8.3: Footnotes - Mathematics


This is an “average” variant. The prices may vary, and also the species (mostly but not always birds are traded). As a rule, however, the problem speaks about 100 animals and 100 monetary units. There are mostly three species, two of which cost more than one unit while the third costs less.


Who wants to, can try to find the full solution with or without negative numbers (which would stand for selling instead of buying), and demonstrate that it does represent an exhaustive solution under the given circumstances. That was done by the Arabic mathematician Abū Kāmil around 900 ce. In the introduction to his treatise about the topic he took the opportunity to mock those practitioners deprived of theoretical insight who gave an arbitrary answer only—and who thus understood the question as a riddle and not as a mathematical problem.


In the Old Babylonian texts, a closed group consisted of the four rectangle problems where the area is given together with the length; the width; the sum of these; or their difference. One may presume that the completion trick was first invented as a way to make this group grow from two to four members.


Who only practices equation algebra for the sake of finding solutions may not think much of coefficients—after all, they are mostly a nuisance to be eliminated. However, Viète and his generation made possible the unfolding of algebraic theory in the seventeenth century by introducing the use of general symbols for the coefficients. Correspondingly, the Old Babylonian teachers, when introducing coefficients, made possible the development of algebraic practice—without the availability and standardized manipulation of coefficients, no free representation is possible.


In order to see that 10 (and 30) had precisely this role one has to show that 10 was not the normal choice in other situations where a parameter was chosen freely. Collation of many sources shows that 10 (respectively 30 in descendants of the school tradition) was the preferred side not only of squares but also of other regular polygons—just as 4, 7, 11, etc. can be seen to have been favorite numbers in the multiplicative-partitive domain but only there, cf. note 4, page 48.


Eshnunna had been subdued by Ur III in 2075 but broke loose already in 2025.


The Liber mensurationum ascribed to an unidentified Abū Bakr “who is called Heus” and translated by Gerard of Cremona.


The quotation is borrowed from this treatise, rendered in “conformal translation” of the Latin twelfth-century version (the best witness of the original wording of the text).

You shouldn't be using eqnarray , first of all, but rather align .

While footnote works in equation , it doesn't in other alignment environments.

According to Kopka and Daly, A Guide to LaTeX, p. 96--97, the footnote command is not allowed in math mode, so you'll have to resort to using footnotemark inside math environments and footnotetext <<Some text>>in normal text mode.

Note that if, as in your example, more than one footnote is needed, you have to adjust the footnote counter by subtracting n-1 ( n being the number of footnotes in math mode) before the first footnotetext and using stepcounter before all other occurences of footnotetext .

If you want to put footnotes in equation environment, just use footnote in math mode:

For eqnarray environment or some other complex environments, use:

(Well, eqnarray is wicked, see eqnarray vs align)

However, it is not a good manner to use footnotes in math equations. The default marks could be considered as exponents. If it is really necessary, it is better to redefine hefootnote to get a different mark style. For example:

Plan Requirements

Applied Mathematics (BS): 120 Total Units

A grade of C- or higher is required.

Students should consult their academic advisors to determine which courses fill this requirement.

Basic Science Electives

Course List
Code Title Hours
BIO 181Introductory Biology: Ecology, Evolution, and Biodiversity4
BIO 183Introductory Biology: Cellular and Molecular Biology4
CH 201
& CH 202
Chemistry - A Quantitative Science
and Quantitative Chemistry Laboratory
PY 202University Physics II4
PY 208
& PY 209
Physics for Engineers and Scientists II
and Physics for Engineers and Scientists II Laboratory

Statistics Electives

Course List
Code Title Hours
Statistics Sequence 1
ST 371
& ST 372
Introduction to Probability and Distribution Theory
and Introduction to Statistical Inference and Regression
Statistics Sequence 2
ST 421
& ST 422
Introduction to Mathematical Statistics I
and Introduction to Mathematical Statistics II
Statistics Sequence 3
MA 421
& ST 380
Introduction to Probability
and Probability and Statistics for the Physical Sciences
Statistics Sequence 4
MA 421
& ST 370
Introduction to Probability
and Probability and Statistics for Engineers
Statistics Sequence 5
MA 421
& ST 422
Introduction to Probability
and Introduction to Mathematical Statistics II

Methods of Applied Math Electives

Course List
Code Title Hours
BMA 573Mathematical Modeling of Physical and Biological Processes I3
BMA 574Mathematical Modeling of Physical and Biological Processes II3
E 531Dynamic Systems and Multivariable Control I3
MA 450Methods of Applied Mathematics I3
MA 451Methods of Applied Mathematics II3
MA 531Dynamic Systems and Multivariable Control I3
MA 573Mathematical Modeling of Physical and Biological Processes I3
MA 574Mathematical Modeling of Physical and Biological Processes II3
OR 531Dynamic Systems and Multivariable Control I3

Math Electives

Course List
Code Title Hours
LOG 335Symbolic Logic3
MA 325Introduction to Applied Mathematics3
MA 335Symbolic Logic3
MA 351Introduction to Discrete Mathematical Models3
BMA 573Mathematical Modeling of Physical and Biological Processes I3
BMA 574Mathematical Modeling of Physical and Biological Processes II3
CSC 416Introduction to Combinatorics3
CSC 427Introduction to Numerical Analysis I3
CSC 428Introduction to Numerical Analysis II3
CSC 565Graph Theory3
CSC 580Numerical Analysis I3
CSC 583Introduction to Parallel Computing3
E 531Dynamic Systems and Multivariable Control I3
ECG 528Options and Derivatives Pricing3
FIM 528Options and Derivatives Pricing3
FIM 548Monte Carlo Methods for Financial Math3
FIM 549Financial Risk Analysis3
ISE 505Linear Programming3
MA 401Applied Differential Equations II3
MA 408Foundations of Euclidean Geometry3
MA 410Theory of Numbers3
MA 412Long-Term Actuarial Models3
MA 413Short-Term Actuarial Models3
MA 416Introduction to Combinatorics3
MA 421Introduction to Probability3
MA 426Mathematical Analysis II3
MA 427Introduction to Numerical Analysis I3
MA 428Introduction to Numerical Analysis II3
MA 430Mathematical Models in the Physical Sciences3
MA 432Mathematical Models in Life and Social Sciences3
MA 437Applications of Algebra3
MA 440Game Theory3
MA 444Problem Solving Strategies for Competitions1
MA 450Methods of Applied Mathematics I3
MA 451Methods of Applied Mathematics II3
MA 491Reading in Honors Mathematics1-6
MA 493Special Topics in Mathematics1-6
MA 499Independent Research in Mathematics1-6
MA 501Advanced Mathematics for Engineers and Scientists I3
MA 502Advanced Mathematics for Engineers and Scientists II3
MA 504Introduction to Mathematical Programming3
MA 505Linear Programming3
MA 512Advanced Calculus II3
MA 513Introduction To Complex Variables3
MA 515Analysis I3
MA 518Geometry of Curves and Surfaces3
MA 520Linear Algebra3
MA 521Abstract Algebra I3
MA 522Computer Algebra3
MA 523Linear Transformations and Matrix Theory3
MA 524Combinatorics I3
MA 526 3
MA 528Options and Derivatives Pricing3
MA 531Dynamic Systems and Multivariable Control I3
MA 532Ordinary Differential Equations I3
MA 534Introduction To Partial Differential Equations3
MA 537Nonlinear Dynamics and Chaos3
MA 540Uncertainty Quantification for Physical and Biological Models3
MA 544Computer Experiments In Mathematical Probability3
MA 546Probability and Stochastic Processes I3
MA 547Stochastic Calculus for Finance3
MA 548Monte Carlo Methods for Financial Math3
MA 549Financial Risk Analysis3
MA 551Introduction to Topology3
MA 555Introduction to Manifold Theory3
MA 561Set Theory and Foundations Of Mathematics3
MA 565Graph Theory3
MA 573Mathematical Modeling of Physical and Biological Processes I3
MA 574Mathematical Modeling of Physical and Biological Processes II3
MA 580Numerical Analysis I3
MA 583Introduction to Parallel Computing3
MA 584Numerical Solution of Partial Differential Equations--Finite Difference Methods3
MA 587Numerical Solution of Partial Differential Equations--Finite Element Method3
MA 591Special Topics1-6
MBA 528 3
OR 504Introduction to Mathematical Programming3
OR 505Linear Programming3
OR 531Dynamic Systems and Multivariable Control I3
OR 565Graph Theory3
ST 412Long-Term Actuarial Models3
ST 413Short-Term Actuarial Models3
ST 546Probability and Stochastic Processes I3

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Primary research articles require a disclosure statement. Review articles present an expert synthesis of evidence and may be treated as an authoritative work on a subject. Review articles therefore require a disclosure statement.Other article types such as editorials, book reviews, comments (amongst others) may, dependent on their content, require a disclosure statement. If you are unclear whether your article type requires a disclosure statement, please contact the Editor-in-Chief.

Please note that, in addition to the above requirements, funding information (given that funding is a potential conflict of interest (as mentioned above)) needs to be disclosed upon submission of the manuscript in the peer review system. This information will automatically be added to the Record of CrossMark, however it is not added to the manuscript itself. Under ‘summary of requirements’ (see below) funding information should be included in the ‘Declarations’ section.

Summary of requirements

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Please see the various examples of wording below and revise/customize the sample statements according to your own needs.

When all authors have the same (or no) conflicts and/or funding it is sufficient to use one blanket statement.

Examples of statements to be used when funding has been received:

Examples of statements to be used when there is no funding:

9.3. Mathematical Functions and Operators

Mathematical operators are provided for many PostgreSQL types. For types without common mathematical conventions for all possible permutations (e.g., date/time types) we describe the actual behavior in subsequent sections.

Table 9-2 shows the available mathematical operators.

Table 9-2. Mathematical Operators

The bitwise operators work only on integral data types, whereas the others are available for all numeric data types. The bitwise operators are also available for the bit string types bit and bit varying, as shown in Table 9-10.

Table 9-3 shows the available mathematical functions. In the table, dp indicates double precision. Many of these functions are provided in multiple forms with different argument types. Except where noted, any given form of a function returns the same data type as its argument. The functions working with double precision data are mostly implemented on top of the host system's C library accuracy and behavior in boundary cases can therefore vary depending on the host system.

Table 9-3. Mathematical Functions

Function Return Type Description Example Result
abs (x) (same as x) absolute value abs(-17.4) 17.4
cbrt (dp) dp cube root cbrt(27.0) 3
ceil (dp or numeric) (same as input) smallest integer not less than argument ceil(-42.8) -42
ceiling (dp or numeric) (same as input) smallest integer not less than argument (alias for ceil ) ceiling(-95.3) -95
degrees (dp) dp radians to degrees degrees(0.5) 28.6478897565412
exp (dp or numeric) (same as input) exponential exp(1.0) 2.71828182845905
floor (dp or numeric) (same as input) largest integer not greater than argument floor(-42.8) -43
ln (dp or numeric) (same as input) natural logarithm ln(2.0) 0.693147180559945
log (dp or numeric) (same as input) base 10 logarithm log(100.0) 2
log (b numeric, x numeric) numeric logarithm to base b log(2.0, 64.0) 6.0000000000
mod (y, x) (same as argument types) remainder of y/x mod(9,4) 1
pi () dp "π" constant pi() 3.14159265358979
power (a dp, b dp) dp a raised to the power of b power(9.0, 3.0) 729
power (a numeric, b numeric) numeric a raised to the power of b power(9.0, 3.0) 729
radians (dp) dp degrees to radians radians(45.0) 0.785398163397448
random () dp random value in the range 0.0 <= x < 1.0 random()
round (dp or numeric) (same as input) round to nearest integer round(42.4) 42
round (v numeric, s int) numeric round to s decimal places round(42.4382, 2) 42.44
setseed (dp) void set seed for subsequent random() calls (value between 0 and 1.0) setseed(0.54823)
sign (dp or numeric) (same as input) sign of the argument (-1, 0, +1) sign(-8.4) -1
sqrt (dp or numeric) (same as input) square root sqrt(2.0) 1.4142135623731
trunc (dp or numeric) (same as input) truncate toward zero trunc(42.8) 42
trunc (v numeric, s int) numeric truncate to s decimal places trunc(42.4382, 2) 42.43
width_bucket (op numeric, b1 numeric, b2 numeric, count int) int return the bucket to which operand would be assigned in an equidepth histogram with count buckets, in the range b1 to b2 width_bucket(5.35, 0.024, 10.06, 5) 3
width_bucket (op dp, b1 dp, b2 dp, count int) int return the bucket to which operand would be assigned in an equidepth histogram with count buckets, in the range b1 to b2 width_bucket(5.35, 0.024, 10.06, 5) 3

Finally, Table 9-4 shows the available trigonometric functions. All trigonometric functions take arguments and return values of type double precision.

PreTeXt Author's Guide

As mentioned in the overview, Section 3.5, we use L a T e X syntax for mathematics. In order to allow for quality display in HTML , and other electronic formats, this limits us to the subset of L a T e X supported by the very capable MathJax Javascript library. Generally this looks like the amsmath package maintained by the American Mathematical Society at their AMS-LaTeX page. For a complete and precise list of what MathJax supports, see the MathJax Supported LaTeX commands page.

Subsection 6.8.1 Inline Mathematics

Use the <m> to place variables or very short expressions within a sentence of a paragraph, the content of a <title> , a <cell> of a table, a footnote, or other similar locations of sentence-like text. You can't cross-reference this text, nor make a knowl with it. Though you can typically cross-reference a containing element.

Do not use L a T e X -isms like displaystyle to try to end-run the inline nature. It will just lead to poor results.

Subsection 6.8.2 One-Line Display Mathematics

The <me> element can be used for longer expressions or a single equation. Typically you will get vertical separation above and below, and the contents will be centered. See below about concluding periods (and other punctuation), and alignment. The <men> variant will produce a numbered equation, and therefore with a provided @xml:id attribute, can be the target of a cross-reference ( <xref> ).

Subsection 6.8.3 Multi-line Display Mathematics

We begin with a pure container, either <md> or <mdn> . The former numbers no lines, the latter numbers every line. Within the container, content, on a per-line basis, goes into a <mrow> element. You can think of <mrow> as being very similar to <me> . If you are tempted to put a L a T e X into an <mrow> , think twice.

On any given <mrow> you can place the @number attribute, with allowable values of yes and no . These will typically be used to override the behavior inherited by the container, but there is no harm if they are redundant. A given line of the display may be the target of a cross-reference, though the numbering flexibility means you can try (and fail) to target an unnumbered equation.

An <mrow> may have a @tag attribute in place of a @number attribute. This will create a “number” on the equation which is just a symbol. This is meant for situations where you do not want to use numbers, and the resulting cross-reference is “local.” In other words, the <xref> and its target are not far apart, such as maybe within the same <example> or the same <proof> . Allowable values for the attribute are: star, dstar, tstar, dagger, ddagger, tdagger, hash, dhash, thash, maltese, dmaltese, tmaltese . These are the names of symbols, with prefixes where the prefix d means “double”, and the prefix t means “triple”. Cross-references to these tagged equations happens in the usual way and should behave as expected. See Section 3.3 and Section 6.6 for more on cross-references.

Subsection 6.8.4 Special Characters

The L a T e X macros, amp , lt , and gt are always available within these mathematics elements, so that you can avoid the special XML characters & , < and > . See Section 3.13 for this same information, but in the broader context of your entire document.

Subsection 6.8.5 Text in Mathematics

Once in a while, you need a little bit of “regular” text within an expression and you do not want it to look like a product of a bunch of one-letter variables. Use the ext<> macro for these. Only. Other ways of switching out of math-mode and into some sort of “regular” text will appear inferior, and can raise errors in certain conversions.

  • Do place surrounding spaces inside the ext<> macro.
  • Do not place any mathematics inside the ext<> macro.
  • Do not use the mbox<> macro as a substitute.
  • Do not use font-changing commands (e.g. m ) as a substitute.

This example amply illustrates the use of macros for XML special characters (twice), appropriate use of the ext<> macro (twice), spaces in the ext<> macro (once), sentence-ending punctuation (see the source, the period is not inside the <me> element) and yes, we did think twice about the (an exception to the rule).

Subsection 6.8.6 Cross-References in Display Mathematics

Subsection 6.8.7 Alignment in Display Mathematics

Displayed mathematics is implemented with the AMS- L a T e X align environment. Ampersands are used to control this, so use the amp macro for these. The first ampersand in a line or row is an alignment point, typically a symbol, like an equality. The next ampersand is a column separator, then the next is an alignment point, then a column separator, then… The moral of the story is you should have (n) alignment points, with (n-1) column separators, for a total of (2n-1) ampersands—always an odd number.

Sometimes you want several short equations on one line. Do not use <me> . Instead use a single <mrow> inside an <md> , and use alignment to spread them out evenly.

For multi-line display mathematics with no ampersands present, each line will be centered. This is implemented with the AMS- L a T e X gather environment.

You can fool the alignment behavior by hiding all your ampersands in macro definitions, so there is the optional @alignment attribute for the <md> or <mdn> element, in order to force the right kind of alignment. Allowable values are gather , align , and alignat . The latter is similar to align , but no space is automatically provided between columns. You can leave it that way, or explicitly add your own. For example, this allows you to precisely arrange individual terms of a system of linear equations, especially when terms with zero coefficients are omitted. When using the alignat option PreTeXt tries to count ampersands to see how many columns you intend, since L a T e X needs this number (we are not sure why). This detection can be fooled too, especially if you have something like a matrix with lots of ampersands for other purposes. So set the @alignat-columns attribute to the number of intended columns, if necessary.

Subsection 6.8.8 Fill-In Blanks in Mathematics

Subsection 6.8.9 Page Breaks for Tall Display Mathematics

For print output, do nothing additional and L a T e X will do its best to break your display between lines. You can turn this behavior off by setting the @break attribute on the <md> or <mdn> to the value no . Once you do this, you can then selectively allow a page break after a given <mrow> by setting the @break attribute on the <mrow> to the value yes .

Subsection 6.8.10 Your Macros

These go in the <docinfo> section, wrapped in a <macros> element. Keep them simple—one or two arguments, and one-line definitions. This is not the place to be fancy, and not the place to try to end-run the structural aspects of PreTeXt. The idea is to define something like adjoint for the matrix A to be a superscript asterisk, and later you can change your mind and use a superscript dagger instead. Keep in the spirit of PreTeXt and use readable, semantic macros. For example, do not use a for the adjoint of A .

Subsection 6.8.11 Punctuation After Display Math

If a chunk of displayed math concludes a sentence, then the sentence-ending punctuation should appear at the conclusion of the display. (And certainly not at the start of the first line after the display!) But do not author the punctuation within the mathematics element, put it afterwards, where it logically belongs.

More specifically, place a sentence-ending period (say) immediately after the closing of an <me> , <men> , <md> , or <mdn> element. PreTeXt will place the period in your output in the right place and in the right way. (By using L a T e X 's ext<> macro, if you are curious to know the details.) Here is an example. The XML source

This all applies more generally to clause-ending punctuation, such as a comma. Take notice of the requirement that the punctuation must be immediately after the closing tag of the math element, otherwise it will not migrate properly. For example, do not interrupt the flow with whitespace, or an XML comment, or anything else.

Subsection 6.8.12 Additional Packages

Generally, you cannot add additional packages for use within mathematics. The exception is a package with support available optionally within MathJax. And it must have the same name as its normal L a T e X version. Then set a docinfo/latex-preamble/package element to be the common name of the package. (The cancel package is one such example.)

Then the supported macros of the package will be available with your mathematics elements, and you can use them within other macro definitions. We do not guarantee the absence of conflicts with other packages in use, even if employed by PreTeXt. Nor do we support debugging such conflicts.

Subsection 6.8.13 Extras

There are two existing additional options, which we might want to remove some day for technical reasons. Macros from the extpfeil extensible arrows package are available by default, and an sfrac<><> macro is available for appealing inline “slanted fractions.”

Teacher voices from an online elementary mathematics community: examining perceptions of professional learning

This study compares web usage data with interviews from 41 participants, who are members of an online professional development site called the Everyday Mathematics Virtual Learning Community (VLC), to explore how elementary school teachers learn from classroom video. Web usage data reveal that the commentary surrounding video posted to the VLC is sparse and surface level, possibly indicating a lack of serious attention to the videos. Interview data, however, indicate that participants felt they learned from this resource. Participants reported that the videos provided them with the opportunity to view and reflect on model lessons, plan curricula, and consider student thinking, among other learning outcomes. Participants also identified key factors that prevented them from posting comments to the site to convey their learning. These results can be used to understand not only how teachers perceive their own learning from classroom video, but also to redesign online professional development experiences to promote expression of that learning.

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In recent years, the study of mathematics textbooks has focused more on their design and usage rather than on the analysis of the textbook itself (Rezat et al., 2018). Many earlier studies investigated how teachers used mathematics textbooks (as well as other curriculum resources) to design and promote their classroom practices (e.g., Haggarty & Pepin, 2002 Pepin, 2018 Pepin et al., 2017 Remillard et al., 2009 Rezat, 2012). In textbook research, it is important to recognise textbooks as ‘potentially implemented curriculums’ (Valverde et al., 2002), which implies that textbooks are vital curriculum resources for classroom teaching, but the same textbook will not necessarily result in the same classroom practice. Thus, although textbooks can be used as agents of instructional change, it is important for researchers to understand the different conditions in which teachers’ work in order to realise change in the classroom. In this paper, we consider ‘teacher design’ using textbooks as key to the implementation of change in an enacted curriculum. Although the implementation of change ranges from daily educational improvement to more drastic instructional innovation for realising curriculum reform, we address instructional change, which refers to change in teachers’ ordinary practices that enriches their instruction by transforming the textbooks they use. This is related to teachers’ design capacity in utilising curriculum materials or creating new materials (Brown, 2009). Since the notion of ‘teacher design’ can be characterised differently in international contexts (Pepin et al., 2017), it is important to take into account local contexts that may affect teachers’ designs and textbook usage in a given country (Rezat et al., 2018).

In the present study we focused on Japanese teachers’ design activities in relation to instructional improvement. We first describe an overview of the Japanese context related to teachers’ use of textbooks and then address international literature reviews. We present two theoretical frameworks in mathematics education that are suitable for approaching such a complex phenomenon, illustrated by two local examples. Thus, in this paper we aim to explore theoretical approaches to Japanese teachers’ lesson designs involving the adaptation of mathematics textbooks for instructional change.

8.8. Coroutines¶

8.8.1. Coroutine function definition¶

Execution of Python coroutines can be suspended and resumed at many points (see coroutine ). Inside the body of a coroutine function, await and async identifiers become reserved keywords await expressions, async for and async with can only be used in coroutine function bodies.

Functions defined with async def syntax are always coroutine functions, even if they do not contain await or async keywords.

It is a SyntaxError to use a yield from expression inside the body of a coroutine function.

An example of a coroutine function:

8.8.2. The async for statement¶

An asynchronous iterable provides an __aiter__ method that directly returns an asynchronous iterator , which can call asynchronous code in its __anext__ method.

The async for statement allows convenient iteration over asynchronous iterables.

Is semantically equivalent to:

See also __aiter__() and __anext__() for details.

It is a SyntaxError to use an async for statement outside the body of a coroutine function.

8.8.3. The async with statement¶

An asynchronous context manager is a context manager that is able to suspend execution in its enter and exit methods.

is semantically equivalent to:

See also __aenter__() and __aexit__() for details.

It is a SyntaxError to use an async with statement outside the body of a coroutine function.

PEP 492 - Coroutines with async and await syntax

The proposal that made coroutines a proper standalone concept in Python, and added supporting syntax.

The exception is propagated to the invocation stack unless there is a finally clause which happens to raise another exception. That new exception causes the old one to be lost.

A string literal appearing as the first statement in the function body is transformed into the function’s __doc__ attribute and therefore the function’s docstring .

A string literal appearing as the first statement in the class body is transformed into the namespace’s __doc__ item and therefore the class’s docstring .