Download How to Write Analytical Expressions and Equations

How to Write Analytical Expressions and Equations
An analytical expression is a combination of numbers, symbols, variables, and operators. An
equation is a statement that two analytical expressions are equivalent.
All equations have their lines indented by 1 cm within the double columns of 8.1 cm width.
Number the equations consecutively starting from (1). Do not use section or sub-section related
numbering, such as (2.1). Enclose the equation numbers in parentheses. Leave one empty row
before and after each equation if the number of equations is small compared to the rest of the paper.
Place the equation numbers at the right side of the double column with the use of the Tab key:
x = vt.
(1)
If the equation length prevents the inclusion of the equation number on the same row, then the
equation number should be placed on the next row at the right side of the double column:
Ax5 + Bx4 + Cx3 + Dx2 + Ex + F = 0. (2)
Separate symbols and variables from the arithmetic operators with the use of the Space key:
x + C – 2 = 0.
(3)
Inclusions of commas, semicolons, or dots should take place as a part of the sentence:
“The problem statement is to obtain the positive
solution(s) of a quadratic equation,
Ax2 + Bx + C = 0, x > 0,
(4)
where A, B, and C are known coefficients.”
Split long equations into several consecutive rows:
Ax9 + Bx8 + Cx7 + Dx6 + Ex5 + Fx4
+ Gx3 + Hx2 + Kx + L = 0
(5)
Equations that cannot easily split into rows may occasionally span both columns:
j
2
∂ 2 Φ mj , n
∂ 2 Φ mj ,n
− h 2 j ∂ Φ m ,n
j
j
[ w yy
+
+
2
] = ( E mj ,n − U eff )Φ mj ,n .
w
w
zz
yz
j 2
j 2
j
j
2m0
∂( y )
∂( z )
∂y ∂z
(6)
Use compact forms of the equations, for example:
∞
( x − a) n
n=0
n!
f ( x) = ∑
f
( n)
(a) ,
(7)
instead of
f ( x) = f (a) + f
(1)
(a )( x − a) +
( x − a) 2
f
2!
( 2)
(a) +
( x − a) 3
f
3!
* By Dobri Atanassov Batovski, Deputy Editor, AU J.T.
( 3)
( x − a) n ( n )
f (a) + ... (8)
n!
(Con’t. on inside back cover)
(a ) + ... +
The compact and readable forms of mathematical writing allow the use of:
a
- solidus ( / ) for the division operator: a / b, instead of
.
b
- exp() for the exponential function: exp(-at), instead of e − at , etc.
The proper inclusion of brackets, {}, [], (), improves the readability of long analytical expressions:
f(x) = log{exp[(Ax + B)(Cx + D)] - exp[(Ex + F)(Gx + H)]},
(9)
as well as removes the uncertainty in the compact form of denominators:
(Ax + B) / [x(Cy + D)], but not (Ax + B) / x(Cy + D).
The compact form is quite useful for short analytical expressions and equations embedded in the
text. In this case, do not use equation numbers:
“When B2 - 4AC < 0, the quadratic equation,
Ax2 + Bx + C = 0, has complex number
solutions.”
The symbols in each equation should be defined in advance or immediately following the
appearance of the equation:
“The relationship between mass, m, and energy,
E, has been obtained by Einstein (1905)
E = mc2,
(10)
where c is the speed of light.”
The following writing styles assist the visual interpretation of the mathematical content:
- text: Throughput, Latency,…
- italic: x, y, z,… (variables)
- bold: k, l, m,…, (vectors, matrices) or
- Greek: α, β, χ,…, etc.
Use italic style to represent the variables (x, y, z,…) in the equations. Note that the various
coefficients (A, B, C,…) are written in italic style as well. However, numerical values are written as
standard text (x9 = 48). Simple analytical expressions can be written with the “Subscript” and
“Superscript” formatting options: Pmax, x(2), nCk,…
References to the equation numbers should include the parentheses: “Substituting (2) in (1)
results in…” The word “Equation” should appear at the beginning of a sentence: “Equation (1) is
derived from...”
Avoid grammatical constructions, such as, “Using (2), the modified equation was obtained.”
[It remains unclear who or what used (2).] Write instead, “The modified equation was obtained by
using (2)…,” or “Using (2), we obtained the modified equation.”
Use the same symbol size in all equations. Avoid reductions of the symbol size to fit within a
single row. The equation length should not exceed the column width. This may result in a
truncation of the right side of the analytical expression or the equation may apparently overlap the
second column.
The importance and the applicability of the equations must be well explained in the text. The
consecutive appearance of multiple equations depends on the author’s point of view, but derivations
of straightforward results should be left for verification to the reader. Long proofs of important
equations should be included in Appendix.