From 0021c782c8988363eb42a86ad1a7faab83a26592 Mon Sep 17 00:00:00 2001
From: Tim Daly
Date: Thu, 14 Jul 2016 01:21:45 0400
Subject: [PATCH] books/axiom.sty add \pct for resizing the % character
Goal: Axiom Literate Programming

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books/bookvolbib Axiom Citations in the Literature
+books/axiom.sty add \pct for resizing the % character
Goal: Axiom Literate Programming

\index{Bradford, Russell}
\begin{chunk}{axiom.bib}
@inproceedings{Brad92,
 author = "Bradford, Russell",
 title = "Algebraic Simplification of MultipleValued Functions",
 booktitle = "Proc. DISCO 92",
 series = "Lecture Notes in Computer Science 721",
 year = "1992",
 paper = "Brad92.pdf",
 abstract =
 "Many current algebra systems have a lax attitude to the
 simplification of expressions involving functions like log and
 $\sqrt{}$, leading the the ability to ``prove'' equalities like $1=1$
 in such systems. In fact, only a little elementary arithmetic is
 needed to devise what the correct simplification should be. We detail
 some of these simplification rules, and outline a method for their
 incorporation into an algebra system."
}

\end{chunk}

\index{Schwarz, Fritz}
\begin{chunk}{axiom.bib}
@article{Schw91,
 author = "Schwarz, Fritz",
 title = "Monomial orderings and Groebner bases",
 journal = "SIGSAM Bulletin",
 volume = "25",
 number = "1",
 pages = "1023",
 keywords = "axiomref",
 abstract =
 "Let there be given a set of monomials in n variables and some order
 relations between them. The following {\sl fundamental problem of
 monomial ordering} is considered. Is it possible to decide whether
 these ordering relations are consistent and if so to extend them to an
 {\sl admissible} ordering for all monomials? The answer is given in
 terms of the algorithm {\sl MACOT} which constructs a matrix of so
 called {\sl cotes} which establishes the desired ordering
 relations. The main area of application of this algorithm, i.e. the
 construction of Groebner bases for different orderings and of
 universal Groebner bases is treated in the last section."
}

\end{chunk}

\index{Bradford, Russell}
\begin{chunk}{axiom.bib}
@inproceedings{Brad92,
 author = "Bradford, Russell",
 title = "Algebraic Simplification of MultipleValued Functions",
 booktitle = "Proc. DISCO 92",
 series = "Lecture Notes in Computer Science 721",
 year = "1992",
 paper = "Brad92.djvu",
 abstract =
 "Many current algebra systems have a lax attitude to the
 simplification of expressions involving functions like log and
 $\sqrt{}$, leading the the ability to ``prove'' equalities like $1=1$
 in such systems. In fact, only a little elementary arithmetic is
 needed to devise what the correct simplification should be. We detail
 some of these simplification rules, and outline a method for their
 incorporation into an algebra system."
}

\end{chunk}

\index{Wang, Dongming}
\begin{chunk}{axiom.bib}
@article{Wang90,
 author = "Wang, Dongming",
 title = "A Class of Cubic Differential Systems with 6tuple Focus",
 journal = "J. Differential Equations",
 publisher = "Academic Press, Inc.",
 volume = "87",
 pages = "305315",
 year = "1990",
 keywords = "axiomref",
 paper = "Wang90.pdf",
 abstract =
 "This paper presents a class of cubic differential systems with the
 origin as a 6tuple focus from which 6 limit cycles may be
 constructed. For this class of differential systems the stability of
 the origin is given."
}

\end{chunk}

\index{Wang, Dongming}
\begin{chunk}{axiom.bib}
@article{Wang91,
 author = "Wang, Dongming",
 title = "Mechanical manipulation for a class of differential systems",
 journal = "Journal of Symbolic Computation",
 volume = "12",
 number = "2",
 pages = "233254",
 year = "1991",
 keywords = "axiomref",
 abstract =
 "The author describes a mechanical procedure for computing the
 Liapunov functions and Liapunov constants for a class of differential
 systems. These functions and constants are used for establishing the
 stability criteria, the conditions for the existence of a center and
 for the investigation of limit cycles. Some problems for handling the
 computer constants, which are usually large polynomials in terms of
 the coefficients of the differential system, and an approach towards
 their solution by using computer algebraic methods are proposed. This
 approach has been successfully applied to check some known results
 mechanically. The author has implemented a system DEMS on an HP1000
 and in Scratchpad II on an IBM4341 for computing and manipulating the
 Liapunov functions and Liapunov constants. As examples, two particular
 cubic systems are discussed in detail. The explicit algebraic
 relations between the computed Liapunov constants and the conditions
 given by Saharnikov are established, which leads to a rediscovery of
 the incompleteness of his conditions. A class of cubic systems with
 6tuple focus is presented to demonstrate the feasibility of the
 approach for finding systems with higher multiple focus."
}

\end{chunk}

\index{Wang, Dongming}
\begin{chunk}{axiom.bib}
@misc{Wang95,
 author = "Wang, Dongming",
 title = "Characteristic Sets and Zero Structure of Polynomial Sets",
 institution = "Johannes Kepler University",
 comment = "Lecture Notes",
 paper = "Wang95.pdf",
 url = "http://wwwpolsys.lip6.fr/~wang/papers/CharSet.ps.gz",
 keywords = "axiomref",
 abtract =
 "This paper provides a tutorial on the theory and method of
 characteristic sets and some relevant topics. The basic algorithms as
 well as their generalization for computing the characteristic set and
 characteristic series of a set of multivariate polynomials are
 presented. The characeristic set, which is of certain triangular form,
 reflects in general the major part of zeros, and the characteristic
 series, which is a sequence of polynomial sets of triangular form,
 furnishes a complete zero decomposition of the given polynomial
 set. Using this decomposition, a complete solution to the algebraic
 decision problem and a method for decomposing any algebraic variety
 into irreducible components are described. Some applications of the
 method are indicated."
}

\end{chunk}


\index{Keady, G.}
\index{Richardson, M.G.}
\begin{chunk}{axiom.bib}
@inproceedings{Kead93a,
 author = "Keady, G. and Richardson, M.G.",
 title = "An application of IRENA to systems of nonlinear equations arising
 in equilibrium flows in networks",
 booktitle = "Proc. ISSAC 1993",
 series = "ISSAC '93",
 year = "1993",
 paper = "Kead93a.pdf",
 keywords = "axiomref",
 abstract =
 "IRENA  an $I$nterface from $RE$DUCE to $NA$G  runs under the REDUCE
 Computer Algebra (CA) system and provides an interactive front end to
 the NAG Fortran Library.

 Here IRENA is tested on a problem closer to an engineering problem
 than previously publised examples. We also illustrate the use of the
 {\tt codeonly} switch, which is relevant to larger scale problems. We
 describe progress on an issue raised in the 'Future Developments'
 section in our {\sl SIGSAM Bulletin} article [2]: the progress improves
 the practical effectiveness of IRENA."
}

\end{chunk}

\index{LeBlanc, S.E.}
\begin{chunk}{axiom.bib}
@inproceedings{LeBl91,
 author = "LeBlanc, S.E.",
 title = "The use of MathCAD and Theorist in the ChE classroom",
 booktitle = "Proc. ASEE Annual Meeting",
 year = "1991",
 pages = "287299",
 keywords = "axiomref"
 abstract =
 "MathCAD and Theorist are two powerful mathematical packages available
 for instruction in the ChE classroom. MathCAD is advertised as an
 `electronic scratchpad' and it certainly lives up to its billing. It
 is an extremely userfriendly collection of numerical routines that
 eliminates the drudgery of solving many of the types of problems
 encountered by undergraduate ChE's (and engineers in general). MathCAD
 is available for both the Macintosh and IBM PC compatibles. The PC
 version is available as a fullfunctioned student version for around
 US\$40 (less than many textbooks). Theorist is a symbolic mathematical
 package for the Macintosh. Many interesting and instructive things can
 be done with it in the ChE curriculum. One of its many attractive
 features includes the ability to generate high quality three
 dimensional plots that can be very instructive in examining the
 behavior of an engineering system. The author discusses the
 application and use of these packages in chemical engineering and give
 example problems and their solutions for a number of courses including
 stoichiometry, unit operations, thermodynamics and design."
}

\end{chunk}

\index{Wang, Dongming}
\begin{chunk}{axiom.bib}
@book{Wang01,
 author = "Wang, Dongming",
 title = "Elimination Methods",
 publisher = "SpringerVerlag",
 isbn = "9783709162026",
 keywords = "axiomref",
 year = "2001",
 abstract =
 "The development of polynomialelimination techniques from classical
 theory to modern algorithms has undergone a tortuous and rugged
 path. This can be observed L. van der Waerden's elimination of the
 ``elimination theory'' chapter from from B. his classic Modern Algebra
 in later editions, A. Weil's hope to eliminate ``from algebraic
 geometry the last traces of elimination theory,'' and S. Abhyankar's
 suggestion to ``eliminate the eliminators of elimination theory.''
 The renaissance and recognition of polynomial elimination owe much to
 the advent and advance of modern computing technology, based on
 which effective algorithms are implemented and applied to diverse
 problems in science and engineering. In the last decade, both
 theorists and practitioners have more and more realized the
 significance and power of elimination methods and their underlying
 theories. Active and extensive research has contributed a great deal
 of new developments on algorithms and softÂ ware tools to the subject,
 that have been widely acknowledged. Their applications have taken
 place from pure and applied mathematics to geometric modeling and
 robotics, and to artificial neural networks. This book provides a
 systematic and uniform treatment of elimination algorithms that
 compute various zero decompositions for systems of multivariate
 polynomials. The central concepts are triangular sets and systems of
 different kinds, in terms of which the decompositions are
 represented. The prerequisites for the concepts and algorithms are
 results from basic algebra and some knowledge of algorithmic
 mathematics."
}

\end{chunk}

\index{Wang, Dongming}
\begin{chunk}{axiom.bib}
@inproceedings{Wang92,
 author = "Wang, Dongming",
 title = "A Method for Factorizing Multivariate Polynomials over Successive
 Algebraic Extension Fields",
 booktitle = "Mathematics and MathematicsMechanization (2001)",
 pages = "138172",
 institution = "Johannes Kepler University",
 url = "http://wwwpolsys.lip6.fr/~wang/papers/Factor.ps.gz",
 paper = "Wang92.pdf",
 year = "1992",
 abstract =
 "We present a method for factorizing multivariate polynomials over
 algebraic fields obtained from successive extensions of the rational
 number field. The basic idea underlying this method is the reduction
 of polynomial factorization over algebraic extension fields to the
 factorization over the rational number vield via linear transformation
 and the computation of characteristic sets with respect to a proper
 variable ordering. The factors over the algebraic extension fields are
 finally determined via GCD (greatest common divisor) computations. We
 have implemented this method in the Maple system. Preliminary
 experiments show that it is rather efficient. We give timing
 statistics in Maple 4.3 on 40 test examples which were partly taken
 from the literature and partly randomly generated. For all those
 examples to which Maple builtin algorithm is applicable, our
 algorithm is always faster."
}

\end{chunk}

\index{Wang, Dongming}
\begin{chunk}{axiom.bib}
@misc{Wang90a,
 author = "Wang, Dongming",
 title = "Some NOtes on Algebraic Methods for Geometry Theorem Proving",
 url = "http://wwwpolsys.lip6.fr/~wang/papers/GTPnote.ps.gz",
 year = "1990",
 paper = "Wang90a.pdf",
 abstract =
 "A new geometry theorem prover which provides the first complete
 implementation of Wu's method and includes several Groebner bases
 based methods is reported. This prover has been used to prove a number
 of nontrivial geometry theorems including several {\sl large} ones
 with less space and time cost than using the existing provers. The
 author presents a new technique by introducing the notion of {\sl
 normal ascending set}. This technique yields in some sense {\sl
 simpler} nondegenerate conditions for Wu's method and allows one to
 prove geometry theorems using characteristic sets but Groeber bases
 type reduction. Parallel variants of Wu's method are discussed; an
 implementation of the parallelized version of his algorithm utilizing
 workstation networks has also been included in our prover. Timing
 statistics for a set of typical examples is given."
}

\end{chunk}

\index{Zhao, Ting}
\index{Wang, Dongming}
\index{Hong, Hoon}
\begin{chunk}{axiom.bib}
@article{Zhao11,
 author = "Zhao, Ting and Wang, Dongming and Hong, Hoon",
 title = "Solution formulats for cubic equations without or with constraints",
 journal = "J. Symbolic Computation",
 volume = "46",
 pages = "904918",
 year = "2011",
 paper = "Zhao11.pdf",
 abstract =
 "We present a convention (for square/cubic roots) which provides
 correct interpretations of the Lagrange formula for all cubic
 polynomial equations with real coefficients. Using this convention, we
 also present a real solution formula for the general cubic equation
 with real coefficients under equality and inequality constraints."
}

\end{chunk}

\index{Li, Xiaoliang}
\index{Mou, Chenqi}
\index{Wang, Dongming}
\begin{chunk}{axiom.bib}
@article{Lixx10,
 author = "Li, Xiaoliang and Mou, Chenqi and Wang, Dongming",
 title = "Decomposing polynomial sets into simple sets over finite fields:
 The zerodimensional case",
 comment = "Provides clear polynomial algorithms",
 journal = "Computers and Mathematics with Applications",
 volume = "60",
 pages = "29832997",
 year = "2010",
 paper = "Lixx10.pdf",
 abstract =
 "This paper presents algorithms for decomposing any zerodimensional
 polynomial set into simple sets over an arbitrary finite field, with
 an associated ideal or zero decomposition. As a key ingredient of
 these algorithms, we generalize the squarefree decomposition approach
 for univariate polynomials over a finite field to that over the field
 product determined by a simple set. As a subprocedure of the
 generalized squarefree decomposition approach, a method is proposed to
 extract the $p$th root of any element in the field
 product. Experiments with a preliminary implementation show the
 effectiveness of our algorithms."
}

\end{chunk}

\index{Wang, Dongming}
\begin{chunk}{axiom.bib}
@article{Wang98,
 author = "Wang, Dongming",
 title = "Decomposing Polynomial Systems into Simple Systems",
 volume = "25",
 number = "3",
 pages = "295314",
 year = "1998",
 paper = "Wang98.pdf",
 abstract =
 "A simple system is a pair of multivariate polynomial sets (one set
 for equations and the other for inequations) ordered in triangular
 form, in which every polynomial is squarefree and has nonvanishing
 leading coefficient with respect to its leading variable. This paper
 presents a method that decomposes any pair of polynomial sets into
 finitely many simple systems with an associated zero decomposition.
 The method employs topdown elimination with splitting and the
 formation of subresultant regular subchains as basic operation."
}

\end{chunk}

\index{Wang, Dongming}
\begin{chunk}{axiom.bib}
@article{Wang94,
 author = "Wang, Dongming",
 title = "Differentiation and Integration of Indefinite Summations with
 Respect to Indexed Variables  Some Rules and Applications",
 journal = "J. Symbolic Computation",
 volume = "18",
 number = "3",
 pages = "249263",
 year = "1994",
 paper = "Wang94.pdf",
 abstract =
 "In this paper we present some rules for the differentiation and
 integration of expressions involving indefinite summations with
 respect to indexed variables which have not yet been taken into
 account of current computer algebra systems. These rules, together
 with several others, have been implemented in MACSYMA and MAPLE as a
 toolkit for manipulating indefinite summations. We discuss some
 implementation issues and report our experiments with a set of typical
 examples. The present work is motivated by our investigation in the
 computeraided analysis and derivation of artificial neural systems.
 The application of our rules to this subject is briefly explained."
}

\end{chunk}

\index{Wang, Dongming}
\begin{chunk}{axiom.bib}
@article{Wang95a,
 author = "Wang, Dongming",
 title = "A Method for Proving Theorems in Differential Geometry and
 Mechanics",
 journal = "J. Universal Computer Science",
 volume = "1",
 number = "9",
 pages = "658673",
 year = "1995",
 url = "http://www.jucs.org/jucs\_1\_9/a\_method\_for\_proving",
 paper = "Wang95a.pdf",
 abstract =
 "A zero decomposition algorithm is presented and used to devise a
 method for proving theorems automatically in differential geometry and
 mechanics. The method has been implemented and its practical
 efficiency is demonstrated by several nontrivial examples including
 Bertrand s theorem, Schell s theorem and KeplerNewton s laws."
}

\end{chunk}

\index{Wang, Dongming}
\begin{chunk}{axiom.bib}
@article{Wang93,
 author = "Wang, Dongming",
 title = "An Elimination Method for Polynomial Systems",
 journal = "J. Symbolic Computation",
 volume = "16",
 number = "2",
 pages = "83114",
 year = "1993",
 paper = "Wang93.pdf",
 abstract =
 "We present an elimination method for polynomial systems, in the form
 of three main algorithms. For any given system [$\mathbb{P}$,$\mathbb{Q}$]
 of two sets of multivariate polynomials, one of the algorithms computes a
 sequence of triangular forms $\mathbb{T}_1,\ldots,\mathbb{T}_e$ and
 polynomial sets $\mathbb{U}_1,\ldots,\mathbb{U}_e$ such that
 Zero($\mathbb{P}$/$\mathbb{Q}$)
 $= \cup_{i=1}^e {\rm\ Zero}(\mathbb{T}_i/\mathbb{U}_i)$,
 where Zero($\mathbb{P}$/$\mathbb{Q}$) denotes the set of common zeros of
 the polynomials in $\mathbb{P}$ which are not zeros of any polynomial in
 $\mathbb{Q}$, and similarly for Zero($\mathbb{T}_i$/$\mathbb{U}_i$).
 The two other algorithms compute the same zero decomposition but with nicer
 properties such as Zero$(\mathbb{T}_i/\mathbb{U}_i) \ne 0$ for each $i$.
 One of them, for which the computed triangular systems
 [$\mathbb{T}_i$, $\mathbb{U}_i$] possess the projection property, provides
 a quantifier elimination procedure for algebraically closed fields.
 For the other, the computed triangular forms $\mathbb{T}_i$ are
 irreducible. The relationship between our method and some existing
 elimination methods is explained. Experimental data for a set of test
 examples by a draft implementation of the method are provided, and show
 that the efficiency of our method is comparable with that of some
 wellknown methods. A few encouraging examples are given in detail for
 illustration."
}

\end{chunk}

\index{Houstis, E.N.}
\index{Gaffney, P.W.}
\begin{chunk}{axiom.bib}
@book{Hous92,
 author = "Houstis, E.N. and Gaffney, P.W.",
 title = "Programming environments for highlevel scientific problem solving",
 year = "1992",
 keywords = "axiomref",
 publisher = "Elsevier",
 isbn = "9780444891761",
 abstract =
 "Programming environments, as the name suggests, are intended to
 provide a unified, extensive range of capabilities for a person
 wishing to solve a problem using a computer. In this particular
 proceedings volume, the problem considered is a highlevel scientific
 computation. In other words, a scientific problem whose solution
 usually requires sophisticated computing techniques and a large
 allocation of computing resources."
}

\end{chunk}

\index{Camion, Paul}
\index{Courteau, Bernard}
\index{Montpetit, Andre}
\begin{chunk}{axiom.bib}
@techreport{Cami92,
 author = "Camion, Paul and Courteau, Bernard and Montpetit, Andre",
 title = "A combinatorial problem in Hamming Graphs and its solution
 in Scratchpad",
 comment = {Un probl\`eme combinatoire dans les graphies de Hamming et sa
 solution en Scratchpad},
 year = "1992",
 month = "January",
 keywords = "axiomref",
 paper = "Cami92.pdf",
 url = "https://hal.inria.fr/inria00074974/document",
 type = "Research report",
 number = "1586",
 institution = "Institut National de Recherche en Informatique et en
 Automatique, Le Chesnay, France",
 abstract =
 "We present a combinatorial problem which arises in the determination
 of the complete weight coset enumerators of errorcorrecting codes
 [1]. In solving this problem by exponential power series with
 coefficients in a ring of multivariate polynomials, we fall on a
 system of differential equations with coefficients in a field of
 rational functions. Thanks to the abstraction capabilities of
 Scratchpad this differential equation may be solved simply and
 naturally, which seems not to be the case for the other computer
 algebra systems now available."
}

\end{chunk}

\index{Dalmas, St\'ephane}
\begin{chunk}{axiom.bib}
 author = "Dalmas, Stephane",
 title = "A polymorphic functional language applied to symbolic computation",
 year = "1992",
 booktitle = "Proc. ISSAC 1992",
 series = "ISSAC 1992",
 pages = "369375",
 isbn = "0897914899 (soft cover) 0897914902 (hard cover)",
 keywords = "axiomref",
 "The programming language in which to describe mathematical objects
 and algorithms is a fundamental issue in the design of a symbolic
 computation system. XFun is a strongly typed functional programming
 language. Although it was not designed as a specialized language, its
 sophisticated type system can be successfully applied to describe
 mathematical objects and structures. After illustrating its main
 features, the author sketches how it could be applied to symbolic
 computation. A comparison with Scratchpad II is attempted. XFun seems
 to exhibit more flexibility simplicity and uniformity."
}

\end{chunk}

\index{OpenMath}
\index{Complex}
\index{DoubleFloat}
\index{Float}
\index{Fraction}
\index{Integer}
\index{List}
\index{SingleInteger}
\index{String}
\index{Symbol}
\index{ExpressionToOpenMath}
\index{OpenMathServerPackage}
\index{Corless, Robert M.}
\index{Jeffrey, David J.}
\index{Watt, Stephen M.}
\index{Davenport, James H.}
\begin{chunk}{axiom.bib}
@article{Corl00,
 author = "Corless, Robert M. and Jeffrey, David J. and Watt, Stephen M. and
 Davenport, James H.",
 title = "``According to Abramowitz and Stegun'' or
 arccoth needn't be Uncouth",
 journal = "SIGSAM Bulletin  Special Issue on OpenMath",
 volume = "34",
 number = "2",
 pages = "5865",
 year = "2000",
 paper = "Corl00.pdf",
 algebra =
 "\newline\refto{category OM OpenMath}
 \newline\refto{domain COMPLEX Complex}
 \newline\refto{domain DFLOAT DoubleFloat}
 \newline\refto{domain FLOAT Float}
 \newline\refto{domain FRAC Fraction}
 \newline\refto{domain INT Integer}
 \newline\refto{domain LIST List}
 \newline\refto{domain SINT SingleInteger}
 \newline\refto{domain STRING String}
 \newline\refto{domain SYMBOL Symbol}
 \newline\refto{package OMEXPR ExpressionToOpenMath}
 \newline\refto{package OMSERVER OpenMathServerPackage}",
 abstract =
 "This paper addresses the definitions in OpenMath of the elementary
 functions. The original OpenMath definitions, like most other sources,
 simply cite [2] as the definition. We show that this is not adequate,
 and propose precise definitions, and explore the relationships between
 these definitions.In particular, we introduce the concept of a couth
 pair of definitions, e.g. of arcsin and arcsinh, and show that the
 pair arccot and {\sl arccoth} can be couth."
}

\end{chunk}

\index{Bronstein, Manuel}
\begin{chunk}{axiom.bib}
@article{Bron90a,
 author = "Bronstein, Manuel",
 title = "Integration of Elementary Functions",
 journal = "J. Symbolic Computation",
 volume = "9"
 pages = "117173",
 year = "1990",
 paper = "Bro90a.pdf",
 abstract =
 "We extend a recent algorithm of Trager to a decision procedure for the
 indefinite integration of elementary functions. We can express the
 integral as an elementary function or prove that it is not
 elementary. We show that if the problem of integration in finite terms
 is solvable on a given elementary function field $k$, then it is
 solvable in any algebraic extension of $k(\theta)$, where $\theta$ is
 a logarithm or exponential of an element of $k$. Our proof considers
 an element of such an extension field to be an algebraic function of
 one variable over $k$.

 In his algorithm for the integration of algebraic functions, Trager
 describes a Hermitetype reduction to reduce the problem to an
 integrand with only simple finite poles on the associated Riemann
 surface. We generalize that technique to curves over liouvillian
 ground fields, and use it to simplify our integrands. Once the
 multipe finite poles have been removed, we use the Puiseux expansions
 of the integrand at infinity and a generalization of the residues to
 compute the integral. We also generalize a result of Rothstein that
 gives us a necessary condition for elementary integrability, and
 provide examples of its use."
}

\end{chunk}

diff git a/src/axiomwebsite/patches.html b/src/axiomwebsite/patches.html
index 4ae06c8..b1eb77e 100644
 a/src/axiomwebsite/patches.html
+++ b/src/axiomwebsite/patches.html
@@ 5468,6 +5468,8 @@ books/bookvolbib Axiom Citations in the Literature
books/bookvolbib Axiom Citations in the Literature
20160712.01.tpd.patch
buglist bug 7305: series should simplify
+20160714.01.tpd.patch
+books/axiom.sty add \pct for resizing the % character

1.7.5.4