition probabilities.
P(t) = exp(tQ) =∞∑
k=0
(tQ)k
k !
This gives the unique solutions to the forward equations
ddt
P(t) = P(t)Q with P(0) = I
and the backward equation
ddt
P(t) = QP(t) with P(0) = I
and fulfills the (semi-)group property:
P(s + t) = P(s)P(t).
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Dynamical Systems in Physics
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Classical Mechanics (in 10min)
Consider point particles (no volume) with mass m. The positionof a particle is given in some coordinates qi .
The velocity of the particle is given by
vi =dqi
dt= qi
its acceleration is given by
ai =dvdt
=d2qi
dt2 = qi
its momentum is defined as
pi = mqi = mdqi
dt
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Classical Mechanics (in 10min)
Consider point particles (no volume) with mass m. The positionof a particle is given in some coordinates qi .
The velocity of the particle is given by
vi =dqi
dt= qi
its acceleration is given by
ai =dvdt
=d2qi
dt2 = qi
its momentum is defined as
pi = mqi = mdqi
dt
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Classical Mechanics (in 10min)
Consider point particles (no volume) with mass m. The positionof a particle is given in some coordinates qi .
The velocity of the particle is given by
vi =dqi
dt= qi
its acceleration is given by
ai =dvdt
=d2qi
dt2 = qi
its momentum is defined as
pi = mqi = mdqi
dt
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Classical Mechanics (in 10min)
Consider point particles (no volume) with mass m. The positionof a particle is given in some coordinates qi .
The velocity of the particle is given by
vi =dqi
dt= qi
its acceleration is given by
ai =dvdt
=d2qi
dt2 = qi
its momentum is defined as
pi = mqi = mdqi
dt
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Lagrange Formalism
Describe the dynamics of a mechanical system via theLagrange function or Lagrangian
L(q1,q2, . . . ,qs, q1, q2, . . . , qs, t)
the action is defined as S =∫ t2
t1L(qi , qi , t)dt .
The Principle of Least Action then implies the Lagrangeequations which give the dynamics:
ddt∂L∂qi− ∂L∂qi
= 0
L.D. Landau and E.M. Lifschitz. Mechanik. Akademie-Verlag,Berlin, 1981.
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Lagrange Formalism
Describe the dynamics of a mechanical system via theLagrange function or Lagrangian
L(q1,q2, . . . ,qs, q1, q2, . . . , qs, t)
the action is defined as S =∫ t2
t1L(qi , qi , t)dt .
The Principle of Least Action then implies the Lagrangeequations which give the dynamics:
ddt∂L∂qi− ∂L∂qi
= 0
L.D. Landau and E.M. Lifschitz. Mechanik. Akademie-Verlag,Berlin, 1981.
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Lagrange Examples
Single Particle:L =
m2
(x2 + y2 + z2)
Pendulum: length l , angle φ, mass m, gravitational constant g
L =m2
l2φ2 + mgl cos(φ)
Double Pendulum: angles φ1 and φ2, lengths l1 and l2, massesm1 and m2 [2, p13]:
L =m1 + m2
2l21 φ
21 +
m2
2l22 φ
22 +
m2l1l2φ1φ2 cos(φ1 − φ2) +
(m1 + m2)gl1 cos(φ1) + m2gl2 cos(φ2)
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Lagrange Examples
Single Particle:L =
m2
(x2 + y2 + z2)
Pendulum: length l , angle φ, mass m, gravitational constant g
L =m2
l2φ2 + mgl cos(φ)
Double Pendulum: angles φ1 and φ2, lengths l1 and l2, massesm1 and m2 [2, p13]:
L =m1 + m2
2l21 φ
21 +
m2
2l22 φ
22 +
m2l1l2φ1φ2 cos(φ1 − φ2) +
(m1 + m2)gl1 cos(φ1) + m2gl2 cos(φ2)
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Lagrange Examples
Single Particle:L =
m2
(x2 + y2 + z2)
Pendulum: length l , angle φ, mass m, gravitational constant g
L =m2
l2φ2 + mgl cos(φ)
Double Pendulum: angles φ1 and φ2, lengths l1 and l2, massesm1 and m2 [2, p13]:
L =m1 + m2
2l21 φ
21 +
m2
2l22 φ
22 +
m2l1l2φ1φ2 cos(φ1 − φ2) +
(m1 + m2)gl1 cos(φ1) + m2gl2 cos(φ2)
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Hamiltonian Formalism
Describe the dynamics of a mechanical system via theHamilton function or Hamiltonian:
H(pi ,qi , t) =∑
i
pi qi − L(qi , qi , t)
The dynamics of the system is then described via theHamiltonian or canonical equations:
qi =dqdt
=∂H∂pi
qi =dqdt
= −∂H∂qi
L.D. Landau and E.M. Lifschitz. Mechanik. Akademie-Verlag,Berlin, 1981.
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Hamiltonian Formalism
Describe the dynamics of a mechanical system via theHamilton function or Hamiltonian:
H(pi ,qi , t) =∑
i
pi qi − L(qi , qi , t)
The dynamics of the system is then described via theHamiltonian or canonical equations:
qi =dqdt
=∂H∂pi
qi =dqdt
= −∂H∂qi
L.D. Landau and E.M. Lifschitz. Mechanik. Akademie-Verlag,Berlin, 1981.
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Hamiltonian Examples
Single Particle:
H =1
2m(p2
x + p2y + p2
z )
Particle in Field:
H =1
2m(p2
x + p2y + p2
z ) + U(x , y , z)
Pendulum: with pφ = ml2φ and φ =pφ
ml2
H =p2φ
2ml2−mgl cos(φ)
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Hamiltonian Examples
Single Particle:
H =1
2m(p2
x + p2y + p2
z )
Particle in Field:
H =1
2m(p2
x + p2y + p2
z ) + U(x , y , z)
Pendulum: with pφ = ml2φ and φ =pφ
ml2
H =p2φ
2ml2−mgl cos(φ)
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Hamiltonian Examples
Single Particle:
H =1
2m(p2
x + p2y + p2
z )
Particle in Field:
H =1
2m(p2
x + p2y + p2
z ) + U(x , y , z)
Pendulum: with pφ = ml2φ and φ =pφ
ml2
H =p2φ
2ml2−mgl cos(φ)
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Quantum Mechanics (in 20min)
Arguably, physics is ultimately about explaining experimentsand forecasting measurement results.
Observables: Entities which are (actually) measured when anexperiment is conducted on a system.
State: Entities which completely describe (or model) thesystem we are interested in.
Measurement establishes a relation between states andobservables of a given system. Dynamics describes howobservables and/or the state changes over time.
Related Questions: What is our knowledge of what? How dowe obtain this information? What is a description on how thesystem changes?
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Quantum Mechanics (in 20min)
Arguably, physics is ultimately about explaining experimentsand forecasting measurement results.
Observables: Entities which are (actually) measured when anexperiment is conducted on a system.
State: Entities which completely describe (or model) thesystem we are interested in.
Measurement establishes a relation between states andobservables of a given system. Dynamics describes howobservables and/or the state changes over time.
Related Questions: What is our knowledge of what? How dowe obtain this information? What is a description on how thesystem changes?
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Quantum Mechanics (in 20min)
Arguably, physics is ultimately about explaining experimentsand forecasting measurement results.
Observables: Entities which are (actually) measured when anexperiment is conducted on a system.
State: Entities which completely describe (or model) thesystem we are interested in.
Measurement establishes a relation between states andobservables of a given system. Dynamics describes howobservables and/or the state changes over time.
Related Questions: What is our knowledge of what? How dowe obtain this information? What is a description on how thesystem changes?
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Quantum Mechanics (in 20min)
Arguably, physics is ultimately about explaining experimentsand forecasting measurement results.
Observables: Entities which are (actually) measured when anexperiment is conducted on a system.
State: Entities which completely describe (or model) thesystem we are interested in.
Measurement establishes a relation between states andobservables of a given system. Dynamics describes howobservables and/or the state changes over time.
Related Questions: What is our knowledge of what? How dowe obtain this information? What is a description on how thesystem changes?
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Quantum Mechanics (in 20min)
Arguably, physics is ultimately about explaining experimentsand forecasting measurement results.
Observables: Entities which are (actually) measured when anexperiment is conducted on a system.
State: Entities which completely describe (or model) thesystem we are interested in.
Measurement establishes a relation between states andobservables of a given system. Dynamics describes howobservables and/or the state changes over time.
Related Questions: What is our knowledge of what? How dowe obtain this information? What is a description on how thesystem changes?
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Postulates for Quantum Mechanics (ca 1950)
The quantum state of a (free) particle is described by a(normalised) complex valued function:
~ψ ∈ L2(x) i.e.∫|ψ(x)|2dx = 1
Two quantum states can be superimposed, i.e.
α1 ~ψ1 + α2 ~ψ2
Any observable A is represented by a linear, self-adjointoperator A on L2(x).Possible measurement results: Eigenvalues of A,representing the observable A:
A~φi = λi ~φi
Probability to measure λn if the system is in state~ψ =
∑i ψi ~φi is
Pr(A = λn, ~ψ) = ‖~ψn‖2
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Postulates for Quantum Mechanics (ca 1950)
The quantum state of a (free) particle is described by a(normalised) complex valued function:
~ψ ∈ L2(x) i.e.∫|ψ(x)|2dx = 1
Two quantum states can be superimposed, i.e.
α1 ~ψ1 + α2 ~ψ2
Any observable A is represented by a linear, self-adjointoperator A on L2(x).Possible measurement results: Eigenvalues of A,representing the observable A:
A~φi = λi ~φi
Probability to measure λn if the system is in state~ψ =
∑i ψi ~φi is
Pr(A = λn, ~ψ) = ‖~ψn‖2
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Postulates for Quantum Mechanics (ca 1950)
The quantum state of a (free) particle is described by a(normalised) complex valued function:
~ψ ∈ L2(x) i.e.∫|ψ(x)|2dx = 1
Two quantum states can be superimposed, i.e.
α1 ~ψ1 + α2 ~ψ2
Any observable A is represented by a linear, self-adjointoperator A on L2(x).Possible measurement results: Eigenvalues of A,representing the observable A:
A~φi = λi ~φi
Probability to measure λn if the system is in state~ψ =
∑i ψi ~φi is
Pr(A = λn, ~ψ) = ‖~ψn‖2
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Postulates for Quantum Mechanics (ca 1950)
The quantum state of a (free) particle is described by a(normalised) complex valued function:
~ψ ∈ L2(x) i.e.∫|ψ(x)|2dx = 1
Two quantum states can be superimposed, i.e.
α1 ~ψ1 + α2 ~ψ2
Any observable A is represented by a linear, self-adjointoperator A on L2(x).Possible measurement results: Eigenvalues of A,representing the observable A:
A~φi = λi ~φi
Probability to measure λn if the system is in state~ψ =
∑i ψi ~φi is
Pr(A = λn, ~ψ) = ‖~ψn‖2
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Postulates for Quantum Mechanics (ca 1950)
The quantum state of a (free) particle is described by a(normalised) complex valued function:
~ψ ∈ L2(x) i.e.∫|ψ(x)|2dx = 1
Two quantum states can be superimposed, i.e.
α1 ~ψ1 + α2 ~ψ2
Any observable A is represented by a linear, self-adjointoperator A on L2(x).Possible measurement results: Eigenvalues of A,representing the observable A:
A~φi = λi ~φi
Probability to measure λn if the system is in state~ψ =
∑i ψi ~φi is
Pr(A = λn, ~ψ) = ‖~ψn‖2
Herbert Wiklicky Dynamical Systems
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Postulates for Quantum Mechanics
Observables and states of a system are represented byhermitian (i.e. self-adjoint) elements a of a C*-algebra A and bystates w (i.e. normalised linear functionals) over this algebra.
Possible results of measurements of an observable a aregiven by the spectrum Sp(a) of an observable. Their probabilitydistribution in a certain state w is given by the probabilitymeasure µ(w) induced by the state w on Sp(a).
Walter Thirring: Quantum Mathematical Physics, 2nd ed.Springer Verlag, 2002
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Postulates for Quantum Mechanics
Observables and states of a system are represented byhermitian (i.e. self-adjoint) elements a of a C*-algebra A and bystates w (i.e. normalised linear functionals) over this algebra.
Possible results of measurements of an observable a aregiven by the spectrum Sp(a) of an observable. Their probabilitydistribution in a certain state w is given by the probabilitymeasure µ(w) induced by the state w on Sp(a).
Walter Thirring: Quantum Mathematical Physics, 2nd ed.Springer Verlag, 2002
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Postulates for Quantum Mechanics
Observables and states of a system are represented byhermitian (i.e. self-adjoint) elements a of a C*-algebra A and bystates w (i.e. normalised linear functionals) over this algebra.
Possible results of measurements of an observable a aregiven by the spectrum Sp(a) of an observable. Their probabilitydistribution in a certain state w is given by the probabilitymeasure µ(w) induced by the state w on Sp(a).
Walter Thirring: Quantum Mathematical Physics, 2nd ed.Springer Verlag, 2002
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Quantum Mathematics
Quantum physics is often/sometimes counter-intuitive.
However, the standard mathematical model of (closed)quantum systems is relatively simple and just requires somebasic (complex) linear algebra.
The information describing the state of an (isolated)quantum mechanical system is representedmathematically by a (normalised) vector in a complexvector Hilbert space H.An observable are represented mathematically by aself-adjoint matrix (operator) A acting on H.
Two states can be combined to form a new state α |x〉+ β |y〉as long as |α|2 + |β|2 = 1 (Superposition).
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Quantum Mathematics
Quantum physics is often/sometimes counter-intuitive.
However, the standard mathematical model of (closed)quantum systems is relatively simple and just requires somebasic (complex) linear algebra.
The information describing the state of an (isolated)quantum mechanical system is representedmathematically by a (normalised) vector in a complexvector Hilbert space H.An observable are represented mathematically by aself-adjoint matrix (operator) A acting on H.
Two states can be combined to form a new state α |x〉+ β |y〉as long as |α|2 + |β|2 = 1 (Superposition).
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Quantum Mathematics
Quantum physics is often/sometimes counter-intuitive.
However, the standard mathematical model of (closed)quantum systems is relatively simple and just requires somebasic (complex) linear algebra.
The information describing the state of an (isolated)quantum mechanical system is representedmathematically by a (normalised) vector in a complexvector Hilbert space H.An observable are represented mathematically by aself-adjoint matrix (operator) A acting on H.
Two states can be combined to form a new state α |x〉+ β |y〉as long as |α|2 + |β|2 = 1 (Superposition).
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Quantum Mathematics
Quantum physics is often/sometimes counter-intuitive.
However, the standard mathematical model of (closed)quantum systems is relatively simple and just requires somebasic (complex) linear algebra.
The information describing the state of an (isolated)quantum mechanical system is representedmathematically by a (normalised) vector in a complexvector Hilbert space H.An observable are represented mathematically by aself-adjoint matrix (operator) A acting on H.
Two states can be combined to form a new state α |x〉+ β |y〉as long as |α|2 + |β|2 = 1 (Superposition).
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Quantum Mathematics
Quantum physics is often/sometimes counter-intuitive.
However, the standard mathematical model of (closed)quantum systems is relatively simple and just requires somebasic (complex) linear algebra.
The information describing the state of an (isolated)quantum mechanical system is representedmathematically by a (normalised) vector in a complexvector Hilbert space H.An observable are represented mathematically by aself-adjoint matrix (operator) A acting on H.
Two states can be combined to form a new state α |x〉+ β |y〉as long as |α|2 + |β|2 = 1 (Superposition).
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Quantum States and Notation
The state of a QM system is usually denoted by |x〉 ∈ H. Theinner product 〈x |y〉 of two vectors in H – which is describing theangle between them – is very important in QM.
P.A.M. Dirac “invented” the Bra-Ket Notation based on thefollowing simple facts:
Typewriters had no sub-scripts ~xiHilbert spaces have inner product
Simply “take inner product appart” to denote vectors in H:⟨xi , yj
⟩=⟨xi |yj
⟩= 〈i | |j〉
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Quantum States and Notation
The state of a QM system is usually denoted by |x〉 ∈ H. Theinner product 〈x |y〉 of two vectors in H – which is describing theangle between them – is very important in QM.
P.A.M. Dirac “invented” the Bra-Ket Notation based on thefollowing simple facts:
Typewriters had no sub-scripts ~xiHilbert spaces have inner product
Simply “take inner product appart” to denote vectors in H:⟨xi , yj
⟩=⟨xi |yj
⟩= 〈i | |j〉
Herbert Wiklicky Dynamical Systems
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Quantum States and Notation
The state of a QM system is usually denoted by |x〉 ∈ H. Theinner product 〈x |y〉 of two vectors in H – which is describing theangle between them – is very important in QM.
P.A.M. Dirac “invented” the Bra-Ket Notation based on thefollowing simple facts:
Typewriters had no sub-scripts ~xiHilbert spaces have inner product
Simply “take inner product appart” to denote vectors in H:⟨xi , yj
⟩=⟨xi |yj
⟩= 〈i | |j〉
Herbert Wiklicky Dynamical Systems
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Quantum States and Notation
The state of a QM system is usually denoted by |x〉 ∈ H. Theinner product 〈x |y〉 of two vectors in H – which is describing theangle between them – is very important in QM.
P.A.M. Dirac “invented” the Bra-Ket Notation based on thefollowing simple facts:
Typewriters had no sub-scripts ~xiHilbert spaces have inner product
Simply “take inner product appart” to denote vectors in H:⟨xi , yj
⟩=⟨xi |yj
⟩= 〈i | |j〉
Herbert Wiklicky Dynamical Systems
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Conventions
Physical Convention:
〈x |αy〉 = α 〈x |y〉
Mathematical Convention:
〈αx , y〉 = α 〈x , y〉
Linear in first or second argument? In mathematics we have:
〈x , αy〉 = 〈αy , x〉 = α〈y , x〉 = α 〈x , y〉
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Conventions
Physical Convention:
〈x |αy〉 = α 〈x |y〉
Mathematical Convention:
〈αx , y〉 = α 〈x , y〉
Linear in first or second argument? In mathematics we have:
〈x , αy〉 = 〈αy , x〉 = α〈y , x〉 = α 〈x , y〉
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Conventions
Physical Convention:
〈x |αy〉 = α 〈x |y〉
Mathematical Convention:
〈αx , y〉 = α 〈x , y〉
Linear in first or second argument? In mathematics we have:
〈x , αy〉 = 〈αy , x〉 = α〈y , x〉 = α 〈x , y〉
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Quantum Measurement
The expected result of measuring A of a system in state|x〉 ∈ H is given by:
〈A〉x = 〈x |A |x〉 = 〈x | |Ax〉
The only possible results are eigenvalues λi of A.The probability of measuring λn in state |x〉 is
Pr(A = λn, x) = 〈x |Pn |x〉
with Pn the (orthogonal) projection (s.t. A =∑
i λiPi )
Pn =
d(n)∑j=1
|λn, j〉 〈λn, j |
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Quantum Measurement
The expected result of measuring A of a system in state|x〉 ∈ H is given by:
〈A〉x = 〈x |A |x〉 = 〈x | |Ax〉
The only possible results are eigenvalues λi of A.The probability of measuring λn in state |x〉 is
Pr(A = λn, x) = 〈x |Pn |x〉
with Pn the (orthogonal) projection (s.t. A =∑
i λiPi )
Pn =
d(n)∑j=1
|λn, j〉 〈λn, j |
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Quantum Measurement
The expected result of measuring A of a system in state|x〉 ∈ H is given by:
〈A〉x = 〈x |A |x〉 = 〈x | |Ax〉
The only possible results are eigenvalues λi of A.The probability of measuring λn in state |x〉 is
Pr(A = λn, x) = 〈x |Pn |x〉
with Pn the (orthogonal) projection (s.t. A =∑
i λiPi )
Pn =
d(n)∑j=1
|λn, j〉 〈λn, j |
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Quantum Dynamics
The dynamics of a closed system is described by theSchrödinger Equation:
i~d |x〉
dt= H |x〉
for the (self-adjoint) Hamiltonian H.The solution is a unitary operator Ut = exp(itH).
TheoremFor any self-adjoint operator A the operator
exp(iA) = eiA =∞∑
n=0
(iA)n
n!
is a unitary operator.
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Quantum Dynamics
The dynamics of a closed system is described by theSchrödinger Equation:
i~d |x〉
dt= H |x〉
for the (self-adjoint) Hamiltonian H.The solution is a unitary operator Ut = exp(itH).
TheoremFor any self-adjoint operator A the operator
exp(iA) = eiA =∞∑
n=0
(iA)n
n!
is a unitary operator.
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Quantum Dynamics
The dynamics of a closed system is described by theSchrödinger Equation:
i~d |x〉
dt= H |x〉
for the (self-adjoint) Hamiltonian H.The solution is a unitary operator Ut = exp(itH).
TheoremFor any self-adjoint operator A the operator
exp(iA) = eiA =∞∑
n=0
(iA)n
n!
is a unitary operator.
Herbert Wiklicky Dynamical Systems
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Adjoint Operator
For a matrix A = (Aij) its transpose matrix AT is defined as
(ATij ) = (Aji)
the conjugate matrix A∗ is defined by
(A∗ij) = (Aij)∗
and the adjoint matrix A† is given via
(A†ij) = (A∗ji) or A† = (A∗)T
Notation: In mathematics the adjoint operator is usuallydenoted by A∗ and defined implicitly via:
〈A(x), y〉 = 〈x ,A∗(y)〉 or 〈A†x |y〉 = 〈x ,Ay〉
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Adjoint Operator
For a matrix A = (Aij) its transpose matrix AT is defined as
(ATij ) = (Aji)
the conjugate matrix A∗ is defined by
(A∗ij) = (Aij)∗
and the adjoint matrix A† is given via
(A†ij) = (A∗ji) or A† = (A∗)T
Notation: In mathematics the adjoint operator is usuallydenoted by A∗ and defined implicitly via:
〈A(x), y〉 = 〈x ,A∗(y)〉 or 〈A†x |y〉 = 〈x ,Ay〉
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Adjoint Operator
For a matrix A = (Aij) its transpose matrix AT is defined as
(ATij ) = (Aji)
the conjugate matrix A∗ is defined by
(A∗ij) = (Aij)∗
and the adjoint matrix A† is given via
(A†ij) = (A∗ji) or A† = (A∗)T
Notation: In mathematics the adjoint operator is usuallydenoted by A∗ and defined implicitly via:
〈A(x), y〉 = 〈x ,A∗(y)〉 or 〈A†x |y〉 = 〈x ,Ay〉
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Adjoint Operator
For a matrix A = (Aij) its transpose matrix AT is defined as
(ATij ) = (Aji)
the conjugate matrix A∗ is defined by
(A∗ij) = (Aij)∗
and the adjoint matrix A† is given via
(A†ij) = (A∗ji) or A† = (A∗)T
Notation: In mathematics the adjoint operator is usuallydenoted by A∗ and defined implicitly via:
〈A(x), y〉 = 〈x ,A∗(y)〉 or 〈A†x |y〉 = 〈x ,Ay〉
Herbert Wiklicky Dynamical Systems
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Unitary Operators
A square matrix/operator U is called unitary if
U†U = I = UU†
That means U’s inverse is U† = U−1. It also implies that U isinvertible and the inverse is easy to compute.
The postulates of Quantum Mechanics require that the timeevolution to a quantum state, e.g. a qubit, are implemented viaa unitary operator (as long as there is no measurement).
The unitary evolution of an (isolated) quantum state/system is amathematical consequence of being a solution of theSchrödinger equation for some Hamiltonian operator H.
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Unitary Operators
A square matrix/operator U is called unitary if
U†U = I = UU†
That means U’s inverse is U† = U−1. It also implies that U isinvertible and the inverse is easy to compute.
The postulates of Quantum Mechanics require that the timeevolution to a quantum state, e.g. a qubit, are implemented viaa unitary operator (as long as there is no measurement).
The unitary evolution of an (isolated) quantum state/system is amathematical consequence of being a solution of theSchrödinger equation for some Hamiltonian operator H.
Herbert Wiklicky Dynamical Systems
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Unitary Operators
A square matrix/operator U is called unitary if
U†U = I = UU†
That means U’s inverse is U† = U−1. It also implies that U isinvertible and the inverse is easy to compute.
The postulates of Quantum Mechanics require that the timeevolution to a quantum state, e.g. a qubit, are implemented viaa unitary operator (as long as there is no measurement).
The unitary evolution of an (isolated) quantum state/system is amathematical consequence of being a solution of theSchrödinger equation for some Hamiltonian operator H.
Herbert Wiklicky Dynamical Systems
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Unitary Operators
A square matrix/operator U is called unitary if
U†U = I = UU†
That means U’s inverse is U† = U−1. It also implies that U isinvertible and the inverse is easy to compute.
The postulates of Quantum Mechanics require that the timeevolution to a quantum state, e.g. a qubit, are implemented viaa unitary operator (as long as there is no measurement).
The unitary evolution of an (isolated) quantum state/system is amathematical consequence of being a solution of theSchrödinger equation for some Hamiltonian operator H.
Herbert Wiklicky Dynamical Systems
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Self Adjoint Operators
An operator A is called self-adjoint or hermitian iff
A = A†
The postulates of Quantum Mechanics require that a quantumobservable A is represented by a self-adjoint operator A.
Possible measurement results are eigenvalues λi of Adefined as
A |i〉 = λi |i〉 or A~ai = λi~ai
Probability to observe λk in state |x〉 =∑
i αi |i〉 is
Pr(A = λk , |x〉) = |αk |2
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Self Adjoint Operators
An operator A is called self-adjoint or hermitian iff
A = A†
The postulates of Quantum Mechanics require that a quantumobservable A is represented by a self-adjoint operator A.
Possible measurement results are eigenvalues λi of Adefined as
A |i〉 = λi |i〉 or A~ai = λi~ai
Probability to observe λk in state |x〉 =∑
i αi |i〉 is
Pr(A = λk , |x〉) = |αk |2
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Self Adjoint Operators
An operator A is called self-adjoint or hermitian iff
A = A†
The postulates of Quantum Mechanics require that a quantumobservable A is represented by a self-adjoint operator A.
Possible measurement results are eigenvalues λi of Adefined as
A |i〉 = λi |i〉 or A~ai = λi~ai
Probability to observe λk in state |x〉 =∑
i αi |i〉 is
Pr(A = λk , |x〉) = |αk |2
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Self Adjoint Operators
An operator A is called self-adjoint or hermitian iff
A = A†
The postulates of Quantum Mechanics require that a quantumobservable A is represented by a self-adjoint operator A.
Possible measurement results are eigenvalues λi of Adefined as
A |i〉 = λi |i〉 or A~ai = λi~ai
Probability to observe λk in state |x〉 =∑
i αi |i〉 is
Pr(A = λk , |x〉) = |αk |2
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Projections
ProjectionsAn operator P on Cn is called projection (or idempotent) iff
P2 = PP = P
Orthogonal ProjectionAn operator P on Cn is called (orthogonal) projection iff
P2 = P = P†
We say that an (orthogonal) projection P projects onto itsimage space P(Cn), which is always a linear sub-spaces of Cn.
Birkhoff-von Neumann: Projection on a Hilbert space form anortho-lattice which gives rise to non-classical a “quantum logic”.
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Projections
ProjectionsAn operator P on Cn is called projection (or idempotent) iff
P2 = PP = P
Orthogonal ProjectionAn operator P on Cn is called (orthogonal) projection iff
P2 = P = P†
We say that an (orthogonal) projection P projects onto itsimage space P(Cn), which is always a linear sub-spaces of Cn.
Birkhoff-von Neumann: Projection on a Hilbert space form anortho-lattice which gives rise to non-classical a “quantum logic”.
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Projections
ProjectionsAn operator P on Cn is called projection (or idempotent) iff
P2 = PP = P
Orthogonal ProjectionAn operator P on Cn is called (orthogonal) projection iff
P2 = P = P†
We say that an (orthogonal) projection P projects onto itsimage space P(Cn), which is always a linear sub-spaces of Cn.
Birkhoff-von Neumann: Projection on a Hilbert space form anortho-lattice which gives rise to non-classical a “quantum logic”.
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Projections
ProjectionsAn operator P on Cn is called projection (or idempotent) iff
P2 = PP = P
Orthogonal ProjectionAn operator P on Cn is called (orthogonal) projection iff
P2 = P = P†
We say that an (orthogonal) projection P projects onto itsimage space P(Cn), which is always a linear sub-spaces of Cn.
Birkhoff-von Neumann: Projection on a Hilbert space form anortho-lattice which gives rise to non-classical a “quantum logic”.
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Spectral Theorem
In the bra-ket notation we can represent a projection onto thesub-space generated by |x〉 by the outer product Px = |x〉〈x |.
TheoremA self-adjoint operator A (on a finite dimensional Hilbert space,e.g. Cn) can be represented uniquely as a linear combination
A =∑
i
λiPi
with λi ∈ R and Pi the (orthogonal) projection onto theeigen-space generated by the eigen-vector |i〉, i.e. Pi = |i〉〈i |
In the degenerate case we had to consider: Pi =∑d(n)
j=1
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Spectral Theorem
In the bra-ket notation we can represent a projection onto thesub-space generated by |x〉 by the outer product Px = |x〉〈x |.
TheoremA self-adjoint operator A (on a finite dimensional Hilbert space,e.g. Cn) can be represented uniquely as a linear combination
A =∑
i
λiPi
with λi ∈ R and Pi the (orthogonal) projection onto theeigen-space generated by the eigen-vector |i〉, i.e. Pi = |i〉〈i |
In the degenerate case we had to consider: Pi =∑d(n)
j=1
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Spectral Theorem
In the bra-ket notation we can represent a projection onto thesub-space generated by |x〉 by the outer product Px = |x〉〈x |.
TheoremA self-adjoint operator A (on a finite dimensional Hilbert space,e.g. Cn) can be represented uniquely as a linear combination
A =∑
i
λiPi
with λi ∈ R and Pi the (orthogonal) projection onto theeigen-space generated by the eigen-vector |i〉, i.e. Pi = |i〉〈i |
In the degenerate case we had to consider: Pi =∑d(n)
j=1
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Measurement Process
If we perform a measurement of the observable represented by:
A =∑
i
λi |i〉〈i |
with eigen-values λi and eigen-vectors |i〉 in a state |x〉 we haveto decompose the state according to the observable, i.e.
|x〉 =∑
i
Pi |x〉 =∑
i
|i〉〈i |x〉 =∑
i
〈i |x〉 |i〉 =∑
i
αi |i〉
With probability |αi |2 = | 〈i |x〉 |2 two things happenThe measurement instrument will the display λi .The state |x〉 collapses to |i〉.
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Measurement Process
If we perform a measurement of the observable represented by:
A =∑
i
λi |i〉〈i |
with eigen-values λi and eigen-vectors |i〉 in a state |x〉 we haveto decompose the state according to the observable, i.e.
|x〉 =∑
i
Pi |x〉 =∑
i
|i〉〈i |x〉 =∑
i
〈i |x〉 |i〉 =∑
i
αi |i〉
With probability |αi |2 = | 〈i |x〉 |2 two things happenThe measurement instrument will the display λi .The state |x〉 collapses to |i〉.
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Measurement Process
If we perform a measurement of the observable represented by:
A =∑
i
λi |i〉〈i |
with eigen-values λi and eigen-vectors |i〉 in a state |x〉 we haveto decompose the state according to the observable, i.e.
|x〉 =∑
i
Pi |x〉 =∑
i
|i〉〈i |x〉 =∑
i
〈i |x〉 |i〉 =∑
i
αi |i〉
With probability |αi |2 = | 〈i |x〉 |2 two things happenThe measurement instrument will the display λi .The state |x〉 collapses to |i〉.
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Measurement Process
If we perform a measurement of the observable represented by:
A =∑
i
λi |i〉〈i |
with eigen-values λi and eigen-vectors |i〉 in a state |x〉 we haveto decompose the state according to the observable, i.e.
|x〉 =∑
i
Pi |x〉 =∑
i
|i〉〈i |x〉 =∑
i
〈i |x〉 |i〉 =∑
i
αi |i〉
With probability |αi |2 = | 〈i |x〉 |2 two things happenThe measurement instrument will the display λi .The state |x〉 collapses to |i〉.
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Spectrum
The set of eigen-values λ1, λ2, . . . of an operator A is calledits spectrum σ(A).
σ(A) = λ | λI− A is not invertible
It is possible that for an eigen-value λi in the equation
A |i〉 = λi |i〉
we may have more than one eigen-vector |i〉, i.e. the dimensionof the eigen-space d(n) > 1. We will not consider thesedegenerate cases here.
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Spectrum
The set of eigen-values λ1, λ2, . . . of an operator A is calledits spectrum σ(A).
σ(A) = λ | λI− A is not invertible
It is possible that for an eigen-value λi in the equation
A |i〉 = λi |i〉
we may have more than one eigen-vector |i〉, i.e. the dimensionof the eigen-space d(n) > 1. We will not consider thesedegenerate cases here.
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Heisenberg and Schrödinger Picture
Describe the dynamics in terms of observables or states.
In particular if we consider not just pure (isolated) states, i.e.vectors in a Hilbert space, but instead probabilistic states whichare repesented by density matrices.
A density matrix ρ ∈ B(H) is a Hermitian semi-positive definitematrix or operator with trace(ρ) = 1. Note that a given purestate |ψ〉 can also be represented with density matrix |ψ〉〈ψ|.
The quantum dynamics can be described as for A observableand ρ state (as density matrix/operator).
Schrödinger Picture: %t = U(t)%0U∗(t).Heisenberg Picture: At = U∗(t)A0U(t).
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Heisenberg and Schrödinger Picture
Describe the dynamics in terms of observables or states.
In particular if we consider not just pure (isolated) states, i.e.vectors in a Hilbert space, but instead probabilistic states whichare repesented by density matrices.
A density matrix ρ ∈ B(H) is a Hermitian semi-positive definitematrix or operator with trace(ρ) = 1. Note that a given purestate |ψ〉 can also be represented with density matrix |ψ〉〈ψ|.
The quantum dynamics can be described as for A observableand ρ state (as density matrix/operator).
Schrödinger Picture: %t = U(t)%0U∗(t).Heisenberg Picture: At = U∗(t)A0U(t).
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Heisenberg and Schrödinger Picture
Describe the dynamics in terms of observables or states.
In particular if we consider not just pure (isolated) states, i.e.vectors in a Hilbert space, but instead probabilistic states whichare repesented by density matrices.
A density matrix ρ ∈ B(H) is a Hermitian semi-positive definitematrix or operator with trace(ρ) = 1. Note that a given purestate |ψ〉 can also be represented with density matrix |ψ〉〈ψ|.
The quantum dynamics can be described as for A observableand ρ state (as density matrix/operator).
Schrödinger Picture: %t = U(t)%0U∗(t).Heisenberg Picture: At = U∗(t)A0U(t).
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Heisenberg and Schrödinger Picture
Describe the dynamics in terms of observables or states.
In particular if we consider not just pure (isolated) states, i.e.vectors in a Hilbert space, but instead probabilistic states whichare repesented by density matrices.
A density matrix ρ ∈ B(H) is a Hermitian semi-positive definitematrix or operator with trace(ρ) = 1. Note that a given purestate |ψ〉 can also be represented with density matrix |ψ〉〈ψ|.
The quantum dynamics can be described as for A observableand ρ state (as density matrix/operator).
Schrödinger Picture: %t = U(t)%0U∗(t).Heisenberg Picture: At = U∗(t)A0U(t).
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Reversibility
Quantum DynamicsFor unitary transformations describing qubit dynamics:
U† = U−1
The quantum dynamics is invertible or reversible
Quantum MeasurementFor projection operators involved in quantum measurement:
P† 6= P−1
The quantum measurement is not reversible. However
P2 = P
The quantum measurement is idempotent.
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Reversibility
Quantum DynamicsFor unitary transformations describing qubit dynamics:
U† = U−1
The quantum dynamics is invertible or reversible
Quantum MeasurementFor projection operators involved in quantum measurement:
P† 6= P−1
The quantum measurement is not reversible. However
P2 = P
The quantum measurement is idempotent.
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Reversibility
Quantum DynamicsFor unitary transformations describing qubit dynamics:
U† = U−1
The quantum dynamics is invertible or reversible
Quantum MeasurementFor projection operators involved in quantum measurement:
P† 6= P−1
The quantum measurement is not reversible. However
P2 = P
The quantum measurement is idempotent.
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Dynamics of Programs
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pWhile – Syntax I
Full programs contain optional variable declarations:
P ::= begin S end| var D begin S end
Declarations are of the form:
r ::= bool| int| c1, . . . , cn | c1 .. cn
D ::= v : r| v : r ; D
with ci (integer) constants and r denoting ranges.
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pWhile – Syntax I
Full programs contain optional variable declarations:
P ::= begin S end| var D begin S end
Declarations are of the form:
r ::= bool| int| c1, . . . , cn | c1 .. cn
D ::= v : r| v : r ; D
with ci (integer) constants and r denoting ranges.
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pWhile – Syntax II
The syntax of statements S is as follows:
S ::= stop| skip| v := a| v ?= r| S1; S2| choose p1 : S1 or p2 : S2 ro| if b then S1 else S2 fi| while b do S od
Where the pi are constants, representing choice probabilities.
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pWhile – Syntax II
The syntax of statements S is as follows:
S ::= [stop]`
| [skip]`
| [v := a]`
| [v ?= r ]`
| S1; S2| choose` p1 : S1 or p2 : S2 ro| if [b]` then S1 else S2 fi| while [b]` do S od
Where the pi are constants, representing choice probabilities.
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Evaluation of Expressions
σ 3 State = Var→ Z ] T
To illustrate approach consider only finite sub-range of Z.Evaluation E of expressions e in state σ:
E(n)σ = nE(v)σ = σ(v)
E(a1 a2)σ = E(a1)σ E(a2)σ
E(true)σ = ttE(false)σ = ffE(not b)σ = ¬E(b)σ
. . . = . . .
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Evaluation of Expressions
σ 3 State = Var→ Z ] T
To illustrate approach consider only finite sub-range of Z.Evaluation E of expressions e in state σ:
E(n)σ = nE(v)σ = σ(v)
E(a1 a2)σ = E(a1)σ E(a2)σ
E(true)σ = ttE(false)σ = ffE(not b)σ = ¬E(b)σ
. . . = . . .
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Evaluation of Expressions
σ 3 State = Var→ Z ] T
To illustrate approach consider only finite sub-range of Z.Evaluation E of expressions e in state σ:
E(n)σ = nE(v)σ = σ(v)
E(a1 a2)σ = E(a1)σ E(a2)σ
E(true)σ = ttE(false)σ = ffE(not b)σ = ¬E(b)σ
. . . = . . .
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pWhile – SOS Semantics I
R0 〈skip, σ〉⇒1〈stop, σ〉
R1 〈stop, σ〉⇒1〈stop, σ〉
R2 〈v:=e, σ〉⇒1〈stop, σ[v 7→ E(e)σ]〉
R3 〈v?=r , σ〉⇒ 1|r|〈stop, σ[v 7→ ri ∈ r ]〉
R41〈S1, σ〉⇒p〈S′1, σ′〉
〈S1; S2, σ〉⇒p〈S′1; S2, σ′〉
R42〈S1, σ〉⇒p〈stop, σ′〉〈S1; S2, σ〉⇒p〈S2, σ
′〉
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pWhile – SOS Semantics II
R51 〈choose p1 : S1 or p2 : S2, σ〉⇒p1〈S1, σ〉
R52 〈choose p1 : S1 or p2 : S2, σ〉⇒p2〈S2, σ〉
R61 〈if b then S1 else S2, σ〉⇒1〈S1, σ〉 if E(b)σ = tt
R62 〈if b then S1 else S2, σ〉⇒1〈S2, σ〉 if E(b)σ = ff
R71 〈while b do S, σ〉⇒1〈S; while b do S, σ〉 if E(b)σ = tt
R72 〈while b do S, σ〉⇒1〈stop, σ〉 if E(b)σ = ff
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Factorial
varm : 0..2;n : 0..2;
beginm := 1;while (n>1) do
m := m*n;n := n-1;
od;stop; # loopingend
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Multi Variable State
The problem we first consider is how to describe distributionsover the cartesian product in order to represent the probabilitiesthat two or more variables have certain values.
As we have D(S) ⊆ V(S) we investigate V(S × S). In order tounderstand the relation between V(S) and V(S × S) and ingeneral V(Sn) we need to consider the tensor product.
Essential for the further treatment is the fact (more later) that
V(S × S) = V(S)⊗ V(S)
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Multi Variable State
The problem we first consider is how to describe distributionsover the cartesian product in order to represent the probabilitiesthat two or more variables have certain values.
As we have D(S) ⊆ V(S) we investigate V(S × S). In order tounderstand the relation between V(S) and V(S × S) and ingeneral V(Sn) we need to consider the tensor product.
Essential for the further treatment is the fact (more later) that
V(S × S) = V(S)⊗ V(S)
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Multi Variable State
The problem we first consider is how to describe distributionsover the cartesian product in order to represent the probabilitiesthat two or more variables have certain values.
As we have D(S) ⊆ V(S) we investigate V(S × S). In order tounderstand the relation between V(S) and V(S × S) and ingeneral V(Sn) we need to consider the tensor product.
Essential for the further treatment is the fact (more later) that
V(S × S) = V(S)⊗ V(S)
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Kronecker Product
Given a n ×m matrix A and a k × l matrix B:
A =
a11 . . . a1m...
. . ....
an1 . . . anm
B =
b11 . . . b1l...
. . ....
bk1 . . . bkl
The tensor or Kronecker product A⊗ B is a nk ×ml matrix:
A⊗ B =
a11B . . . a1mB...
. . ....
an1B . . . anmB
Special cases are square matrices (n = m and k = l) andvectors (row n = k = 1, column m = l = 1).
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Kronecker Product
Given a n ×m matrix A and a k × l matrix B:
A =
a11 . . . a1m...
. . ....
an1 . . . anm
B =
b11 . . . b1l...
. . ....
bk1 . . . bkl
The tensor or Kronecker product A⊗ B is a nk ×ml matrix:
A⊗ B =
a11B . . . a1mB...
. . ....
an1B . . . anmB
Special cases are square matrices (n = m and k = l) andvectors (row n = k = 1, column m = l = 1).
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Kronecker Product
Given a n ×m matrix A and a k × l matrix B:
A =
a11 . . . a1m...
. . ....
an1 . . . anm
B =
b11 . . . b1l...
. . ....
bk1 . . . bkl
The tensor or Kronecker product A⊗ B is a nk ×ml matrix:
A⊗ B =
a11B . . . a1mB...
. . ....
an1B . . . anmB
Special cases are square matrices (n = m and k = l) andvectors (row n = k = 1, column m = l = 1).
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Tensor Product Properties
The tensor product of n linear operators A1, A2, . . . , An isassociative (but in general not commutative) and has e.g. thefollowing properties:
1 (A1 ⊗ . . .⊗ An) · (B1 ⊗ . . .⊗ Bn) = A1 · B1 ⊗ . . .⊗ An · Bn
2 A1 ⊗ . . .⊗ (αAi)⊗ . . .⊗ An = α(A1 ⊗ . . .⊗ Ai ⊗ . . .⊗ An)
3 A1 ⊗ . . .⊗ (Ai + Bi)⊗ . . .⊗ An == (A1 ⊗ . . .⊗Ai ⊗ . . .⊗An) + (A1 ⊗ . . .⊗Bi ⊗ . . .⊗An)
4 (A1 ⊗ . . .⊗ An)∗ = A∗1 ⊗ . . .⊗ A∗n5 ‖A1 ⊗ . . .⊗ An‖ = ‖A1‖ . . . ‖An‖
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Tensor Product Properties
The tensor product of n linear operators A1, A2, . . . , An isassociative (but in general not commutative) and has e.g. thefollowing properties:
1 (A1 ⊗ . . .⊗ An) · (B1 ⊗ . . .⊗ Bn) = A1 · B1 ⊗ . . .⊗ An · Bn
2 A1 ⊗ . . .⊗ (αAi)⊗ . . .⊗ An = α(A1 ⊗ . . .⊗ Ai ⊗ . . .⊗ An)
3 A1 ⊗ . . .⊗ (Ai + Bi)⊗ . . .⊗ An == (A1 ⊗ . . .⊗Ai ⊗ . . .⊗An) + (A1 ⊗ . . .⊗Bi ⊗ . . .⊗An)
4 (A1 ⊗ . . .⊗ An)∗ = A∗1 ⊗ . . .⊗ A∗n5 ‖A1 ⊗ . . .⊗ An‖ = ‖A1‖ . . . ‖An‖
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Tensor Product Properties
The tensor product of n linear operators A1, A2, . . . , An isassociative (but in general not commutative) and has e.g. thefollowing properties:
1 (A1 ⊗ . . .⊗ An) · (B1 ⊗ . . .⊗ Bn) = A1 · B1 ⊗ . . .⊗ An · Bn
2 A1 ⊗ . . .⊗ (αAi)⊗ . . .⊗ An = α(A1 ⊗ . . .⊗ Ai ⊗ . . .⊗ An)
3 A1 ⊗ . . .⊗ (Ai + Bi)⊗ . . .⊗ An == (A1 ⊗ . . .⊗Ai ⊗ . . .⊗An) + (A1 ⊗ . . .⊗Bi ⊗ . . .⊗An)
4 (A1 ⊗ . . .⊗ An)∗ = A∗1 ⊗ . . .⊗ A∗n5 ‖A1 ⊗ . . .⊗ An‖ = ‖A1‖ . . . ‖An‖
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Tensor Product Properties
The tensor product of n linear operators A1, A2, . . . , An isassociative (but in general not commutative) and has e.g. thefollowing properties:
1 (A1 ⊗ . . .⊗ An) · (B1 ⊗ . . .⊗ Bn) = A1 · B1 ⊗ . . .⊗ An · Bn
2 A1 ⊗ . . .⊗ (αAi)⊗ . . .⊗ An = α(A1 ⊗ . . .⊗ Ai ⊗ . . .⊗ An)
3 A1 ⊗ . . .⊗ (Ai + Bi)⊗ . . .⊗ An == (A1 ⊗ . . .⊗Ai ⊗ . . .⊗An) + (A1 ⊗ . . .⊗Bi ⊗ . . .⊗An)
4 (A1 ⊗ . . .⊗ An)∗ = A∗1 ⊗ . . .⊗ A∗n5 ‖A1 ⊗ . . .⊗ An‖ = ‖A1‖ . . . ‖An‖
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Tensor Product Properties
The tensor product of n linear operators A1, A2, . . . , An isassociative (but in general not commutative) and has e.g. thefollowing properties:
1 (A1 ⊗ . . .⊗ An) · (B1 ⊗ . . .⊗ Bn) = A1 · B1 ⊗ . . .⊗ An · Bn
2 A1 ⊗ . . .⊗ (αAi)⊗ . . .⊗ An = α(A1 ⊗ . . .⊗ Ai ⊗ . . .⊗ An)
3 A1 ⊗ . . .⊗ (Ai + Bi)⊗ . . .⊗ An == (A1 ⊗ . . .⊗Ai ⊗ . . .⊗An) + (A1 ⊗ . . .⊗Bi ⊗ . . .⊗An)
4 (A1 ⊗ . . .⊗ An)∗ = A∗1 ⊗ . . .⊗ A∗n5 ‖A1 ⊗ . . .⊗ An‖ = ‖A1‖ . . . ‖An‖
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Tensor Product Properties
The tensor product of n linear operators A1, A2, . . . , An isassociative (but in general not commutative) and has e.g. thefollowing properties:
1 (A1 ⊗ . . .⊗ An) · (B1 ⊗ . . .⊗ Bn) = A1 · B1 ⊗ . . .⊗ An · Bn
2 A1 ⊗ . . .⊗ (αAi)⊗ . . .⊗ An = α(A1 ⊗ . . .⊗ Ai ⊗ . . .⊗ An)
3 A1 ⊗ . . .⊗ (Ai + Bi)⊗ . . .⊗ An == (A1 ⊗ . . .⊗Ai ⊗ . . .⊗An) + (A1 ⊗ . . .⊗Bi ⊗ . . .⊗An)
4 (A1 ⊗ . . .⊗ An)∗ = A∗1 ⊗ . . .⊗ A∗n5 ‖A1 ⊗ . . .⊗ An‖ = ‖A1‖ . . . ‖An‖
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Tensor Product Base
Every vector space has an algebraic base ei
x = x1e1 + x2e2 + . . .
This allows to specify vectors via coordinates x = (x1, x2, . . .).Base vectors are in this context simply of the form
ei = (ei1,ei2, . . .) with eij =
1 for i = j0 otherwise
The tensor product space V ⊗W can be seen as generated by(formal) tensors of the form vi ⊗wj with in vi ∈ V and w ∈ Wbase vectors.
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Tensor Product Base
Every vector space has an algebraic base ei
x = x1e1 + x2e2 + . . .
This allows to specify vectors via coordinates x = (x1, x2, . . .).Base vectors are in this context simply of the form
ei = (ei1,ei2, . . .) with eij =
1 for i = j0 otherwise
The tensor product space V ⊗W can be seen as generated by(formal) tensors of the form vi ⊗wj with in vi ∈ V and w ∈ Wbase vectors.
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Tensor Product Base
Every vector space has an algebraic base ei
x = x1e1 + x2e2 + . . .
This allows to specify vectors via coordinates x = (x1, x2, . . .).Base vectors are in this context simply of the form
ei = (ei1,ei2, . . .) with eij =
1 for i = j0 otherwise
The tensor product space V ⊗W can be seen as generated by(formal) tensors of the form vi ⊗wj with in vi ∈ V and w ∈ Wbase vectors.
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Tensor Product Base
Every vector space has an algebraic base ei
x = x1e1 + x2e2 + . . .
This allows to specify vectors via coordinates x = (x1, x2, . . .).Base vectors are in this context simply of the form
ei = (ei1,ei2, . . .) with eij =
1 for i = j0 otherwise
The tensor product space V ⊗W can be seen as generated by(formal) tensors of the form vi ⊗wj with in vi ∈ V and w ∈ Wbase vectors.
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Classical and Probabilistic State
We have (always) a finite number v of variables.
Classical state σ ∈ State given by:
σ ∈ State = (Var→ Value) = Valuev
For each variable we assume also a finite range of values.
Probabilistic state d of a single variable is a distribution overpossible values of the variable.
d ∈ V(Value) = (xc)c∈Value | xi ∈ R
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Classical and Probabilistic State
We have (always) a finite number v of variables.
Classical state σ ∈ State given by:
σ ∈ State = (Var→ Value) = Valuev
For each variable we assume also a finite range of values.
Probabilistic state d of a single variable is a distribution overpossible values of the variable.
d ∈ V(Value) = (xc)c∈Value | xi ∈ R
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Classical and Probabilistic State
We have (always) a finite number v of variables.
Classical state σ ∈ State given by:
σ ∈ State = (Var→ Value) = Valuev
For each variable we assume also a finite range of values.
Probabilistic state d of a single variable is a distribution overpossible values of the variable.
d ∈ V(Value) = (xc)c∈Value | xi ∈ R
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Classical and Probabilistic State
We have (always) a finite number v of variables.
Classical state σ ∈ State given by:
σ ∈ State = (Var→ Value) = Valuev
For each variable we assume also a finite range of values.
Probabilistic state d of a single variable is a distribution overpossible values of the variable.
d ∈ V(Value) = (xc)c∈Value | xi ∈ R
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States and Tensor Products
For finite ranges we can represent distributions over cartesianproduct as an element in the tensor product in V(Value)⊗v .
Probabilistic state d of a all variables together
d ∈ V(Var→ Value) =
= V(Value1 × Value2 × . . .× Valuev ) =
= V(Value1)⊗ V(Value2)⊗ . . .⊗ V(Valuev )
For infinite value ranges we would need to consider measures.Product measures exist, for example, by Fubini’s Theorem.
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States and Tensor Products
For finite ranges we can represent distributions over cartesianproduct as an element in the tensor product in V(Value)⊗v .
Probabilistic state d of a all variables together
d ∈ V(Var→ Value) =
= V(Value1 × Value2 × . . .× Valuev ) =
= V(Value1)⊗ V(Value2)⊗ . . .⊗ V(Valuev )
For infinite value ranges we would need to consider measures.Product measures exist, for example, by Fubini’s Theorem.
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States and Tensor Products
For finite ranges we can represent distributions over cartesianproduct as an element in the tensor product in V(Value)⊗v .
Probabilistic state d of a all variables together
d ∈ V(Var→ Value) =
= V(Value1 × Value2 × . . .× Valuev ) =
= V(Value1)⊗ V(Value2)⊗ . . .⊗ V(Valuev )
For infinite value ranges we would need to consider measures.Product measures exist, for example, by Fubini’s Theorem.
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States and Tensor Products
For finite ranges we can represent distributions over cartesianproduct as an element in the tensor product in V(Value)⊗v .
Probabilistic state d of a all variables together
d ∈ V(Var→ Value) =
= V(Value1 × Value2 × . . .× Valuev ) =
= V(Value1)⊗ V(Value2)⊗ . . .⊗ V(Valuev )
For infinite value ranges we would need to consider measures.Product measures exist, for example, by Fubini’s Theorem.
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States and Tensor Products
For finite ranges we can represent distributions over cartesianproduct as an element in the tensor product in V(Value)⊗v .
Probabilistic state d of a all variables together
d ∈ V(Var→ Value) =
= V(Value1 × Value2 × . . .× Valuev ) =
= V(Value1)⊗ V(Value2)⊗ . . .⊗ V(Valuev )
For infinite value ranges we would need to consider measures.Product measures exist, for example, by Fubini’s Theorem.
Herbert Wiklicky Dynamical Systems
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States and Tensor Products
For finite ranges we can represent distributions over cartesianproduct as an element in the tensor product in V(Value)⊗v .
Probabilistic state d of a all variables together
d ∈ V(Var→ Value) =
= V(Value1 × Value2 × . . .× Valuev ) =
= V(Value1)⊗ V(Value2)⊗ . . .⊗ V(Valuev )
For infinite value ranges we would need to consider measures.Product measures exist, for example, by Fubini’s Theorem.
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Probabilistic Control Flow
Consider the following (labelled) program:
1: while [z < 100]1 do2: choose2 1
3 : [x:=3]3 or 23 : [x:=1]4 ro
3: od4: [stop]5
Its probabilistic control flow is given by:
flow(P) = 〈1,1,2〉, 〈1,1,5〉, 〈2, 13,3〉, 〈2, 2
3,4〉, 〈3,1,1〉, 〈4,1,1〉.
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Probabilistic Control Flow
Consider the following (labelled) program:
1: while [z < 100]1 do2: choose2 1
3 : [x:=3]3 or 23 : [x:=1]4 ro
3: od4: [stop]5
Its probabilistic control flow is given by:
flow(P) = 〈1,1,2〉, 〈1,1,5〉, 〈2, 13,3〉, 〈2, 2
3,4〉, 〈3,1,1〉, 〈4,1,1〉.
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Init — First Statement
init([skip]`) = `
init([stop]`) = `
init([v:=e]`) = `
init([v?=e]`) = `
init(S1; S2) = init(S1)
init(choose` p1 : S1 or p2 : S2) = `
init(if [b]` then S1 else S2) = `
init(while [b]` do S) = `
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Final — Last Statements
final([skip]`) = `final([stop]`) = `final([v:=e]`) = `final([v?=e]`) = `final(S1; S2) = final(S2)
final(choose` p1 : S1 or p2 : S2) = final(S1) ∪ final(S2)
final(if [b]` then S1 else S2) = final(S1) ∪ final(S2)
final(while [b]` do S) = `
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Flow I — Control Transfer
flow([skip]`) = ∅flow([stop]`) = 〈`,1, `〉flow([v:=e]`) = ∅flow([v?=e]`) = ∅flow(S1; S2) = flow(S1) ∪ flow(S2) ∪
∪ (`,1, init(S2)) | ` ∈ final(S1)
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Flow ||— Control Transfer
flow(choose` p1 : S1 or p2 : S2) = flow(S1) ∪ flow(S2) ∪∪ (`,p1, init(S1)), (`,p2, init(S2))
flow(if [b]` then S1 else S2) = flow(S1) ∪ flow(S2) ∪∪ (`,1, init(S1)), (`,1, init(S2))
flow(while [b]` do S) = flow(S) ∪∪ (`,1, init(S))∪ (`′,1, `) | `′ ∈ final(S)
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Collecting Semantics
The collecting semantics of a program P is given by:
T(P) =∑
〈i,pij ,j〉∈F(P)
pij · T(`i , `j)
i.e. as a linear operator on V(Value)⊗v ⊗ V(Lab).
Local effects T(`i , `j): Data Update N + Control Step M
T(`i , `j) = Ni ⊗Mij = Ni1 ⊗ Ni2 ⊗ . . .⊗ Niv ⊗Mij
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Collecting Semantics
The collecting semantics of a program P is given by:
T(P) =∑
〈i,pij ,j〉∈F(P)
pij · T(`i , `j)
i.e. as a linear operator on V(Value)⊗v ⊗ V(Lab).
Local effects T(`i , `j): Data Update N + Control Step M
T(`i , `j) = Ni ⊗Mij = Ni1 ⊗ Ni2 ⊗ . . .⊗ Niv ⊗Mij
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Local Transfer Operators
T(`1, `2) = I⊗ E(`1, `2) for [skip]`1
T(`, `) = I⊗ E(`, `) for [stop]`
T(`1, `2) = U(v ← e)⊗ E(`1, `2) for [v:=e]`1
T(`1, `2) =(
1|r |∑
c∈r U(v := c))⊗ E(`1, `2) for [v?=r ]`1
T(`, `k ) = I⊗ E(`, `k ) for [choose]`
T(`, `t ) = P(b = tt)⊗ E(`, `t ) for [b]`
T(`, `f ) = P(b = ff)⊗ E(`, `f ) for [b]`
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Trivial Operators
Matrix Units – Represent a single transition
(E(m,n))ij =
1 if m = i ∧ n = j0 otherwise.
Identity – Represents “no change” transition
(I)ij =
1 if i = j0 otherwise.
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Tests Operators and Filters
Select a certain value c ∈ Value:
(P(c))ij =
1 if i = c = j0 otherwise.
Select a certain classical state σ ∈ State:
P(σ) =v⊗
i=1
P(σ(vi))
Select states where expression e = a | b evaluates to c:
P(e = c) =∑E(e)σ=c
P(σ)
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Tests Operators and Filters
Select a certain value c ∈ Value:
(P(c))ij =
1 if i = c = j0 otherwise.
Select a certain classical state σ ∈ State:
P(σ) =v⊗
i=1
P(σ(vi))
Select states where expression e = a | b evaluates to c:
P(e = c) =∑E(e)σ=c
P(σ)
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Tests Operators and Filters
Select a certain value c ∈ Value:
(P(c))ij =
1 if i = c = j0 otherwise.
Select a certain classical state σ ∈ State:
P(σ) =v⊗
i=1
P(σ(vi))
Select states where expression e = a | b evaluates to c:
P(e = c) =∑E(e)σ=c
P(σ)
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Selection via Projections
Filtering out relevant configurations, i.e. only those which fulfilla certain condition. Use diagonal matrix P:
(P)ii =
1 if condition holds for Ci0 otherwise.
C1C2C3C4C5C6
t
·
0 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 0
=
0C2C30
C50
t
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Selection via Projections
Filtering out relevant configurations, i.e. only those which fulfilla certain condition. Use diagonal matrix P:
(P)ii =
1 if condition holds for Ci0 otherwise.
C1C2C3C4C5C6
t
·
0 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0 1 00 0 0 0 0 0
=
0C2C30
C50
t
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Update Operators
For all initial values change to constant c ∈ Value:
(U(c))ij =
1 if j = c0 otherwise.
Set value of the k th variable vk ∈ Var to constant c ∈ Value:
U(vk ← c) =
(k−1⊗i=1
I
)⊗ U(c)⊗
(v⊗
i=k+1
I
)
Set value of variable vk ∈ Var to value given by e = a | b:
U(vk ← e) =∑
c
P(e = c)U(vk ← c)
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Update Operators
For all initial values change to constant c ∈ Value:
(U(c))ij =
1 if j = c0 otherwise.
Set value of the k th variable vk ∈ Var to constant c ∈ Value:
U(vk ← c) =
(k−1⊗i=1
I
)⊗ U(c)⊗
(v⊗
i=k+1
I
)
Set value of variable vk ∈ Var to value given by e = a | b:
U(vk ← e) =∑
c
P(e = c)U(vk ← c)
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Update Operators
For all initial values change to constant c ∈ Value:
(U(c))ij =
1 if j = c0 otherwise.
Set value of the k th variable vk ∈ Var to constant c ∈ Value:
U(vk ← c) =
(k−1⊗i=1
I
)⊗ U(c)⊗
(v⊗
i=k+1
I
)
Set value of variable vk ∈ Var to value given by e = a | b:
U(vk ← e) =∑
c
P(e = c)U(vk ← c)
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Program Approximation
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Problems and Solutions
A general approach towards problems and attempts to solvethem could be described as follows:
If the problem is to difficultformulate a simplified version,try to solve this easy problem.
Investigate the realtion between the true and theapproximate solution.
We know that program analysis is a hard (undecidible) problem.
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Problems and Solutions
A general approach towards problems and attempts to solvethem could be described as follows:
If the problem is to difficultformulate a simplified version,try to solve this easy problem.
Investigate the realtion between the true and theapproximate solution.
We know that program analysis is a hard (undecidible) problem.
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Problems and Solutions
A general approach towards problems and attempts to solvethem could be described as follows:
If the problem is to difficultformulate a simplified version,try to solve this easy problem.
Investigate the realtion between the true and theapproximate solution.
We know that program analysis is a hard (undecidible) problem.
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Problems and Solutions
A general approach towards problems and attempts to solvethem could be described as follows:
If the problem is to difficultformulate a simplified version,try to solve this easy problem.
Investigate the realtion between the true and theapproximate solution.
We know that program analysis is a hard (undecidible) problem.
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Problems and Solutions
A general approach towards problems and attempts to solvethem could be described as follows:
If the problem is to difficultformulate a simplified version,try to solve this easy problem.
Investigate the realtion between the true and theapproximate solution.
We know that program analysis is a hard (undecidible) problem.
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Problems and Solutions
A general approach towards problems and attempts to solvethem could be described as follows:
If the problem is to difficultformulate a simplified version,try to solve this easy problem.
Investigate the realtion between the true and theapproximate solution.
We know that program analysis is a hard (undecidible) problem.
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Probabilistic Program Analysis
Possible aims of Static Program Analysis:
Safe Approximations:Correct under all circumstances.Good/Close Estimates:Fix it (at runtime) if there is a problem.
With modern computer architectures some compile time tasks(type checking, threading, etc.) become runtime features.
A possible application could support Speculative Evaluation.
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Probabilistic Program Analysis
Possible aims of Static Program Analysis:
Safe Approximations:Correct under all circumstances.Good/Close Estimates:Fix it (at runtime) if there is a problem.
With modern computer architectures some compile time tasks(type checking, threading, etc.) become runtime features.
A possible application could support Speculative Evaluation.
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Probabilistic Program Analysis
Possible aims of Static Program Analysis:
Safe Approximations:Correct under all circumstances.Good/Close Estimates:Fix it (at runtime) if there is a problem.
With modern computer architectures some compile time tasks(type checking, threading, etc.) become runtime features.
A possible application could support Speculative Evaluation.
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Probabilistic Program Analysis
Possible aims of Static Program Analysis:
Safe Approximations:Correct under all circumstances.Good/Close Estimates:Fix it (at runtime) if there is a problem.
With modern computer architectures some compile time tasks(type checking, threading, etc.) become runtime features.
A possible application could support Speculative Evaluation.
Herbert Wiklicky Dynamical Systems
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Probabilistic Program Analysis
Possible aims of Static Program Analysis:
Safe Approximations:Correct under all circumstances.Good/Close Estimates:Fix it (at runtime) if there is a problem.
With modern computer architectures some compile time tasks(type checking, threading, etc.) become runtime features.
A possible application could support Speculative Evaluation.
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Semantical Abstraction
Consider a Concrete Domain C and an Abstract Domain D:
C A //
T
D
T#
C
A// D
With an abstraction A : C→ D and a concretisation G : D→ C:
T# = GTA
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Semantical Abstraction
Consider a Concrete Domain C and an Abstract Domain D:
C A //
T
D
T#
C
A// D
With an abstraction A : C→ D and a concretisation G : D→ C:
T# = GTA
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Semantical Abstraction
Consider a Concrete Domain C and an Abstract Domain D:
C A //
T
D
T#
C
A// D
With an abstraction A : C→ D and a concretisation G : D→ C:
T# = GTA
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Semantical Abstraction
Consider a Concrete Domain C and an Abstract Domain D:
C A //
T
D
T#
C
A// D
With an abstraction A : C→ D and a concretisation G : D→ C:
T# = GTA
Abstract Interpretation: (A,G) form a Galois Connection.
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Semantical Abstraction
Consider a Concrete Domain C and an Abstract Domain D:
C A //
T
D
T#
C
A// D
With an abstraction A : C→ D and a concretisation G : D→ C:
T# = GTA
Probabilistic Abst.Int.: (A,G) Moore-Penrose Pseudo-Inverse.
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Galois Connections
DefinitionLet C = (C,≤) and D = (D,v) be two partially ordered set. Ifthere are two functions α : C → D and γ : D → C such that forall c ∈ C and all d ∈ D:
c ≤C γ(d) iff α(c) v d ,
then (C, α, γ,D) form a Galois connection.
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Moore-Penrose Pseudo-Inverse I
DefinitionLet C and D be two Hilbert spaces and A : C → D a boundedlinear map. A bounded linear map A† = G : D → C is theMoore-Penrose pseudo-inverse of A iff
(i) A G = PA,(ii) G A = PG,
where PA and PG denote orthogonal projections onto theranges of A and G.
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(Orthogonal) Projections – Idempotents
On finite dimensional vector (Hilbert) spaces we have an innerproduct 〈., .〉. This allows us to define an adjoint via:
〈A(x), y〉 = 〈x ,A∗(y)〉
An operator A is self-adjoint if A = A∗.An operator A is positive, i.e. A w 0, if there exists anoperator B such that A = B∗B.An (orthogonal) projection is a self-adjoint E with EE = E.
Projections identify (closed) sub-spaces YE = Ex | x ∈ V.
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(Orthogonal) Projections – Idempotents
On finite dimensional vector (Hilbert) spaces we have an innerproduct 〈., .〉. This allows us to define an adjoint via:
〈A(x), y〉 = 〈x ,A∗(y)〉
An operator A is self-adjoint if A = A∗.An operator A is positive, i.e. A w 0, if there exists anoperator B such that A = B∗B.An (orthogonal) projection is a self-adjoint E with EE = E.
Projections identify (closed) sub-spaces YE = Ex | x ∈ V.
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(Orthogonal) Projections – Idempotents
On finite dimensional vector (Hilbert) spaces we have an innerproduct 〈., .〉. This allows us to define an adjoint via:
〈A(x), y〉 = 〈x ,A∗(y)〉
An operator A is self-adjoint if A = A∗.An operator A is positive, i.e. A w 0, if there exists anoperator B such that A = B∗B.An (orthogonal) projection is a self-adjoint E with EE = E.
Projections identify (closed) sub-spaces YE = Ex | x ∈ V.
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(Orthogonal) Projections – Idempotents
On finite dimensional vector (Hilbert) spaces we have an innerproduct 〈., .〉. This allows us to define an adjoint via:
〈A(x), y〉 = 〈x ,A∗(y)〉
An operator A is self-adjoint if A = A∗.An operator A is positive, i.e. A w 0, if there exists anoperator B such that A = B∗B.An (orthogonal) projection is a self-adjoint E with EE = E.
Projections identify (closed) sub-spaces YE = Ex | x ∈ V.
Herbert Wiklicky Dynamical Systems
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(Orthogonal) Projections – Idempotents
On finite dimensional vector (Hilbert) spaces we have an innerproduct 〈., .〉. This allows us to define an adjoint via:
〈A(x), y〉 = 〈x ,A∗(y)〉
An operator A is self-adjoint if A = A∗.An operator A is positive, i.e. A w 0, if there exists anoperator B such that A = B∗B.An (orthogonal) projection is a self-adjoint E with EE = E.
Projections identify (closed) sub-spaces YE = Ex | x ∈ V.
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Moore-Penrose Pseudo-Inverse II
DefinitionAn operator A ∈ B(H) is Moore-Penrose invertible if thereexists an element G ∈ B(H) such that:
(i) AGA = A,(ii) GAG = G,(iii) (AG)∗ = AG,(iv) (GA)∗ = GA.
If it exists G = A† is called Moore-Penrose pseudo-inverse.
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Galois Connection II
DefinitionLet C = (C,≤C) and D = (D,≤D) be two partially ordered setswith two order-preserving functions α : C 7→ D and γ : D 7→ C.Then (C, α, γ,D) form a Galois connection iff
(i) α γ is reductive i.e. ∀d ∈ D, α γ(d) ≤D d ,(ii) γ α is extensive i.e. ∀c ∈ C, c ≤C γ α(c).
Proposition
Let (C, α, γ,D) be a Galois connection. Then α and γ arequasi-inverse, i.e.
(i) α γ α = α
(ii) γ α γ = γ
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Galois Connection II
DefinitionLet C = (C,≤C) and D = (D,≤D) be two partially ordered setswith two order-preserving functions α : C 7→ D and γ : D 7→ C.Then (C, α, γ,D) form a Galois connection iff
(i) α γ is reductive i.e. ∀d ∈ D, α γ(d) ≤D d ,(ii) γ α is extensive i.e. ∀c ∈ C, c ≤C γ α(c).
Proposition
Let (C, α, γ,D) be a Galois connection. Then α and γ arequasi-inverse, i.e.
(i) α γ α = α
(ii) γ α γ = γ
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Examples of Abstractions
Parity Abstraction operator on V(1, . . . ,n) (with n even):
Ap =
1 00 11 00 1...
...0 1
A†p =
( 2n 0 2
n 0 . . . 00 2
n 0 2n . . . 2
n
)
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Examples of Abstractions
Parity Abstraction operator on V(1, . . . ,n) (with n even):
Ap =
1 00 11 00 1...
...0 1
A†p =
( 2n 0 2
n 0 . . . 00 2
n 0 2n . . . 2
n
)
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Examples of Abstractions
Sign Abstraction operator on V(−n, . . . ,0, . . . ,n):
As =
1 0 0...
......
1 0 00 1 00 0 1...
......
0 0 1
A†s =
1n . . . 1
n 0 0 . . . 00 . . . 0 1 0 . . . 00 . . . 0 0 1
n . . . 1n
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Examples of Abstractions
Sign Abstraction operator on V(−n, . . . ,0, . . . ,n):
As =
1 0 0...
......
1 0 00 1 00 0 1...
......
0 0 1
A†s =
1n . . . 1
n 0 0 . . . 00 . . . 0 1 0 . . . 00 . . . 0 0 1
n . . . 1n
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Lifting of an extraction function α : C 7→ D
Power Set lifting to an abstraction function α : P(C)→ P(D)
α(c1, c2, . . .) = α(c1), α(c2), . . .
Vector Space lifting to an abstraction function ~α : V(C)→ V(D)
~α(p1 · ~c1 + p2 · ~c2 + . . .) = pi · α(c1) + p2 · α(c2) . . .
Support Set: supp : V(C)→ P(C)
supp(~x) =
ci | 〈ci ,pi〉 ∈ ~x and pi 6= 0
Uniform Distribution: vec : P(C)→ V(C)
vec(x) = 〈ci ,1/|x |〉
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Lifting of an extraction function α : C 7→ D
Power Set lifting to an abstraction function α : P(C)→ P(D)
α(c1, c2, . . .) = α(c1), α(c2), . . .
Vector Space lifting to an abstraction function ~α : V(C)→ V(D)
~α(p1 · ~c1 + p2 · ~c2 + . . .) = pi · α(c1) + p2 · α(c2) . . .
Support Set: supp : V(C)→ P(C)
supp(~x) =
ci | 〈ci ,pi〉 ∈ ~x and pi 6= 0
Uniform Distribution: vec : P(C)→ V(C)
vec(x) = 〈ci ,1/|x |〉
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Relation between Abstractions [PPDP00]
PropositionLet ~α be a probabilistic abstraction function and let ~γ be itsMoore-Penrose pseudo-inverse.
Then ~γ ~α is extensive with respect to the inclusion on thesupport sets of vectors in V(C), i.e. ∀~x ∈ V(C),
supp(~x) ⊆ supp(~γ ~α(~x)).
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Least Square Approximation
Given a linear equationxA = b
it has either (i) a (unique) solution x , or (ii) the residual
rx = b − xA
is non-zero for all x .
The (unique) least-square solution x , i.e. for which the residual‖b − xA‖ is minimal, can be obtained using the Moore-Penrosepseudo-inverse:
x = bA†
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Least Square Approximation
Given a linear equationxA = b
it has either (i) a (unique) solution x , or (ii) the residual
rx = b − xA
is non-zero for all x .
The (unique) least-square solution x , i.e. for which the residual‖b − xA‖ is minimal, can be obtained using the Moore-Penrosepseudo-inverse:
x = bA†
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Abstract LOS Semantics
Moore-Penrose Pseudo-Inverse of a Tensor Product is simply
(A1 ⊗ A2 ⊗ . . .⊗ An)† = A†1 ⊗ A†2 ⊗ . . .⊗ A†n
Via linearity we can construct T# in the same way as T, i.e
T#(P) =∑
〈i,pij ,j〉∈F(P)
pij · T#(`i , `j)
with local abstraction of individual variables:
T#(`i , `j) = (A†1Ni1A1)⊗ (A†2Ni2A2)⊗ . . .⊗ (A†v Niv Av )⊗Mij
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Abstract LOS Semantics
Moore-Penrose Pseudo-Inverse of a Tensor Product is simply
(A1 ⊗ A2 ⊗ . . .⊗ An)† = A†1 ⊗ A†2 ⊗ . . .⊗ A†n
Via linearity we can construct T# in the same way as T, i.e
T#(P) =∑
〈i,pij ,j〉∈F(P)
pij · T#(`i , `j)
with local abstraction of individual variables:
T#(`i , `j) = (A†1Ni1A1)⊗ (A†2Ni2A2)⊗ . . .⊗ (A†v Niv Av )⊗Mij
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Proof Argument
T# = A†TA= A†(
∑i,j
T(i , j))A
=∑i,j
A†T(i , j)A
=∑i,j
(⊗
k
Ak ⊗ I)†T(i , j)(⊗
k
Ak ⊗ I)
=∑i,j
(⊗
k
Ak ⊗ I)†(⊗
k
Nik ⊗Mij)(⊗
k
Ak ⊗ I)
=∑i,j
(⊗
k
(A†kNikAk )⊗Mij)
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Proof Argument
T# = A†TA= A†(
∑i,j
T(i , j))A
=∑i,j
A†T(i , j)A
=∑i,j
(⊗
k
Ak ⊗ I)†T(i , j)(⊗
k
Ak ⊗ I)
=∑i,j
(⊗
k
Ak ⊗ I)†(⊗
k
Nik ⊗Mij)(⊗
k
Ak ⊗ I)
=∑i,j
(⊗
k
(A†kNikAk )⊗Mij)
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Proof Argument
T# = A†TA= A†(
∑i,j
T(i , j))A
=∑i,j
A†T(i , j)A
=∑i,j
(⊗
k
Ak ⊗ I)†T(i , j)(⊗
k
Ak ⊗ I)
=∑i,j
(⊗
k
Ak ⊗ I)†(⊗
k
Nik ⊗Mij)(⊗
k
Ak ⊗ I)
=∑i,j
(⊗
k
(A†kNikAk )⊗Mij)
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Proof Argument
T# = A†TA= A†(
∑i,j
T(i , j))A
=∑i,j
A†T(i , j)A
=∑i,j
(⊗
k
Ak ⊗ I)†T(i , j)(⊗
k
Ak ⊗ I)
=∑i,j
(⊗
k
Ak ⊗ I)†(⊗
k
Nik ⊗Mij)(⊗
k
Ak ⊗ I)
=∑i,j
(⊗
k
(A†kNikAk )⊗Mij)
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Proof Argument
T# = A†TA= A†(
∑i,j
T(i , j))A
=∑i,j
A†T(i , j)A
=∑i,j
(⊗
k
Ak ⊗ I)†T(i , j)(⊗
k
Ak ⊗ I)
=∑i,j
(⊗
k
Ak ⊗ I)†(⊗
k
Nik ⊗Mij)(⊗
k
Ak ⊗ I)
=∑i,j
(⊗
k
(A†kNikAk )⊗Mij)
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Proof Argument
T# = A†TA= A†(
∑i,j
T(i , j))A
=∑i,j
A†T(i , j)A
=∑i,j
(⊗
k
Ak ⊗ I)†T(i , j)(⊗
k
Ak ⊗ I)
=∑i,j
(⊗
k
Ak ⊗ I)†(⊗
k
Nik ⊗Mij)(⊗
k
Ak ⊗ I)
=∑i,j
(⊗
k
(A†kNikAk )⊗Mij)
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Example: Factorial
1: [m← 1]1;2: while [n > 1]2 do3: [m← m × n]3;4: [n← n − 1]4
5: od6: [stop]5
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Example: Factorial
1: [m← 1]1;2: while [n > 1]2 do3: [m← m × n]3;4: [n← n − 1]4
5: od6: [stop]5
T = U(m← 1)⊗ E(1,2)
+ P(n > 1)⊗ E(2,3)
+ P(n ≤ 1)⊗ E(2,5)
+ U(m← m × n)⊗ E(3,4)
+ U(n← n − 1)⊗ E(4,2)
+ I⊗ E(5,5)
The abstract versions of the local filters and updates, e.g.P#(n > 1),U#(m← m × n),U#(n← n − 1) etc. justify ourprevious ad hoc analysis.
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Example: Factorial
1: [m← 1]1;2: while [n > 1]2 do3: [m← m × n]3;4: [n← n − 1]4
5: od6: [stop]5
T# = U#(m← 1)⊗ E(1,2)
+ P#(n > 1)⊗ E(2,3)
+ P#(n ≤ 1)⊗ E(2,5)
+ U#(m← m × n)⊗ E(3,4)
+ U#(n← n − 1)⊗ E(4,2)
+ I# ⊗ E(5,5)
The abstract versions of the local filters and updates, e.g.P#(n > 1),U#(m← m × n),U#(n← n − 1) etc. justify ourprevious ad hoc analysis.
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Example: Factorial
1: [m← 1]1;2: while [n > 1]2 do3: [m← m × n]3;4: [n← n − 1]4
5: od6: [stop]5
T# = U#(m← 1)⊗ E(1,2)
+ P#(n > 1)⊗ E(2,3)
+ P#(n ≤ 1)⊗ E(2,5)
+ U#(m← m × n)⊗ E(3,4)
+ U#(n← n − 1)⊗ E(4,2)
+ I# ⊗ E(5,5)
The abstract versions of the local filters and updates, e.g.P#(n > 1),U#(m← m × n),U#(n← n − 1) etc. justify ourprevious ad hoc analysis.
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Abstract Semantics
Abstraction: A = Ap ⊗ I, i.e. m abstract (parity) but n concrete.
T# = U#(m← 1)⊗ E(1,2)
+ P#(n > 1)⊗ E(2,3)
+ P#(n ≤ 1)⊗ E(2,5)
+ U#(m← m × n)⊗ E(3,4)
+ U#(n← n − 1)⊗ E(4,2)
+ I# ⊗ E(5,5)
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Abstract Semantics
U#(m← i) =
=
(0 10 1
)⊗
1 0 0 0 . . . 00 1 0 0 . . . 00 0 1 0 . . . 00 0 0 1 . . . 0...
......
.... . .
...0 0 0 . . . 1
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Abstract Semantics
U#(n← n − 1) =
=
(1 00 1
)⊗
0 0 0 0 . . . 01 0 0 0 . . . 00 1 0 0 . . . 00 0 1 0 . . . 0...
......
.... . .
...0 0 0 0 . . . 0
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Abstract Semantics
P#(n > 1))
=
(1 00 1
)⊗
0 0 0 0 . . . 00 0 0 0 . . . 00 0 1 0 . . . 00 0 0 1 . . . 0...
......
.... . .
...0 0 0 0 . . . 1
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Abstract Semantics
P#(n ≤ 1) =
=
(1 00 1
)⊗
1 0 0 0 . . . 00 1 0 0 . . . 00 0 0 0 . . . 00 0 0 0 . . . 0...
......
.... . .
...0 0 0 0 . . . 0
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Abstract Semantics
U#(m← m × n) =
(1 00 0
)⊗
1 0 0 0 . . . 00 1 0 0 . . . 00 0 1 0 . . . 00 0 0 1 . . . 0...
......
.... . .
...0 0 0 0 . . . 1
+
+
(0 01 0
)⊗
1 0 0 0 . . . 00 0 0 0 . . . 00 0 1 0 . . . 00 0 0 0 . . . 0...
......
.... . .
...
0 0 0 0 . . .. . .
+
(0 00 1
)⊗
0 0 0 0 . . . 00 1 0 0 . . . 00 0 0 0 . . . 00 0 0 1 . . . 0...
......
.... . .
...
0 0 0 0 . . .. . .
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Implementation
Implementation of concrete and abstract semantics of Factorialusing octave. Ranges: n ∈ 1,2,max and m ∈ 1,2,max!.
n dim(T(F )) dim(T#(F ))
2 45 303 140 404 625 505 3630 606 25235 707 201640 808 1814445 909 18144050 100
Using uniform initial distributions d0 for n and m.
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Implementation
Implementation of concrete and abstract semantics of Factorialusing octave. Ranges: n ∈ 1,2,max and m ∈ 1,2,max!.
n dim(T(F )) dim(T#(F ))
2 45 303 140 404 625 505 3630 606 25235 707 201640 808 1814445 909 18144050 100
Using uniform initial distributions d0 for n and m.
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Scaleablity
The abstract probabilities for m being even or odd when weexecute the abstract program for various maximal n values are:
n even odd10 0.81818 0.18182
100 0.98019 0.0198021000 0.99800 0.0019980
10000 0.99980 0.00019998
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The End
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Bibliography
Morris W. Hirsch, Stephen Smale, and Robert L. Devaney.Differential Equations, Dynamical Systems and AnIntroduction to Chaos.Elsevier, 2004.
L.D. Landau and E.M. Lifschitz.Mechanik.Akademie-Verlag, Berlin, 1981.
Barry Simon.Representation of Finite and Compact Groups, volume 10of Graduate Studies in Mathematics.AMS, 1996.
Herbert Wiklicky Dynamical Systems