# /docs/math/ Directory Listing

## Files

• 2018-pieronkiewicz.pdf: “Mathematicians Who Never Were”⁠, Barbara Pieronkiewicz (backlinks)

• 2014-schmid.pdf: “Two curious integrals and a graphic proof”⁠, Hanspeter Schmid (backlinks)

• 2012-gowers.pdf: “Vividness in Mathematics and Narrative”⁠, Timothy Gowers (backlinks)

• 2009-birtwistle-com3200programminglanguagesemantics-ch5.5-backwardinductionandpetardsbghtheorem.pdf: “COM3200: Programming Language Semantics: Chapter 5. Induction Techniques. 5.5. Backward Induction and Petard's BGH Theorem”⁠, Graham Birtwistle (backlinks)

• 2009-athreya.pdf: “Big Game Hunting for Graduate Students in Mathematics”⁠, Jayadev Athreya, Apoorva Khare (backlinks)

• 2007-colton.pdf: ⁠, (2007-01-01; backlinks):

We discuss what constitutes knowledge in pure mathematics and how new advances are made and communicated. We describe the impact of computer algebra systems, automated theorem provers, programs designed to generate examples, mathematical databases, and theory formation programs on the body of knowledge in pure mathematics. We discuss to what extent the output from certain programs can be considered a discovery in pure mathematics. This enables us to assess the state of the art with respect to Newell and Simon’s prediction that a computer would discover and prove an important mathematical theorem.

• 2002-dijkstra.pdf: ⁠, (2002-12-01):

At a given moment, the concept of polite mathematics emerged, the underlying idea of which is that, even if you have only 60 readers, it pays to spend an hour if by doing so you can save your average reader a minute. By inventing an idealized ‘average reader’, we could translate most of the lofty, human goal of politeness into more or less formal criteria we could apply to our texts. This note is devoted to the resulting notational and stylistic conventions that were adopted as the years went by. We don’t want to baffle or puzzle our reader, in particular it should be clear what has to be done to check our argument and it should be possible to do so without pencil and paper. This dictates small, explicit steps. On the other hand it is well known that brevity is the leading characteristic of mathematical elegance, and some fear that this ideal excludes the small, explicit steps, but one of the joys of my professional life has been the discovery that this fear is unfounded, for brevity can be achieved without committing the sin of omission. I should point out that my ideal of crisp clarity is not universally shared. Some consider the puzzles that are created by their omissions as spicy challenges, without which their texts would be boring; others shun clarity lest their work is considered trivial.

• 2002-descartes.pdf: “Hymne to Hymen”⁠, B. Descartes, C. A. B. Smith

• 2001-borwein.pdf: ⁠, (2001-03; backlinks):

Using Fourier transform techniques, we establish inequalities for integrals of the form

We then give quite striking closed form evaluations of such integrals and finish by discussing various extensions and applications.

[Keywords: sinc integrals, Fourier transforms, convolution, Parseval’s theorem]

• 1996-hoare.pdf: ⁠, (1996-03; backlinks):

By surveying current software engineering practice, this paper reveals that the techniques employed to achieve reliability are little different from those which have proved effective in all other branches of modern engineering: rigorous management of procedures for design inspection and review; quality assurance based on a wide range of targeted tests; continuous evolution by removal of errors from products already in widespread use; and defensive programming, among other forms of deliberate over-engineering. Formal methods and proof play a small direct role in large scale programming; but they do provide a conceptual framework and basic understanding to promote the best of current practice, and point directions for future improvement.

• 1988-brockett.pdf: ⁠, (1988-12-07):

We establish a number of properties associated with the dynamical system ̇H = [H, [H, N]], where H and N are symmetric n by n matrices and [A, B] = AB − BA. The most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices.

Particular emphasis is placed on the role of this equation as an analog computer. For example, it is shown how to map the data associated with a linear programming problem into H(0) and N in such a way as to have ̇H = [H[H, N]] evolve to a solution of the linear programming problem.

This result can be applied to find systems that solve a variety of generic combinatorial optimization problems, and it also provides an algorithm for diagonalizing symmetric matrices.

• 1986-boolos.pdf: “Review of Manin Yu. I.. _A course in mathematical logic_”⁠, George Boolos (backlinks)

• 1985-euler.pdf: “Lion-Hunting with Logic”⁠, Houston Euler (backlinks)

• 1983-bracewell.pdf: ⁠, (1983-12-01):

The (DHT) resembles the (DFT) but is free from 2 characteristics of the DFT that are sometimes computationally undesirable.

The inverse DHT is identical with the direct transform, and so it is not necessary to keep track of the +i and −i versions as with the DFT. Also, the DHT has real rather than complex values and thus does not require provision for complex arithmetic or separately managed storage for real and imaginary parts. Nevertheless, the DFT is directly obtainable from the DHT by a simple additive operation.

In most image-processing applications the convolution of 2 data sequences f1 and f2 is given by DHT of [(DHT of f1) × (DHT of f2)], which is a rather simpler algorithm than the DFT permits, especially if images are to be manipulated in 2 dimensions. It permits faster computing. Since the speed of the fast Fourier transform depends on the number of multiplications, and since one complex multiplication equals 4 real multiplications, a fast Hartley transform also promises to speed up Fourier-transform calculations.

The name “discrete Hartley transform” is proposed because the DHT bears the same relation to an integral transform described by [] as the DFT bears to the Fourier transform.

• 1982-pondiczery.pdf: “Letters [Mathematical Intelligencer, Volume 4, issue 1, March 1982]”⁠, Peter M. Neumann, E. S. Pondiczery, Guy Boillat, W. Nowacki, editors (backlinks)

• 1980-farlow.pdf: “A rebuke of A. B. Smith's paper, 'A Note on Piffles'”⁠, S. J. Farlow (backlinks)

• 1979-lutzen.pdf: “Heaviside's Operational Calculus and the Attempts to Rigorise It”⁠, Jesper Lützen

• 1979-hersh.pdf: “Some Proposals for Reviving the Philosophy of mathematics”⁠, Reuben Hersh (backlinks)

• 1979-demillo.pdf: ⁠, (1979; backlinks):

Many people have argued that computer programming should strive to become more like mathematics. Maybe so, but not in the way they seem to think. The aim of program verification, an attempt to make programming more mathematics-like, is to increase dramatically one’s confidence in the correct functioning of a piece of software, and the device that verifiers use to achieve this goal is a long chain of formal, deductive logic. In mathematics, the aim is to increase one’s confidence in the correctness of a theorem, and it’s true that one of the devices mathematicians could in theory use to achieve this goal is a long chain of formal logic. But in fact they don’t. What they use is a proof, a very different animal. Nor does the proof settle the matter; contrary to what its name suggests, a proof is only one step in the direction of confidence. We believe that, in the end, it is a social process that determines whether mathematicians feel confident about a theorem—and we believe that, because no comparable social process can take place among program verifiers, program verification is bound to fail. We can’t see how it’s going to be able to affect anyone’s confidence about programs.

• 1976-barrington.pdf: “15 New Ways To Catch A Lion”⁠, John Barrington [Ian Stewart] (backlinks)

• 1975-chaitin.pdf: “Randomness and Mathematical Proof”⁠, Gregory Chatin (backlinks)

• 1973-knuth.pdf: ⁠, (1973; backlinks):

This chapter discusses the difficulties associated with the computer-science theories.

The theory of automata is slowly changing to a study of random-access computations, and this work promises to be more useful. Any algorithm programmable on a certain kind of pushdown automaton can be performed efficiently on a random-access machine, no matter how slowly the pushdown program runs.

Another difficulty with the theory of languages is that it has led to an overemphasis on syntax as opposed to semantics. For many years there was much light on syntax and very little on semantics; so simple semantic constructions were unnaturally grafted onto syntactic definitions, making rather unwieldy grammars, instead of searching for theories more appropriate to semantics.

Theories are often more structured and more interesting when they are based on real problems; somehow they are more exciting than completely abstract theories will ever be.

• 1972-davis.pdf: “Fidelity in Mathematical Discourse: Is One and One Really Two?”⁠, Philip J. Davis (backlinks)

• 1968-dudley.pdf: “Further Techniques in the Theory of Big Game Hunting”⁠, Patricia L. Dudley, G. T. Evans, K. D. Hansen, I. D. Richardson (backlinks)

• 1968-morphy.pdf: “Some Modern Mathematical Methods in the Theory of Lion Hunting”⁠, Otto Morphy (backlinks)

• 1968-hammersley.pdf: “On the enfeeblement of mathematical skills by 'Modern Mathematics' and by similar soft intellectual trash in schools and universities”⁠, J. M. Hammersley (backlinks)

• 1967-austin.pdf: “A Note On Piffles, By A. B. Smith”⁠, A. K. Austin (backlinks)

• 1967-roselius.pdf: “On a Theorem of H. Pétard”⁠, Christian Roselius (backlinks)

• 1966-scott.pdf: ⁠, (1966):

Probability concepts nowadays are presented in the standard framework of the Kolmogorov axioms. A sample space is given together with an σ-field of subsets, the events, and an σ-additive probability measure defined on this σ-field.

It is more natural in many situations to assign probabilities to statements rather than sets. It may be mathematically useful to translate everything into a set-theoretical formulation, but the step is not always necessary or even helpful. The main task is to carry over the standard concepts from ordinary logic to what might be called “probability logic.” Indeed ordinary logic is a special case: the assignment of truth values to formulas can be viewed as assigning probabilities that are either 0 (for false) or 1 (for true).

In a sense, the symmetric probability systems are opposite to ordinary relational systems.

• 1965-good.pdf: “A New Method of Catching a Lion”⁠, I. J. Good (backlinks)

• 1942-hartley.pdf: ⁠, (1942-03-01; backlinks):

The is here expressed in a more symmetrical form which leads to certain analogies between the function of the original variable and its transform. Also it permits a function of time, for example, to be analyzed into 2 independent sets of sinusoidal components, one of which is represented in terms of positive frequencies, and the other of negative.

The steady-state treatment of transmission problems in terms of this analysis is similar to the familiar ones and may be carried out either in terms of real quantities or of complex exponentials. In the transient treatment, use is made of the analogies referred to above, and their relation to the method of “paired echoes” is discussed.

A restatement is made of the condition which is known to be necessary in order that a given steady-state characteristic may represent a passive or stable active system (actual or ideal).

A particular necessary condition is deduced from this as an illustration.

• 1930-ramsey.pdf: “On a Problem of Formal Logic”⁠, Frank P. Ramsey (backlinks)

• 1928-carslaw.pdf: ⁠, (1928-10-01; backlinks):

This essay-review of Jeffreys’ very welcome and valuable Tract with the above title has been written at the editor’s request. Many readers of the Gazette must have heard of Heaviside’s operational method of solving the equations of dynamics and mathematical physics. If they have tried to learn about them from Heaviside’s own works, they have attempted a difficult task. Nothing more obscure than his mathematical writings is known to me. A Cambridge Tract is now at their disposal. Prom it much may be learned; but the air of mystery still—at least in part—remains.

• 1926-ramsey.pdf: “The Foundations of Mathematics”⁠, Frank P. Ramsey (backlinks)

• 1892-heaviside.pdf: “On Operators in Physical Mathematics. Part I”⁠, Oliver Heaviside

• 1980-euler-rationalmechanicsflexibleelasticbodies16381788.pdf

• 1956-wiener-iamamathematician.pdf

• 1953-wiener-exprodigymychildhoodyouth.pdf