This is a work in progress

This is copyrighted 2004 Springer publishers. I am the author of chapter 13 on mathematics as well as chapter 15 on physics, which will follow later. This chapter refers to earlier parts of the style guide. I am not sure if I will node these. This is a work in progress, some formatting needs to be worked on to get it to show up nicely here. Some equations in the original document were rendered as images and I will need to re-create them as entities, however the substance of this chapter is mostly complete in the form you have it now.

Remeber, this is a style guide. That means this is what Springer wants from authors, but it is not defnitive. It is as consistent as is possible. Any feedback would be very welcome.

13 Mathematics

Contents

• 13.1 General

• 13.2 Numbers

• 13.3 Formatting

• 13.3.1 Upright Characters

• 13.3.2 Variables

• 13.3.3 Functions

• 13.5 Displayed Equations

• 13.5.1 General

• 13.5.2 Punctuation

• 13.6 Inline Equations

• 13.7 Theorems, Definitions, and Proofs

• 13.8 Other Issues

13.1 General

This chapter deals with the representation of mathematical content in Springer's scientific, technical, and medical books and journals. It does not specifically cover the style requirements of the Springer book and journal programs in the discipline of mathematics.

Mathematics is a language unto itself. Every mathematical object that exists has a clear and unambiguous meaning. When choosing a style of notation for depicting mathematics the most important consideration is how to retain the clarity of meaning of the mathematical objects. In plain language the symbols we use often have more than one meaning. The sentence "Virginia is in America" may refer to both a place "Virginia" and a person "Virginia." With no further information the sentence is ambiguous. When reading a mathematical expression there must be no room for such confusion. Though the objects of mathematical discourse have unambiguous meanings the long history of the subject has led to the use of many different notations. In many instances there is no international standard for mathematical notation. Springer has adopted a best use policy and has chosen a style which we hope aids in the clear presentation of mathematics. When copyediting mathematics one should always ask "Is the author consistent? Is the meaning clear?"

For some product types, there is a defined style requirement for mathematical content, while for other products only consistency in representation is required within each book, book chapter, or journal article. For the bulk of the products, we define two levels of math markup: standard and intensive (see Sect.]9.5.2), which corrspond to the copy editing levels described in Sect. 1.3. This chapter provides detailed information on the correct handling of this content.

13.2 Numbers

As described in Sect. 6.3.1, the numbers "one" to "ten" should be in words when not used in conjunction with units, but larger numbers (e.g., 11, 12, etc.) should always be digits. A common alternative in the "technical" disciplines is to represent numbers up to nine using words, and then use numerals, i.e., to use "one" to "nine" and then "10," "11," etc. This also applies to ordinals, e.g. "10th." Either scheme is acceptable, but a mixture of styles should be resolved.

Decimals The decimal point should be represented using a period, so correct any numbers in the non-English-language form, e.g., change "10.000,00" to the "10,000.00" form. The center dot is not recommended for decimal points. Two different styles are used for separating each bank of three digits to the left of the decimal point. The one given in the bottom row here is common in TeX-based and camera-ready products.
Commas 9,999 or 9999 (no separator) 10,000 10,000,000
Spaces 9 999 or 9999 (no separator) 10 000 10 000 000
For digits to the right of the decimal point you may use the thousands-space style applied for each bank of three digits (counted from the decimal point), e.g., 10 000.000 00; alternatively authors often do not apply any thousands separators to the right of the decimal point, and this is acceptable.

Exponential Notation The "×" sign is preferred to center dots "·" in exponential (or "scientific") notation, e.g., 2.99×108 not 2.99·108. The 1.2E03 form (sometimes 1.2e03) is not preferred but is acceptable. A mixture of styles should be resolved, but an author is allowed to use the 1.2×103 form in the text and the 1.2E03 form in tables to save space.

Logarithms Use the following forms:
log a x log to the base a of x
ln x or loge x natural log of x
lg x or log10 x

 common log of x

Extra Zeros Be very careful when inserting or removing zeros, whether to the right or left of decimal points; they often contain information. For example, the values 6 mV, 6.0 mV, and 6.00 mV are not equivalent, they imply different measurement precisions. Although, ideally, comparable entries in a column of a table should usually have the same precision, i.e., the same number of digits after the decimal point, leave the usage provided by the author.

Billions Most English-language texts in physics, engineering, and computer science now use the "American" system, where "billion" is 109 and "trillion" is 1012, not the "British" system's 1012 and 1018, respectively; see Merriam–Webster's under "number." The spelled-out versions are discouraged because of the potential ambiguity; advise the author to change to numerals.

Number Systems Various number systems appear in scientific texts.

Binary Use spaces to separate bytes or nibbles (half-bytes), e.g., 00111010 or 0011 1010. In long lists or tables, advise the author to use nibble separation. An author may use a base subscript, e.g., 001110102; leave it even though it may be redundant.

Hexadecimal Impose consistency using a recognized scheme; the "0×" prefix is preferred. Use upright uppercase letters A–F. Example: 0×003A = 3A16 = 3A H = 3A Hex = 5810

Octal Use a base-8 subscript. For binary equivalents, use banks of 3 digits. Example: 100 101 = 458 = 3710

SI Prefixes Use the standard prefixes for decimal multiples of units. These are written in upright font and are closed up to the unit. Positive
Value Prefix Abbreviation

1024 yotta Y
10–1 deci d
1021 zetta Z
10–2 centi c
1018 exa E
10–3 milli m
1015 peta P
10–6 micro µ (upright)
1012 tera T
10–9 nano n
109 giga G
10–12 pico p
106 mega M
10–15 femto f
103 kilo k
10–18 atto a
102 hecto h
10–21 zepto z
101 deca (or deka) da
10–24 yocto y

13.3 Formatting

Mathematical expressions are frequently complex with many components, for example in a given expression some symbols may represent operators, some represent functions and some represent variables. The most common question when typesetting mathematical notation is, "Should symbol "x" be set to upright or set in italic?"

13.3.1 Upright Characters

Do not italicize numbers, punctuation, parentheses, common function names (see Sect. 13.3.3), units, or mathematical signs (e.g., "+") wherever they occur (regardless of whether or not they appear in an equation or as a superscript or subscript).

13.3.2 Variables

We have decided upon the following rule of thumb:

manually converted up to here

– If a symbol in an equation takes a value which needs to be substituted into it before the result of the equation can be calculated, then it should be set in italic. This symbol is a container for a value. The contents of the container can either be a constant value or it can vary, but the important point is that the symbol, at the final reckoning, will be replaced by something else.

– If the symbol is a label, either for the name of a particle for example, or attached to a variable giving information about that variable, then it should be set in upright. This label does not act as a container for a value.

Here are some examples. When talking about the mass of the electron we use m e. The "m" will eventually be replaced by the number for the electron mass, the "e" simply labels this mass as being that of the electron. When we talk about the electronic charge of the electron we use "e". Here, when calculating, "e" will be replaced by the number for the electronic charge!

If the author is talking about some point x, some distance d, or some time t, all of these are set to italic in the expectation that later they will be replaced be the value of the coordinate, distance or time respectively. If the author wishes to label these as being in some positive region of some space then he might mark them as x pos, d pos, or t pos . Here the "pos" is set to be upright as it denotes "positive."

If a variable has a running index such as x i , where i runs between 0 and 3 for example, then this index i, though a label, is set to italic. In this instance the label i is a container for a set of values and in each instance x 0, x 1, x 2, and x 3 each take an independent value.

The numbers themselves do not act as containers for other values and so are set to upright. The complex number "i" we consider to be a creature of the same genus as the numbers and so we set it upright. Often it is represented as "j." Upright presentation is also a great aid in differentiating the complex number "i" from "i" used as a running index. The mixed use of "i" is very common.

The Roman and lowercase Greek alphabets provide the most common examples of variables. The use of upright and italic display for Greek letters follows the same convention as for Latin letters. Sometimes the Hebrew alphabet is used and sometimes the Frankfurter alphabet is used (mathematicians and physicists are fond of using many symbols).

An exception to these rules is made for a multiletter abbreviaton. It is usually written upright rather than in italics even when it represents a value, for example, IP (for ionization potential) and BER (for bit error rate). (An exception is the dimensionless Reynolds number "Re" (note that "Re" commonly denotes the "Real" part of a complex number). Use pH, pI, and pKa.)

Here are some examples of commonly used combinations:

? B("Bohr"), T C("Curie"), m p("proton"), m e("electron"), e (value of the elementary charge), N A("Avogadro"), k B("Boltzmann"), c or c 0(speed of light in vacuum), h and (Planck's constant), ? 0and ? 0(magnetic and electric constants), n-ary, p-adic, etc. Note "e" is upright when it stands for the word "electron."

13.3.3 Functions

In an equation with an arbitrary function the function has to be replaced by the operation that it represents before a final value for the equation can be obtained. General functions are therefore set in italic. In the equation before calculating the value of y we need to know what the functions f(x) and g(x) are. These functions act as containers for some proper given operator on x and therefore are set in italic. When the function is named explicitly, it does not act as a container for some arbitrary operation but tells us exactly what the operations to be performed on the argument is. In this case these function names should be set upright.

The most common functions that are set upright are:

C.C., c.c. for complex conjugate

cos for cosine

cot for cotangent

csc, cosec for cosecant

curl for curl

d for differential

delta (greek "D")

det for determinant

div for divergence

erf for error function

exp or e for exponential

H.C., h.c. for hermetian conjugate

Im for imaginary part

Log or log for logorithim

Ln, ln for natural logorithim

max for maximum

min for minimum

mod for modulo

∏ for product (the mathematical form of the greek letter "P")

Re for real part

sgn for sign

sec for secant

∑ for summation (greek "S")

sin for sine

sqrt for square root

tan for tangent

tr, Tr for trace

Here are several equations containing such functions:

13.4 Vectors, Tensors, and Matrices

These are multidimensional objects. They have more than one component and so some way is needed to distinguish them from simple objects.

Vectors Vectors are represented differently in different fields. For products in the physics program, use bold and italics for characters representing vectors (e.g., v). In the mathematics program, use bold and upright (e.g., v). For other disciplines, check with your Springer contact. Vectors are often represented using other styles, such as an overhead arrow, ; this is acceptable but deprecated. Any style mixtures should be corrected. Vector components should be in italics, not in bold, e.g., . For the norm of a vector, follow the author's usage of or , or simply v.

Matrices The characters representing matrices may be bold, bold and italic, or italic. Matrix elements should be in italics and nonbold. The superscripts "T" or "t" (transpose) and "H" (Hermitian) should be upright. Full matrices should be written as displayed equations. For matrix dimensions use "×," e.g., "a 3×3 matrix" or "an n×m matrix." Matrix determinants can be represented using straight lines, e.g., , or as "det B ." Follow the author's style for representing omitted elements (e.g., use ellipses). Examples:

Tensors The characters representing tensors are generally displayed using a sans serif font (e.g., Arial).

13.5 Displayed Equations

13.5.1 General

Copy edtiors are, as a rule, not responsible for making improvements with regard to issues such as page breaks and equation positioning. They are responsible, however, for some matters of presentation and for the text. For example, phrases such as "the following equation" or "the equation below" in the text may frequently be removed as redundant or superfluous. For citation rules see Sect. 9.5.1

The alignment of parts of an equation may be important. The analogy of "verbs" and "conjunctions," such as described in the Chicago Manual of Style, may be helpful in this context. "Verb" characters include

= ≠ < > ≤ ≥ ⊃ ⊂ ∈ ∉ ≅ ≡ ≈

while "conjunctions" are

+ – × ± ∪ ∩

"Verbs" in successive lines should be aligned beneath each other. A "conjunction" at the start of a line should be aligned under the first element to the right of the "verb" in the line above it. Example:

A "verb" or "conjunction" should not be repeated across a linebreak; it should appear once only as the first element of the following line. Example:

13.5.2 Punctuation

It is useful to assume that a displayed equation is part of a sentence and that punctuation follows normal language rules, which is demonstrated in the following examples showing combinations of text and equations (see Chap. 5). Unless otherwise instructed, follow this usage unless an author has been consistent in not punctuating displayed equations.

Equation ends the sentence:

... where z is fixed and

 (13.1)

From this relation ...

Sentence continues after equation:

... where z is fixed,

 (13.2)

and w takes the value 1.

Combined example:

... because

 (13.3)

where x refers to ..., and

 (13.4)

In this example, the addition of punctuation would be incorrect:

... where

and b are known.

Punctuation within an equation environment:

... because

 (13.5)

and

 (13.6)

Another example of punctuation within an equation environment:

... where

 (13.7)

Taking ...

Sometimes it is useful to consider equations to be the items in a list. Here is such an example:

... our list of results:

 (13.8)

where x, y, and a refer to ..., respectively,

 (13.9)

where w and z are defined elsewhere, and

 (13.10)

As in any list, be careful not to misplace commas or other punctuation.

13.6 Inline Equations

Grammatically, an inline equation is a normal part of a paragraph and sentence. It follows normal punctuation rules (see Sect. 13.5.2 and Chap. 5). It is also never numbered; if the author has numbered an inline equation, either it must be changed to a displayed equation (to retain the numbering) or the numbering must be dropped. Because of the consequent renumbering of following equations, such an equation should as a rule be changed from inline to displayed.

You may suggest to an author that certain mathematical content be changed from inline to displayed for better presentation. Examples are fractions and content with "tall" symbols. Inline fractions should employ a solidus (virgule or slash) instead of a built-up construction wherever sensible, for example using instead of the built-up construction

If tall symobls are used inline, suggest to the author that the more convenient inline forms be used; for example, use for , use for , or use for .

RPN, or reverse Polish (postfix) notation, cannot be changed to infix form; do not change "a b +" to "a + b." RPN is sometimes used in texts dealing with microprocessors, stack-based programs, etc.

13.7 Theorems, Definitions, and Proofs

Theorem Style In styling a theorem, the word "Theorem" and the number are in bold, and the following text in italics. Example:

Theorem 12 (Fermat's Little Theorem 28) Let p be a prime and assume that , then

Numbers, punctuation, parentheses, etc. should not be in italics. Edit the text if the author has used a "dangling" theorem:

... according to the following

Theorem 12 Let x be a prime ...

Proof Style In styling a proof, the word "Proof" is in italics and the text follows in upright. An end-of-proof box may be used, the preferred position being flush right. Example:

Proof  From (12.9) we know that a polynomial in x with

... and the rest follows by induction.         ?

Definition Style In styling a definition, the word "Definition" is in bold, the text is upright, and the terms being defined are italic.

Other Authors may use the theorem, definition, or proof styles for algorithms, assumptions, axioms, cases, claims, conjectures, corollaries, demonstrations, examples, exercises, hypotheses, lemmas, notes, problems, properties, propositions, questions, remarks, rules, solutions, etc. Furthermore, an author may use offset text, or different or smaller fonts for such environments. This should be done consistently.

13.8 Other Issues

Functions and Symbols For clarity, functions and operands are usually distinguished using parentheses, e.g., "f(x)" and "sin (?+?)." A symbol may act as the verb in a sentence where it can be translated into an English-language phrase such as "is equal to" or "is an element of." Examples:

Here the function

The modulus is large and

Multiplication Symbol Center dots (·) or line dots (.) are often used for multiplication. In many cases they are redundant and can be removed, i.e.., "a·b" or "a.b" be changed to the "ab" form. For a text involving vectors, an author may have used the center dot to indicate scalar (inner) multiplication as distinct from vector multiplication u×v, and in such cases the center dot must be retained. Use the "times" (×) symbol, rather than an asterisk (*) or the letter "x," in constructions such as "10 mm×0.7 mm×0.3 mm," "10,000×g centrifuge," and "100× lens."

Similar Symbols There are many symbols that are very similar in meaning. Be extremely careful in changing an author's usage of such symbols, but query any apparent misuse. For example, some scientists distinguish between "≈" meaning "is approximately equal to" and "~" meaning "is similar to," "is of the order of," or "is asymptotically equal to," while others consider "≈," "~," "≅," and "?" to be interchangeable, each meaning "is approximately equal to." The symbols "≅" and "≡" are often used to mean "is congruent to," a specific mathematical relation. Many references avoid defining these symbols because they vary in usage, even within disciplines. It's the author's responsibility to use the correct symbols consistently. Take into account the scale and measurement precision implied in the text and query the author if you suspect a mistake or inconsistent usage.

Equation References On general usage, see Sect. 9.5.1. It may sometimes be useful to retain specific qualifiers such as "congruence" or "inequality" but remove normally redundant qualifiers such as "equation," "relation," or "expression." Some people, however, distinguish between a "relationship" and an "equation" while others distinguish the term "relationship" (e.g., "in this chapter we examine the relationship between x and y") and a specific instance of a relationship at some point in time, a "relation" (e.g., "here the relation applies"). Accept an author's usage; do not change it.

Derivatives An author may use schemes such as dots and double dots above characters to represent derivatives, e.g., double derivative .

Ellipses Centered dots are preferred for "=" and for operators such as "+," "–," and "×," and line dots are preferred for commas. Alternatively, line dots may be used throughout. Symbols and commas should precede and follow ellipses. Examples:

Parentheses In general, apply the hierarchy , e.g., . If necessary, this nesting hierarchy may be repeated in bold, working outwards. Alternatively the parentheses may be nested using "taller" symbols, i.e., . Parentheses of different style are not required in unambiguous cases, e.g., . Unmatched ("lonely") parentheses in equations often reveal a typographical error; sometimes the required correction is obvious, but more often it's safer to ask the author to correct the problem. Parentheses should be "taller" than the enclosed content and most software packages automatically handle this; such issues are usually more efficiently corrected by the author or typesetter, so where possible use "global" instructions with specific examples. The "less than" and "greater than" signs ("<" and ">") should not be mixed up with angle brackets ("⟨" and "⟩"). Note any special cases. In many scientific treatments parentheses have strict meanings, and the style should not be changed, e.g., {111} vs ⟨111⟩ vs (111) in solid-state physics. Some styles involve deliberately asymmetrical notation. Examples: the mathematical intervals (a,b] and " vs "O III." Query the author if you suspect an error.

Signs In descriptive treatments, terms such as "greater than" are preferred. For example, use "the number of patients was greater than 30," not "... was > 30." Where the text is more "mathematical" than descriptive, or there are numerous relations, signs are more efficient for the reader and should be used.

Prime A prime should appear adjacent to the relevant character. In terms of meaning, the character and its prime form a unit, which itself may, for example, have a superscript attached to it. The usages and are not equivalent.

Special Fonts Some fonts have specific mathematical meaning and usage. Examples include number sets ( , , , , , etc.) and symbols with defined or "local" meaning.

Abbreviations The term "iff" (if and only if) is in Merriam–Webster's and does not require explanation. For abbreviations that do not appear in this dictionary, such as "wlog" (without loss of generality) and "wrt" (with respect to), add a first-use definition. Confirm proper use using a suitable dictionary. The abbreviations for Hermitian conjugate ("H.c."), Hermitian adjoint ("H.a."), and complex conjugate ("c.c.") should be upright, as should terms such as "const." for "constant".

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