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1,500 |
Editors' remarks (on two complexity theory surveys in the Bulletin)
|
math.HO
|
The authors discuss the role of controversy in mathematics as a preface to
two opposing articles on computational complexity theory: "Some basic
information on information-based complexity theory" by Beresford Parlett
[math.NA/9201266] and "Perspectives on information-based complexity" by J. F.
Traub and Henryk Wo\'zniakowski [math.NA/9201269].
|
math
|
1,501 |
Two notes on notation
|
math.HO
|
The author advocates two specific mathematical notations from his popular
course and joint textbook, "Concrete Mathematics". The first of these,
extending an idea of Iverson, is the notation "[P]" for the function which is 1
when the Boolean condition P is true and 0 otherwise. This notation can
encourage and clarify the use of characteristic functions and Kronecker deltas
in sums and integrals.
The second notation puts Stirling numbers on the same footing as binomial
coefficients. Since binomial coefficients are written on two lines in
parentheses and read "n choose k", Stirling numbers of the first kind should be
written on two lines in brackets and read "n cycle k", while Stirling numbers
of the second kind should be written in braces and read "n subset k". (I might
say "n partition k".) The written form was first suggested by Imanuel Marx. The
virtues of this notation are that Stirling partition numbers frequently appear
in combinatorics, and that it more clearly presents functional relations
similar to those satisfied by binomial coefficients.
|
math
|
1,502 |
``Theoretical mathematics'': Toward a cultural synthesis of mathematics and theoretical physics
|
math.HO
|
Is speculative mathematics dangerous? Recent interactions between physics and
mathematics pose the question with some force: traditional mathematical norms
discourage speculation, but it is the fabric of theoretical physics. In
practice there can be benefits, but there can also be unpleasant and
destructive consequences. Serious caution is required, and the issue should be
considered before, rather than after, obvious damage occurs. With the hazards
carefully in mind, we propose a framework that should allow a healthy and
positive role for speculation.
|
math
|
1,503 |
Responses to ``Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics'', by A. Jaffe and F. Quinn
|
math.HO
|
This article is a collection of letters solicited by the editors of the
Bulletin in response to a previous article by Jaffe and Quinn
[math.HO/9307227]. The authors discuss the role of rigor in mathematics and the
relation between mathematics and theoretical physics.
|
math
|
1,504 |
Response to comments on ``Theoretical mathematics''
|
math.HO
|
The authors discuss various objections and rejoinders in the collected
responses [math.HO/9404229,math.HO/9404236] to their original article on the
relationship between mathematics and theoretical physics [math.HO/9307227].
|
math
|
1,505 |
Editor's column (on an article by Jaffe and Quinn)
|
math.HO
|
This note is a preface to various responses [math.HO/9404229,math.HO/9404236]
to an opinion piece by Jaffe and Quinn [math.HO/9307227] on the relationship
between mathematics and theoretical physics.
|
math
|
1,506 |
On proof and progress in mathematics
|
math.HO
|
In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms
of progress in mathematics that are not captured by formal proofs of theorems,
especially in his own work in the theory of foliations and geometrization of
3-manifolds and dynamical systems.
|
math
|
1,507 |
Electronic Mathematics Journals
|
math.HO
|
In the Forum section of the November, 1993 Notices of the American
Mathematical Society, John Franks discussed the electronic journal of the
future. Since then, the New York Journal of Mathematics, the first electronic
general mathematics journal, has begun publication. In this article, we explore
the issues of electronic journal publishing in the context of this new project.
We also discuss future developments.
|
math
|
1,508 |
Mathematics Journals Should Be Electronic and Free
|
math.HO
|
Many important journal functions would be lost if the mathematical community
replaced all paper journals with electronic media. Electronic media are useful
for some purposes, but they will not be the basis for a publishing revolution
in the near future.
|
math
|
1,509 |
Interactive games, dialogues and the verbalization
|
math.HO
|
The note is devoted to an interactive game theoretic formalization of
dialogues as psycholinguistic phenomena and the unraveling of a hidden dialogue
structure of 2-person differential interactive games. In the field-theoretic
description of interactive games the dialogues are defined naively as
interactive games of discrete time with intention fields of continuous time;
the correct mathematical formulation is proposed. The states and the controls
of a dialogue correspond to the speech whereas the intention fields describe
the understanding. In the case of dialogues the main inverse problem is to
describe geometrical and algebraical properties of the understanding. On the
other hand, a precise mathematical definition of dialogues allows to formulate
a problem of the unraveling of a hidden dialogue structure of any 2-person
differential interactive game. Such procedure is called the verbalization. It
means that the states of a differential interactive game are interpreted as
intention fields of a hidden dialogue and the problem is to describe such
dialogue completely. If a 2-person differential interactive game is
verbalizable one is able to consider many linguistic (e.g. the formal grammar
of a related hidden dialogue) or psycholinguistic (e.g. the dynamical
correlation of various implications) aspects of it.
|
math
|
1,510 |
On the section of a cone
|
math.HO
|
A problem from Democritus is used to illustrate the building, and use, of
infinitesimal covectors from its regularized, finite, counterpart.
|
math
|
1,511 |
Caleidoscope-roulette: the resonance phenomena in perception games
|
math.HO
|
Kaleidoscope-roulettes, a proper class of perception games, is described.
Kaleidoscope-roulette is defined as a perception and, hence, verbalizable
interactive game, whose hidden dialogue consists of quasirandom sequences of
``words''. The resonance phenomena in such games and their controlling are
discussed.
|
math
|
1,512 |
On the babylonian method of extracting root squares
|
math.HO
|
We discuss the babylonian method of extracting the root square of a number,
from the point of view of modern mathematics. We also speculate that the
babylonian mathematics was rich enough for a generalization of this method,
despite the lack of general statements and justified procedures in their
mathematics.
|
math
|
1,513 |
Poincaré's Proof of the so-called Birkhoff-Witt Theorem
|
math.HO
|
A methodical analysis of the research related to the article, ``Sur les
groupes continus'', of Henri Poincar\'{e} reveals many historical
misconceptions and inaccuracies regarding his contribution to Lie theory. A
thorough reading of this article confirms the precedence of his discovery of
many important concepts, especially that of the \textit{universal enveloping
algebra} of a Lie algebra over a field of characteristic zero, the \textit{%
canonical map} (\textit{symmetrization}) of the symmetric algebra onto the
universal enveloping algebra. The essential part of this article consists of a
detailed discussion of his rigorous, complete, and enlightening proof of the
so-called Birkhoff-Witt theorem.
|
math
|
1,514 |
Representative dynamics
|
math.HO
|
This short note is devoted to the representative dynamics, which realizes a
link between the theory of controlled systems and representation theory.
Dynamical inverse problem of representation theory for controlled systems is
considered: to solve it means to correspond a representative dynamics to the
controlled system.
|
math
|
1,515 |
A Mathematical Theory of Origami Numbers and Constructions
|
math.HO
|
We give a hierarchial set of axioms for mathematical origami. The hierachy
gives the fields of Pythagorean numbers, first discussed by Hilbert, the field
of Euclidean constructible numbers which are obtained by the usual
constructions of straightedge and compass, and the Origami numbers, which is
also the field generated from the intersections of conics or equivalently the
marked ruler.
|
math
|
1,516 |
q-Newton binomial: from Euler to Gauss
|
math.HO
|
A counter-intuitive result of Gauss (formulae (1.6), (1.7) below) is made
less mysterious by virtue of being generalized through the introduction of an
additional parameter.
|
math
|
1,517 |
The Modular Tree of Pythagorus
|
math.HO
|
The Pythagorean triples have the structure of a ternary rooted tree; the tree
is based on the Cayley graph of a free subgroup of the modular group
|
math
|
1,518 |
On Fermat's marginal note: a suggestion
|
math.HO
|
A suggestion is put forward regarding a partial proof of FLT(case1), which is
elegant and simple enough to have caused Fermat's enthusiastic remark in the
margin of his Bachet edition of Diophantus' "Arithmetica". It is based on an
extension of Fermat's Small Theorem (FST) to mod p^k for any k>0, and the cubic
roots of 1 mod p^k for primes p=1 mod 6. For this solution in residues the
exponent p distributes over a sum, which blocks extension to equality for
integers, providing a partial proof of FLT case1 for all p=1 mod 6. This simple
solution begs the question why it was not found earlier. Some mathematical,
historical and psychological reasons are presented. . . . . In a companion
paper, on the triplet structure of Arithmetic mod p^k, this cubic root solution
is extended to the general rootform of FLT (mod p^k) (case1), called "triplet".
While the cubic root solution (a^3=1 mod p^k) involves one inverse pair:
a+a^{-1} = -1 mod p^k, a triplet has three inverse pairs in a 3-loop: a+b^{-1}
= b+c^{-1} = c+a^{-1} = -1 (mod p^k) where abc = 1 (mod p^k), which reduces to
the cubic root form if a=b=c (\neq 1) mod p^k. The triplet structure is not
restricted to p-th power residues (for some p \geq 59) but applies to all
residues in the group G_k(.) of units in the semigroup of multiplication mod
p^k.
|
math
|
1,519 |
Notes to the early history of the Knot Theory in Japan
|
math.HO
|
We give a description of the growth of research on Knot Theory in Japan. We
place our report in a general historical context. In particular, we compare the
development of research on mathematical topology in Japan with that in Poland
and USA, observing several similarities. Toward the end of XIX century and at
the beginning of XX century several young mathematicians, educated in Germany,
France or England were returning to their native countries and building, almost
from scratch, schools of modern mathematics. After a general description of the
growth of topology in Japan between the World Wars, we describe the beginning
of Knot Theory in Japan. Gaisi Takeuti, later a famous logician, conducted the
first Knot Theory seminar in Japan in 1952 or 1953. Kunio Murasugi (later a
prominent knot theoretist) was the only student who attended it. In Osaka, a
Knot Theory seminar started in 1955, initiated by Hidetaka Terasaka and his
students Shin'ichi Kinoshita and Takeshi Yajima. We complete the paper by
listing 70 Japanese topologists born before 1946, and by sketching the
biography of Fox.
|
math
|
1,520 |
Tolstoy's Mathematics in "War and Peace"
|
math.HO
|
The nineteenth century Russian author Leo Tolstoy based his egalitarian views
on sociology and history on mathematical and probabilistic views, and he also
proposed a mathematical theory of waging war.
|
math
|
1,521 |
Short Calculus
|
math.HO
|
This paper has been withdrawn by the author.
|
math
|
1,522 |
The Rational Cuboid Table of Maurice Kraitchik
|
math.HO
|
The original tables of body cuboids by Maurice Kraitchik are corrected,
restoring 159 missing cuboids. His table range is then extended for all odd
sides less than 1,000,000 to a new limit of 4,294,967,295. Over this new range,
12,517 unique body cuboids are listed, from the original 416.
|
math
|
1,523 |
Kepler's Area Law in the Principia: Filling in some details in Newton's proof of Prop. 1
|
math.HO
|
During the past 25 years there has been a controversy regarding the adequacy
of Newton's proof of Prop. 1 in Book 1 of the {\it Principia}. This proposition
is of central importance because its proof of Kepler's area law allowed Newton
to introduce a geometric measure for time to solve problems in orbital dynamics
in the {\it Principia}. It is shown here that the critics of Prop. 1 have
misunderstood Newton's fundamental limit argument by neglecting to consider the
justification for this limit which he gave in Lemma 3. We clarify the proof of
Prop. 1 by filling in some details left out by Newton which show that his proof
of this proposition was adequate and well grounded.
|
math
|
1,524 |
The Life and Works of Raoul Bott
|
math.HO
|
a 10-page biography of Raoul Bott followed by a 25-page discussion of his
major papers
|
math
|
1,525 |
Weierstraß
|
math.HO
|
We give a short biographical sketch of Karl Weierstrass.
|
math
|
1,526 |
Foundations of Mathematics
|
math.HO
|
This article discusses what can be proved about the foundations of
mathematics using the notions of algorithm and information. The first part is
retrospective, and presents a beautiful antique, Godel's proof, the first
modern incompleteness theorem, Turing's halting problem, and a piece of
postmodern metamathematics, the halting probability Omega. The second part
looks forward to the new century and discusses the convergence of theoretical
physics and theoretical computer science and hopes for a theoretical biology,
in which the notions of algorithm and information are again crucial.
|
math
|
1,527 |
Noether
|
math.HO
|
We give a short biographical sketch of Emmy Noether.
|
math
|
1,528 |
Hypothesis of the Functional Semantic Constructions and Mathematics in the Functional Semantic Aspect
|
math.HO
|
This essay contains three parts. The first part of essay focuses on the
hypothesis of the functional semantic constructions (FSC-Hypothesis). This
hypothesis explains that a language, a number, a money are the functional
semantic constructions. In the second part the author considers the Mathematics
with respect to the FSC-Hypothesis. Author turns in the solution for the
following problems: Ontology of Mathematics, Objects of Mathematics, Number,
Classification of the numbers. Last part contains the critical remarks to the
axiomatic allocation of the real numbers to the linear point continuum and to
the countability / uncountability of the set of rational and of the set of real
numbers.
|
math
|
1,529 |
On the intelligibility of the universe and the notions of simplicity, complexity and irreducibility
|
math.HO
|
We discuss views about whether the universe can be rationally comprehended,
starting with Plato, then Leibniz, and then the views of some distinguished
scientists of the previous century. Based on this, we defend the thesis that
comprehension is compression, i.e., explaining many facts using few theoretical
assumptions, and that a theory may be viewed as a computer program for
calculating observations. This provides motivation for defining the complexity
of something to be the size of the simplest theory for it, in other words, the
size of the smallest program for calculating it. This is the central idea of
algorithmic information theory (AIT), a field of theoretical computer science.
Using the mathematical concept of program-size complexity, we exhibit
irreducible mathematical facts, mathematical facts that cannot be demonstrated
using any mathematical theory simpler than they are. It follows that the world
of mathematical ideas has infinite complexity and is therefore not fully
comprehensible, at least not in a static fashion. Whether the physical world
has finite or infinite complexity remains to be seen. Current science believes
that the world contains randomness, and is therefore also infinitely complex,
but a deterministic universe that simulates randomness via pseudo-randomness is
also a possibility, at least according to recent highly speculative work of S.
Wolfram.
|
math
|
1,530 |
Scholarly mathematical communication at a crossroads
|
math.HO
|
This essay was invited for publication in Nieuw Archief voor Wiskunde; it
will also appear in translation in the SMF Gazette and in the DMV Mitteilungen.
I discuss the recent trends in scholarly communication in mathematics, the
current state and intentions of the arXiv, and a proposal to reform peer review
with the arXiv as a foundation.
|
math
|
1,531 |
On the information-theoretic approach to Gödel's incompleteness theorem
|
math.HO
|
In this paper we briefly review and analyze three published proofs of
Chaitin's theorem, the celebrated information-theoretic version of G\"odel's
incompleteness theorem. Then, we discuss our main perplexity concerning a key
step common to all these demonstrations.
|
math
|
1,532 |
Some non-conventional ideas about algorithmic complexity
|
math.HO
|
In this paper the author presents some non-conventional thoughts on the
complexity of the Universe and the algorithmic reproducibility of the human
brain, essentially sparked off by the notion of algorithmic complexity. We must
warn that though they evoke suggestive scenarios, they are still quite
speculative.
|
math
|
1,533 |
International comparisons in mathematics education: an overview
|
math.HO
|
The paper opens with an overview of the discussion of international
comparisons (including goals) in mathematics education. Afterwards, the two
most important recent international studies, the PISA Study and TIMSS-Repeat,
are described. After a short description of the qualitative-quantitative
debate, a qualitatively oriented small-scale study is described. The paper
closes with reflection on the possibilities and limitations of such studies.
|
math
|
1,534 |
Graphical explanation for the speed of the Fast Fourier Transform
|
math.HO
|
For a sample set of 1024 values, the FFT is 102.4 times faster than the
discrete Fourier transform (DFT). The basis for this remarkable speed advantage
is the `bit-reversal' scheme of the Cooley-Tukey algorithm. Eliminating the
burden of `degeneracy' by this means is readily understood using vector
graphics.
|
math
|
1,535 |
Two philosophical applications of algorithmic information theory
|
math.HO
|
Two philosophical applications of the concept of program-size complexity are
discussed. First, we consider the light program-size complexity sheds on
whether mathematics is invented or discovered, i.e., is empirical or is a
priori. Second, we propose that the notion of algorithmic independence sheds
light on the question of being and how the world of our experience can be
partitioned into separate entities.
|
math
|
1,536 |
First Case of Fermat's Last Theorem
|
math.HO
|
In this paper two conjectures are proposed based on which we can prove the
first case of Fermat's Last Theorem(FLT) for all primes $p \equiv -1
(\bmod~6)$. With Pollaczek's result {\bf [1]} and the conjectures the first
case of FLT can be proved for all primes greater than 3. With a computer
Conjecture1 was verified to be true for primes $\leq 2437$ and Conjecture2 for
primes $\leq 100003$.
|
math
|
1,537 |
From Philosophy to Program Size
|
math.HO
|
Most work on computational complexity is concerned with time. However this
course will try to show that program-size complexity, which measures
algorithmic information, is of much greater philosophical significance. I'll
discuss how one can use this complexity measure to study what can and cannot be
achieved by formal axiomatic mathematical theories. In particular, I'll show
(a) that there are natural information-theoretic constraints on formal
axiomatic theories, and that program-size complexity provides an alternative
path to incompleteness from the one originally used by Kurt Godel. Furthermore,
I'll show (b) that in pure mathematics there are mathematical facts that are
true for no reason, that are true by accident. These have to do with
determining the successive binary digits of the precise numerical value of the
halting probability Omega for a "self-delimiting" universal Turing machine. I
believe that these meta-theorems (a,b) showing (a) that the complexity of
axiomatic theories can be characterized information-theoretically and (b) that
God plays dice in pure mathematics, both strongly suggest a quasi-empirical
view of mathematics. I.e., math is different from physics, but perhaps not as
different as people usually think. I'll also discuss the convergence of
theoretical computer science with theoretical physics, Leibniz's ideas on
complexity, Stephen Wolfram's book A New Kind of Science, and how to attempt to
use information theory to define what a living being is.
|
math
|
1,538 |
Sur l'origine des chiffres arabes
|
math.HO
|
Sur l'origine des chiffres arabes A. Boucenna 1 From the pagination of an
Algerian Arabic manuscript of the beginning of the 19th century,we rediscover
the original shape that the Arabic numerals had before passing in Europe and
underwent the transformation that gave the modern Arabic numerals. This
original shape,whose use disappeared completely, proves that these numerals
have their origin in the Arabic letters. Contrary to what some hypotheses
pretend, particularly those that present them as drifting of Indian characters,
the 10 Arabic numerals that we use are, nothing else, 10 Arabic letters more or
less modified and taken in the "Abjadi" order. The hypothesis of the Indian
origin of the Arabic numerals is revealed a mistake denied by the shape of the
Arabic numerals and by the logic of the right to left representation of the
numbers and the algorithm of the elementary operations. The Arabic numerals
that simplified the writing of the numbers and the algorithms of the elementary
operations are believed to be born in the Maghreb(North Africa). From Bejaia
(Bougie) they passed in to Europe to give, after evolution, the modern Arabic
numerals : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. they also migrate to the Middle East
(the Mashrek) to give, after transformations in shape and adding two Hebrew
letters, the Arabic numerals, "Mashrekis", that are used currently in Middle
East.
|
math
|
1,539 |
Teaching linear algebra at university
|
math.HO
|
Linear algebra represents, with calculus, the two main mathematical subjects
taught in science universities. However this teaching has always been
difficult. In the last two decades, it became an active area for research works
in mathematics education in several countries. Our goal is to give a synthetic
overview of the main results of these works focusing on the most recent
developments. The main issues we will address concern: the epistemological
specificity of linear algebra and the interaction with research in history of
mathematics; the cognitive flexibility at stake in learning linear algebra;
three principles for the teaching of linear algebra as postulated by G. Harel;
the relation between geometry and linear algebra; an original teaching design
experimented by M. Rogalsk.
|
math
|
1,540 |
Popularizing mathematics: from eight to infinity
|
math.HO
|
It is rare to succeed in getting mathematics into ordinary conversation
without meeting all kinds of reservations. In order to raise public awareness
of mathematics effectively, it is necessary to modify such attitudes. In this
paper, we point to some possible topics for general mathematical conversation.
|
math
|
1,541 |
Reforms of the university mathematics education for non-mathematical specialties
|
math.HO
|
This article is a part of the report for the research project ``Reform of the
Course System and Teaching Content of Higher Mathematics (For Non-Mathematical
Specialties)'' in 1995, supported by the National Ministry of Education. There
are thirteen universities participated in this project. The Report not only
reflects results of our participants, but also includes valuable opinions of
many colleagues in the mathematical education circles. In this article, after a
brief description on the history and reform situation of the higher mathematics
education in China, attention concentrates to three aspects. They are: main
problems in this field existing; the functions of mathematics accomplishment
for college students; these concern course system, teaching and learning
philosophy, such as overemphasized specialty education, overlooking to arouse
rational thinking and aesthetic conceptions, etc. The last aspect contains a
discussion on several important relationships, such as: knowledge impartment
and quality cultivation, inherence and modernization of the mathematical
knowledge, teacher's guidance and students' initiative, mathematical basic
training and mathematical application consciousness and ability cultivation,
etc.
|
math
|
1,542 |
The teaching of proof
|
math.HO
|
This panel draws on research of the teaching of mathematical proof, conducted
in five countries at different levels of schooling. With a shared view of proof
as essential to the teaching and learning of mathematics, the authors present
results of studies that explore the challenges for teachers in helping students
learn to reason in disciplined ways about mathematical claims.
|
math
|
1,543 |
"Algebraic truths" vs "geometric fantasies": Weierstrass' Response to Riemann
|
math.HO
|
In the 1850s Weierstrass succeeded in solving the Jacobi inversion problem
for the hyper-elliptic case, and claimed he was able to solve the general
problem. At about the same time Riemann successfully applied the geometric
methods that he set up in his thesis (1851) to the study of Abelian integrals,
and the solution of Jacobi inversion problem. In response to Riemann's
achievements, by the early 1860s Weierstrass began to build the theory of
analytic functions in a systematic way on arithmetical foundations, and to
present it in his lectures. According to Weierstrass, this theory provided the
foundations of the whole of both elliptic and Abelian function theory, the
latter being the ultimate goal of his mathematical work. Riemann's theory of
complex functions seems to have been the background of Weierstrass's work and
lectures. Weierstrass' unpublished correspondence with his former student
Schwarz provides strong evidence of this. Many of Weierstrass' results,
including his example of a continuous non-differentiable function as well as
his counter-example to Dirichlet principle, were motivated by his criticism of
Riemann's methods, and his distrust in Riemann's ``geometric fantasies''.
Instead, he chose the power series approach because of his conviction that the
theory of analytic functions had to be founded on simple "algebraic truths".
Even though Weierstrass failed to build a satisfactory theory of functions of
several complex variables, the contradiction between his and Riemann's
geometric approach remained effective until the early decades of the 20$^{th}$
century.
|
math
|
1,544 |
From quaternions to cosmology: spaces of constant curvature, ca. 1873-1925
|
math.HO
|
After mathematicians and physicists had learned that the structure of
physical space was not necessarily Euclidean, it became conceivable that the
global topological structure of space was non-trivial. In the context of the
late 19th century debates on physical space this speculation gave rise to the
problem of classifying spaces of constant curvature from a topological point of
view. William Kingdon Clifford, Felix Klein and Wilhelm Killing, the latter of
whom devoted a substantial amount of work to the topic in the early 1890s,
clearly perceived this problem as relevant for both mathematics and natural
philosophy (i.e., physics or cosmology). To some extent, a cosmological
interest may even be found among those authors who restated the space form
problem in more modern terms in the early 20th century, such as Heinz Hopf.
|
math
|
1,545 |
The third approach to the history of mathematics in China
|
math.HO
|
The first approach to the history of mathematics in China led by Li Yan
(1892--1963) and Qian Baocong (1892--1974) featured discovering {\it what}
mathematics had been done in China's past. From the 1970s on, Wu Wen-tsun and
others shifted this research paradigm to one of recovering {\it how}
mathematics was done in ancient China. Both approaches, however, focus on the
same problem, that is mathematics in history. The theme of the third approach
is supposed to be {\it why} mathematics was done. Combining this approach with
the former two, the research paradigm will be improved from one of mathematics
in history to that of the history of mathematics.
|
math
|
1,546 |
Passages of Proof
|
math.HO
|
In this paper we propose a new perspective on the evolution and history of
the idea of mathematical proof. Proofs will be studied at three levels:
syntactical, semantical and pragmatical. Computer-assisted proofs will be give
a special attention. Finally, in a highly speculative part, we will anticipate
the evolution of proofs under the assumption that the quantum computer will
materialize. We will argue that there is little `intrinsic' difference between
traditional and `unconventional' types of proofs.
|
math
|
1,547 |
Thoughts on the Riemann hypothesis
|
math.HO
|
The simultaneous appearance in May 2003 of four books on the Riemann
hypothesis (RH) provoked these reflections. We briefly discuss whether the RH
should be added as a new axiom, or whether a proof of the RH might involve the
notion of randomness.
|
math
|
1,548 |
The Informal Logic of Mathematical Proof
|
math.HO
|
Informal logic is a method of argument analysis which is complementary to
that of formal logic, providing for the pragmatic treatment of features of
argumentation which cannot be reduced to logical form. The central claim of
this paper is that a more nuanced understanding of mathematical proof and
discovery may be achieved by paying attention to the aspects of mathematical
argumentation which can be captured by informal, rather than formal, logic. Two
accounts of argumentation are considered: the pioneering work of Stephen
Toulmin [The uses of argument, Cambridge University Press, 1958] and the more
recent studies of Douglas Walton, [e.g. The new dialectic: Conversational
contexts of argument, University of Toronto Press, 1998]. The focus of both of
these approaches has largely been restricted to natural language argumentation.
However, Walton's method in particular provides a fruitful analysis of
mathematical proof. He offers a contextual account of argumentational
strategies, distinguishing a variety of different types of dialogue in which
arguments may occur. This analysis represents many different fallacious or
otherwise illicit arguments as the deployment of strategies which are sometimes
admissible in contexts in which they are inadmissible. I argue that
mathematical proofs are deployed in a greater variety of types of dialogue than
has commonly been assumed. I proceed to show that many of the important
philosophical and pedagogical problems of mathematical proof arise from a
failure to make explicit the type of dialogue in which the proof is introduced.
|
math
|
1,549 |
Leibniz, Information, Math and Physics
|
math.HO
|
The information-theoretic point of view proposed by Leibniz in 1686 and
developed by algorithmic information theory (AIT) suggests that mathematics and
physics are not that different. This will be a first-person account of some
doubts and speculations about the nature of mathematics that I have entertained
for the past three decades, and which have now been incorporated in a digital
philosophy paradigm shift that is sweeping across the sciences.
|
math
|
1,550 |
Embodied Mathematics and the Origins of Geometry
|
math.HO
|
In this paper, we propose that 'embodied mathematics' should be studied not
only by reduction to the present individual bodily experience but in an
historical context as well, as far as the origins of mathematics are concerned.
Some early mathematical results are the Theorems of Geometry and arose as
attempts to objectively render the main perceptual categories such as
verticality, horizontality, similarity (or its varieties). Inasmuch as these
are of a qualitative nature, it was required that they be expressed in a
quantitative way in order to be objectified.
The first form of this objectification occurred in the case of 'archetypal
results', namely the Pythagorean triads and the internal ratio of the legs in
the right triangles. In the next stage, a 'scientific' treatment would come
from a shift of objectification and descriptions inside an abstract theory,
which would constitute the first logicomathematical knowledge. In this theory,
the 'archetypal results' were incorporated, generalized and acquired their
unquestionable, supertemporal validity. The study presents a particular
epistemological analysis of some of the main terms used in the beginnings of
Geometrical Thought and Euclid's Elements, utilizing the theoretical apparatus
of the theory of 'embodied mathematics'. It also traces models of
objectification for the 'archetypal results' and indicates their diffusion in
later mathematical developments.
|
math
|
1,551 |
Der rechnende Dichter (The calculating poet)
|
math.HO
|
A small and unsystematic selection of my favorite appearances of
mathematicians and mathematics in German literature. It includes classic and
romantic (Lessing, Goethe, Wezel, F. Schlegel, Kleist, Novalis, Grillparzer,
Heine), modern (Wedekind, Doeblin, Morgenstern, Musil, Brecht, Eich), as well
as post-modern authors (Hans-Magnus Enzensberger, Volker Braun, Botho Strauss).
|
math
|
1,552 |
The Number System of the Old European Script
|
math.HO
|
The oldest (c. 4000 BC) undeciphered language is the Old European Script
known from approximately 940 inscribed objects (82% of inscriptions on pottery)
found in excavations in the Vinca-Tordos region Transylvania. Also, it is not
known for what the script was used, but the prevailing theory is that it had a
religious purpose. We note that more than a quarter of the inscriptions are on
the bottom of a pot-seemingly a most illogical and inglorious place to pay
homage to a deity. Consistent with this, in a survey we performed of pots and
containers in modern locations, we could not find a single religious
inscription. Here we delineate the number system of the Old European Script,
and propose that the Old European Script may have been used for economic
purposes. The delineation of the number signs of the Old European Script should
facilitate further understanding of the rest of the script and of the Old
European culture, especially as new archaeologic findings emerge.
|
math
|
1,553 |
Why I don't like "pure mathematics"
|
math.HO
|
An opiniated essay on what pure mathematics is and why the adjective "pure"
in "pure mathematics" is not a good choice.
|
math
|
1,554 |
Solving the quartic with a pencil
|
math.HO
|
This expository paper presents the general solution of a quartic equation as
a jump off point to introduce Lefschetz fibrations. It should be accessible to
a broad audience.
|
math
|
1,555 |
Smarandache Sequences: Explorations and Discoveries with a Computer Algebra System
|
math.HO
|
We study Smarandache sequences of numbers, and related problems, via a
Computer Algebra System. Solutions are discovered, and some conjectures
presented.
|
math
|
1,556 |
A short constructive proof of Jordan's decomposition theorem
|
math.HO
|
Although there are many simple proofs of Jordan's decomposition theorem in
the literature (see [1], the references mentioned there, and [2]), our proof
seems to be even more elementary. In fact, all we need is the theorem on the
dimensions of rang and kernel and the existence of eigenvalues of a linear
transformation on a nontrivial finite dimensional complex vector space.
|
math
|
1,557 |
Biography of John Rainwater
|
math.HO
|
The following paragraphs will describe the origins of John Rainwater, the
impact of his work, the motivations for various parts of it and the prospects
for his future.
|
math
|
1,558 |
A Brief Survey of the History of the Calculus of Variations and its Applications
|
math.HO
|
In this paper, we trace the development of the theory of the calculus of
variations. From its roots in the work of Greek thinkers and continuing through
to the Renaissance, we see that advances in physics serve as a catalyst for
developments in the mathematical theory. From the 18th century onwards, the
task of establishing a rigourous framework of the calculus of variations is
studied, culminating in Hilbert's work on the Dirichlet problem and the
development of optimal control theory. Finally, we make a brief tour of some
applications of the theory to diverse problems.
|
math
|
1,559 |
Role of Mathematics in Physical Sciences
|
math.HO
|
The role of mathematics in physical sciences is discussed, particularly how
higher mathematics found applications in empirical problems. Several examples
are given to illustrate this role.
|
math
|
1,560 |
Meta Math! The Quest for Omega
|
math.HO
|
This book presents a personal account of the mathematics and metamathematics
of the 20th century leading up to the discovery of the halting probability
Omega. The emphasis is on history of ideas and philosophical implications.
|
math
|
1,561 |
Fibonacci Rectangles
|
math.HO
|
Is there any other proportion for a rectangle, other than the Golden
Proportion, that will allow the process of cutting off successive squares to
produce an infinite paving of the original rectangle by squares of different
sizes? The answer is: No. The only proportion that allows this pattern is the
Golden Ratio. Two proofs are given.
|
math
|
1,562 |
Course of linear algebra and multidimensional geometry
|
math.HO
|
This is a standard textbook for the course of linear algebra and
multidimensional geometry as it was taught in 1991-1998 at Mathematical
Department of Bashkir State University. Both coordinate and invariant
approaches are used, but invariant approach is preferred.
|
math
|
1,563 |
The Eudoxus Real Numbers
|
math.HO
|
This note describes a representation of the real numbers due to Schanuel. The
representation lets us construct the real numbers from first principles. Like
the well-known construction of the real numbers using Dedekind cuts, the idea
is inspired by the ancient Greek theory of proportion, due to Eudoxus. However,
unlike the Dedekind construction, the construction proceeds directly from the
integers to the real numbers bypassing the intermediate construction of the
rational numbers.
The construction of the additive group of the reals depends on rather simple
algebraic properties of the integers. This part of the construction can be
generalised in several natural ways, e.g., with an arbitrary abelian group
playing the role of the integers. This raises the question: what does the
construction construct? In an appendix we sketch some generalisations and
answer this question in some simple cases.
The treatment of the main construction is intended to be self-contained and
assumes familiarity only with elementary algebra in the ring of integers and
with the definitions of the abstract algebraic notions of group, ring and
field.
|
math
|
1,564 |
Leibniz, Randomness and the Halting Probability
|
math.HO
|
This paper, which is dedicated to Alan Turing on the 50th anniversary of his
death, gives an overview and discusses the philosophical implications of
incompleteness, uncomputability and randomness.
|
math
|
1,565 |
Areal Optimization of Polygons
|
math.HO
|
We will first solve the following problem analytically: given a piece of wire
of specified length, we will find where the wire should be cut and bent to form
two regular polygons not necessarily having the same number of sides, so that
the combined area of the polygons thus formed is maximized, minimized, greater
than, and less than a specified area. We will extend the results to the cases
where the wire is divided into three and finally into an arbitrary number of
segments. The second problem we will solve is as follows: two wires of
specified length are to be bent into two regular polygons whose total number of
sides is fixed. We will determine how the total number of polygonal sides are
to be allocated between the wires so that the total area of the polygons is
maximized. We will extend the results found here to the case where we are given
any number of wires of specified length.
|
math
|
1,566 |
Consecutive, Reversed, Mirror, and Symmetric Smarandache Sequences of Triangular Numbers
|
math.HO
|
We use the Maple system to check the investigations of S. S. Gupta regarding
the Smarandache consecutive and the reversed Smarandache sequences of
triangular numbers [Smarandache Notions Journal, Vol. 14, 2004, pp. 366-368].
Furthermore, we extend previous investigations to the mirror and symmetric
Smarandache sequences of triangular numbers.
|
math
|
1,567 |
Totally real origami and impossible paper folding
|
math.HO
|
This paper gives one set of axioms for origami constructions, and describes
the set of constructible points under these axioms. The determination of the
set of cunstructible points for this particular set of axioms is related to
Hilbert's 17 th problem.
|
math
|
1,568 |
On Bernoulli Numbers and Its Properties
|
math.HO
|
In this survey paper, I first review the history of Bernoulli numbers, then
examine the modern definition of Bernoulli numbers and the appearance of
Bernoulli numbers in expansion of functions. I revisit some properties of
Bernoulli numbers and the history of the computation of big Bernoulli numbers.
|
math
|
1,569 |
Philosophy as a cultural resource and medium of reflection for Hermann Weyl
|
math.HO
|
Here we review a kind of post-World-War-II "Nachtrag" to H. Weyl's
philosophical comments on mathematics and the natural sciences published in the
middle of the 1920s. In a talk given at Z\"urich in the late 1940s, Weyl
discussed F.Gonseth's dialectical epistemology and considered it as being
restricted too strictly to aspects of historical change. His own experiences
with post-Kantian dialectical philosophy, in particular J.G. Fichte's
derivation of the concept of space and matter, had been a stronger dialectical
background for his own 1918 studies in purely infintitesimal geometry and the
early geometrically unified field theory of matter (extending the Mie-Hilbert
program). Although now Weyl distantiated himself from the speculative features
of his youthful philosophizing and in particular from his earlier enthusiasm
for Fichte, he again had deep doubts as to the cultural foundations of modern
mathematical sciences and its role in material culture of high modernity. For
Weyl, philosophical "reflection" was a cultural necessity; he now turned
towards K. Jasper's and M. Heidegger's existentialism to find deeper grounds,
similar to his turn towards Fichte's philosophy after World War I.
|
math
|
1,570 |
Irreducible Complexity in Pure Mathematics
|
math.HO
|
By using ideas on complexity and randomness originally suggested by the
mathematician-philosopher Gottfried Leibniz in 1686, the modern theory of
algorithmic information is able to show that there can never be a "theory of
everything" for all of mathematics.
|
math
|
1,571 |
How real are real numbers?
|
math.HO
|
We discuss mathematical and physical arguments against continuity and in
favor of discreteness, with particular emphasis on the ideas of Emile Borel
(1871-1956).
|
math
|
1,572 |
On Amicable Numbers With Different Parity
|
math.HO
|
In this paper we provide a straightforward proof that if a pair of amicable
numbers with different parity exists (one number odd and the other one even),
then the odd amicable number must be a perfect square, while the even amicable
number has to be equal to the product of a power of 2 and an odd perfect
square.
|
math
|
1,573 |
Looking through newly to the amazing irrationals
|
math.HO
|
We survay some nice result concerning the irrationals with a metric space
point of view.Here is ofcourse nothing new may be or an expert in this field.
|
math
|
1,574 |
Mathematical Education
|
math.HO
|
This essay, originally published in the Sept 1990 Notices of the AMS,
discusses problems of our mathematical education system that often stem from
widespread misconceptions by well-meaning people of the process of learning
mathematics. The essay also discusses ideas for fixing some of the problems.
Most of what I wrote in 1990 remains equally applicable today.
|
math
|
1,575 |
The Uses of Argument in Mathematics
|
math.HO
|
Stephen Toulmin once observed that `it has never been customary for
philosophers to pay much attention to the rhetoric of mathematical debate'.
Might the application of Toulmin's layout of arguments to mathematics remedy
this oversight?
Toulmin's critics fault the layout as requiring so much abstraction as to
permit incompatible reconstructions. Mathematical proofs may indeed be
represented by fundamentally distinct layouts. However, cases of genuine
conflict characteristically reflect an underlying disagreement about the nature
of the proof in question.
|
math
|
1,576 |
Notes on Theory of Quadratic Residues
|
math.HO
|
The Law of Quadratic Reciprocity was conjectured by Euler and Legendre who
both found an incomplete proof. Gauss called this law "Theorema Fundamentale",
and he was the first who gave a complete proof, he also highlighted the
equivalence of his formulation with those of Euler and Legrendre. Hereby notes
gives a overview of the Theory of Quadratic Residues using a classical approach
with some application to Diophantine Equations, such as Two Square Theorem and
Pythagorean Quadruplets.
|
math
|
1,577 |
Epistemology as Information Theory: From Leibniz to Omega
|
math.HO
|
In 1686 in his Discours de Metaphysique, Leibniz points out that if an
arbitrarily complex theory is permitted then the notion of "theory" becomes
vacuous because there is always a theory. This idea is developed in the modern
theory of algorithmic information, which deals with the size of computer
programs and provides a new view of Godel's work on incompleteness and Turing's
work on uncomputability. Of particular interest is the halting probability
Omega, whose bits are irreducible, i.e., maximally unknowable mathematical
facts. More generally, these ideas constitute a kind of "digital philosophy"
related to recent attempts of Edward Fredkin, Stephen Wolfram and others to
view the world as a giant computer. There are also connections with recent
"digital physics" speculations that the universe might actually be discrete,
not continuous. This systeme du monde is presented as a coherent whole in my
book Meta Math!, which will be published this fall.
|
math
|
1,578 |
Laplace transformation updated
|
math.HO
|
The traditional theory of Laplace transformation in its currently prevalent
form is unsatisfactory. Its deficiencies can be traced back to a mismatch of
the definition intervals of the original function and of the inverse
L-transform. A new approach is outlined by which Laplace transformation becomes
liberated from its inconsistencies.
|
math
|
1,579 |
Saunders Mac Lane, the Knight of Mathematics
|
math.HO
|
This is a short obituary of Saunders Mac Lane (1909--2005).
|
math
|
1,580 |
Traits
|
math.HO
|
Reminiscences about Alexandr Danilovich Alexandrov (1912--1999)
|
math
|
1,581 |
Group actions in number theory
|
math.HO
|
Students having had a semester course in abstract algebra are exposed to the
elegant way in which finite group theory leads to proofs of familiar facts in
elementary number theory. In this note we offer two examples of such group
theoretical proofs using the action of a group on a set. The first is Fermat's
little theorem and the second concerns a well known identity involving the
famous Euler phi function. The tools that we use to establish both results are
sometimes seen in a second semester algebra course in which group actions are
studied. Specifically, we will use the class equation of a group action and
Burnside's theorem.
|
math
|
1,582 |
Counting the Positive Rationals: A Brief Survey
|
math.HO
|
We discuss some examples that illustrate the countability of the positive
rational numbers and related sets. Techniques include radix representations,
Godel numbering, the fundamental theorem of arithmetic, continued fractions,
Egyptian fractions, and the sequence of ratios of successive hyperbinary
representation numbers.
|
math
|
1,583 |
Asymptotic behaviour of Turing Machines
|
math.HO
|
This paper has been withdrawn. See published paper
http://arxiv.org/math.HO/0512390
|
math
|
1,584 |
Asymptotic behavior and halting probability of Turing Machines
|
math.HO
|
Through a straightforward Bayesian approach we show that under some general
conditions a maximum running time, namely the number of discrete steps
performed by a computer program during its execution, can be defined such that
the probability that such a program will halt after that time is smaller than
any arbitrary fixed value. Consistency with known results and consequences are
also discussed.
|
math
|
1,585 |
How to axiomatize school geometry
|
math.HO
|
This is an attempt to present axioms for Euclidean geometry, aiming at the
following goals: to work with geometric notions (thus not merely identify
points with pairs of numbers, giving a special status to a particular
coordinate system); to be appropriate to the way geometry is done in science
and engineering - not to conceal its algebraic nature; to respond to the desire
that one would accept intuitively/empirically that the axioms are valid in our
physical everyday world (or rather in the idealization that geometry is) - that
seemingly disfavoring taking the theorem of Pythagoras as an axiom; to have
accessible the rigor and standards of "pure" mathematics. The style in this
note is that of usual mathematical writings - for an unsophisticated audience
the style of the presentation must surely be quite different.
|
math
|
1,586 |
Zeno's Paradoxes. A Cardinal Problem 1. On Zenonian Plurality
|
math.HO
|
In this paper the claim that Zeno's paradoxes have been solved is contested.
Although no one has ever touched Zeno without refuting him (Whitehead), it will
be our aim to show that, whatever it was that was refuted, it was certainly not
Zeno. The paper is organised in two parts. In the first part we will
demonstrate that upon direct analysis of the Greek sources, an underlying
structure common to both the Paradoxes of Plurality and the Paradoxes of Motion
can be exposed. This structure bears on a correct - Zenonian - interpretation
of the concept of division through and through. The key feature, generally
overlooked but essential to a correct understanding of all his arguments, is
that they do not presuppose time. Division takes place simultaneously. This
holds true for both PP and PM. In the second part a mathematical representation
will be set up that catches this common structure, hence the essence of all
Zeno's arguments, however without refuting them. Its central tenet is an
aequivalence proof for Zeno's procedure and Cantor's Continuum Hypothesis. Some
number theoretic and geometric implications will be shortly discussed.
Furthermore, it will be shown how the Received View on the motion-arguments can
easely be derived by the introduction of time as a (non-Zenonian) premiss, thus
causing their collapse into arguments which can be approached and refuted by
Aristotle's limit-like concept of the potentially infinite, which remained -
though in different disguises - at the core of the refutational strategies that
have been in use up to the present. Finally, an interesting link to Newtonian
mechanics via Cremona geometry can be established.
|
math
|
1,587 |
Good reduction, bad reduction
|
math.HO
|
We give some general properties of good and bad reduction, and some recent
examples (worked out with Dipendra Prasad) of varieties having bad reduction
not accounted for by their cohomology. We include some consequences of our
remarks for varieties over number fields having good reduction everywhere.
|
math
|
1,588 |
The Tao of Mathematics, and Think Locally
|
math.HO
|
An informal discussion of Serre's conjecture on the modularity of odd
irreducible representations of Gal(\bar Q|Q) into GL_2(\bar F_p), using
Ramanujan's tau-function as an illustrative example. Also, a word about the
importance of thinking locally.
|
math
|
1,589 |
Numbers and periods
|
math.HO
|
A somewhat pretentious presentation of number systems (N, Z, Q, R, C, Q_p,
>...). The problem of a p-adic characterisation of good-reduction p-adic curves
is posed.
|
math
|
1,590 |
Variations on an inequality from IMO'2001
|
math.HO
|
Some extensions of an inequality from IMO'2001 are proven by means of the
Lagrange multiplier criterion.
|
math
|
1,591 |
On projective two-dimensional Finsler spaces with special metric
|
math.HO
|
We present the English translation of the paper where one special class of
Finsler spaces was introduced. Now this class is known as so called "Kropina
spaces". The article was written in 1958 and published in Russian in "Trudy
seminara po vektornomu i tenzornomu analizu" ("Workshops of the Seminar in
vector and tensor Analysis"), vol. XI, 1961.
|
math
|
1,592 |
Lanchester combat models
|
math.HO
|
An overview of Lanchester combat models, emphasising their pedagogical
possibilities. After a description of the aimed-fire model and comments on the
literature, we introduce briefly a range of further topics: a discrete
equivalent, the unaimed-fire model, mixed forces, the meaning of a 'unit',
support troops, Bracken's generalization and an asymmetric model.
|
math
|
1,593 |
Origin of the numerals
|
math.HO
|
Through the pagination of an Arabian Algerian manuscript of the beginning of
the 19th century, we rediscover the original shape, the "Ghubari" shape, of the
numerals. Contrary to some assumptions, particularly those which claim that
they are derived from Indian characters, this "Ghubari" shape, whose use has
completely disappeared, shows that the ten modern numerals derive from ten
Arabic letters. The symbol of a "Ghubari" numeral corresponds to the Arabic
letter whose "Abjadi" numerical value is equal to this numeral. The assumption
of the Indian origin of the numerals is denied by the shape of the numerals and
by the right left sociological logic of the representation of the numerals and
the algorithms of the basic operations. The numerals are born in Maghreb or in
Spain. In Europe, the "Ghubari" numerals became the modern numerals: 0, 1, 2,
3, 4, 5, 6, 7, 8, 9 and in the Middle East, borrowing two Hebrew letters, they
gave the "Mashriki" numerals: ۰ ۱ ۲ ۳ ٩ ٨
٧ ٦ ٥ ٤ .
|
math
|
1,594 |
Max Dehn, un mathématicien aux préoccupations universelles
|
math.HO
|
This is a free summary of a much longer article published in german in
"Forschung Frankfurt". This article presents facts concerning the works of Max
Dehn and the history of Frankfurt University.
E. Hellinger, R. Moufang, C. L. Siegel, A. Weil, P. Epstein, W. Hartner are
named among others. Of course these texts are centered around the critical
period 1933-1941. Comments, references, and acknowledgements have been added by
the translator, which does not claim any competence. In fact he attempted this
translation mainly for the sake of human rights enforcement.
Please look at web pages at uni-frankfurt.de and propose translations into
other languages to professors Wolfart et al !
|
math
|
1,595 |
The History of Barbilian's Metrization Procedure
|
math.HO
|
Barbilian spaces are metric spaces with a metric induced by a special
procedure of metrization which is inspired by the study of the models of
non-Euclidean geometry. In the present material we discuss the history of
Barbilian spaces and the evolution of the theory. We point out that some of the
current references to the work done in Barbilian spaces refer to Barbilian's
contribution from 1934, while his construction has been largely extended in
four works published in Romanian in 1959-1962.
|
math
|
1,596 |
Euler and magic squares (De quadratis magicis)
|
math.HO
|
Magic squares have always been and are still fascinating for many people, be
it only because of their mathematical properties. Their origin is still but
certain : we find no magic squares in Greece, and only a 3x3 one in China at
the beginning of our era. Most of their development was made in islamic
countries. In Europe, Euler wrote two memoirs and numerous pages on magic
squares. One of his problems is the famous "officer problem".
(In french : Les carres magiques ont toujours fascine la plupart des gens,
tant par leur apparente simplicite que par leur etonnante propriete. Leur
origine est toutefois assez lointaine et incertaine : il n'y a pas de traces de
carres magiques en Grece et on trouve seulement un carre de 3x3 en Chine vers
le debut de notre ere. Euler a consacre deux memoires et de nombreuses pages de
ses carnets a l'etude des carres magiques. L'un des problemes enonce est le
fameux "probleme des officiers".)
|
math
|
1,597 |
The prime analog of the Kepler-Bouwkamp constant
|
math.HO
|
The prime analog of the Kepler-Bouwkamp constant is evaluated.
|
math
|
1,598 |
The regularized product of the Fibonacci numbers
|
math.HO
|
The regularized product of the Fibonacci numbers is evaluated.
|
math
|
1,599 |
A geometric method to compute some elementary integrals
|
math.HO
|
An elementary, albeit higher dimensional, argument is used to compute the
area under the power function curve between 0 and 1.
|
math
|
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