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SPRINGER STYLE GUIDE STYLING OF HEADINGS In books, all major words in a heading are capitalized. This means: capitalize any word except articles, conjunctions, or prepositions that are four letters or less in length. The following paragraphs cover specific questions regarding the styling of headings.

Abbreviations Although abbreviations that have been defined in the text may be used in headings, it is just as acceptable to use the written-out forms instead of the abbreviations in headings. Follow author’s usage, but ensure that either written-out forms or abbreviations are used in headings throughout. A displayed heading should not consist solely of an abbreviation.

Capitalization of Latin Phrases Capitalize all words of commonly used Latin phrases—words appendix of Webster’s New Collegiate Dictionary appearing in titles and headings (e.g., Review of In Vivo Studies). Latin phrases not in Webster’s are italicized and lowercase. Species and genus names are styled as in the text.

Capitalization of Compound Terms Always capitalize the first element of a hyphenated compound. But: •	Do not capitalize the second element(s) if one of the elements is normally only used as a combining form (e.g., Pre-urethritis and the Third Trimester). •	Capitalize both elements if the combination is temporary (e.g., Dose-Dependent Effects; Augmentation of Input-Output Devices). •	Capitalize both elements if the second is a proper noun (e.g., Study of Non-Hamiltonian Functions). •	Always capitalize the main element of a technical term, regardless of prefixes, but do not capitalize the second (or further) element(s) of a hyphenated technical term (e.g., L-Dopa and Kidney Reinfection; Analysis of threo-1-Phenyl-1-hydroxy-2-aminopropane). •	Avoid the use of suspended hyphens; repeat the letter, as in “Success of Treatment with β2-Agonists and β2-Antagonists.” CAPITALIZATION As a rule, capitalize proper nouns (names, brand names), epithets, and some geographical and historical terms. Other terms should not be capitalized. Parts of a Publication The rules on capitalization differ in some respects for books and journals. In a journal article, titles and headings have only an initial capital, whereas in a book chapter all the words (except for conjunctions, articles, and prepositions up to four letters in length) in a title and headings are capitalized.

Within the text of a chapter or article, capitalize nouns and abbreviations used to refer to a specific part of a book or a journal article:

General: Table 4, Fig. 6, Eq. 3, Sect. 4.37, Introduction, Appendix Book specific: Contents, Preface, Foreword, Chap. 5, Vol. 4, Index

When any of these terms are used generically, they are not capitalized or abbreviated. Furthermore, do not capitalize “title page,” “the table of contents,” “page,” and “cover.” Names Proprietary or brand names are capitalized. e.g., Valium, Xerox, Vaseline, Seconal, Teflon, Dacron, Sephadex, Plexiglas Retain the symbols ®, ©, or ™ that may come after a brand or proper name if they are provided by the author. Do not add them where they are not present.

Proper names are usually capitalized when used as eponyms or possessive adjectives: Parkinson’s disease, Erlenmeyer flask, Gram’s stain, Cushing syndrome but follow the author’s usage of lowercase if this is accepted alternate usage: petri dish, gram-negative bacteria (acceptable according to Webster’s) Do not capitalize verbs, adjectives, or units of measure based on proper nouns; in some disciplines, however, such nouns are capitalized e.g., parkinsonism, pasteurize, mendelian traits, dalton, Hamiltonian, Gaussian

For author’s names, always follow the author’s capitalization and spacing e.g., particles such as “van”

If usage of other names is inconsistent, make them consistent if you are reasonably certain which version is correct. Specific Terms The practice of treating “specific” terms as names, and therefore capitalizing them, should be applied sparingly. For instance, references to “the editor,” “the author,” and “the editorial board” should be lowercase. Do not capitalize general terms appearing with numbers or letters designating a specific part of a series e.g., day 1, grade II, group A or modified by proper names (e.g., the Wistar rat) Official titles should, of course, be capitalized. Historical and Geographical Terms Capitalize historical periods and events: the Middle Ages, the Crusades, the Industrial Revolution, the Stone Age, the Second World War (or World War II)

but do not capitalize centuries (e.g., in the eighteenth century).

Be careful when capitalizing geographical terms. Merriam-Webster’s Geographical Dictionary is our standard reference. Here are some examples: •	Central America vs central Europe •	Southeast Asia vs the southeast of England •	South Africa vs southern Africa, southern Asia •	North Pole, North Africa vs northern Germany •	the Continent (i.e., vs the UK or Ireland) •	Near East, Middle East ABBREVIATIONS Unit Abbreviations Use the accepted abbreviations of units. The use of SI units is preferred but do not make any general changes in an author’s use of other units without checking the information in the science-specific chapters or consulting your Springer copy editing contact. Units are written lowercase and in full when used independently of values. They are abbreviated when used with values (the well-known exceptions regarding capitalization are Celsius, Fahrenheit, and Ångstrom).

When a unit measure appears with a number, use a numeral and the appropriate abbreviation: 10 mg (not ten milligrams) 3 weeks. (not three) If 10 ml of 15% dextrose solution is... This also applies when a virgule is used: ...yields 3 mg/ml Units are separated from each other by spaces: 5 mA h or 3 mg ml–1 Units of measure are not abbreviated when they are not preceded by a numeral: ...is measured in milligrams per liter. Never pluralize a unit abbreviation: 4 g, 300 ml, 15 mol, 3.5 M, 45 µg/1,000 ml Unit abbreviations are always spaced from the number (as in the above examples) and are not followed by a period. Some units are represented by a symbol that is closed up to the number; the common ones are percent and per mil 30%, 2‰ degrees Celsius and Fahrenheit 34°C, 90°F And minutes and seconds (, , but not for time) Note that in chemistry, a space is inserted between a number and °C. Otherwise, unit abbreviations are not closed up.

Unit abbreviations are not repeated in a range (e.g., 30–40%, 100–400°C). Note: K for kelvin is not used with a degree sign (e.g., 300 K).

The item being measured should follow the correct unit e.g., either “an aqueous solution of 0.3 g NaCl/ml” or “an aqueous solution of NaCl (0.3 g/ml).” An exception can be made for drug dosages in superficial editing, such as “15 mg/kg kanamycin” (which means “15 mg kanamycin per kilogram body weight”).

The use of “of” with a unit abbreviation and a quantity is not necessary but it may occasionally be used for clarity (3 g of chloride, 30,000 IU of penicillin). It is not used in the phrases such as “44 mmHg.” D not delete “of” following a percentage: 30% of the patients. Some abbreviations may be either upper- or lowercase, such as N/n and P/p. Follow the author’s usage. Nonunit Abbreviations and Acronyms Nonunit abbreviations should be spelled out at first use, the abbreviation following in parentheses. Some that occur very frequently need not be explained; of course, any definition given should not be deleted. Once introduced, an abbreviation should be used throughout a manuscript. There are a number of common-sense limitations, however: •	Do not insist on the use of an abbreviation or acronym if the author has written the term out in the majority of cases. •	If a considerable amount of text separates uses, the abbreviation may be re-explained. •	An otherwise abbreviated term may also be written out in full when it is used as the first word in a major section. •	An abbreviation or acronym may also be mentioned once (but not “used” in the sense of introduced), normally at first occurrence, and then not used any more.

Abbreviations and acronyms do not need to be used consistently in every article in a journal or in every chapter in a multiauthor book; generally, each article or chapter should be viewed independently. Consistency is required within one manuscript. Some publications may contain a list of abbreviations. The abbreviations included there may be defined in the text at first mention. The addition of s to pluralize an acronym or nonunit abbreviation is optional. Commonly used abbreviations of foreign words and expressions, e.g., those found in the body of Merriam-Webster’s, are not italicized (examples are et al., etc., cf., viz., i.e., and e.g.). Abbreviations are frequently used if another part of the same work or a specific part of another publication is referred to. e.g., Vol., p., Chap., Sect., Fig., Eq. (Do not abbreviate “Table.”) Do not abbreviate the units “day,” “week,” “month,” and “year.” The names of months may be abbreviated. Do not abbreviate the names of countries in running text; the only exceptions are USA (noun) or US (adjective) for United States and UK for United Kingdom. Abbreviate US state names using the two-letter postal abbreviations

A genus name should be written out in full at first use, but may be abbreviated to its first letter (capital, with a period) when subsequently used in a binomial.

UNITS AND QUANTITIES When a cardinal number is used with a unit, use the numeral (and unit abbreviation if applicable): 30 ml, 44%, 7 years but with ordinals below 11 either usage is acceptable: the fourth day or the 4th day, but the 11th day Always use numerals with decimals: 0.03, 33.3, 4.5-fold In decimals less than one, a zero precedes the decimal point: 0.33, 0.50 except in special cases where the quantity can never be greater than unity and customary usage allows its omission (if this is the author’s preferred style): p < .05 or p < 0.05, r = .86 or r = 0.86 (accept either lowercase or uppercase p and r) Use numerals when referring to something that is numbered: p. 33, step 4, day 7, point 5, group 3, phase II A specific case is the use of numerals when referring to specific numbered parts of a publication: Part III, Vol. 7, Chap. 4 When one number immediately follows another with no intervening punctuation, and both would normally be written in numerals, one may be written as a word for the sake of clarity: 11 40-ml flasks or eleven 40-ml flasks OTHER USES Time Use numerals for the time of day: 12 noon, 6 p.m., 7:55 a.m. (US) or 7.55 a.m. (UK), 1930 hours, 0715 hours Dates Use numerals for years and decades: 1964, the 1960s (not the 1960’s, the ’60s, or the sixties) but not for centuries, which are always spelled out and lowercase (e.g., the eighteenth century). Use numerals for the day of the month and the year, but use letters for the month: 3 January 1977 (preferred form) or January 3, 1977 3 Jan. 1977 Money Use numerals for money, specifying the country of origin where ambiguous (e.g., for dollars): US $6.60, Australian $3 million, 100 Swiss francs, £10, 5000 euros Names Spell out numbers that are part of a proper name (e.g., Fifth Symphony).

EN DASH En dash is longer than a hyphen. It can be used between numbers, between words, and as a placeholder. Be careful to distinguish it from a minus sign.

In Numbers The most common use of an en dash is to indicate a range (only with numerals, not with numbers which are written as words): The years 1960–1977 saw... measures 300–450 cm in length... An en dash is not used with “from” or “between.” Thus, change “between 75–100 kg” to “between 75 and 100 kg”, and “ranged from 3,000–4,000 IU/kg body wt” to “ranged from 3,000 to 4,000 IU/kg body wt.” Do not use an en dash when a plus or minus appears with either of the numbers (e.g., use “−100 to +110°C” rather than “−100–+110°C”).

In Text En dashes are used to link combinations or to indicate relations in adjectival compounds consisting of two or more parts of equal status, i.e., when the first part of the compound does not modify the second, but rather the noun. They often stand for “and” or “to.” Examples are: oil–water interface, dose–response curve, Bose–Einstein condensation, He–Ne laser, space–time continuum, I–V plot, solid–liquid interface, spin–orbit coupling Do not insist on this usage if an author has consistently used hyphens since it is uncommon even though correct

Other Uses An en dash is used to indicate a single chemical bond (CH3–CH2COOH). In tables, an en dash can be used to indicate that data are missing or unavailable; however, do not add dashes if the author has not provided them. MATHEMATICS Variables We have decided upon the following rule of thumb: •	If a symbol in an equation takes a value that needs to be substituted, then it should be set in italic. This symbol is a container for a value. •	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 roman. e.g., when talking about the mass of the electron we use me. “m” will be replaced by the number for the electron mass, the “e” simply labels this mass as being that of the electron.

If a variable has a running index such as xi, 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 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 abbreviation. It is usually in roman than in italics even when it represents a value, e.g., 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: •	μB (“Bohr”), •	TC (“Curie”), •	mp (“proton”), •	me (“electron”), •	e (value of the elementary charge), •	NA (“Avogadro”), kB (“Boltzmann”), •	c or c0 (speed of light in vacuum), •	h and (Planck’s constant), •	μ0 and ε0 (magnetic and electric constants), •	n-ary, p-adic, etc. Note “e” is upright when it stands for the word “electron.”

Functions In an equation general functions are set in italic. e.g., 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 •	grad for gradient •	H.C., h.c. for hermetian conjugate •	Im for imaginary part •	Log or log for logarithm •	Ln, ln for natural logarithm •	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

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. •	Products in the physics program, use bold and italics for characters representing vectors (e.g., v). •	Mathematics program, use bold and upright (e.g., v). •	Vectors are often represented using other styles, such as an overhead arrow. 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, 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).

Displayed Equations General Copyedtiors are responsible for some matters of presentation and for the text. e.g., phrases such as “the following equation” or “the equation below” in the text may frequently be removed as redundant or superfluous

The alignment of parts of an equation may be important. The analogy of “verbs” and “conjunctions,” 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.

Punctuation It is useful to assume that a displayed equation is part of a sentence and that punctuation follows normal language rules. Unless otherwise instructed, follow this usage unless an author has been consistent in not punctuating displayed equations.

Inline Equations Grammatically, an inline equation is a normal part of a paragraph and sentence. It follows normal punctuation rules. 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

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.

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 roman-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.

Others 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.

Functions and Symbols For clarity, functions and operands are usually distinguished using parentheses, e.g., “f(x)” and “sin (φ + θ).” 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.”

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.” 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. 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. Note any special cases. In many scientific treatments parentheses have strict meanings, and the style should not be changed, e.g., {111} vs (111) in solid-state physics. Some styles involve deliberately asymmetrical notation. e.g., the mathematical intervals (a,b] and [a,b) have different meanings

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, e.g., “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”. QUOTATIONS

In American usage periods and other punctuation appear before closing quotation marks, e.g., “The character change in a tumor is usually called ‘tumor progression.’ ” In British usage period and other punctuations appear before the closing quotation marks. if it is part of the original quotation (i.e., if a complete sentence is being quoted) and after if it is not. UK/US SPELLINGS British usage

Rule

Both -ise and -ize endings are accepted for many words, but not for all. Use either -ise or -ize throughout a manuscript, as far as possible. Example

realise or realize 	American Usage

Rule

In most instances the -ize ending is used 	Example

realize, recognize recognise or recognize 	But beware of exceptions 	advertise, surmise, compromise Use -yse 	analyse (not analyze) 	Use -yze 	analyze (but lyse) When the past tense, gerund, etc. is formed from a verb, the final single consonant is often doubled, regardless of stress. preferred, occurring, labelled, marvellous, worshipper but benefited 	A final consonant is doubled only if the stress is on the final syllable. preferred, occurring, labeled, marvelous, worshiper, benefited, but transferred Past tenses of some verbs may end in -t or -ed. spelled or spelt 	Use -ed in such cases. spelled Use -ce endings for some nouns. defence, licence 	Use -se in such cases. defense, license Some -ce nouns take an -se ending as verbs. practice (n), practise (v) 	Both forms have the same spelling 	practice (n or v) 	licence (n), license (v) 		license (n or v) The suffix -wards is more common. forwards, backwards, towards 	The suffix -ward is more common. forward, backward, toward Some prepositions may end in -st. while or whilst 	These prepositions do not end in -st. while amid or amidst 		amid, but in the midst of Use ou 	colour, behaviour, mould 	Use o 	color, behavior, mold but coloration, tumorous Use -re 	centre, fibre 	Use -er 	center, fiber Use ph 	sulphur 	Use f 	sulfur Use ae 	haematology, leukaemia, faecal 	Use e 	hematology, leukemia, fecal (but leukopenia) Generally use oe 	oedema, oesophagus, manoeuvre (but fetus is preferred to foetus) 	Use e 	edema, esophagus, maneuver Other examples 	aluminium 	Other examples 	aluminum artefact 		artifact fulfil 		fulfill grey 		gray Leucocyte 		leukocyte sceptical 		skeptical skilful 		skillful

HYPHENATION When a Hyphen is Used Hyphens are frequently used to connect prefixes to words if as a result the same vowel would appear twice together. However, there are numerous frequently used words for which this is not the case, e.g., coordinate or microorganism. Check Webster’s if you are uncertain. Compound adjectives are generally hyphenated when they precede a noun in the following cases: •	Combinations of adjective and past participle hard-driven man (but that man is hard driven) •	Combinations of noun and past participle handkerchief was hand-stitched •	Combinations of noun and present participle eye-opening experience •	Combinations of adverb and past participle (unless the adverb ends in “ly”) well-planned robbery, but poorly executed heist •	Compounds of degree short-range telescope, high-energy propellant •	Numeral–unit combinations 3-ml flask, 10-kg weight, 30-year-old patient, 5-mm-diameter catheter •	When using unit abbreviations with ranges, the correct style is: 3- to 10-ml flask, 20- to 30-year-old patients •	Generally avoid the following usage (although it is correct) since it can be confusing: 3–10-ml flask, 20–30-year-old patients •	When there is a danger of confusion: A “small-bowel constriction” is a constriction of the small bowel. A “small bowel constriction” might be a small constriction of the bowel. recover, re-cover; recreation, re-creation; unionized, un-ionized •	When a prefix or suffix modifies a two-word adjectival phrase, all three parts are hyphenated both prior to and following a noun: non-antigen-specific reaction (in the predicate, this should read “was not antigen specific”) Hyphenate chemical names according to the Merck Index; otherwise leave as submitted if the style is consistent. Use a hyphen with a prefix before a proper noun (e.g., non-Hodgkin’s lymphoma). Repeat the prefix in hyphenated compound terms if the prefix is constant but the main word changes (e.g., “β2-agonists and β2-antagonists are”). When a Hyphen is not Used Do not put a hyphen after “-ly” adverbs (as in the fourth example in the previous list). Also, do not hyphenate compound modifiers that are commonly read as a unit: small cell carcinoma, soft tissue lesions, open heart surgery, medical school students, in vivo cultures, birth control method, amino acid solution, high molecular weight substance. The use of multiple hyphens may obstruct understanding and is usually unnecessary. Finally, do not add hyphens to proper nouns (for example, do not put a hyphen in “Monte Carlo method”).