SQL Tables – Physicspupil

May 26, 2022July 24, 2022

202205261415 Arithmetic in Lisp

Given two operands $a$ and $b$ which are real numbers, define the operators,

$a+b$ , $a-b$ , $a\times b$ , $a\div b$ , $a^{b}$ , $\sqrt[b]{a}$ , $a\bmod b$ ,

with respect to arithmetic operations such as addition, subtraction, multiplication, division, exponentiation, rooting, and Modulo.

+------+------------+-----+-----+-------------------------+
|  ID  |  operator  |  a  |  b  |  instance for example   |
+------+------------+-----+-----+-------------------------+
|  1   |  +         |  2  |  3  |  a + b  = 2 + 3    = 5  |
|  2   |  -         |  5  |  2  |  a - b  = 5 - 2    = 3  |
|  3   |  ×         |  3  |  2  |  a * b  = 3 * 2    = 6  |
|  4   |  ÷         |  9  |  3  |  a / b  = 9 / 3    = 3  |
|  5   |  ^         |  3  |  2  |  a ^ b  = 3 ^ 2    = 9  |
|  6   |  √         |  3  |  8  |  a √ b  = 3 √ 8    = 2  |
|  7   |  mod       |  7  |  4  |  a mod b = 7 mod 4 = 3  |
+------+------ -----+-----+-----+-------------------------+

+------+------------+-----+-----+-------------------------+

| ID | operator | a | b | instance for example |

+------+------------+-----+-----+-------------------------+

| 1 | + | 2 | 3 | a + b = 2 + 3 = 5 |

| 2 | - | 5 | 2 | a - b = 5 - 2 = 3 |

| 3 | × | 3 | 2 | a * b = 3 * 2 = 6 |

| 4 | ÷ | 9 | 3 | a / b = 9 / 3 = 3 |

| 5 | ^ | 3 | 2 | a ^ b = 3 ^ 2 = 9 |

| 6 | √ | 3 | 8 | a √ b = 3 √ 8 = 2 |

| 7 | mod | 7 | 4 | a mod b = 7 mod 4 = 3 |

+------+------ -----+-----+-----+-------------------------+

Not so accustomed to these, Lisp cannot interpret or compile them until we have defined those seven primitive functions, better known as symbolic computations; as were newborn infants taught human intelligence by grownup adults between homo sapiens.

To Lisp, arithmetic operations are usually $\textrm{\scriptsize{NOT}}$ predefined. We have in our toolkit $\textrm{\scriptsize{ONLY}}$ three mathematical functors, i.e., list, iteration, and recursion.

The roots of quadratic equation are given by the formula:

$\displaystyle{x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$

the expected outcome of which is calculated by parts, first the discriminant ‘delta’:

(define (delta a b c) (- (square b) (* 4 a c)))

1	(define (delta a b c) (- (square b) (* 4 a c)))

$\Delta = b^2-4ac$ ;

then the plus-minus ‘pm’:

(define (pm u v)
	(values (cons (+ u v) (- u v))))

1 2	(define (pm u v) (values (cons (+ u v) (- u v))))

$\pm (x)=(+x,-x)$ ;

and hence

(/ (pm (- b) (sqrt (delta a b c))) (* a 2))

1	(/ (pm (- b) (sqrt (delta a b c))) (* a 2))

will solve for real roots, ignoring the complex though.

$\therefore$ Predefining the seven primitive functions is in order for solving any computational problems by Lisp.

(to be continued)

May 25, 2022May 26, 2022

202205251335 SQL Table 005

CREATE TABLE SQL_Table_005
    (ID int,
    Function varchar(255),
    Derivative varchar(255));
INSERT INTO SQL_Table_005
    (Function, Derivative)
    VALUES
    ('position', 'velocity')
    ('velocity', 'acceleration')
    ('momentum', 'force')
    ('work', 'power')
    ('angular momentum', 'torque')
    ('electric charge', 'electric current')
    ('magnetic flux', 'induced voltage')

CREATE TABLE SQL_Table_005

(ID int,

Function varchar(255),

Derivative varchar(255));

INSERT INTO SQL_Table_005

(Function, Derivative)

VALUES

('position', 'velocity')

('velocity', 'acceleration')

('momentum', 'force')

('work', 'power')

('angular momentum', 'torque')

('electric charge', 'electric current')

('magnetic flux', 'induced voltage')

Output.
+------+--------------------+--------------------+
|  ID  |  Function          |  Derivative        |
+------+--------------------+--------------------+
|  1   |  position          |  velocity          |
|  2   |  velocity          |  acceleration      |
|  3   |  momentum          |  force             |
|  4   |  work              |  power             |
|  5   |  angular momentum  |  torque            |
|  6   |  electric charge   |  electric current  |
|  7   |  magnetic flux     |  induced voltage   |
+------+--------------------+--------------------+

Output.

+------+--------------------+--------------------+

| ID | Function | Derivative |

+------+--------------------+--------------------+

| 1 | position | velocity |

| 2 | velocity | acceleration |

| 3 | momentum | force |

| 4 | work | power |

| 5 | angular momentum | torque |

| 6 | electric charge | electric current |

| 7 | magnetic flux | induced voltage |

+------+--------------------+--------------------+

December 2, 2021October 24, 2022

202112021138 SQL Table 004

Background. (Equilibrium)

Equilibrium is established by the exchange of energy, volume, or particle number between different systems or subsystems.

_{Section 2.2. Arovas. (2019). Thermodynamics and Statistical Mechanics}

CREATE TABLE SQL_Table_004
    (ID int,
    Object_of_exchange varchar(255),
    Constancy varchar(255)
    Subject_of_equilibrium varchar(255)
    );
INSERT INTO SQL_Table_004
    (Object_of_exchange, Constancy, Subject_of_equilibrium)
    VALUES
    ('energy', 'temperature (T)', 'thermal')
    ('volume', 'momentum/temperature (p/T)', 'mechanical')
    ('particle', 'chemical potential/temperature (\mu /T)', 'chemical')

CREATE TABLE SQL_Table_004

(ID int,

Object_of_exchange varchar(255),

Constancy varchar(255)

Subject_of_equilibrium varchar(255)

);

INSERT INTO SQL_Table_004

(Object_of_exchange, Constancy, Subject_of_equilibrium)

VALUES

('energy', 'temperature (T)', 'thermal')

('volume', 'momentum/temperature (p/T)', 'mechanical')

('particle', 'chemical potential/temperature (\mu /T)', 'chemical')

Output.
+------+---------------------+-------------------------------+--------------------------+
|  ID  | Object_of_exchange  |  Constancy                    |  Subject_of_equilibrium  |
+------+---------------------+-------------------------------+--------------------------+
|  1   |  energy             |  temperature (T)              |  thermal                 |
|  2   |  volume             |  momentum/temperature (p/T)   |  mechanical              |
|  3   |  particle           |  chemical potential/          |  chemical                |
|      |                     |    temperature (\mu /T)       |                          |
+------+---------------------+-------------------------------+--------------------------+

Output.

+------+---------------------+-------------------------------+--------------------------+

+------+---------------------+-------------------------------+--------------------------+

+------+---------------------+-------------------------------+--------------------------+

September 29, 2021August 11, 2022

202109291137 SQL Table 003

CREATE TABLE SQL_Table_003
	(ID int,
	Formulations_of_quantum_mechanics varchar(255),
	Proposer varchar(255),
	Time_(Yr) varchar(255)
	);
INSERT INTO SQL_Table_003
	(Formulations_of_quantum_mechanics, Proposer, Time_(Yr))
	VALUES
	('transformation theory', 'Paul Dirac', '~1927 ')
	('matrix mechanics', 'Werner Heisenberg, Max Born, Pascual Jordan', '~1925')
	('wave mechanics', 'Erwin Schrödinger', '~1925')

CREATE TABLE SQL_Table_003

(ID int,

Formulations_of_quantum_mechanics varchar(255),

Proposer varchar(255),

Time_(Yr) varchar(255)

);

INSERT INTO SQL_Table_003

(Formulations_of_quantum_mechanics, Proposer, Time_(Yr))

VALUES

('transformation theory', 'Paul Dirac', '~1927 ')

('matrix mechanics', 'Werner Heisenberg, Max Born, Pascual Jordan', '~1925')

('wave mechanics', 'Erwin Schrödinger', '~1925')

Output.
+------+-------------------------------------+--------------------------------+-------------+
|  ID  |  Formulations_of_quantum_mechanics  |  Proposer                      |  Time_(Yr)  |
+------+-------------------------------------+--------------------------------+-------------+
|  1   |  transformation theory              |  Paul Dirac                    |  ~1927      |
|  2   |  matrix mechanics                   |  Werner Heisenberg, Max Born,  |  ~1925      |
|      |                                     |    Pascual Jordan              |             |
|  3   |  wave mechanics                     |  Erwin Schrödinger             |  ~1925      |
|  4   |  path integral formulation          |  Richard Feynman               |  ~1948      |
+------+-------------------------------------+--------------------------------+-------------+

Output.

+------+-------------------------------------+--------------------------------+-------------+

+------+-------------------------------------+--------------------------------+-------------+

| 1 | transformation theory | Paul Dirac | ~1927 |

| 2 | matrix mechanics | Werner Heisenberg, Max Born, | ~1925 |

| 3 | wave mechanics | Erwin Schrödinger | ~1925 |

| 4 | path integral formulation | Richard Feynman | ~1948 |

+------+-------------------------------------+--------------------------------+-------------+

September 28, 2021November 29, 2021

202109281720 SQL Table 002

CREATE TABLE SQL_Table_002
    (ID int,
    Velocity varchar(255),
    Nonquantum varchar(255),
    Quantum varchar(255)
    );
INSERT INTO SQL_Table_002
    (Velocity, Nonquantum, Quantum)
    VALUES
    ('v << c', 'Classical Mechanics (Newton)', 'Quantum Mechanics (Bohr, Heisenberg, Schrödinger, et al.)')
    ('0 < |v-c| << 1', 'Special Relativity (Einstein)', 'Quantum Field Theory (Dirac, Pauli, Feynman, Schwinger, et al.)')

CREATE TABLE SQL_Table_002

(ID int,

Velocity varchar(255),

Nonquantum varchar(255),

Quantum varchar(255)

);

INSERT INTO SQL_Table_002

(Velocity, Nonquantum, Quantum)

VALUES

('v << c', 'Classical Mechanics (Newton)', 'Quantum Mechanics (Bohr, Heisenberg, Schrödinger, et al.)')

('0 < |v-c| << 1', 'Special Relativity (Einstein)', 'Quantum Field Theory (Dirac, Pauli, Feynman, Schwinger, et al.)')

Output.
+------+------------------+-----------------------+----------------------------+
|  ID  |  Velocity        |  Nonquantum           |  Quantum                   |
+------+------------------+-----------------------+----------------------------+
|  1   |  v << c          |  Classical Mechanics  |  Quantum Mechanics         |
|      |                  |   (Newton)            |   (Bohr, Heisenberg,       |
|      |                  |                       |   Schrödinger, et al.)     |
|  2   |  0 < |v-c| << 1  |  Special Relativity   |  Quantum Field Theory      |
|      |                  |   (Einstein)          |   (Dirac, Pauli, Feynman,  |
|      |                  |                       |    Schwinger, et al.)      |
+------+------------------+-----------------------+----------------------------+
(D. J. Griffiths on Introduction to Electrodynamics (1999), pg. xi)

Output.

+------+------------------+-----------------------+----------------------------+

+------+------------------+-----------------------+----------------------------+

| 1 | v << c | Classical Mechanics | Quantum Mechanics |

| 2 | 0 < |v-c| << 1 | Special Relativity | Quantum Field Theory |

+------+------------------+-----------------------+----------------------------+

(D. J. Griffiths on Introduction to Electrodynamics (1999), pg. xi)

September 28, 2021November 29, 2021

202109281636 SQL Table 001

CREATE TABLE SQL_Table_001
    (ID int,
    Subject varchar(255),
    Object varchar(255),
    Category varchar(255)
    );
INSERT INTO SQL_Table_001
    (Subject, Object, Category)
    VALUES
    ('algebraic geometry', 'regular (polynomial) functions', 'algebraic varieties')
    ('topology', 'continuous functions', 'topological spaces')
    ('differential topology', 'differentiable functions', 'differentiable manifolds')
    ('complex analysis', 'analytic (power series) functions', 'complex manifolds');

CREATE TABLE SQL_Table_001

(ID int,

Subject varchar(255),

Object varchar(255),

Category varchar(255)

);

INSERT INTO SQL_Table_001

(Subject, Object, Category)

VALUES

('algebraic geometry', 'regular (polynomial) functions', 'algebraic varieties')

('topology', 'continuous functions', 'topological spaces')

('differential topology', 'differentiable functions', 'differentiable manifolds')

('complex analysis', 'analytic (power series) functions', 'complex manifolds');

Output.
+------+-------------------------+-------------------------------------+----------------------------+
|  ID  |  Subject                |  Object                             |  Category                  |
+------+-------------------------+-------------------------------------+----------------------------+
|  1   |  algebraic geometry     |  regular (polynomial) functions     |  algebraic varieties       |
|  2   |  topology               |  continuous functions               |  topological spaces        |
|  3   |  differential topology  |  differentiable functions           |  differentiable manifolds  |
|  4   |  complex analysis       |  analytic (power series) functions  |  complex manifolds         |
+------+-------------------------+-------------------------------------+----------------------------+
(J.S. Milne on Algebraic Geometry (2017), pg. 7)

Output.

+------+-------------------------+-------------------------------------+----------------------------+

+------+-------------------------+-------------------------------------+----------------------------+

+------+-------------------------+-------------------------------------+----------------------------+

(J.S. Milne on Algebraic Geometry (2017), pg. 7)