Symmetry Preserving The Scope of Problems Discretization Schemes - - PowerPoint PPT Presentation

SLIDE 1

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Symmetry Preserving Discretization Schemes through Hypercomplex Variables

Nelson Faustino

Center of Mathematics, Computation and Cognition, UFABC nelson.faustino@ufabc.edu.br

ICNAAM 2017, Thessaloniki, Greece, 25–30 September 2017

1 / 34

SLIDE 2

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

1

The Scope of Problems Function Theoretical Methods in Numerical Analysis Motivation behind this talk

2

Lie-algebraic discretizations Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

3

Discretization of Operators of Sturm-Liouville type Discrete Electromagnetic Schr¨

• dinger operators

Interplay with Bayesian Statistics

2 / 34

SLIDE 3

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

’(. . . ) When Columbus set sail, he was like an applied mathematician, 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 the work 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/dual graphs vs. 7−point representation of the ’discrete’ Laplacian ∆h.

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SLIDE 4

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Overview

Why should we use Finite Difference Dirac Operators?

Hypercomplex analysis approach:

1

Useful to rewrite our main problem in a more compact form (e.g. Lam´ e/Navier-Stokes/Schr¨

• dinger 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 in comparison with the usual convergence conditions associated to standard finite difference schemes.

4 / 34

SLIDE 5

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Overview

Why should we use Finite Difference Dirac Operators?

Hypercomplex analysis approach:

1

Useful to rewrite our main problem in a more compact form (e.g. Lam´ e/Navier-Stokes/Schr¨

• dinger 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 in comparison with the usual convergence conditions associated to standard finite difference schemes.

4 / 34

SLIDE 6

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Overview

Why should we use Finite Difference Dirac Operators?

Hypercomplex analysis approach:

1

Useful to rewrite our main problem in a more compact form (e.g. Lam´ e/Navier-Stokes/Schr¨

• dinger 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 in comparison with the usual convergence conditions associated to standard finite difference schemes.

4 / 34

SLIDE 7

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Overview

Some references

1

Boundary value problems: G¨ urlebeck and Spr¨

• ßig -

Quaternionic and Clifford calculus for Engineers and Physicists (1997).

2

Discrete Fundamental solutions for Difference Dirac

• perators: G¨

urlebeck and Hommel, On finite difference Dirac operators and their fundamental solutions, Adv. Appl. Clifford Algebras, 11, 89 – 106 (2003).

3

Numerical implementation using discrete counterparts: Faustino, G¨ urlebeck, Hommel, and K¨ ahler - Difference Potentials for the Navier-Stokes equations in unbounded domains, J. Diff. Eq. & Appl., Journal of Difference Equations and Applications, 12(6), 577-595.

4

To take a look for further progresses on this direction, attend tomorrow (September 26) the morning talks of the 13th Symposium on Clifford Analysis and Applications (ROOM 3).

5 / 34

SLIDE 8

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Overview

Some references

1

Boundary value problems: G¨ urlebeck and Spr¨

• ßig -

Quaternionic and Clifford calculus for Engineers and Physicists (1997).

2

Discrete Fundamental solutions for Difference Dirac

• perators: G¨

urlebeck and Hommel, On finite difference Dirac operators and their fundamental solutions, Adv. Appl. Clifford Algebras, 11, 89 – 106 (2003).

3

Numerical implementation using discrete counterparts: Faustino, G¨ urlebeck, Hommel, and K¨ ahler - Difference Potentials for the Navier-Stokes equations in unbounded domains, J. Diff. Eq. & Appl., Journal of Difference Equations and Applications, 12(6), 577-595.

4

To take a look for further progresses on this direction, attend tomorrow (September 26) the morning talks of the 13th Symposium on Clifford Analysis and Applications (ROOM 3).

5 / 34

SLIDE 9

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Overview

Some references

1

Boundary value problems: G¨ urlebeck and Spr¨

• ßig -

Quaternionic and Clifford calculus for Engineers and Physicists (1997).

2

Discrete Fundamental solutions for Difference Dirac

• perators: G¨

urlebeck and Hommel, On finite difference Dirac operators and their fundamental solutions, Adv. Appl. Clifford Algebras, 11, 89 – 106 (2003).

3

Numerical implementation using discrete counterparts: Faustino, G¨ urlebeck, Hommel, and K¨ ahler - Difference Potentials for the Navier-Stokes equations in unbounded domains, J. Diff. Eq. & Appl., Journal of Difference Equations and Applications, 12(6), 577-595.

4

To take a look for further progresses on this direction, attend tomorrow (September 26) the morning talks of the 13th Symposium on Clifford Analysis and Applications (ROOM 3).

5 / 34

SLIDE 10

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Overview

Some references

1

Boundary value problems: G¨ urlebeck and Spr¨

• ßig -

Quaternionic and Clifford calculus for Engineers and Physicists (1997).

2

Discrete Fundamental solutions for Difference Dirac

• perators: G¨

urlebeck and Hommel, On finite difference Dirac operators and their fundamental solutions, Adv. Appl. Clifford Algebras, 11, 89 – 106 (2003).

3

Numerical implementation using discrete counterparts: Faustino, G¨ urlebeck, Hommel, and K¨ ahler - Difference Potentials for the Navier-Stokes equations in unbounded domains, J. Diff. Eq. & Appl., Journal of Difference Equations and Applications, 12(6), 577-595.

4

To take a look for further progresses on this direction, attend tomorrow (September 26) the morning talks of the 13th Symposium on Clifford Analysis and Applications (ROOM 3).

5 / 34

SLIDE 11

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Lie algebraic discretization schemes

A rigorous way to preserve (continuous) symmetries under discretization.

The roots of Lie algebraic discretization schemes: Discretization of finite difference operators: A.Dimakis, M¨ uller-Hoissen and T.Striker (1996), Journal of Physics A: Mathematical and General, 29(21), 6861 Finite difference operators vs. symplectic solvers: Dattoli, G., Ottaviani, P. L., Torre, A., & V´ azquez, L. (1997). La Rivista del Nuovo Cimento (1978-1999), 20(2), 3-133. Motivation for the approach enclosed on this talk: Umbral calculus: Finite difference operators as convergent power series determined in terms of partial derivatives. Quantum Field Theory: Provides a way to represent R−polynomial algebra as the Bose algebra. Classes of Wigner Quantal Systems: Hypercomplex analysis in its minimal form corresponds to a realization of the Lie superalgebra

• sp(1|2) (a refinement of the Lie algebra sl(2, R) ∼

= su(1, 1)).

6 / 34

SLIDE 12

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Lie algebraic discretization schemes

A rigorous way to preserve (continuous) symmetries under discretization.

The roots of Lie algebraic discretization schemes: Discretization of finite difference operators: A.Dimakis, M¨ uller-Hoissen and T.Striker (1996), Journal of Physics A: Mathematical and General, 29(21), 6861 Finite difference operators vs. symplectic solvers: Dattoli, G., Ottaviani, P. L., Torre, A., & V´ azquez, L. (1997). La Rivista del Nuovo Cimento (1978-1999), 20(2), 3-133. Motivation for the approach enclosed on this talk: Umbral calculus: Finite difference operators as convergent power series determined in terms of partial derivatives. Quantum Field Theory: Provides a way to represent R−polynomial algebra as the Bose algebra. Classes of Wigner Quantal Systems: Hypercomplex analysis in its minimal form corresponds to a realization of the Lie superalgebra

• sp(1|2) (a refinement of the Lie algebra sl(2, R) ∼

= su(1, 1)).

6 / 34

SLIDE 13

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Lie algebraic discretization schemes

A rigorous way to preserve (continuous) symmetries under discretization.

The roots of Lie algebraic discretization schemes: Discretization of finite difference operators: A.Dimakis, M¨ uller-Hoissen and T.Striker (1996), Journal of Physics A: Mathematical and General, 29(21), 6861 Finite difference operators vs. symplectic solvers: Dattoli, G., Ottaviani, P. L., Torre, A., & V´ azquez, L. (1997). La Rivista del Nuovo Cimento (1978-1999), 20(2), 3-133. Motivation for the approach enclosed on this talk: Umbral calculus: Finite difference operators as convergent power series determined in terms of partial derivatives. Quantum Field Theory: Provides a way to represent R−polynomial algebra as the Bose algebra. Classes of Wigner Quantal Systems: Hypercomplex analysis in its minimal form corresponds to a realization of the Lie superalgebra

• sp(1|2) (a refinement of the Lie algebra sl(2, R) ∼

= su(1, 1)).

6 / 34

SLIDE 14

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Lie algebraic discretization schemes

A rigorous way to preserve (continuous) symmetries under discretization.

The roots of Lie algebraic discretization schemes: Discretization of finite difference operators: A.Dimakis, M¨ uller-Hoissen and T.Striker (1996), Journal of Physics A: Mathematical and General, 29(21), 6861 Finite difference operators vs. symplectic solvers: Dattoli, G., Ottaviani, P. L., Torre, A., & V´ azquez, L. (1997). La Rivista del Nuovo Cimento (1978-1999), 20(2), 3-133. Motivation for the approach enclosed on this talk: Umbral calculus: Finite difference operators as convergent power series determined in terms of partial derivatives. Quantum Field Theory: Provides a way to represent R−polynomial algebra as the Bose algebra. Classes of Wigner Quantal Systems: Hypercomplex analysis in its minimal form corresponds to a realization of the Lie superalgebra

• sp(1|2) (a refinement of the Lie algebra sl(2, R) ∼

= su(1, 1)).

6 / 34

SLIDE 15

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Lie algebraic discretization schemes

A rigorous way to preserve (continuous) symmetries under discretization.

The roots of Lie algebraic discretization schemes: Discretization of finite difference operators: A.Dimakis, M¨ uller-Hoissen and T.Striker (1996), Journal of Physics A: Mathematical and General, 29(21), 6861 Finite difference operators vs. symplectic solvers: Dattoli, G., Ottaviani, P. L., Torre, A., & V´ azquez, L. (1997). La Rivista del Nuovo Cimento (1978-1999), 20(2), 3-133. Motivation for the approach enclosed on this talk: Umbral calculus: Finite difference operators as convergent power series determined in terms of partial derivatives. Quantum Field Theory: Provides a way to represent R−polynomial algebra as the Bose algebra. Classes of Wigner Quantal Systems: Hypercomplex analysis in its minimal form corresponds to a realization of the Lie superalgebra

• sp(1|2) (a refinement of the Lie algebra sl(2, R) ∼

= su(1, 1)).

6 / 34

SLIDE 16

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Multivariate polynomials

Basic setting

Monomial over x = (x1, x2, . . . , xn) ∈ Rn: xα = xα1

1 xα2 2

. . . xαn

n

Ring of polynomials R[x]: Each P(x) ∈ R[x] is a linear combination of monomials xα. Multi-index derivatives for ∂x := (∂x1, ∂x2, . . . , ∂xn): ∂α

x := ∂α1 x1 ∂α2 x2 . . . ∂αn xn ∈ End(R[x]).

Multi-index enumerative notation: α! = α1!α2! . . . αn!, β α

• =

β! α!(β−α)!

Binomial formula: (x + y)β =

|β|

• |α|=0

β α

• xαyβ−α =

|β|

• |α|=0

[∂α

x xβ]x=y

α! xα

7 / 34

SLIDE 17

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Multivariate polynomials

Basic setting

Monomial over x = (x1, x2, . . . , xn) ∈ Rn: xα = xα1

1 xα2 2

. . . xαn

n

Ring of polynomials R[x]: Each P(x) ∈ R[x] is a linear combination of monomials xα. Multi-index derivatives for ∂x := (∂x1, ∂x2, . . . , ∂xn): ∂α

x := ∂α1 x1 ∂α2 x2 . . . ∂αn xn ∈ End(R[x]).

Multi-index enumerative notation: α! = α1!α2! . . . αn!, β α

• =

β! α!(β−α)!

Binomial formula: (x + y)β =

|β|

• |α|=0

β α

• xαyβ−α =

|β|

• |α|=0

[∂α

x xβ]x=y

α! xα

7 / 34

SLIDE 18

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Multivariate polynomials

Basic setting

Monomial over x = (x1, x2, . . . , xn) ∈ Rn: xα = xα1

1 xα2 2

. . . xαn

n

Ring of polynomials R[x]: Each P(x) ∈ R[x] is a linear combination of monomials xα. Multi-index derivatives for ∂x := (∂x1, ∂x2, . . . , ∂xn): ∂α

x := ∂α1 x1 ∂α2 x2 . . . ∂αn xn ∈ End(R[x]).

Multi-index enumerative notation: α! = α1!α2! . . . αn!, β α

• =

β! α!(β−α)!

Binomial formula: (x + y)β =

|β|

• |α|=0

β α

• xαyβ−α =

|β|

• |α|=0

[∂α

x xβ]x=y

α! xα

7 / 34

SLIDE 19

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Multivariate polynomials

Basic setting

Monomial over x = (x1, x2, . . . , xn) ∈ Rn: xα = xα1

1 xα2 2

. . . xαn

n

Ring of polynomials R[x]: Each P(x) ∈ R[x] is a linear combination of monomials xα. Multi-index derivatives for ∂x := (∂x1, ∂x2, . . . , ∂xn): ∂α

x := ∂α1 x1 ∂α2 x2 . . . ∂αn xn ∈ End(R[x]).

Multi-index enumerative notation: α! = α1!α2! . . . αn!, β α

• =

β! α!(β−α)!

Binomial formula: (x + y)β =

|β|

• |α|=0

β α

• xαyβ−α =

|β|

• |α|=0

[∂α

x xβ]x=y

α! xα

7 / 34

SLIDE 20

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Multivariate polynomials

Basic setting

Monomial over x = (x1, x2, . . . , xn) ∈ Rn: xα = xα1

1 xα2 2

. . . xαn

n

Ring of polynomials R[x]: Each P(x) ∈ R[x] is a linear combination of monomials xα. Multi-index derivatives for ∂x := (∂x1, ∂x2, . . . , ∂xn): ∂α

x := ∂α1 x1 ∂α2 x2 . . . ∂αn xn ∈ End(R[x]).

Multi-index enumerative notation: α! = α1!α2! . . . αn!, β α

• =

β! α!(β−α)!

Binomial formula: (x + y)β =

|β|

• |α|=0

β α

• xαyβ−α =

|β|

• |α|=0

[∂α

x xβ]x=y

α! xα

7 / 34

SLIDE 21

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Multivariate polynomials

Shift-invariant operators

Taylor series representation in R[x]: P(x + y) = exp(y · ∂x)P(x). Shift-invariant operator: Q(∂x) is shift-invariant iff Q(∂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 be expressed (as a convergent series) in the gradient ∂x, that is Q =

∞

• |α|=0

aα α! ∂α

x ,

where aα = [Qxα]x=0.

8 / 34

SLIDE 22

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Multivariate polynomials

Shift-invariant operators

Taylor series representation in R[x]: P(x + y) = exp(y · ∂x)P(x). Shift-invariant operator: Q(∂x) is shift-invariant iff Q(∂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 be expressed (as a convergent series) in the gradient ∂x, that is Q =

∞

• |α|=0

aα α! ∂α

x ,

where aα = [Qxα]x=0.

8 / 34

SLIDE 23

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Multivariate polynomials

Shift-invariant operators

Taylor series representation in R[x]: P(x + y) = exp(y · ∂x)P(x). Shift-invariant operator: Q(∂x) is shift-invariant iff Q(∂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 be expressed (as a convergent series) in the gradient ∂x, that is Q =

∞

• |α|=0

aα α! ∂α

x ,

where aα = [Qxα]x=0.

8 / 34

SLIDE 24

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Basic polynomial sequences

A way to construct polynomial sequences towards interpolation theory

Definition (Basic polynomial sequence) A polynomial sequence {Vα}α, where Vα is a multivariate polynomial of degree |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 delta

• perator Ox = (Ox1, Ox2, . . . , Oxn).

9 / 34

SLIDE 25

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Basic polynomial sequences

A way to construct polynomial sequences towards interpolation theory

Definition (Basic polynomial sequence) A polynomial sequence {Vα}α, where Vα is a multivariate polynomial of degree |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 delta

• perator Ox = (Ox1, Ox2, . . . , Oxn).

9 / 34

SLIDE 26

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Basic polynomial sequences

A way to construct polynomial sequences towards interpolation theory

Definition (Basic polynomial sequence) A polynomial sequence {Vα}α, where Vα is a multivariate polynomial of degree |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 delta

• perator Ox = (Ox1, Ox2, . . . , Oxn).

9 / 34

SLIDE 27

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Basic polynomial sequences

A way to construct polynomial sequences towards interpolation theory

Definition (Basic polynomial sequence) A polynomial sequence {Vα}α, where Vα is a multivariate polynomial of degree |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 delta

• perator Ox = (Ox1, Ox2, . . . , Oxn).

9 / 34

SLIDE 28

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Some Results

Results 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 sequence

• f binomial type, i.e.

Vβ(x + y) =

|β|

• |α|=0

β α

• Vα(x)Vβ−α(y).

Theorem (A. Di Bucchianico, 1999) Let Q = Q(∂x) be a shift invariant operator. Let Ox be a multivariate delta operator with basic polynomial sequence {Vα}α. Then Q =

• |α|≥0

aα α! Oα

x ,

with aα = [QVα(x)]x=0.

10 / 34

SLIDE 29

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Some Results

Results 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 sequence

• f binomial type, i.e.

Vβ(x + y) =

|β|

• |α|=0

β α

• Vα(x)Vβ−α(y).

Theorem (A. Di Bucchianico, 1999) Let Q = Q(∂x) be a shift invariant operator. Let Ox be a multivariate delta operator with basic polynomial sequence {Vα}α. Then Q =

• |α|≥0

aα α! Oα

x ,

with aα = [QVα(x)]x=0.

10 / 34

SLIDE 30

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Some Results

Results 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)Oα

x

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

SLIDE 31

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Finite difference toolbox

1

Equidistant lattice with mesh width h > 0: hZn =

• x = (x1, . . . , xn) ∈ Rn : x

h ∈ Zn

2

Forward/backward finite difference operators (∂+j

h f)(x) = f(x + hej) − f(x)

h , (∂−j

h f)(x) = f(x) − f(x − hej)

h .

3

Translation property: ∂+j

h and ∂−j h

are interrelated by (T ±j

h f)(x) = f(x ± hej) i.e.

T −j

h (∂+j h f)(x) = (∂−j h f)(x)

and T +j

h (∂−j h f)(x) = (∂+j h f)(x).

4

Product rules for finite difference operators: ∂+j

h (g(x)f(x))

= (∂+j

h g)(x)f(x + hej) + g(x)(∂+j h f)(x)

∂−j

h (g(x)f(x))

= (∂−j

h g)(x)f(x − hej) + g(x)(∂−j h f)(x).

12 / 34

SLIDE 32

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Finite difference toolbox

1

Equidistant lattice with mesh width h > 0: hZn =

• x = (x1, . . . , xn) ∈ Rn : x

h ∈ Zn

2

Forward/backward finite difference operators (∂+j

h f)(x) = f(x + hej) − f(x)

h , (∂−j

h f)(x) = f(x) − f(x − hej)

h .

3

Translation property: ∂+j

h and ∂−j h

are interrelated by (T ±j

h f)(x) = f(x ± hej) i.e.

T −j

h (∂+j h f)(x) = (∂−j h f)(x)

and T +j

h (∂−j h f)(x) = (∂+j h f)(x).

4

Product rules for finite difference operators: ∂+j

h (g(x)f(x))

= (∂+j

h g)(x)f(x + hej) + g(x)(∂+j h f)(x)

∂−j

h (g(x)f(x))

= (∂−j

h g)(x)f(x − hej) + g(x)(∂−j h f)(x).

12 / 34

SLIDE 33

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Finite difference toolbox

1

Equidistant lattice with mesh width h > 0: hZn =

• x = (x1, . . . , xn) ∈ Rn : x

h ∈ Zn

2

Forward/backward finite difference operators (∂+j

h f)(x) = f(x + hej) − f(x)

h , (∂−j

h f)(x) = f(x) − f(x − hej)

h .

3

Translation property: ∂+j

h and ∂−j h

are interrelated by (T ±j

h f)(x) = f(x ± hej) i.e.

T −j

h (∂+j h f)(x) = (∂−j h f)(x)

and T +j

h (∂−j h f)(x) = (∂+j h f)(x).

4

Product rules for finite difference operators: ∂+j

h (g(x)f(x))

= (∂+j

h g)(x)f(x + hej) + g(x)(∂+j h f)(x)

∂−j

h (g(x)f(x))

= (∂−j

h g)(x)f(x − hej) + g(x)(∂−j h f)(x).

12 / 34

SLIDE 34

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Finite difference toolbox

1

Equidistant lattice with mesh width h > 0: hZn =

• x = (x1, . . . , xn) ∈ Rn : x

h ∈ Zn

2

Forward/backward finite difference operators (∂+j

h f)(x) = f(x + hej) − f(x)

h , (∂−j

h f)(x) = f(x) − f(x − hej)

h .

3

Translation property: ∂+j

h and ∂−j h

are interrelated by (T ±j

h f)(x) = f(x ± hej) i.e.

T −j

h (∂+j h f)(x) = (∂−j h f)(x)

and T +j

h (∂−j h f)(x) = (∂+j h f)(x).

4

Product rules for finite difference operators: ∂+j

h (g(x)f(x))

= (∂+j

h g)(x)f(x + hej) + g(x)(∂+j h f)(x)

∂−j

h (g(x)f(x))

= (∂−j

h g)(x)f(x − hej) + g(x)(∂−j h f)(x).

12 / 34

SLIDE 35

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Radial-type discretization

Lie-algebraic formulation

Radial-type approach: Study of finite difference operators belonging to the 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] = δjkI

2

e1, e2, . . . , en are the generators of the Clifford algebra of signature (0, n). Multivector operators: Basic left endomorphisms acting that act on functions with values on Cℓ0,n. Multivector derivative: L = n

j=1 ejLj stands the Lie-algebraic

counterpart of the Dirac operator D = n

j=1 ej∂xj .

Multivector multiplication: M = n

j=1 ejMj stands the

Lie-algebraic counterpart for the left multiplication of f(x) by a Clifford vector X = n

j=1 xjej.

13 / 34

SLIDE 36

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Radial-type discretization

Lie-algebraic formulation

Radial-type approach: Study of finite difference operators belonging to the 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] = δjkI

2

e1, e2, . . . , en are the generators of the Clifford algebra of signature (0, n). Multivector operators: Basic left endomorphisms acting that act on functions with values on Cℓ0,n. Multivector derivative: L = n

j=1 ejLj stands the Lie-algebraic

counterpart of the Dirac operator D = n

j=1 ej∂xj .

Multivector multiplication: M = n

j=1 ejMj stands the

Lie-algebraic counterpart for the left multiplication of f(x) by a Clifford vector X = n

j=1 xjej.

13 / 34

SLIDE 37

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Radial-type discretization

Lie-algebraic formulation

Radial-type approach: Study of finite difference operators belonging to the 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] = δjkI

2

e1, e2, . . . , en are the generators of the Clifford algebra of signature (0, n). Multivector operators: Basic left endomorphisms acting that act on functions with values on Cℓ0,n. Multivector derivative: L = n

j=1 ejLj stands the Lie-algebraic

counterpart of the Dirac operator D = n

j=1 ej∂xj .

Multivector multiplication: M = n

j=1 ejMj stands the

Lie-algebraic counterpart for the left multiplication of f(x) by a Clifford vector X = n

j=1 xjej.

13 / 34

SLIDE 38

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Radial-type discretization

Lie-algebraic formulation

Radial-type approach: Study of finite difference operators belonging to the 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] = δjkI

2

e1, e2, . . . , en are the generators of the Clifford algebra of signature (0, n). Multivector operators: Basic left endomorphisms acting that act on functions with values on Cℓ0,n. Multivector derivative: L = n

j=1 ejLj stands the Lie-algebraic

counterpart of the Dirac operator D = n

j=1 ej∂xj .

Multivector multiplication: M = n

j=1 ejMj stands the

Lie-algebraic counterpart for the left multiplication of f(x) by a Clifford vector X = n

j=1 xjej.

13 / 34

SLIDE 39

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

An Intermezzo

Quantum Field Theory (QFT) setting

Basic polynomial sequences (or quasi-monomials) may be constructed explicitly by a direct application of the Quantum Field Lemma. Fock space: Vector space (F, ·|·) such that

1

F: Free algebra generated by the elements a−

j

and a+

j from

the 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 the

Heisenberg-Weyl relations [a+

j , a+ k ] = 0,

[a−

j , a− k ] = 0,

[a−

j , a+ k ] = δjkI.

Standard lemma in QFT: All the basic vectors in F have the following form ηα :=  

n

• j=1

(a†

j )αj

  Φ

14 / 34

SLIDE 40

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

An Intermezzo

Quantum Field Theory (QFT) setting

Basic polynomial sequences (or quasi-monomials) may be constructed explicitly by a direct application of the Quantum Field Lemma. Fock space: Vector space (F, ·|·) such that

1

F: Free algebra generated by the elements a−

j

and a+

j from

the 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 the

Heisenberg-Weyl relations [a+

j , a+ k ] = 0,

[a−

j , a− k ] = 0,

[a−

j , a+ k ] = δjkI.

Standard lemma in QFT: All the basic vectors in F have the following form ηα :=  

n

• j=1

(a†

j )αj

  Φ

14 / 34

SLIDE 41

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

An Intermezzo

Quantum Field Theory (QFT) setting

Basic polynomial sequences (or quasi-monomials) may be constructed explicitly by a direct application of the Quantum Field Lemma. Fock space: Vector space (F, ·|·) such that

1

F: Free algebra generated by the elements a−

j

and a+

j from

the 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 the

Heisenberg-Weyl relations [a+

j , a+ k ] = 0,

[a−

j , a− k ] = 0,

[a−

j , a+ k ] = δjkI.

Standard lemma in QFT: All the basic vectors in F have the following form ηα :=  

n

• j=1

(a†

j )αj

  Φ

14 / 34

SLIDE 42

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

An Intermezzo

Quantum Field Theory (QFT) setting

Basic polynomial sequences (or quasi-monomials) may be constructed explicitly by a direct application of the Quantum Field Lemma. Fock space: Vector space (F, ·|·) such that

1

F: Free algebra generated by the elements a−

j

and a+

j from

the 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 the

Heisenberg-Weyl relations [a+

j , a+ k ] = 0,

[a−

j , a− k ] = 0,

[a−

j , a+ k ] = δjkI.

Standard lemma in QFT: All the basic vectors in F have the following form ηα :=  

n

• j=1

(a†

j )αj

  Φ

14 / 34

SLIDE 43

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

An Intermezzo

Quantum Field Theory (QFT) setting

Basic polynomial sequences (or quasi-monomials) may be constructed explicitly by a direct application of the Quantum Field Lemma. Fock space: Vector space (F, ·|·) such that

1

F: Free algebra generated by the elements a−

j

and a+

j from

the 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 the

Heisenberg-Weyl relations [a+

j , a+ k ] = 0,

[a−

j , a− k ] = 0,

[a−

j , a+ k ] = δjkI.

Standard lemma in QFT: All the basic vectors in F have the following form ηα :=  

n

• j=1

(a†

j )αj

  Φ

14 / 34

SLIDE 44

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Canonical discretization of finite difference

• perators

Forward/Backward differences:

First Expansion Theorem: ∂±j

h

= ± 1

h

• exp(±h∂xj ) − I
• .

Falling/Rising factorials: n

j=1

• xjT ∓j

h

αj 1 = n

j=1 xj(xj ∓ h) . . . (xj ∓ (αj − 1)h).

Figure: Chromatic polynomial: The falling factorial counts the number

• f ways to color the complete graph of order |α| with exactly n

j=1 xj

colors.

15 / 34

SLIDE 45

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Canonical discretization of finite difference

• perators

Forward/Backward differences:

First Expansion Theorem: ∂±j

h

= ± 1

h

• exp(±h∂xj ) − I
• .

Falling/Rising factorials: n

j=1

• xjT ∓j

h

αj 1 = n

j=1 xj(xj ∓ h) . . . (xj ∓ (αj − 1)h).

Figure: Chromatic polynomial: The falling factorial counts the number

• f ways to color the complete graph of order |α| with exactly n

j=1 xj

colors.

15 / 34

SLIDE 46

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Canonical discretization of finite difference

• perators of hypercomplex type

A list of examples

1

Forward finite differences: The set of operators ∂+j

h and

xjT −j

h

: f(x) → xjf(x − hej) span the Weyl-Heisenberg algebra of dimension 2n + 1. Moreover D+

h = n j=1 ej∂+j h and

Xh = n

j=1 ejxjT −j h

are the corresponding multivector ladder

• perators on the lattice hZn.

2

Backward finite differences: ∂−j

h

and xjT +j

h

: f(x) → xjf(x + hej) also span the Weyl-Heisenberg algebra of dimension 2n + 1. This turns out D−

h = n j=1 ej∂−j h

and X−h = n

j=1 ejxjT +j h

as the corresponding multivector ladder operators on the lattice hZn.

3

Discretization of the Hermite operator: D+

h and Xh − D− h are

• btained from the set of ladder operators Lj = ∂+j

h and

Lj = xjT −j

h

− ∂−j

h . 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

SLIDE 47

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Canonical discretization of finite difference

• perators of hypercomplex type

A list of examples

1

Forward finite differences: The set of operators ∂+j

h and

xjT −j

h

: f(x) → xjf(x − hej) span the Weyl-Heisenberg algebra of dimension 2n + 1. Moreover D+

h = n j=1 ej∂+j h and

Xh = n

j=1 ejxjT −j h

are the corresponding multivector ladder

• perators on the lattice hZn.

2

Backward finite differences: ∂−j

h

and xjT +j

h

: f(x) → xjf(x + hej) also span the Weyl-Heisenberg algebra of dimension 2n + 1. This turns out D−

h = n j=1 ej∂−j h

and X−h = n

j=1 ejxjT +j h

as the corresponding multivector ladder operators on the lattice hZn.

3

Discretization of the Hermite operator: D+

h and Xh − D− h are

• btained from the set of ladder operators Lj = ∂+j

h and

Lj = xjT −j

h

− ∂−j

h . 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

SLIDE 48

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Canonical discretization of finite difference

• perators of hypercomplex type

A list of examples

1

Forward finite differences: The set of operators ∂+j

h and

xjT −j

h

: f(x) → xjf(x − hej) span the Weyl-Heisenberg algebra of dimension 2n + 1. Moreover D+

h = n j=1 ej∂+j h and

Xh = n

j=1 ejxjT −j h

are the corresponding multivector ladder

• perators on the lattice hZn.

2

Backward finite differences: ∂−j

h

and xjT +j

h

: f(x) → xjf(x + hej) also span the Weyl-Heisenberg algebra of dimension 2n + 1. This turns out D−

h = n j=1 ej∂−j h

and X−h = n

j=1 ejxjT +j h

as the corresponding multivector ladder operators on the lattice hZn.

3

Discretization of the Hermite operator: D+

h and Xh − D− h are

• btained from the set of ladder operators Lj = ∂+j

h and

Lj = xjT −j

h

− ∂−j

h . 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

SLIDE 49

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Quasi-Monomiality Approach

The 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) →

• xj + h

2

• f(x + hej) and
• xj − h

2

• T −j

h

: f(x) →

• xj − h

2

• f(x − hej) satisfy
• ∂−j

h ,

• xk + h

2

• T +k

h

• =
• ∂+j

h ,

• xk − h

2

• T −k

h

• = δjkI
• cf. N. F. Appl. Math. Comp., 2014

The EGF of the form Gh(x, y; κ) = n

j=1

1 κ 1

h log (1 + hyj)

(1 + hyj)

xj h

yield the set of operators Lj = ∂+j

h and Mj =

• xj − κ′(∂xj )κ
• ∂xj

−1 T −j

h

as generators of the Weyl-Heisenberg algebra of dimension 2n + 1. Moreover, they are unique.

17 / 34

SLIDE 50

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Quasi-Monomiality Approach

The 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 a multi-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 one can construct each Clifford-vector-valued polynomial wk(x; h; λ) of order k by means of the operational rule wk(x; h; λ) = µk (Λh)k a, a ∈ Cℓ0,n.

18 / 34

SLIDE 51

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Quasi-Monomiality Approach

The Exponential Generating Function (EGF) approach

Appell set definition: {wk(x; h; λ) : k ∈ N0} is an Appell set carrying 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 satisfying D+

h wk(x; h; λ) = kwk−1(x; h; λ).

Appell set equivalent formulation: Find for each (x, t) ∈ hZn × R a 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

n 2; −t2 4 (Λh)2

• a + tΛh 0F1

n 2 + 1; −t2 4 (Λh)2

• a

= Γ n 2 tΛh 2 − n

2 +1

J n

2 −1(tΛh)a + n J n 2 (tΛh)

• a

19 / 34

SLIDE 52

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Quasi-Monomiality Approach

The Exponential Generating Function (EGF) approach

Appell set definition: {wk(x; h; λ) : k ∈ N0} is an Appell set carrying 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 satisfying D+

h wk(x; h; λ) = kwk−1(x; h; λ).

Appell set equivalent formulation: Find for each (x, t) ∈ hZn × R a 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

n 2; −t2 4 (Λh)2

• a + tΛh 0F1

n 2 + 1; −t2 4 (Λh)2

• a

= Γ n 2 tΛh 2 − n

2 +1

J n

2 −1(tΛh)a + n J n 2 (tΛh)

• a

19 / 34

SLIDE 53

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Quasi-Monomiality Approach

The Exponential Generating Function (EGF) approach

Appell set definition: {wk(x; h; λ) : k ∈ N0} is an Appell set carrying 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 satisfying D+

h wk(x; h; λ) = kwk−1(x; h; λ).

Appell set equivalent formulation: Find for each (x, t) ∈ hZn × R a 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

n 2; −t2 4 (Λh)2

• a + tΛh 0F1

n 2 + 1; −t2 4 (Λh)2

• a

= Γ n 2 tΛh 2 − n

2 +1

J n

2 −1(tΛh)a + n J n 2 (tΛh)

• a

19 / 34

SLIDE 54

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian 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 = ∂+j

h 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 , ∂−j h , W −j h

= µ−1w

• xj + h

2

• T −j

h , W +j h

= µ−1w

• xj + h

2

• T +j

h

and I, with j = 1, 2, . . . , n, do not endow a canonical realization of an Weyl-Heisenberg type algebra of dimension 4n + 1. Fill the Weyl-Heisenberg gap: The set of operators W −j

h

= µ−1w

• xj + h

2

• T −j

h , W +j h

= µ−1w

• xj + h

2

• T +j

h

and Wj = µ−1w (xj) I generate a Lie algebra isomorphic to sl(2n, R) (N.F., SIGMA 9 (2013), 065– Lemma 1).

20 / 34

SLIDE 55

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian 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 = ∂+j

h 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 , ∂−j h , W −j h

= µ−1w

• xj + h

2

• T −j

h , W +j h

= µ−1w

• xj + h

2

• T +j

h

and I, with j = 1, 2, . . . , n, do not endow a canonical realization of an Weyl-Heisenberg type algebra of dimension 4n + 1. Fill the Weyl-Heisenberg gap: The set of operators W −j

h

= µ−1w

• xj + h

2

• T −j

h , W +j h

= µ−1w

• xj + h

2

• T +j

h

and Wj = µ−1w (xj) I generate a Lie algebra isomorphic to sl(2n, R) (N.F., SIGMA 9 (2013), 065– Lemma 1).

20 / 34

SLIDE 56

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian 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 = ∂+j

h 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 , ∂−j h , W −j h

= µ−1w

• xj + h

2

• T −j

h , W +j h

= µ−1w

• xj + h

2

• T +j

h

and I, with j = 1, 2, . . . , n, do not endow a canonical realization of an Weyl-Heisenberg type algebra of dimension 4n + 1. Fill the Weyl-Heisenberg gap: The set of operators W −j

h

= µ−1w

• xj + h

2

• T −j

h , W +j h

= µ−1w

• xj + h

2

• T +j

h

and Wj = µ−1w (xj) I generate a Lie algebra isomorphic to sl(2n, R) (N.F., SIGMA 9 (2013), 065– Lemma 1).

20 / 34

SLIDE 57

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Application to Cauchy problems

N.F . SIGMA, 2013

Homogeneous Cauchy problem in [0, ∞) × hZn:    ∂tg(t, x) + E+

h g(t, x) − E− h g(t, x) = 0

, t > 0 g(0, x) = f(x) , t = 0 E+

h g(t, x) = E− h g(t, x)

, t ≥ 0. Semigroup action: The one-parameter representation Eh(t) = exp(tE−

h − tE+ h ) of the Lie group SU(1, 1) yields

g(t, x) = Eh(t)f(x) as a polynomial solution of the above homogeneous Cauchy problem. Discrete series connection: One can see that the semigroup (Eh(t))t≥0 gives, in particular, a direct link between the positive series representation of SU(1, 1) with the negative ones.

21 / 34

SLIDE 58

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Application to Cauchy problems

N.F . SIGMA, 2013

Homogeneous Cauchy problem in [0, ∞) × hZn:    ∂tg(t, x) + E+

h g(t, x) − E− h g(t, x) = 0

, t > 0 g(0, x) = f(x) , t = 0 E+

h g(t, x) = E− h g(t, x)

, t ≥ 0. Semigroup action: The one-parameter representation Eh(t) = exp(tE−

h − tE+ h ) of the Lie group SU(1, 1) yields

g(t, x) = Eh(t)f(x) as a polynomial solution of the above homogeneous Cauchy problem. Discrete series connection: One can see that the semigroup (Eh(t))t≥0 gives, in particular, a direct link between the positive series representation of SU(1, 1) with the negative ones.

21 / 34

SLIDE 59

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Application to Cauchy problems

N.F . SIGMA, 2013

Homogeneous Cauchy problem in [0, ∞) × hZn:    ∂tg(t, x) + E+

h g(t, x) − E− h g(t, x) = 0

, t > 0 g(0, x) = f(x) , t = 0 E+

h g(t, x) = E− h g(t, x)

, t ≥ 0. Semigroup action: The one-parameter representation Eh(t) = exp(tE−

h − tE+ h ) of the Lie group SU(1, 1) yields

g(t, x) = Eh(t)f(x) as a polynomial solution of the above homogeneous Cauchy problem. Discrete series connection: One can see that the semigroup (Eh(t))t≥0 gives, in particular, a direct link between the positive series representation of SU(1, 1) with the negative ones.

21 / 34

SLIDE 60

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Discrete electromagnetic Schr¨

• dinger
• perators on the lattice

Discrete electromagnetic Schr¨

• dinger 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

SLIDE 61

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Discrete electromagnetic Schr¨

• dinger
• perators on the lattice

Discrete electromagnetic Schr¨

• dinger 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

SLIDE 62

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Discrete electromagnetic Schr¨

• dinger
• perators on the lattice

Discrete electromagnetic Schr¨

• dinger 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

SLIDE 63

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Discrete electromagnetic Schr¨

• dinger
• perators on the lattice

Discrete electromagnetic Schr¨

• dinger 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

SLIDE 64

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Discrete electromagnetic Schr¨

• dinger
• perators on the lattice

Discrete electromagnetic Schr¨

• dinger 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

SLIDE 65

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Asymptotic approximation of a Sturm-Liouville problem

Discrete electromagnetic Schr¨

• dinger operators Lh on hZn

Lhf(x) = − h2 2µ

n

• j=1

∂ ∂xj

• w

xj qh ∂f ∂xj (x)

• + V

x h

• 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 1 q xj h

• (1 + O (h)) .

2

Asymptotic constraint associated to the discrete magnetic potential: qΦh(x) + 1 2µ

n

• j=1

2 qh − ah(xj) − ah(xj − h)

• = V

x h

• + O
• h3

.

23 / 34

SLIDE 66

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Asymptotic approximation of a Sturm-Liouville problem

Discrete electromagnetic Schr¨

• dinger operators Lh on hZn

Lhf(x) = − h2 2µ

n

• j=1

∂ ∂xj

• w

xj qh ∂f ∂xj (x)

• + V

x h

• 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 1 q xj h

• (1 + O (h)) .

2

Asymptotic constraint associated to the discrete magnetic potential: qΦh(x) + 1 2µ

n

• j=1

2 qh − ah(xj) − ah(xj − h)

• = V

x h

• + O
• h3

.

23 / 34

SLIDE 67

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Asymptotic approximation of a Sturm-Liouville problem

Discrete electromagnetic Schr¨

• dinger operators Lh on hZn

Lhf(x) = − h2 2µ

n

• j=1

∂ ∂xj

• w

xj qh ∂f ∂xj (x)

• + V

x h

• 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 1 q xj h

• (1 + O (h)) .

2

Asymptotic constraint associated to the discrete magnetic potential: qΦh(x) + 1 2µ

n

• j=1

2 qh − ah(xj) − ah(xj − h)

• = V

x h

• + O
• h3

.

23 / 34

SLIDE 68

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

The factorization approach

Isospectral 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+j

h

with A+j

h =

• qh

4µ

• ah(xj)T +j

h

− 2 qhI

• A−

h = n

• j=1

ejA−j

h

with A−j

h

=

• qh

4µ 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∂+j

h ,

Mh =

n

• j=1

ej

• hah(xj − h)2T −j

h

− 4 q2hI

• .

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SLIDE 69

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

The factorization approach

quasi-monomials vs. bound states of the discrete electromagnetic Schr¨

• dinger
• perators

Moreover, in case where the discrete electric and magnetic potentials are given by Φh(x) = h 8µ

n

• j=1

4 q2h2

• φ(x; h)2

φ(x + hej; h)2 + φ(x − hej; h)2 φ(x; h)2

• ah(x)

=

n

• j=1

ej 2 qh φ(x; h) φ(x + hej; h) it follows straightforwardly from the factorization property Lh = 1

2(A+ h A− h + A− h A+ h ) that Lh is isospectral equivalent to the

anti-commutator MhD+

h + D+ h Mh.

Indeed, the isospectral formula φ(x; h)−1Lh(φ(x; h)f(x) = − q 4µh(MhD+

h + D+ h Mh)

yields naturally from the combination of the factorization property with the aforementioned isospectral relations.

25 / 34

SLIDE 70

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

A sketch of a recent paper of mine

published recently at Applied Mathematics and Computation

We have shown that Lh is asymptotically equivalent to the discrete harmonic oscillator − 1

2m∆h + qΦh(x) with mass m ∼ µq

h , whose kinetic term is written in terms of the ’discrete’ Laplacian ∆h. For the particular choice ah(xj) = 1 q

• 1

h + µ xj h + µ

2

• it readily

follows that that the asymptotic expansion of Lh reduces to Lhf(x) = − 1 2µq (E+

h f(x) − E− h f(x)) + V

x h

• 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

SLIDE 71

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

A sketch of a recent paper of mine

published recently at Applied Mathematics and Computation

We have shown that Lh is asymptotically equivalent to the discrete harmonic oscillator − 1

2m∆h + qΦh(x) with mass m ∼ µq

h , whose kinetic term is written in terms of the ’discrete’ Laplacian ∆h. For the particular choice ah(xj) = 1 q

• 1

h + µ xj h + µ

2

• it readily

follows that that the asymptotic expansion of Lh reduces to Lhf(x) = − 1 2µq (E+

h f(x) − E− h f(x)) + V

x h

• 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

SLIDE 72

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

A sketch of a recent paper of mine

published recently at Applied Mathematics and Computation

From the Energy condition associated to the Fock space Fh endowed by the Clifford module ℓ2(hZn; Cℓ0,n) := ℓ2(hZn) ⊗ Cℓ0,n, the vacuum vectors of the form ψ0(x; h) = φ(x; h)s (s ∈ Pin(n)), the quantity 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 towards Dirac’s insight on quasi-probabilities (they may take negative values) to compute some examples involving the well-known Poisson and hypergeometric distributions, likewise quasi-probability distributions involving the generalized Mittag-Leffler/Wright functions. In some of the cases they may take negative values.

27 / 34

SLIDE 73

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

A sketch of a recent paper of mine

published recently at Applied Mathematics and Computation

From the Energy condition associated to the Fock space Fh endowed by the Clifford module ℓ2(hZn; Cℓ0,n) := ℓ2(hZn) ⊗ Cℓ0,n, the vacuum vectors of the form ψ0(x; h) = φ(x; h)s (s ∈ Pin(n)), the quantity 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 towards Dirac’s insight on quasi-probabilities (they may take negative values) to compute some examples involving the well-known Poisson and hypergeometric distributions, likewise quasi-probability distributions involving the generalized Mittag-Leffler/Wright functions. In some of the cases they may take negative values.

27 / 34

SLIDE 74

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Examples

Generalization of the so-called Poisson distribution

hnφ(x; h)2 =           

n

• j=1

Eα,β

• 4

q2−αh2 −1 4

xj h q (2−α)xj h

h−

2xj h

Γ

• β + α

xj h

• , if x ∈ hZn

≥0

, otherwise As a matter of fact, the Mittag-Leffler function Eα,β(λ) = ∞

m=0

λm Γ(β + αm) is well defined for Re(α) > 0, Re(β) > 0.

1

Discrete Electric Potential: Φh(x) = h 8µ

n

• j=1

1 qα   Γ

• α + β + α

xj h

• Γ
• β + α

xj h

• +

Γ

• β + α

xj h

• Γ
• β − α + α

xj h

• 

.

2

Discrete Magnetic Potential: ah(x) =

n

• j=1

ej

• 1

qα Γ

• α + β + α

xj h

• Γ
• β + α

xj h

• .

28 / 34

SLIDE 75

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Examples

Generalization of the so-called Poisson distribution

hnφ(x; h)2 =           

n

• j=1

Eα,β

• 4

q2−αh2 −1 4

xj h q (2−α)xj h

h−

2xj h

Γ

• β + α

xj h

• , if x ∈ hZn

≥0

, otherwise As a matter of fact, the Mittag-Leffler function Eα,β(λ) = ∞

m=0

λm Γ(β + αm) is well defined for Re(α) > 0, Re(β) > 0.

1

Discrete Electric Potential: Φh(x) = h 8µ

n

• j=1

1 qα   Γ

• α + β + α

xj h

• Γ
• β + α

xj h

• +

Γ

• β + α

xj h

• Γ
• β − α + α

xj h

• 

.

2

Discrete Magnetic Potential: ah(x) =

n

• j=1

ej

• 1

qα Γ

• α + β + α

xj h

• Γ
• β + α

xj h

• .

28 / 34

SLIDE 76

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Appell Set Property

The Poisson-Charlier connection

1

Remark: For the special choices q = 2 and λ =

1 h2 , the resulting

ladder operator that yields from the Mittag-Leffler distribution Mh =

n

• j=1

ej

• xj + 1

h

• T −j

h

− 1 hI

• 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 operational formula mk(x; h) = µk(Mh)ks (s ∈ Pin(n)), carrying the constants µ2m = (−1)m 1

2

• m

n

2

• m

(k = 2m) and µ2m+1 = (−1)m 3

2

• 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-Charlier polynomials in disguise.

29 / 34

SLIDE 77

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Appell Set Property

The Poisson-Charlier connection

1

Remark: For the special choices q = 2 and λ =

1 h2 , the resulting

ladder operator that yields from the Mittag-Leffler distribution Mh =

n

• j=1

ej

• xj + 1

h

• T −j

h

− 1 hI

• 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 operational formula mk(x; h) = µk(Mh)ks (s ∈ Pin(n)), carrying the constants µ2m = (−1)m 1

2

• m

n

2

• m

(k = 2m) and µ2m+1 = (−1)m 3

2

• 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-Charlier polynomials in disguise.

29 / 34

SLIDE 78

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Examples

The Generalized Wright distribution

                    

n

• j=1

1Ψ1

(δ, γ) (β, α) γγ αα 4 q1+γ−αh2 −1 × Γ

• δ + γ

xj h

• Γ
• β + α

xj h

α

αxj h γ− γxj h 4 xj h q− (1+γ−α)xj h

h−

2xj h

Γ xj

h + 1

• ,

x ∈ hZn

≥0

, otherwise .

1

Notice that the Wright series

1Ψ1

(δ, γ) (β, α) λ

• is absolutely

convergent for |λ| < αα γγ and of |λ| = αα γγ , Re(β) − Re(δ) > 1

2 for

h2 > γ2γ α2α 4 q1+γ−α and of h2 = γ2γ α2α 4 q1+γ−α , Re(β) − Re(δ) > 1

2

whenever α − γ = −1.

30 / 34

SLIDE 79

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Generalized Wright distributions

Some additional remarks

For γ = δ = 1, the aforementioned likelihood function is the Mittag-Leffler distribution in disguise. Moreover, if α = Re(α) > 0, α → 0+ and h > 2 q , the previous distribution simplifies to hnφ(x; h)2 =         

n

• j=1
• 1 −

4 q2h2 −1 q−

2xj h h− 2xj h

, if x ∈ hZn

≥0

, otherwise . For β = δ, the likelihood function amalgamates the Poisson distribution (α = γ = 1) as well as the orthogonal measure that gives rise, up to the constant

• 1 −

4 q2h2

−βn , to the hypergeometric distribution on hZn

≥0, carrying the parameter

λ =

4 q2h2 (α → 0+, γ = 1 and h > 2 q ).

31 / 34

SLIDE 80

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Generalized Wright distributions

Some additional remarks

For γ = δ = 1, the aforementioned likelihood function is the Mittag-Leffler distribution in disguise. Moreover, if α = Re(α) > 0, α → 0+ and h > 2 q , the previous distribution simplifies to hnφ(x; h)2 =         

n

• j=1
• 1 −

4 q2h2 −1 q−

2xj h h− 2xj h

, if x ∈ hZn

≥0

, otherwise . For β = δ, the likelihood function amalgamates the Poisson distribution (α = γ = 1) as well as the orthogonal measure that gives rise, up to the constant

• 1 −

4 q2h2

−βn , to the hypergeometric distribution on hZn

≥0, carrying the parameter

λ =

4 q2h2 (α → 0+, γ = 1 and h > 2 q ).

31 / 34

SLIDE 81

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Bibliography

Main references used in my talk

Faustino, Nelson (2013). Special Functions of Hypercomplex Variable on the Lattice Based on SU (1, 1). Symmetry, Integrability and Geometry: Methods and Applications 9, no. 0 : 65-18. Faustino, N. (2014). Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle. Applied Mathematics and Computation, 247, 607-622. Faustino, N. (2017). Hypercomplex Fock states for discrete electromagnetic Schr¨

• dinger operators: A Bayesian

probability perspective. Applied Mathematics and Computation, 315, 531-548. Faustino, Nelson (2017). Symmetry Preserving Discretization Schemes through Hypercomplex Variables, Conference: 15th International Conference of Numerical Analysis and Applied Mathematics , DOI: 10.13140/RG.2.2.35900.74882

32 / 34

SLIDE 82

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Bibliography

Main references used in my talk

Faustino, Nelson (2013). Special Functions of Hypercomplex Variable on the Lattice Based on SU (1, 1). Symmetry, Integrability and Geometry: Methods and Applications 9, no. 0 : 65-18. Faustino, N. (2014). Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle. Applied Mathematics and Computation, 247, 607-622. Faustino, N. (2017). Hypercomplex Fock states for discrete electromagnetic Schr¨

• dinger operators: A Bayesian

probability perspective. Applied Mathematics and Computation, 315, 531-548. Faustino, Nelson (2017). Symmetry Preserving Discretization Schemes through Hypercomplex Variables, Conference: 15th International Conference of Numerical Analysis and Applied Mathematics , DOI: 10.13140/RG.2.2.35900.74882

32 / 34

SLIDE 83

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Bibliography

Main references used in my talk

Faustino, Nelson (2013). Special Functions of Hypercomplex Variable on the Lattice Based on SU (1, 1). Symmetry, Integrability and Geometry: Methods and Applications 9, no. 0 : 65-18. Faustino, N. (2014). Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle. Applied Mathematics and Computation, 247, 607-622. Faustino, N. (2017). Hypercomplex Fock states for discrete electromagnetic Schr¨

• dinger operators: A Bayesian

probability perspective. Applied Mathematics and Computation, 315, 531-548. Faustino, Nelson (2017). Symmetry Preserving Discretization Schemes through Hypercomplex Variables, Conference: 15th International Conference of Numerical Analysis and Applied Mathematics , DOI: 10.13140/RG.2.2.35900.74882

32 / 34

SLIDE 84

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Bibliography

Main references used in my talk

Faustino, Nelson (2013). Special Functions of Hypercomplex Variable on the Lattice Based on SU (1, 1). Symmetry, Integrability and Geometry: Methods and Applications 9, no. 0 : 65-18. Faustino, N. (2014). Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle. Applied Mathematics and Computation, 247, 607-622. Faustino, N. (2017). Hypercomplex Fock states for discrete electromagnetic Schr¨

• dinger operators: A Bayesian

probability perspective. Applied Mathematics and Computation, 315, 531-548. Faustino, Nelson (2017). Symmetry Preserving Discretization Schemes through Hypercomplex Variables, Conference: 15th International Conference of Numerical Analysis and Applied Mathematics , DOI: 10.13140/RG.2.2.35900.74882

32 / 34

SLIDE 85

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

A final thought

”To first approximation, the human brain is a harmonic oscillator.” Barry Simon to Charles Fefferman in a private conversation as they walked around the Princeton campus. Figure: Barry Simon Figure: Charles Fefferman SimonFest 2006: Barry Stories– http://math.caltech.edu/SimonFest/stories.html♯fefferman

33 / 34

SLIDE 86

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

A final thought

”To first approximation, the human brain is a harmonic oscillator.” Barry Simon to Charles Fefferman in a private conversation as they walked around the Princeton campus. Figure: Barry Simon Figure: Charles Fefferman SimonFest 2006: Barry Stories– http://math.caltech.edu/SimonFest/stories.html♯fefferman

33 / 34

SLIDE 87

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

A final thought

”To first approximation, the human brain is a harmonic oscillator.” Barry Simon to Charles Fefferman in a private conversation as they walked around the Princeton campus. Figure: Barry Simon Figure: Charles Fefferman SimonFest 2006: Barry Stories– http://math.caltech.edu/SimonFest/stories.html♯fefferman

33 / 34

SLIDE 88

Symmetry Pre- serving Dis- cretiza- tion Schemes Nelson Faustino The Scope of Problems

Function Theoretical Methods in Numerical Analysis Motivation behind this talk

Lie-algebraic discretizations

Umbral Calculus Revisited Radial algebra approach Appell Sets su(1, 1) symmetries

Discretization of Operators of Sturm-Liouville type

Discrete Electromagnetic Schr¨

• dinger
• perators

Interplay with Bayesian Statistics

Thank you for your attention!

The author would like to thank the organizers of the ICNAAM 2017 for their kind invitation and for the financial support as well.

Figure: Pictures from my university, located at ABC Paulista (S˜ ao Paulo, Brazil)

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