Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Symmetry PreservingDiscretization Schemes through
Hypercomplex Variables
Nelson Faustino
Center of Mathematics, Computation and Cognition, UFABC
ICNAAM 2017, Thessaloniki, Greece, 25–30 September 2017
1 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
1 The Scope of ProblemsFunction Theoretical Methods in Numerical AnalysisMotivation behind this talk
2 Lie-algebraic discretizationsUmbral Calculus RevisitedRadial algebra approachAppell Setssu(1,1) symmetries
3 Discretization of Operators of Sturm-Liouville typeDiscrete Electromagnetic Schrodinger operatorsInterplay with Bayesian Statistics
2 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
’(. . . ) When Columbus set sail, he was like an appliedmathematician, paid for the search of the solution of a concrete problem:find a way to India. His discovery of the New World was similar to thework of a pure mathematician (. . . )”Vladimir Arnol’d, Notices of AMS, Volume 44, Number 4 (1997)
Figure: From left to right: Discrete Dirac operators on graphs/dualgraphs vs. 7−point representation of the ’discrete’ Laplacian ∆h.
3 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
OverviewWhy should we use Finite Difference Dirac Operators?
Hypercomplex analysis approach:1 Useful to rewrite our main problem in a more compact form
(e.g. Lame/Navier-Stokes/Schrodinger equations);2 Get exact representation formulae to solve vector-field
problems numerically (discrete counterparts);3 The regularity conditions that we need to impose on the
design of convergence schemes are quite lower incomparison with the usual convergence conditionsassociated to standard finite difference schemes.
4 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
OverviewWhy should we use Finite Difference Dirac Operators?
Hypercomplex analysis approach:1 Useful to rewrite our main problem in a more compact form
(e.g. Lame/Navier-Stokes/Schrodinger equations);2 Get exact representation formulae to solve vector-field
problems numerically (discrete counterparts);3 The regularity conditions that we need to impose on the
design of convergence schemes are quite lower incomparison with the usual convergence conditionsassociated to standard finite difference schemes.
4 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
OverviewWhy should we use Finite Difference Dirac Operators?
Hypercomplex analysis approach:1 Useful to rewrite our main problem in a more compact form
(e.g. Lame/Navier-Stokes/Schrodinger equations);2 Get exact representation formulae to solve vector-field
problems numerically (discrete counterparts);3 The regularity conditions that we need to impose on the
design of convergence schemes are quite lower incomparison with the usual convergence conditionsassociated to standard finite difference schemes.
4 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
OverviewSome references
1 Boundary value problems: Gurlebeck and Sproßig -Quaternionic and Clifford calculus for Engineers andPhysicists (1997).
2 Discrete Fundamental solutions for Difference Diracoperators: Gurlebeck and Hommel, On finite differenceDirac operators and their fundamental solutions, Adv. Appl.Clifford Algebras, 11, 89 – 106 (2003).
3 Numerical implementation using discrete counterparts:Faustino, Gurlebeck, Hommel, and Kahler - DifferencePotentials for the Navier-Stokes equations in unboundeddomains, J. Diff. Eq. & Appl., Journal of Difference Equationsand Applications, 12(6), 577-595.
4 To take a look for further progresses on this direction, attendtomorrow (September 26) the morning talks of the13th Symposium on Clifford Analysis and Applications(ROOM 3).
5 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
OverviewSome references
1 Boundary value problems: Gurlebeck and Sproßig -Quaternionic and Clifford calculus for Engineers andPhysicists (1997).
2 Discrete Fundamental solutions for Difference Diracoperators: Gurlebeck and Hommel, On finite differenceDirac operators and their fundamental solutions, Adv. Appl.Clifford Algebras, 11, 89 – 106 (2003).
3 Numerical implementation using discrete counterparts:Faustino, Gurlebeck, Hommel, and Kahler - DifferencePotentials for the Navier-Stokes equations in unboundeddomains, J. Diff. Eq. & Appl., Journal of Difference Equationsand Applications, 12(6), 577-595.
4 To take a look for further progresses on this direction, attendtomorrow (September 26) the morning talks of the13th Symposium on Clifford Analysis and Applications(ROOM 3).
5 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
OverviewSome references
1 Boundary value problems: Gurlebeck and Sproßig -Quaternionic and Clifford calculus for Engineers andPhysicists (1997).
2 Discrete Fundamental solutions for Difference Diracoperators: Gurlebeck and Hommel, On finite differenceDirac operators and their fundamental solutions, Adv. Appl.Clifford Algebras, 11, 89 – 106 (2003).
3 Numerical implementation using discrete counterparts:Faustino, Gurlebeck, Hommel, and Kahler - DifferencePotentials for the Navier-Stokes equations in unboundeddomains, J. Diff. Eq. & Appl., Journal of Difference Equationsand Applications, 12(6), 577-595.
4 To take a look for further progresses on this direction, attendtomorrow (September 26) the morning talks of the13th Symposium on Clifford Analysis and Applications(ROOM 3).
5 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
OverviewSome references
1 Boundary value problems: Gurlebeck and Sproßig -Quaternionic and Clifford calculus for Engineers andPhysicists (1997).
2 Discrete Fundamental solutions for Difference Diracoperators: Gurlebeck and Hommel, On finite differenceDirac operators and their fundamental solutions, Adv. Appl.Clifford Algebras, 11, 89 – 106 (2003).
3 Numerical implementation using discrete counterparts:Faustino, Gurlebeck, Hommel, and Kahler - DifferencePotentials for the Navier-Stokes equations in unboundeddomains, J. Diff. Eq. & Appl., Journal of Difference Equationsand Applications, 12(6), 577-595.
4 To take a look for further progresses on this direction, attendtomorrow (September 26) the morning talks of the13th Symposium on Clifford Analysis and Applications(ROOM 3).
5 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Lie algebraic discretization schemesA rigorous way to preserve (continuous) symmetries under discretization.
The roots of Lie algebraic discretization schemes:
Discretization of finite difference operators:A.Dimakis, Muller-Hoissen and T.Striker (1996), Journal ofPhysics A: Mathematical and General, 29(21), 6861
Finite difference operators vs. symplectic solvers:Dattoli, G., Ottaviani, P. L., Torre, A., & Vazquez, L. (1997). LaRivista del Nuovo Cimento (1978-1999), 20(2), 3-133.
Motivation for the approach enclosed on this talk:
Umbral calculus: Finite difference operators as convergentpower series determined in terms of partial derivatives.
Quantum Field Theory: Provides a way to representR−polynomial algebra as the Bose algebra.
Classes of Wigner Quantal Systems: Hypercomplex analysis inits minimal form corresponds to a realization of the Lie superalgebraosp(1|2) (a refinement of the Lie algebra sl(2,R) ∼= su(1, 1)).
6 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Lie algebraic discretization schemesA rigorous way to preserve (continuous) symmetries under discretization.
The roots of Lie algebraic discretization schemes:
Discretization of finite difference operators:A.Dimakis, Muller-Hoissen and T.Striker (1996), Journal ofPhysics A: Mathematical and General, 29(21), 6861
Finite difference operators vs. symplectic solvers:Dattoli, G., Ottaviani, P. L., Torre, A., & Vazquez, L. (1997). LaRivista del Nuovo Cimento (1978-1999), 20(2), 3-133.
Motivation for the approach enclosed on this talk:
Umbral calculus: Finite difference operators as convergentpower series determined in terms of partial derivatives.
Quantum Field Theory: Provides a way to representR−polynomial algebra as the Bose algebra.
Classes of Wigner Quantal Systems: Hypercomplex analysis inits minimal form corresponds to a realization of the Lie superalgebraosp(1|2) (a refinement of the Lie algebra sl(2,R) ∼= su(1, 1)).
6 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Lie algebraic discretization schemesA rigorous way to preserve (continuous) symmetries under discretization.
The roots of Lie algebraic discretization schemes:
Discretization of finite difference operators:A.Dimakis, Muller-Hoissen and T.Striker (1996), Journal ofPhysics A: Mathematical and General, 29(21), 6861
Finite difference operators vs. symplectic solvers:Dattoli, G., Ottaviani, P. L., Torre, A., & Vazquez, L. (1997). LaRivista del Nuovo Cimento (1978-1999), 20(2), 3-133.
Motivation for the approach enclosed on this talk:
Umbral calculus: Finite difference operators as convergentpower series determined in terms of partial derivatives.
Quantum Field Theory: Provides a way to representR−polynomial algebra as the Bose algebra.
Classes of Wigner Quantal Systems: Hypercomplex analysis inits minimal form corresponds to a realization of the Lie superalgebraosp(1|2) (a refinement of the Lie algebra sl(2,R) ∼= su(1, 1)).
6 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Lie algebraic discretization schemesA rigorous way to preserve (continuous) symmetries under discretization.
The roots of Lie algebraic discretization schemes:
Discretization of finite difference operators:A.Dimakis, Muller-Hoissen and T.Striker (1996), Journal ofPhysics A: Mathematical and General, 29(21), 6861
Finite difference operators vs. symplectic solvers:Dattoli, G., Ottaviani, P. L., Torre, A., & Vazquez, L. (1997). LaRivista del Nuovo Cimento (1978-1999), 20(2), 3-133.
Motivation for the approach enclosed on this talk:
Umbral calculus: Finite difference operators as convergentpower series determined in terms of partial derivatives.
Quantum Field Theory: Provides a way to representR−polynomial algebra as the Bose algebra.
Classes of Wigner Quantal Systems: Hypercomplex analysis inits minimal form corresponds to a realization of the Lie superalgebraosp(1|2) (a refinement of the Lie algebra sl(2,R) ∼= su(1, 1)).
6 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Lie algebraic discretization schemesA rigorous way to preserve (continuous) symmetries under discretization.
The roots of Lie algebraic discretization schemes:
Discretization of finite difference operators:A.Dimakis, Muller-Hoissen and T.Striker (1996), Journal ofPhysics A: Mathematical and General, 29(21), 6861
Finite difference operators vs. symplectic solvers:Dattoli, G., Ottaviani, P. L., Torre, A., & Vazquez, L. (1997). LaRivista del Nuovo Cimento (1978-1999), 20(2), 3-133.
Motivation for the approach enclosed on this talk:
Umbral calculus: Finite difference operators as convergentpower series determined in terms of partial derivatives.
Quantum Field Theory: Provides a way to representR−polynomial algebra as the Bose algebra.
Classes of Wigner Quantal Systems: Hypercomplex analysis inits minimal form corresponds to a realization of the Lie superalgebraosp(1|2) (a refinement of the Lie algebra sl(2,R) ∼= su(1, 1)).
6 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Multivariate polynomialsBasic setting
Monomial over x = (x1, x2, . . . , xn) ∈ Rn: xα = xα11 xα2
2 . . . xαnn
Ring of polynomials R[x ]: Each P(x) ∈ R[x ] is a linearcombination of monomials xα.
Multi-index derivatives for ∂x := (∂x1 , ∂x2 , . . . , ∂xn ):
∂αx := ∂α1x1 ∂
α2x2 . . . ∂
αnxn ∈ End(R[x ]).
Multi-index enumerative notation:
α! = α1!α2! . . . αn!,(βα
)= β!
α!(β−α)!
Binomial formula:
(x + y)β =
|β|∑|α|=0
(βα
)xαyβ−α =
|β|∑|α|=0
[∂αx xβ ]x=y
α!xα
7 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Multivariate polynomialsBasic setting
Monomial over x = (x1, x2, . . . , xn) ∈ Rn: xα = xα11 xα2
2 . . . xαnn
Ring of polynomials R[x ]: Each P(x) ∈ R[x ] is a linearcombination of monomials xα.
Multi-index derivatives for ∂x := (∂x1 , ∂x2 , . . . , ∂xn ):
∂αx := ∂α1x1 ∂
α2x2 . . . ∂
αnxn ∈ End(R[x ]).
Multi-index enumerative notation:
α! = α1!α2! . . . αn!,(βα
)= β!
α!(β−α)!
Binomial formula:
(x + y)β =
|β|∑|α|=0
(βα
)xαyβ−α =
|β|∑|α|=0
[∂αx xβ ]x=y
α!xα
7 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Multivariate polynomialsBasic setting
Monomial over x = (x1, x2, . . . , xn) ∈ Rn: xα = xα11 xα2
2 . . . xαnn
Ring of polynomials R[x ]: Each P(x) ∈ R[x ] is a linearcombination of monomials xα.
Multi-index derivatives for ∂x := (∂x1 , ∂x2 , . . . , ∂xn ):
∂αx := ∂α1x1 ∂
α2x2 . . . ∂
αnxn ∈ End(R[x ]).
Multi-index enumerative notation:
α! = α1!α2! . . . αn!,(βα
)= β!
α!(β−α)!
Binomial formula:
(x + y)β =
|β|∑|α|=0
(βα
)xαyβ−α =
|β|∑|α|=0
[∂αx xβ ]x=y
α!xα
7 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Multivariate polynomialsBasic setting
Monomial over x = (x1, x2, . . . , xn) ∈ Rn: xα = xα11 xα2
2 . . . xαnn
Ring of polynomials R[x ]: Each P(x) ∈ R[x ] is a linearcombination of monomials xα.
Multi-index derivatives for ∂x := (∂x1 , ∂x2 , . . . , ∂xn ):
∂αx := ∂α1x1 ∂
α2x2 . . . ∂
αnxn ∈ End(R[x ]).
Multi-index enumerative notation:
α! = α1!α2! . . . αn!,(βα
)= β!
α!(β−α)!
Binomial formula:
(x + y)β =
|β|∑|α|=0
(βα
)xαyβ−α =
|β|∑|α|=0
[∂αx xβ ]x=y
α!xα
7 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Multivariate polynomialsBasic setting
Monomial over x = (x1, x2, . . . , xn) ∈ Rn: xα = xα11 xα2
2 . . . xαnn
Ring of polynomials R[x ]: Each P(x) ∈ R[x ] is a linearcombination of monomials xα.
Multi-index derivatives for ∂x := (∂x1 , ∂x2 , . . . , ∂xn ):
∂αx := ∂α1x1 ∂
α2x2 . . . ∂
αnxn ∈ End(R[x ]).
Multi-index enumerative notation:
α! = α1!α2! . . . αn!,(βα
)= β!
α!(β−α)!
Binomial formula:
(x + y)β =
|β|∑|α|=0
(βα
)xαyβ−α =
|β|∑|α|=0
[∂αx xβ ]x=y
α!xα
7 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Multivariate polynomialsShift-invariant operators
Taylor series representation in R[x ]: P(x + y) = exp(y · ∂x )P(x).
Shift-invariant operator: Q(∂x ) is shift-invariant iffQ(∂x ) exp(y · ∂x ) = exp(y · ∂x )Q(∂x ).
Theorem (First expansion theorem)
A linear operator Q : R[x ]→ R[x ] is shift-invariant if and only if it can beexpressed (as a convergent series) in the gradient ∂x , that is
Q =∞∑|α|=0
aαα!∂αx ,
where aα = [Qxα]x=0.
8 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Multivariate polynomialsShift-invariant operators
Taylor series representation in R[x ]: P(x + y) = exp(y · ∂x )P(x).
Shift-invariant operator: Q(∂x ) is shift-invariant iffQ(∂x ) exp(y · ∂x ) = exp(y · ∂x )Q(∂x ).
Theorem (First expansion theorem)
A linear operator Q : R[x ]→ R[x ] is shift-invariant if and only if it can beexpressed (as a convergent series) in the gradient ∂x , that is
Q =∞∑|α|=0
aαα!∂αx ,
where aα = [Qxα]x=0.
8 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Multivariate polynomialsShift-invariant operators
Taylor series representation in R[x ]: P(x + y) = exp(y · ∂x )P(x).
Shift-invariant operator: Q(∂x ) is shift-invariant iffQ(∂x ) exp(y · ∂x ) = exp(y · ∂x )Q(∂x ).
Theorem (First expansion theorem)
A linear operator Q : R[x ]→ R[x ] is shift-invariant if and only if it can beexpressed (as a convergent series) in the gradient ∂x , that is
Q =∞∑|α|=0
aαα!∂αx ,
where aα = [Qxα]x=0.
8 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Basic polynomial sequencesA way to construct polynomial sequences towards interpolation theory
Definition (Basic polynomial sequence)
A polynomial sequence {Vα}α, where Vα is a multivariate polynomial ofdegree |α| such that
1 V0(x) = 1 (Initial condition);2 Vα(0) = δα,0 (Interpolating property);3 Oxj Vα(x) = αjVα−vj (x).(Delta operator)
is called basic polynomial sequence of the multivariate deltaoperator Ox = (Ox1 ,Ox2 , . . . ,Oxn ).
9 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Basic polynomial sequencesA way to construct polynomial sequences towards interpolation theory
Definition (Basic polynomial sequence)
A polynomial sequence {Vα}α, where Vα is a multivariate polynomial ofdegree |α| such that
1 V0(x) = 1 (Initial condition);2 Vα(0) = δα,0 (Interpolating property);3 Oxj Vα(x) = αjVα−vj (x).(Delta operator)
is called basic polynomial sequence of the multivariate deltaoperator Ox = (Ox1 ,Ox2 , . . . ,Oxn ).
9 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Basic polynomial sequencesA way to construct polynomial sequences towards interpolation theory
Definition (Basic polynomial sequence)
A polynomial sequence {Vα}α, where Vα is a multivariate polynomial ofdegree |α| such that
1 V0(x) = 1 (Initial condition);2 Vα(0) = δα,0 (Interpolating property);3 Oxj Vα(x) = αjVα−vj (x).(Delta operator)
is called basic polynomial sequence of the multivariate deltaoperator Ox = (Ox1 ,Ox2 , . . . ,Oxn ).
9 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Basic polynomial sequencesA way to construct polynomial sequences towards interpolation theory
Definition (Basic polynomial sequence)
A polynomial sequence {Vα}α, where Vα is a multivariate polynomial ofdegree |α| such that
1 V0(x) = 1 (Initial condition);2 Vα(0) = δα,0 (Interpolating property);3 Oxj Vα(x) = αjVα−vj (x).(Delta operator)
is called basic polynomial sequence of the multivariate deltaoperator Ox = (Ox1 ,Ox2 , . . . ,Oxn ).
9 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Some ResultsResults may be found on my PhD dissertation entitled Discrete Clifford analysis,Universidade de Aveiro (2009)
Theorem (Basic polynomial sequences vs polynomials of binomial type)
{Vα(x)}α is a basic polynomial sequence if and only if is a sequenceof binomial type, i.e.
Vβ(x + y) =
|β|∑|α|=0
(βα
)Vα(x)Vβ−α(y).
Theorem (A. Di Bucchianico, 1999)
Let Q = Q(∂x ) be a shift invariant operator. Let Ox be a multivariatedelta operator with basic polynomial sequence {Vα}α. Then
Q =∑|α|≥0
aαα!
Oαx , with aα = [QVα(x)]x=0.
10 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Some ResultsResults may be found on my PhD dissertation entitled Discrete Clifford analysis,Universidade de Aveiro (2009)
Theorem (Basic polynomial sequences vs polynomials of binomial type)
{Vα(x)}α is a basic polynomial sequence if and only if is a sequenceof binomial type, i.e.
Vβ(x + y) =
|β|∑|α|=0
(βα
)Vα(x)Vβ−α(y).
Theorem (A. Di Bucchianico, 1999)
Let Q = Q(∂x ) be a shift invariant operator. Let Ox be a multivariatedelta operator with basic polynomial sequence {Vα}α. Then
Q =∑|α|≥0
aαα!
Oαx , with aα = [QVα(x)]x=0.
10 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Some ResultsResults may be found on my PhD dissertation entitled Discrete Clifford analysis,Universidade de Aveiro (2009)
Theorem (Expansion theorem)
Q = Q(∂x ) is uniquely determined by
Q =∞∑|α|=0
aα(x)Oαx
where the polynomials aα(x) are given by
∞∑|α|=0
aα(x)tα =QV (x , t)V (x , t)
, with V (x , t) =∑∞|α|=0
Vα(x)α!
tα.
11 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Finite difference toolbox
1 Equidistant lattice with mesh width h > 0:
hZn ={
x = (x1, . . . , xn) ∈ Rn :xh∈ Zn
}2 Forward/backward finite difference operators
(∂+jh f)(x) =
f(x + hej )− f(x)
h, (∂−j
h f)(x) =f(x)− f(x − hej )
h.
3 Translation property: ∂+jh and ∂−j
h are interrelated by(T±j
h f)(x) = f(x ± hej ) i.e.
T−jh (∂+j
h f)(x) = (∂−jh f)(x) and T+j
h (∂−jh f)(x) = (∂+j
h f)(x).
4 Product rules for finite difference operators:
∂+jh (g(x)f(x)) = (∂+j
h g)(x)f(x + hej ) + g(x)(∂+jh f )(x)
∂−jh (g(x)f(x)) = (∂−j
h g)(x)f(x − hej ) + g(x)(∂−jh f)(x).
12 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Finite difference toolbox
1 Equidistant lattice with mesh width h > 0:
hZn ={
x = (x1, . . . , xn) ∈ Rn :xh∈ Zn
}2 Forward/backward finite difference operators
(∂+jh f)(x) =
f(x + hej )− f(x)
h, (∂−j
h f)(x) =f(x)− f(x − hej )
h.
3 Translation property: ∂+jh and ∂−j
h are interrelated by(T±j
h f)(x) = f(x ± hej ) i.e.
T−jh (∂+j
h f)(x) = (∂−jh f)(x) and T+j
h (∂−jh f)(x) = (∂+j
h f)(x).
4 Product rules for finite difference operators:
∂+jh (g(x)f(x)) = (∂+j
h g)(x)f(x + hej ) + g(x)(∂+jh f )(x)
∂−jh (g(x)f(x)) = (∂−j
h g)(x)f(x − hej ) + g(x)(∂−jh f)(x).
12 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Finite difference toolbox
1 Equidistant lattice with mesh width h > 0:
hZn ={
x = (x1, . . . , xn) ∈ Rn :xh∈ Zn
}2 Forward/backward finite difference operators
(∂+jh f)(x) =
f(x + hej )− f(x)
h, (∂−j
h f)(x) =f(x)− f(x − hej )
h.
3 Translation property: ∂+jh and ∂−j
h are interrelated by(T±j
h f)(x) = f(x ± hej ) i.e.
T−jh (∂+j
h f)(x) = (∂−jh f)(x) and T+j
h (∂−jh f)(x) = (∂+j
h f)(x).
4 Product rules for finite difference operators:
∂+jh (g(x)f(x)) = (∂+j
h g)(x)f(x + hej ) + g(x)(∂+jh f )(x)
∂−jh (g(x)f(x)) = (∂−j
h g)(x)f(x − hej ) + g(x)(∂−jh f)(x).
12 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Finite difference toolbox
1 Equidistant lattice with mesh width h > 0:
hZn ={
x = (x1, . . . , xn) ∈ Rn :xh∈ Zn
}2 Forward/backward finite difference operators
(∂+jh f)(x) =
f(x + hej )− f(x)
h, (∂−j
h f)(x) =f(x)− f(x − hej )
h.
3 Translation property: ∂+jh and ∂−j
h are interrelated by(T±j
h f)(x) = f(x ± hej ) i.e.
T−jh (∂+j
h f)(x) = (∂−jh f)(x) and T+j
h (∂−jh f)(x) = (∂+j
h f)(x).
4 Product rules for finite difference operators:
∂+jh (g(x)f(x)) = (∂+j
h g)(x)f(x + hej ) + g(x)(∂+jh f )(x)
∂−jh (g(x)f(x)) = (∂−j
h g)(x)f(x − hej ) + g(x)(∂−jh f)(x).
12 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Radial-type discretizationLie-algebraic formulation
Radial-type approach: Study of finite difference operators belonging tothe algebra
Alg {Lj ,Mj , ej : j = 1, . . . , n},
1 Lj and Mj are position and momentum operators, respectively,satisfying the set of Weyl-Heisenberg algebra relations[Lj , Lk ] = [Mj ,Mk ] = 0 and [Lj ,Mk ] = δjk I
2 e1, e2, . . . , en are the generators of the Clifford algebra of signature(0, n).
Multivector operators: Basic left endomorphisms acting that act onfunctions with values on C`0,n.
Multivector derivative: L =∑n
j=1 ejLj stands the Lie-algebraiccounterpart of the Dirac operator D =
∑nj=1 ej∂xj .
Multivector multiplication: M =∑n
j=1 ejMj stands theLie-algebraic counterpart for the left multiplication of f(x) by aClifford vector X =
∑nj=1 xj ej .
13 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Radial-type discretizationLie-algebraic formulation
Radial-type approach: Study of finite difference operators belonging tothe algebra
Alg {Lj ,Mj , ej : j = 1, . . . , n},
1 Lj and Mj are position and momentum operators, respectively,satisfying the set of Weyl-Heisenberg algebra relations[Lj , Lk ] = [Mj ,Mk ] = 0 and [Lj ,Mk ] = δjk I
2 e1, e2, . . . , en are the generators of the Clifford algebra of signature(0, n).
Multivector operators: Basic left endomorphisms acting that act onfunctions with values on C`0,n.
Multivector derivative: L =∑n
j=1 ejLj stands the Lie-algebraiccounterpart of the Dirac operator D =
∑nj=1 ej∂xj .
Multivector multiplication: M =∑n
j=1 ejMj stands theLie-algebraic counterpart for the left multiplication of f(x) by aClifford vector X =
∑nj=1 xj ej .
13 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Radial-type discretizationLie-algebraic formulation
Radial-type approach: Study of finite difference operators belonging tothe algebra
Alg {Lj ,Mj , ej : j = 1, . . . , n},
1 Lj and Mj are position and momentum operators, respectively,satisfying the set of Weyl-Heisenberg algebra relations[Lj , Lk ] = [Mj ,Mk ] = 0 and [Lj ,Mk ] = δjk I
2 e1, e2, . . . , en are the generators of the Clifford algebra of signature(0, n).
Multivector operators: Basic left endomorphisms acting that act onfunctions with values on C`0,n.
Multivector derivative: L =∑n
j=1 ejLj stands the Lie-algebraiccounterpart of the Dirac operator D =
∑nj=1 ej∂xj .
Multivector multiplication: M =∑n
j=1 ejMj stands theLie-algebraic counterpart for the left multiplication of f(x) by aClifford vector X =
∑nj=1 xj ej .
13 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Radial-type discretizationLie-algebraic formulation
Radial-type approach: Study of finite difference operators belonging tothe algebra
Alg {Lj ,Mj , ej : j = 1, . . . , n},
1 Lj and Mj are position and momentum operators, respectively,satisfying the set of Weyl-Heisenberg algebra relations[Lj , Lk ] = [Mj ,Mk ] = 0 and [Lj ,Mk ] = δjk I
2 e1, e2, . . . , en are the generators of the Clifford algebra of signature(0, n).
Multivector operators: Basic left endomorphisms acting that act onfunctions with values on C`0,n.
Multivector derivative: L =∑n
j=1 ejLj stands the Lie-algebraiccounterpart of the Dirac operator D =
∑nj=1 ej∂xj .
Multivector multiplication: M =∑n
j=1 ejMj stands theLie-algebraic counterpart for the left multiplication of f(x) by aClifford vector X =
∑nj=1 xj ej .
13 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
An IntermezzoQuantum Field Theory (QFT) setting
Basic polynomial sequences (or quasi-monomials) may be constructedexplicitly by a direct application of the Quantum Field Lemma.
Fock space: Vector space (F , 〈·|·〉) such that1 F : Free algebra generated by the elements a−j and a+
j fromthe vacuum vector Φ such that a−j Φ = 0.
2 〈·|·〉: Euclidean inner product in F such that 〈Φ|Φ〉 = 1 and a+j
are adjoint to a−j , i.e. 〈a+j x |y〉 = 〈x |a−j y〉.
Bose algebra: Fock space F whose generators a±j a†j satisfy theHeisenberg-Weyl relations
[a+j , a
+k ] = 0, [a−j , a
−k ] = 0, [a−j , a
+k ] = δjk I.
Standard lemma in QFT: All the basic vectors in F have thefollowing form
ηα :=
n∏j=1
(a†j )αj
Φ
14 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
An IntermezzoQuantum Field Theory (QFT) setting
Basic polynomial sequences (or quasi-monomials) may be constructedexplicitly by a direct application of the Quantum Field Lemma.
Fock space: Vector space (F , 〈·|·〉) such that1 F : Free algebra generated by the elements a−j and a+
j fromthe vacuum vector Φ such that a−j Φ = 0.
2 〈·|·〉: Euclidean inner product in F such that 〈Φ|Φ〉 = 1 and a+j
are adjoint to a−j , i.e. 〈a+j x |y〉 = 〈x |a−j y〉.
Bose algebra: Fock space F whose generators a±j a†j satisfy theHeisenberg-Weyl relations
[a+j , a
+k ] = 0, [a−j , a
−k ] = 0, [a−j , a
+k ] = δjk I.
Standard lemma in QFT: All the basic vectors in F have thefollowing form
ηα :=
n∏j=1
(a†j )αj
Φ
14 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
An IntermezzoQuantum Field Theory (QFT) setting
Basic polynomial sequences (or quasi-monomials) may be constructedexplicitly by a direct application of the Quantum Field Lemma.
Fock space: Vector space (F , 〈·|·〉) such that1 F : Free algebra generated by the elements a−j and a+
j fromthe vacuum vector Φ such that a−j Φ = 0.
2 〈·|·〉: Euclidean inner product in F such that 〈Φ|Φ〉 = 1 and a+j
are adjoint to a−j , i.e. 〈a+j x |y〉 = 〈x |a−j y〉.
Bose algebra: Fock space F whose generators a±j a†j satisfy theHeisenberg-Weyl relations
[a+j , a
+k ] = 0, [a−j , a
−k ] = 0, [a−j , a
+k ] = δjk I.
Standard lemma in QFT: All the basic vectors in F have thefollowing form
ηα :=
n∏j=1
(a†j )αj
Φ
14 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
An IntermezzoQuantum Field Theory (QFT) setting
Basic polynomial sequences (or quasi-monomials) may be constructedexplicitly by a direct application of the Quantum Field Lemma.
Fock space: Vector space (F , 〈·|·〉) such that1 F : Free algebra generated by the elements a−j and a+
j fromthe vacuum vector Φ such that a−j Φ = 0.
2 〈·|·〉: Euclidean inner product in F such that 〈Φ|Φ〉 = 1 and a+j
are adjoint to a−j , i.e. 〈a+j x |y〉 = 〈x |a−j y〉.
Bose algebra: Fock space F whose generators a±j a†j satisfy theHeisenberg-Weyl relations
[a+j , a
+k ] = 0, [a−j , a
−k ] = 0, [a−j , a
+k ] = δjk I.
Standard lemma in QFT: All the basic vectors in F have thefollowing form
ηα :=
n∏j=1
(a†j )αj
Φ
14 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
An IntermezzoQuantum Field Theory (QFT) setting
Basic polynomial sequences (or quasi-monomials) may be constructedexplicitly by a direct application of the Quantum Field Lemma.
Fock space: Vector space (F , 〈·|·〉) such that1 F : Free algebra generated by the elements a−j and a+
j fromthe vacuum vector Φ such that a−j Φ = 0.
2 〈·|·〉: Euclidean inner product in F such that 〈Φ|Φ〉 = 1 and a+j
are adjoint to a−j , i.e. 〈a+j x |y〉 = 〈x |a−j y〉.
Bose algebra: Fock space F whose generators a±j a†j satisfy theHeisenberg-Weyl relations
[a+j , a
+k ] = 0, [a−j , a
−k ] = 0, [a−j , a
+k ] = δjk I.
Standard lemma in QFT: All the basic vectors in F have thefollowing form
ηα :=
n∏j=1
(a†j )αj
Φ
14 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Canonical discretization of finite differenceoperatorsForward/Backward differences:
First Expansion Theorem: ∂±jh = ± 1
h
(exp(±h∂xj )− I
).
Falling/Rising factorials:∏nj=1
(xjT∓jh
)αj1 =
∏nj=1 xj (xj ∓ h) . . . (xj ∓ (αj − 1)h).
Figure: Chromatic polynomial: The falling factorial counts the numberof ways to color the complete graph of order |α| with exactly
∏nj=1 xj
colors.
15 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Canonical discretization of finite differenceoperatorsForward/Backward differences:
First Expansion Theorem: ∂±jh = ± 1
h
(exp(±h∂xj )− I
).
Falling/Rising factorials:∏nj=1
(xjT∓jh
)αj1 =
∏nj=1 xj (xj ∓ h) . . . (xj ∓ (αj − 1)h).
Figure: Chromatic polynomial: The falling factorial counts the numberof ways to color the complete graph of order |α| with exactly
∏nj=1 xj
colors.
15 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Canonical discretization of finite differenceoperators of hypercomplex typeA list of examples
1 Forward finite differences: The set of operators ∂+jh and
xjT−jh : f(x) 7→ xj f(x − hej ) span the Weyl-Heisenberg algebra of
dimension 2n + 1. Moreover D+h =
∑nj=1 ej∂
+jh and
Xh =∑n
j=1 ejxjT−jh are the corresponding multivector ladder
operators on the lattice hZn.2 Backward finite differences: ∂−j
h and xjT+jh : f(x) 7→ xj f(x + hej )
also span the Weyl-Heisenberg algebra of dimension 2n + 1. Thisturns out D−h =
∑nj=1 ej∂
−jh and X−h =
∑nj=1 ejxjT
+jh as the
corresponding multivector ladder operators on the lattice hZn.3 Discretization of the Hermite operator: D+
h and Xh − D−h areobtained from the set of ladder operators Lj = ∂+j
h andLj = xjT
−jh − ∂
−jh . Moreover Xh − D−h generate hypercomplex
extensions of the Poisson-Charlier polynomials (cf. N.F., Appl.Math. Comp., Vol. 247, pp. 607-622, 2014).
16 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Canonical discretization of finite differenceoperators of hypercomplex typeA list of examples
1 Forward finite differences: The set of operators ∂+jh and
xjT−jh : f(x) 7→ xj f(x − hej ) span the Weyl-Heisenberg algebra of
dimension 2n + 1. Moreover D+h =
∑nj=1 ej∂
+jh and
Xh =∑n
j=1 ejxjT−jh are the corresponding multivector ladder
operators on the lattice hZn.2 Backward finite differences: ∂−j
h and xjT+jh : f(x) 7→ xj f(x + hej )
also span the Weyl-Heisenberg algebra of dimension 2n + 1. Thisturns out D−h =
∑nj=1 ej∂
−jh and X−h =
∑nj=1 ejxjT
+jh as the
corresponding multivector ladder operators on the lattice hZn.3 Discretization of the Hermite operator: D+
h and Xh − D−h areobtained from the set of ladder operators Lj = ∂+j
h andLj = xjT
−jh − ∂
−jh . Moreover Xh − D−h generate hypercomplex
extensions of the Poisson-Charlier polynomials (cf. N.F., Appl.Math. Comp., Vol. 247, pp. 607-622, 2014).
16 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Canonical discretization of finite differenceoperators of hypercomplex typeA list of examples
1 Forward finite differences: The set of operators ∂+jh and
xjT−jh : f(x) 7→ xj f(x − hej ) span the Weyl-Heisenberg algebra of
dimension 2n + 1. Moreover D+h =
∑nj=1 ej∂
+jh and
Xh =∑n
j=1 ejxjT−jh are the corresponding multivector ladder
operators on the lattice hZn.2 Backward finite differences: ∂−j
h and xjT+jh : f(x) 7→ xj f(x + hej )
also span the Weyl-Heisenberg algebra of dimension 2n + 1. Thisturns out D−h =
∑nj=1 ej∂
−jh and X−h =
∑nj=1 ejxjT
+jh as the
corresponding multivector ladder operators on the lattice hZn.3 Discretization of the Hermite operator: D+
h and Xh − D−h areobtained from the set of ladder operators Lj = ∂+j
h andLj = xjT
−jh − ∂
−jh . Moreover Xh − D−h generate hypercomplex
extensions of the Poisson-Charlier polynomials (cf. N.F., Appl.Math. Comp., Vol. 247, pp. 607-622, 2014).
16 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Quasi-Monomiality ApproachThe Exponential Generating Function (EGF) approach
Many degrees of freedom for choose discretization operators:(cf. N.F., SIGMA 9 (2013), 065) The set of operators(xj + h
2
)T+j
h : f(x) 7→(xj + h
2
)f(x + hej ) and(
xj − h2
)T−j
h : f(x) 7→(xj − h
2
)f(x − hej ) satisfy[
∂−jh ,
(xk +
h2
)T+k
h
]=
[∂+j
h ,
(xk −
h2
)T−k
h
]= δjk I
cf. N. F. Appl. Math. Comp., 2014
The EGF of the form
Gh(x , y ;κ) =∏n
j=11
κ( 1
h log (1 + hyj )) (1 + hyj )
xjh
yield the set of operators Lj = ∂+jh and Mj =
(xj − κ′(∂xj )κ
(∂xj
)−1)
T−jh
as generators of the Weyl-Heisenberg algebra of dimension 2n + 1.Moreover, they are unique.
17 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Quasi-Monomiality ApproachThe Exponential Generating Function (EGF) approach
Proposition (N.F., Appl. Math. Comp., 2014)
Let κ(t) defined as above and Xh the multiplication operator. If there is amulti-variable function λ(y) (y ∈ Rn) such that
λ
(D+
h exp(x · y)
exp(x · y)
)=
n∏j=1
κ(yj )
then the Fourier dual Λh of D+h is given by
Λh = Xh −[log λ
(D+
h
), x].
Quasi-Monomiality formulation: Based on Fock space formalism onecan construct each Clifford-vector-valued polynomial wk (x ; h;λ) of orderk by means of the operational rule
wk (x ; h;λ) = µk (Λh)k a, a ∈ C`0,n.
18 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Quasi-Monomiality ApproachThe Exponential Generating Function (EGF) approach
Appell set definition: {wk (x ; h;λ) : k ∈ N0} is an Appell setcarrying D+
h if w0(x ; h;λ) = a is a Clifford number and D+h wk (x ; h;λ)
is a Clifford-vector-valued polynomial of degree k − 1 satisfyingD+
h wk (x ; h;λ) = kwk−1(x ; h;λ).
Appell set equivalent formulation: Find for each (x , t) ∈ hZn × Ra EGF Gh(x , t ;λ) satisfying the set of equations
D+h Gh(x , t ;λ) = tGh(x , t ;λ) for (x , t) ∈ hZn × R \ {0}
Gh(x , 0;λ) = a for x ∈ hZn.
Bessel type hypergeometric functions:
Gh(x , t ;λ) = 0F1
(n2
;− t2
4(Λh)2
)a + tΛh 0F1
(n2
+ 1;− t2
4(Λh)2
)a
= Γ(n
2
)( tΛh
2
)− n2 +1 (
J n2−1(tΛh)a + n J n
2(tΛh)
)a
19 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Quasi-Monomiality ApproachThe Exponential Generating Function (EGF) approach
Appell set definition: {wk (x ; h;λ) : k ∈ N0} is an Appell setcarrying D+
h if w0(x ; h;λ) = a is a Clifford number and D+h wk (x ; h;λ)
is a Clifford-vector-valued polynomial of degree k − 1 satisfyingD+
h wk (x ; h;λ) = kwk−1(x ; h;λ).
Appell set equivalent formulation: Find for each (x , t) ∈ hZn × Ra EGF Gh(x , t ;λ) satisfying the set of equations
D+h Gh(x , t ;λ) = tGh(x , t ;λ) for (x , t) ∈ hZn × R \ {0}
Gh(x , 0;λ) = a for x ∈ hZn.
Bessel type hypergeometric functions:
Gh(x , t ;λ) = 0F1
(n2
;− t2
4(Λh)2
)a + tΛh 0F1
(n2
+ 1;− t2
4(Λh)2
)a
= Γ(n
2
)( tΛh
2
)− n2 +1 (
J n2−1(tΛh)a + n J n
2(tΛh)
)a
19 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Quasi-Monomiality ApproachThe Exponential Generating Function (EGF) approach
Appell set definition: {wk (x ; h;λ) : k ∈ N0} is an Appell setcarrying D+
h if w0(x ; h;λ) = a is a Clifford number and D+h wk (x ; h;λ)
is a Clifford-vector-valued polynomial of degree k − 1 satisfyingD+
h wk (x ; h;λ) = kwk−1(x ; h;λ).
Appell set equivalent formulation: Find for each (x , t) ∈ hZn × Ra EGF Gh(x , t ;λ) satisfying the set of equations
D+h Gh(x , t ;λ) = tGh(x , t ;λ) for (x , t) ∈ hZn × R \ {0}
Gh(x , 0;λ) = a for x ∈ hZn.
Bessel type hypergeometric functions:
Gh(x , t ;λ) = 0F1
(n2
;− t2
4(Λh)2
)a + tΛh 0F1
(n2
+ 1;− t2
4(Λh)2
)a
= Γ(n
2
)( tΛh
2
)− n2 +1 (
J n2−1(tΛh)a + n J n
2(tΛh)
)a
19 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Why one needs su(1,1) based symmetries?The Weyl-Heisenberg symmetry breaking
Main Goal:
For a given polynomial w(t) of degree 1, with µ = ∂+jh w(xj ) = ∂−j
h w(xj ),study the spectra of the coupled eigenvalue problem
E+h f(x) = E−h f(x) = εf(x)
carrying E±h =∑n
j=1 µ−1w
(xj ± h
2
)∂±j
h .
Drawback: The set of operators∂+j
h , ∂−jh ,W−j
h = µ−1w(xj + h
2
)T−j
h ,W+jh = µ−1w
(xj + h
2
)T+j
h and I,with j = 1, 2, . . . , n, do not endow a canonical realization of anWeyl-Heisenberg type algebra of dimension 4n + 1.
Fill the Weyl-Heisenberg gap: The set of operatorsW−j
h = µ−1w(xj + h
2
)T−j
h ,W+jh = µ−1w
(xj + h
2
)T+j
h andWj = µ−1w (xj ) I generate a Lie algebra isomorphic to sl(2n,R)(N.F., SIGMA 9 (2013), 065– Lemma 1).
20 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Why one needs su(1,1) based symmetries?The Weyl-Heisenberg symmetry breaking
Main Goal:
For a given polynomial w(t) of degree 1, with µ = ∂+jh w(xj ) = ∂−j
h w(xj ),study the spectra of the coupled eigenvalue problem
E+h f(x) = E−h f(x) = εf(x)
carrying E±h =∑n
j=1 µ−1w
(xj ± h
2
)∂±j
h .
Drawback: The set of operators∂+j
h , ∂−jh ,W−j
h = µ−1w(xj + h
2
)T−j
h ,W+jh = µ−1w
(xj + h
2
)T+j
h and I,with j = 1, 2, . . . , n, do not endow a canonical realization of anWeyl-Heisenberg type algebra of dimension 4n + 1.
Fill the Weyl-Heisenberg gap: The set of operatorsW−j
h = µ−1w(xj + h
2
)T−j
h ,W+jh = µ−1w
(xj + h
2
)T+j
h andWj = µ−1w (xj ) I generate a Lie algebra isomorphic to sl(2n,R)(N.F., SIGMA 9 (2013), 065– Lemma 1).
20 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Why one needs su(1,1) based symmetries?The Weyl-Heisenberg symmetry breaking
Main Goal:
For a given polynomial w(t) of degree 1, with µ = ∂+jh w(xj ) = ∂−j
h w(xj ),study the spectra of the coupled eigenvalue problem
E+h f(x) = E−h f(x) = εf(x)
carrying E±h =∑n
j=1 µ−1w
(xj ± h
2
)∂±j
h .
Drawback: The set of operators∂+j
h , ∂−jh ,W−j
h = µ−1w(xj + h
2
)T−j
h ,W+jh = µ−1w
(xj + h
2
)T+j
h and I,with j = 1, 2, . . . , n, do not endow a canonical realization of anWeyl-Heisenberg type algebra of dimension 4n + 1.
Fill the Weyl-Heisenberg gap: The set of operatorsW−j
h = µ−1w(xj + h
2
)T−j
h ,W+jh = µ−1w
(xj + h
2
)T+j
h andWj = µ−1w (xj ) I generate a Lie algebra isomorphic to sl(2n,R)(N.F., SIGMA 9 (2013), 065– Lemma 1).
20 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Application to Cauchy problemsN.F. SIGMA, 2013
Homogeneous Cauchy problem in [0,∞)× hZn:
∂tg(t , x) + E+
h g(t , x)− E−h g(t , x) = 0 , t > 0g(0, x) = f(x) , t = 0E+
h g(t , x) = E−h g(t , x) , t ≥ 0.
Semigroup action: The one-parameter representationEh(t) = exp(tE−h − tE+
h ) of the Lie group SU(1, 1) yieldsg(t , x) = Eh(t)f(x) as a polynomial solution of the abovehomogeneous Cauchy problem.
Discrete series connection: One can see that the semigroup(Eh(t))t≥0 gives, in particular, a direct link between the positiveseries representation of SU(1, 1) with the negative ones.
21 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Application to Cauchy problemsN.F. SIGMA, 2013
Homogeneous Cauchy problem in [0,∞)× hZn:
∂tg(t , x) + E+
h g(t , x)− E−h g(t , x) = 0 , t > 0g(0, x) = f(x) , t = 0E+
h g(t , x) = E−h g(t , x) , t ≥ 0.
Semigroup action: The one-parameter representationEh(t) = exp(tE−h − tE+
h ) of the Lie group SU(1, 1) yieldsg(t , x) = Eh(t)f(x) as a polynomial solution of the abovehomogeneous Cauchy problem.
Discrete series connection: One can see that the semigroup(Eh(t))t≥0 gives, in particular, a direct link between the positiveseries representation of SU(1, 1) with the negative ones.
21 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Application to Cauchy problemsN.F. SIGMA, 2013
Homogeneous Cauchy problem in [0,∞)× hZn:
∂tg(t , x) + E+
h g(t , x)− E−h g(t , x) = 0 , t > 0g(0, x) = f(x) , t = 0E+
h g(t , x) = E−h g(t , x) , t ≥ 0.
Semigroup action: The one-parameter representationEh(t) = exp(tE−h − tE+
h ) of the Lie group SU(1, 1) yieldsg(t , x) = Eh(t)f(x) as a polynomial solution of the abovehomogeneous Cauchy problem.
Discrete series connection: One can see that the semigroup(Eh(t))t≥0 gives, in particular, a direct link between the positiveseries representation of SU(1, 1) with the negative ones.
21 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Discrete electromagnetic Schrodingeroperators on the lattice
Discrete electromagnetic Schrodinger operators Lh on hZn
Lhf(x) =1
2µ
n∑j=1
(2
qhf(x)− ah(xj )f(x + hej )− ah(xj − h)f(x − hej )
)+ q Φh(x)f(x).
µ - mass
q- electric charge
ah(x) =n∑
j=1
ejah(xj ) - discrete magnetic potential.
Φh(x)- electric potential.
22 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Discrete electromagnetic Schrodingeroperators on the lattice
Discrete electromagnetic Schrodinger operators Lh on hZn
Lhf(x) =1
2µ
n∑j=1
(2
qhf(x)− ah(xj )f(x + hej )− ah(xj − h)f(x − hej )
)+ q Φh(x)f(x).
µ - mass
q- electric charge
ah(x) =n∑
j=1
ejah(xj ) - discrete magnetic potential.
Φh(x)- electric potential.
22 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Discrete electromagnetic Schrodingeroperators on the lattice
Discrete electromagnetic Schrodinger operators Lh on hZn
Lhf(x) =1
2µ
n∑j=1
(2
qhf(x)− ah(xj )f(x + hej )− ah(xj − h)f(x − hej )
)+ q Φh(x)f(x).
µ - mass
q- electric charge
ah(x) =n∑
j=1
ejah(xj ) - discrete magnetic potential.
Φh(x)- electric potential.
22 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Discrete electromagnetic Schrodingeroperators on the lattice
Discrete electromagnetic Schrodinger operators Lh on hZn
Lhf(x) =1
2µ
n∑j=1
(2
qhf(x)− ah(xj )f(x + hej )− ah(xj − h)f(x − hej )
)+ q Φh(x)f(x).
µ - mass
q- electric charge
ah(x) =n∑
j=1
ejah(xj ) - discrete magnetic potential.
Φh(x)- electric potential.
22 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Discrete electromagnetic Schrodingeroperators on the lattice
Discrete electromagnetic Schrodinger operators Lh on hZn
Lhf(x) =1
2µ
n∑j=1
(2
qhf(x)− ah(xj )f(x + hej )− ah(xj − h)f(x − hej )
)+ q Φh(x)f(x).
µ - mass
q- electric charge
ah(x) =n∑
j=1
ejah(xj ) - discrete magnetic potential.
Φh(x)- electric potential.
22 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Asymptotic approximation of a Sturm-Liouvilleproblem
Discrete electromagnetic Schrodinger operators Lh on hZn
Lhf(x) = − h2
2µ
n∑j=1
∂
∂xj
(w(
xj
qh
)∂f∂xj
(x)
)+ V
(xh
)f(x) + O
(h3).
The above asymptotic approximation is satisfied whenever:1 Asymptotic constraint associated to the discrete magnetic
potential:
ah(x) =n∑
j=1
ej w(
1q
xj
h
)(1 + O (h)) .
2 Asymptotic constraint associated to the discrete magneticpotential:
qΦh(x) +1
2µ
n∑j=1
(2
qh− ah(xj )− ah(xj − h)
)= V
(xh
)+ O
(h3).
23 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Asymptotic approximation of a Sturm-Liouvilleproblem
Discrete electromagnetic Schrodinger operators Lh on hZn
Lhf(x) = − h2
2µ
n∑j=1
∂
∂xj
(w(
xj
qh
)∂f∂xj
(x)
)+ V
(xh
)f(x) + O
(h3).
The above asymptotic approximation is satisfied whenever:1 Asymptotic constraint associated to the discrete magnetic
potential:
ah(x) =n∑
j=1
ej w(
1q
xj
h
)(1 + O (h)) .
2 Asymptotic constraint associated to the discrete magneticpotential:
qΦh(x) +1
2µ
n∑j=1
(2
qh− ah(xj )− ah(xj − h)
)= V
(xh
)+ O
(h3).
23 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Asymptotic approximation of a Sturm-Liouvilleproblem
Discrete electromagnetic Schrodinger operators Lh on hZn
Lhf(x) = − h2
2µ
n∑j=1
∂
∂xj
(w(
xj
qh
)∂f∂xj
(x)
)+ V
(xh
)f(x) + O
(h3).
The above asymptotic approximation is satisfied whenever:1 Asymptotic constraint associated to the discrete magnetic
potential:
ah(x) =n∑
j=1
ej w(
1q
xj
h
)(1 + O (h)) .
2 Asymptotic constraint associated to the discrete magneticpotential:
qΦh(x) +1
2µ
n∑j=1
(2
qh− ah(xj )− ah(xj − h)
)= V
(xh
)+ O
(h3).
23 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
The factorization approachIsospectral relations
Pair of ladder operators
Consider now the pair (A+h ,A
−h ) of ladder operators defined
componentwise by
A+h =
n∑j=1
ejA+jh with A+j
h =
√qh4µ
(ah(xj )T
+jh −
2qh
I)
A−h =n∑
j=1
ejA−jh with A−j
h =
√qh4µ
(2
qhI − ah(xj − h)T−j
h
).
From the assumption that the vacuum vector ψ0(x ; h) = φ(x ; h)s(s ∈ Pin(n)) annihilated by A+
h , it readily follows that the pair (A+h ,A
−h ) is
isospectral equivalent to the pair (D+h ,Mh), with
D+h f(x) =
n∑j=1
ej∂+jh , Mh =
n∑j=1
ej
(hah(xj − h)2T−j
h −4
q2hI).
24 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
The factorization approachquasi-monomials vs. bound states of the discrete electromagnetic Schrodingeroperators
Moreover, in case where the discrete electric and magnetic potentials aregiven by
Φh(x) =h
8µ
n∑j=1
4q2h2
(φ(x ; h)2
φ(x + hej ; h)2 +φ(x − hej ; h)2
φ(x ; h)2
)
ah(x) =n∑
j=1
ej2
qhφ(x ; h)
φ(x + hej ; h)
it follows straightforwardly from the factorization propertyLh = 1
2 (A+h A−h + A−h A+
h ) that Lh is isospectral equivalent to theanti-commutator MhD+
h + D+h Mh.
Indeed, the isospectral formula
φ(x ; h)−1Lh(φ(x ; h)f(x) = − q4µh
(MhD+h + D+
h Mh)
yields naturally from the combination of the factorization property withthe aforementioned isospectral relations.
25 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
A sketch of a recent paper of minepublished recently at Applied Mathematics and Computation
We have shown that Lh is asymptotically equivalent to thediscrete harmonic oscillator − 1
2m ∆h + qΦh(x) with mass m ∼ µqh
,whose kinetic term is written in terms of the ’discrete’ Laplacian ∆h.
For the particular choice ah(xj ) =1q
(1h + µ
xjh +
µ
2
)it readily
follows that that the asymptotic expansion of Lh reduces to
Lhf(x) = − 12µq
(E+h f(x)− E−h f(x)) + V
(xh
)f(x),
with V(x
h
)= −
n∑j=1
xj
h+ qΦh(x). Hereby E±h corresponds to the
forward/backward counterpart of the radial derivative
E =n∑
j=1
xj∂xj , carrying the polynomial w(xj ) = 1 + µxj .
26 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
A sketch of a recent paper of minepublished recently at Applied Mathematics and Computation
We have shown that Lh is asymptotically equivalent to thediscrete harmonic oscillator − 1
2m ∆h + qΦh(x) with mass m ∼ µqh
,whose kinetic term is written in terms of the ’discrete’ Laplacian ∆h.
For the particular choice ah(xj ) =1q
(1h + µ
xjh +
µ
2
)it readily
follows that that the asymptotic expansion of Lh reduces to
Lhf(x) = − 12µq
(E+h f(x)− E−h f(x)) + V
(xh
)f(x),
with V(x
h
)= −
n∑j=1
xj
h+ qΦh(x). Hereby E±h corresponds to the
forward/backward counterpart of the radial derivative
E =n∑
j=1
xj∂xj , carrying the polynomial w(xj ) = 1 + µxj .
26 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
A sketch of a recent paper of minepublished recently at Applied Mathematics and Computation
From the Energy condition associated to the Fock space Fh
endowed by the Clifford module `2(hZn; C`0,n) := `2(hZn)⊗ C`0,n,the vacuum vectors of the form ψ0(x ; h) = φ(x ; h)s (s ∈ Pin(n)), thequantity
Pr
n∑j=1
ejXj = x
= hnψ0(x ; h)†ψ0(x ; h)
may be regarded as a discrete quasi-probability law on hZn,carrying a set of independent and identically distributed (i.i.d.)random variables X1,X2, . . . ,Xn.
We have used of the Bayesian probability framework towardsDirac’s insight on quasi-probabilities (they may take negative values)to compute some examples involving the well-known Poisson andhypergeometric distributions, likewise quasi-probability distributionsinvolving the generalized Mittag-Leffler/Wright functions. In some ofthe cases they may take negative values.
27 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
A sketch of a recent paper of minepublished recently at Applied Mathematics and Computation
From the Energy condition associated to the Fock space Fh
endowed by the Clifford module `2(hZn; C`0,n) := `2(hZn)⊗ C`0,n,the vacuum vectors of the form ψ0(x ; h) = φ(x ; h)s (s ∈ Pin(n)), thequantity
Pr
n∑j=1
ejXj = x
= hnψ0(x ; h)†ψ0(x ; h)
may be regarded as a discrete quasi-probability law on hZn,carrying a set of independent and identically distributed (i.i.d.)random variables X1,X2, . . . ,Xn.
We have used of the Bayesian probability framework towardsDirac’s insight on quasi-probabilities (they may take negative values)to compute some examples involving the well-known Poisson andhypergeometric distributions, likewise quasi-probability distributionsinvolving the generalized Mittag-Leffler/Wright functions. In some ofthe cases they may take negative values.
27 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
ExamplesGeneralization of the so-called Poisson distribution
hnφ(x ; h)2 =
n∏
j=1
Eα,β(
4q2−αh2
)−1 4xjh q
(2−α)xjh h−
2xjh
Γ(β + α
xjh
) , if x ∈ hZn≥0
0 , otherwise
As a matter of fact, the Mittag-Leffler function
Eα,β(λ) =∑∞
m=0λm
Γ(β + αm)is well defined for Re(α) > 0, Re(β) > 0.
1 Discrete Electric Potential:
Φh(x) =h
8µ
n∑j=1
1qα
Γ(α + β + α
xjh
)Γ(β + α
xjh
) +Γ(β + α
xjh
)Γ(β − α + α
xjh
).
2 Discrete Magnetic Potential:
ah(x) =n∑
j=1
ej
√√√√√ 1qα
Γ(α + β + α
xjh
)Γ(β + α
xjh
) .
28 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
ExamplesGeneralization of the so-called Poisson distribution
hnφ(x ; h)2 =
n∏
j=1
Eα,β(
4q2−αh2
)−1 4xjh q
(2−α)xjh h−
2xjh
Γ(β + α
xjh
) , if x ∈ hZn≥0
0 , otherwise
As a matter of fact, the Mittag-Leffler function
Eα,β(λ) =∑∞
m=0λm
Γ(β + αm)is well defined for Re(α) > 0, Re(β) > 0.
1 Discrete Electric Potential:
Φh(x) =h
8µ
n∑j=1
1qα
Γ(α + β + α
xjh
)Γ(β + α
xjh
) +Γ(β + α
xjh
)Γ(β − α + α
xjh
).
2 Discrete Magnetic Potential:
ah(x) =n∑
j=1
ej
√√√√√ 1qα
Γ(α + β + α
xjh
)Γ(β + α
xjh
) .
28 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Appell Set PropertyThe Poisson-Charlier connection
1 Remark: For the special choices q = 2 and λ = 1h2 , the resulting
ladder operator that yields from the Mittag-Leffler distribution
Mh =n∑
j=1
ej
((xj +
1h
)T−j
h −1h
I)
corresponds to a finite difference
approximation of the Clifford-Hermite operator
xI − D = − exp
(|x |2
2
)D exp
(−|x |
2
2
).
2 The resulting quasi-monomials generated from the operationalformula mk (x ; h) = µk (Mh)k s (s ∈ Pin(n)), carrying the constants
µ2m = (−1)m
( 12
)m( n
2
)m
(k = 2m) and µ2m+1 =
(−1)m
( 32
)m( n
2 + 1)
m
(k = 2m + 1)
possess the Appell set property D+h mk (x ; h) = kmk−1(x ; h). They
correspond to an hypercomplex extension of the Poisson-Charlierpolynomials in disguise.
29 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Appell Set PropertyThe Poisson-Charlier connection
1 Remark: For the special choices q = 2 and λ = 1h2 , the resulting
ladder operator that yields from the Mittag-Leffler distribution
Mh =n∑
j=1
ej
((xj +
1h
)T−j
h −1h
I)
corresponds to a finite difference
approximation of the Clifford-Hermite operator
xI − D = − exp
(|x |2
2
)D exp
(−|x |
2
2
).
2 The resulting quasi-monomials generated from the operationalformula mk (x ; h) = µk (Mh)k s (s ∈ Pin(n)), carrying the constants
µ2m = (−1)m
( 12
)m( n
2
)m
(k = 2m) and µ2m+1 =
(−1)m
( 32
)m( n
2 + 1)
m
(k = 2m + 1)
possess the Appell set property D+h mk (x ; h) = kmk−1(x ; h). They
correspond to an hypercomplex extension of the Poisson-Charlierpolynomials in disguise.
29 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
ExamplesThe Generalized Wright distribution
n∏j=1
1Ψ1
[(δ, γ)(β, α)
γγ
αα4
q1+γ−αh2
]−1
×
Γ(δ + γ
xjh
)Γ(β + α
xjh
) ααxj
h γ−γxj
h 4xjh q−
(1+γ−α)xjh h−
2xjh
Γ(
xjh + 1
) , x ∈ hZn≥0
0 , otherwise
.
1 Notice that the Wright series 1Ψ1
[(δ, γ)(β, α)
λ
]is absolutely
convergent for |λ| < αα
γγand of |λ| =
αα
γγ, Re(β)− Re(δ) > 1
2 for
h2 >γ2γ
α2α
4q1+γ−α and of h2 =
γ2γ
α2α
4q1+γ−α , Re(β)− Re(δ) > 1
2
whenever α− γ = −1.
30 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Generalized Wright distributionsSome additional remarks
For γ = δ = 1, the aforementioned likelihood function is theMittag-Leffler distribution in disguise. Moreover, if α = Re(α) > 0,
α→ 0+ and h >2q
, the previous distribution simplifies to
hnφ(x ; h)2 =
n∏
j=1
(1− 4
q2h2
)−1
q−2xjh h−
2xjh , if x ∈ hZn
≥0
0 , otherwise
.
For β = δ, the likelihood function amalgamates the Poissondistribution (α = γ = 1) as well as the orthogonal measure that
gives rise, up to the constant(
1− 4q2h2
)−βn, to the
hypergeometric distribution on hZn≥0, carrying the parameter
λ = 4q2h2 (α→ 0+, γ = 1 and h > 2
q ).
31 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Generalized Wright distributionsSome additional remarks
For γ = δ = 1, the aforementioned likelihood function is theMittag-Leffler distribution in disguise. Moreover, if α = Re(α) > 0,
α→ 0+ and h >2q
, the previous distribution simplifies to
hnφ(x ; h)2 =
n∏
j=1
(1− 4
q2h2
)−1
q−2xjh h−
2xjh , if x ∈ hZn
≥0
0 , otherwise
.
For β = δ, the likelihood function amalgamates the Poissondistribution (α = γ = 1) as well as the orthogonal measure that
gives rise, up to the constant(
1− 4q2h2
)−βn, to the
hypergeometric distribution on hZn≥0, carrying the parameter
λ = 4q2h2 (α→ 0+, γ = 1 and h > 2
q ).
31 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
BibliographyMain references used in my talk
Faustino, Nelson (2013). Special Functions ofHypercomplex Variable on the Lattice Based on SU (1, 1).Symmetry, Integrability and Geometry: Methods andApplications 9, no. 0 : 65-18.
Faustino, N. (2014). Classes of hypercomplex polynomials ofdiscrete variable based on the quasi-monomiality principle.Applied Mathematics and Computation, 247, 607-622.
Faustino, N. (2017). Hypercomplex Fock states for discreteelectromagnetic Schrodinger operators: A Bayesianprobability perspective. Applied Mathematics andComputation, 315, 531-548.
Faustino, Nelson (2017). Symmetry PreservingDiscretization Schemes through Hypercomplex Variables,Conference: 15th International Conference of NumericalAnalysis and Applied Mathematics , DOI:10.13140/RG.2.2.35900.74882
32 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
BibliographyMain references used in my talk
Faustino, Nelson (2013). Special Functions ofHypercomplex Variable on the Lattice Based on SU (1, 1).Symmetry, Integrability and Geometry: Methods andApplications 9, no. 0 : 65-18.
Faustino, N. (2014). Classes of hypercomplex polynomials ofdiscrete variable based on the quasi-monomiality principle.Applied Mathematics and Computation, 247, 607-622.
Faustino, N. (2017). Hypercomplex Fock states for discreteelectromagnetic Schrodinger operators: A Bayesianprobability perspective. Applied Mathematics andComputation, 315, 531-548.
Faustino, Nelson (2017). Symmetry PreservingDiscretization Schemes through Hypercomplex Variables,Conference: 15th International Conference of NumericalAnalysis and Applied Mathematics , DOI:10.13140/RG.2.2.35900.74882
32 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
BibliographyMain references used in my talk
Faustino, Nelson (2013). Special Functions ofHypercomplex Variable on the Lattice Based on SU (1, 1).Symmetry, Integrability and Geometry: Methods andApplications 9, no. 0 : 65-18.
Faustino, N. (2014). Classes of hypercomplex polynomials ofdiscrete variable based on the quasi-monomiality principle.Applied Mathematics and Computation, 247, 607-622.
Faustino, N. (2017). Hypercomplex Fock states for discreteelectromagnetic Schrodinger operators: A Bayesianprobability perspective. Applied Mathematics andComputation, 315, 531-548.
Faustino, Nelson (2017). Symmetry PreservingDiscretization Schemes through Hypercomplex Variables,Conference: 15th International Conference of NumericalAnalysis and Applied Mathematics , DOI:10.13140/RG.2.2.35900.74882
32 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
BibliographyMain references used in my talk
Faustino, Nelson (2013). Special Functions ofHypercomplex Variable on the Lattice Based on SU (1, 1).Symmetry, Integrability and Geometry: Methods andApplications 9, no. 0 : 65-18.
Faustino, N. (2014). Classes of hypercomplex polynomials ofdiscrete variable based on the quasi-monomiality principle.Applied Mathematics and Computation, 247, 607-622.
Faustino, N. (2017). Hypercomplex Fock states for discreteelectromagnetic Schrodinger operators: A Bayesianprobability perspective. Applied Mathematics andComputation, 315, 531-548.
Faustino, Nelson (2017). Symmetry PreservingDiscretization Schemes through Hypercomplex Variables,Conference: 15th International Conference of NumericalAnalysis and Applied Mathematics , DOI:10.13140/RG.2.2.35900.74882
32 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
A final thought
”To first approximation, the human brain is a harmonic oscillator.”
Barry Simon to Charles Fefferman in a private conversation as theywalked around the Princeton campus.
Figure: Barry SimonFigure: Charles Fefferman
SimonFest 2006: Barry Stories–http://math.caltech.edu/SimonFest/stories.html]fefferman
33 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
A final thought
”To first approximation, the human brain is a harmonic oscillator.”
Barry Simon to Charles Fefferman in a private conversation as theywalked around the Princeton campus.
Figure: Barry SimonFigure: Charles Fefferman
SimonFest 2006: Barry Stories–http://math.caltech.edu/SimonFest/stories.html]fefferman
33 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
A final thought
”To first approximation, the human brain is a harmonic oscillator.”
Barry Simon to Charles Fefferman in a private conversation as theywalked around the Princeton campus.
Figure: Barry SimonFigure: Charles Fefferman
SimonFest 2006: Barry Stories–http://math.caltech.edu/SimonFest/stories.html]fefferman
33 / 34
Symmetry Pre-serving Dis-
cretiza-tion Schemes
Nelson Faustino
The Scope ofProblemsFunction TheoreticalMethods inNumerical Analysis
Motivation behindthis talk
Lie-algebraicdiscretizationsUmbral CalculusRevisited
Radial algebraapproach
Appell Sets
su(1, 1)symmetries
Discretization ofOperators ofSturm-LiouvilletypeDiscreteElectromagneticSchrodingeroperators
Interplay withBayesian Statistics
Thank you for your attention!
The author would like to thank the organizers of the ICNAAM2017 for their kind invitation and for the financial support as well.
Figure: Pictures from my university, located at ABC Paulista (Sao Paulo,Brazil)
34 / 34