OPTICS

• Study of laser optics for Thomson scattering at plasmas
• Analysis of a periscope system
• Design and assembly of a fluorescent position sensor
• Design and assembly of Nd-YAG laser beam profiler
• Design of a colorimeter
• Design and assembly of an apparatus for contour determination of colored fabrics
• Design of a slit projector for visual inspection of welds
• Study of an optical system to detect transition radiation
• Design of a telescope for plasma diagnostics
• Design and assembly of a laser diode reflectometer for writing and inspection of WORM’s
• Design of an in-situ optical inspection system for boron nitride deposition on silicon wafers
• “Fundamentals in optical engineering”, von Karman Institute for Fluid Dynamics, 08/04/2011. Publication is part of VKI LS 2011-04 “Recent developments in unmanned aircraft systems”

MATHEMATICAL PHYSICS

PhD title: “Dimensional exploration of the SI integer lattice.”

“On a Mathematical Method for Discovering Relations Between Physical Quantities: a Photonics Case Study.” https://www.researchgate.net/profile/Philippe_Chevalier2/publications/ .Quantity calculus defines the rules that apply to SI physical quantities used in physics and engineering. This research aims at the construction of a rigorous mathematical framework explaining the selection rule resulting in constitutive equations. Here, we show that each SI physical quantity, that is represented by a lattice point in a seven dimensional integer lattice, has a unique $7D$-hypersphere. The lattice points incident on the $7D$-hypersphere are forming rectangles containing the origin $\vt{o}$, the lattice point $\vt{z}$ representing the selected physical quantity and the lattice point representations $\vt{x},\vt{y}$ of a pair of distinguishable physical quantities $[x],[y]$ where $\vt{z}=\vt{x}+\vt{y}$\;. The resulting rectangles are the geometric representations of realizable constitutive equations for the selected physical quantity $[z]$\;. We apply the “$nD$-hypersphere´´ method on the physical quantities $\vt{E},\vt{H},\vt{D},\vt{B}$ and find the integral forms of Maxwell’s equations. We find an integer sequence of non-degenerated unique rectangles formed by 4 lattice points $\vt{o},\vt{x},\vt{y},\vt{z}$ in $\mZ^7$ as function of the infinity norm $\norm{\vt{z}}_{\infty}=s$\;

A “table of Mendeleev” for physical quantities? https://www.researchgate.net/publication/262067273_A_table_of_Mendeleev_for_physical_quantities; Seminar KUL, Leuven, Belgium, 06/05/2014. We know from chemistry the celebrated table of Mendeleev, that organizes the chemical elements. That discovery was a major step in the development of chemistry. The mathematical structure classifying the physical quantities is presently unknown. We follow a bottom-up approach starting from the building blocks of the physics language, that are the physical quantities. Here we show that classes of physical quantities, that are expressed according to the SI convention, are mathematically equivalent to integer lattice points of the seven dimensional integer lattice. The integer lattice points are mathematically connected through signed permutations of the integer lattice point coordinates. These connections result in leader classes, known from coding theory. The assignment of a Gödel number to each leader class generates a well-ordered set, in analogy with the assignment of the atomic number to the chemical elements. The proposed mathematical structure, a measure polytope, should provide the physicist with a useful framework in his search for describing the physics of nature.

MATHEMATICS

Integer sequences submitted to the On-Line Encyclopedia of Integer Sequences

https://oeis.org/A240934; Number of rectangles formed by the absolute leader classes of the seven dimensional integer lattice as function of the infinity norm n, where the rectangles have one common lattice point being the origin of the seven dimensional integer lattice.

A240934 sequence is a(n)=[120, 7196, 162554, 1341957, 9255603, 40532530, 168302117, 523421602, 1637895896, 4129547423]

http://oeis.org/A247557; Number of rectangles formed by the absolute leader classes of the seven-dimensional integer lattice as a function of the infinity norm n and having a unique perimeter, where the rectangles have one common lattice point being the origin of the seven-dimensional integer lattice.

A247557 sequence is a(n)=[1, 7, 26, 79, 182, 333, 693, 1180, 1999, 3247]

http://oeis.org/A128891 ; The constant A128891 and the constant of A128892 are connected by the equation Sum_{n>=0}S_n – 2*Pi*Sum{n>=0}V_n = 2, where S_n and V_n are respectively the area and volume of a n-dimensional sphere of unit radius.

http://oeis.org/A128892 ; The constant is equal to Sum_{n>=0} S_n, where S_n is the area of an n-dimensional sphere of unit radius. This constant and the constant of A128891 are connected by the equation Sum_{n>=0}S_n – 2*Pi*Sum{n>=0}V_n = 2, where V_n is the volume of an n-dimensional sphere of unit radius.

http://oeis.org/A266387; Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 322560.

http://oeis.org/A266398; Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 13440.

http://oeis.org/A266397; Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 26880.

http://oeis.org/A266396; Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 80640.

http://oeis.org/A266395; Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 161280.

http://oeis.org/A008586; The number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 2688.

http://oeis.org/A001477; The number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 8960 or 168.

http://oeis.org/A045943 ; Number of orbits of Aut(Z^7) as function of the infinity norm (n+1) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 5376 or 17920 or 20160.

http://oeis.org/A102860 ; Number of orbits of Aut(Z^7) as function of the infinity norm (n+2) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 53760.

http://oeis.org/A154286 ; Number of orbits of Aut(Z^7) as function of the infinity norm (n+4) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 107520.

http://oeis.org/A000579 ; Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 645120.

http://oeis.org/A115067 ; Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 6720.

http://oeis.org/A002412 ; Number of orbits of Aut(Z^7) as function of the infinity norm (n+1) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 40320.

http://oeis.org/A270950 ; Number of distinct cardinalities of orbits of lattice points under the automorphism group of the n-dimensional integer lattice.

A270950 sequence is a(n)=1, 1, 2, 5, 9, 12, 20, 29, 40, 53, 76, 99, 132, 172, 216, 270, 341, 750,…

For n=0 the a(0)=1. For n=3 we have the following distinct cardinalities of the orbits 6, 8, 12, 24, 48 and thus a(3)=5. For n=4 we have the distinct cardinalities of the orbits 8, 16, 24, 32, 48, 64, 96, 192, 384 and thus a(4)=9. For n=5 we have the distinct cardinalities of the orbits 10, 32, 40, 160, 240, 320, 480, 640, 960, 1920, 3840 and thus a(5)=12.

We define the subset of the hypercubic shell as the set consisting of all elements xz (with x an element of Aut(Z^n)) and call it an orbit of z under Aut(Z^n). We are interested in the cardinalities of these orbits as function of the dimension n of the integer lattice. The number of orbits for a selected hypercubic shell with infinity norm s is given by C(n+s-1,s). Let q_i be the number of characters of type i of the alphabet A. Suppose that the characters occurring in the n-tuple z are subjected to signed permutations and that we denote the set containing the generated lattice points as [z]. The cardinality of the set [z] is given by the equation: card{[z]}=2^{q-q_0} {q!}/{q_0!q_1!q_2!… q_s!}. The number of distinct cardinalities card{[z]} depends on the infinity norm s and on the dimension n of the integer lattice. The maximum number of possible cardinalities is obtained by setting s=n. What we count now is the number of distinct cardinalities as function of the dimension n of the integer lattice.