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This is an overview (in french) of the Theory of Species for a general audience. Basic notions are introduced in a non too technical manner, with an explanation of why should one approach the notion of discrete structures in this particular way.
We emphasize the dominance in the Monty Hall problem, both in the classical scenario and its multi-door generalization. This is used to show optimality of the class of always-switching strategies for nonuniform allocation of the prize and arbitrary door-revealing mechanism in the event of match.
In this lecture notes we present the equations and the physics involved in the dynamic of incompressible fluids. We present the mathematical techniques needed in order to prove the existence and uniqueness result for the case where we consider Burgers equation. We also explain an useful numerical method when dealing with this kind of equations. These lecture notes were written for the 2010 JAE-Intro Summer School. This Summer School was organized by ICMAT-CSIC and takes place in Madrid.
Consider a periodical (in two independent directions) tiling of the plane with polygons (faces). In this article we shall only give examples using squares, regular hexagons, equilateral triangles and parallelograms ("unions" of two equilateral triangles). We shall call some "multiple" of the fundamental region "the board". We naturally identify pairs of corresponding edges of the the board. Figures 9 and 19-29, in this article, show different boards. The "border" of the board is represented by a yellow thick line, unless part of it or all of it is the edge of a face. The board is tiled by a finite number of polygons. Construct polygonal plates in the same number, shape and size as the polygons of the board. Adjacent to each side of each plate draw a number, or two numbers, like it is shown in Figures 1 and 18-29. Figure 1 shows the obvious possibility of having plates with simple drawings, coloured drawings, etc. Now the game is to put the plates over the board polygons in such a way that the numbers near each board edge are equal. If there is at least one solution of this puzzle one says that we have a periodical plane puzzle with numbers. These puzzles are a tool in teaching and learning mathematics. For those that already have some mathematical knowledge, they are a source for many examples and exercises, that go from the elementary to complex ones, in combinatorics, group theory (including symmetry and permutation groups), programming, and so on. The object of this work is to point out some possibilities by giving simple examples. The computer is the only practical way of "materializing" infinite periodical plane puzzles. Hence, these puzzles can be very well put into practice as computer games. This article follows some others on puzzles with numbers.
Annotated parallel text in Latin and English of the classic paper of Adam Adamandy Kochański "Observationes Cyclometricae", published in Acta Eruditorum 4:394--398 (1685), in which he presented approximate geometric method of rectification of the circle, as well as a series of rational approximations of π. The included Latin text of Observationes closely follows the original text published in Acta Eruditorum. Punctuation, capitalization, and mathematical notation have been preserved. Several misprints which appeared in the original are also reproduced unchanged, but with a footnote indicating correction. Every effort has been made to preserve the layout of original tables. The translation is as faithful as possible, often literal, and it is mainly intended to be of help to those who wish to study the original Latin text.
The year 2017 marks the 80th anniversary of Witt's famous paper containing key results, including the Witt cancellation theorem, which form the foundation for the algebraic theory of quadratic forms. We pay homage to this paper by presenting a transparent and algebraic proof of the Witt cancellation theorem, which itself is based on a cancellation. We also present an overview of some recent spectacular work which is still building on Witt's original creation of the algebraic theory of quadratic forms.
Incorporating designs into the tiles that form tessellations presents an interesting challenge for artists. Creating a viable MC Escher like image that works esthetically as well as functionally requires resolving incongruencies at a tile's edge while constrained by its shape. Escher was the most well known practitioner in this style of mathematical visualization, but there are significant mathematical shapes to which he never applied his artistry. These shapes can incorporate designs that form images as appealing as those produced by Escher, and our paper explores this for traditional tessellations, Penrose Tilings, fractals, and fractal/tessellation combinations. To illustrate the versatility of tiling art, images were created with multiple figures and negative space leading to patterns distinct from the work of others.
Expository notes on Clifford algebras and spinors with a detailed discussion of Majorana, Weyl, and Dirac spinors. The paper is meant as a review of background material, needed, in particular, in now fashionable theoretical speculations on neutrino masses. It has a more mathematical flavour than the over twenty-seven-year-old "Introduction to Majorana masses" by P.D. Mannheim and includes historical notes and biographical data on past participants in the story.
Given a sequence converging to zero, we consider the set of numbers which are sums of (infinite, finite, or empty) subsequences. When the original sequence is not absolutely summable, the subsum set is an unbounded closed interval which includes zero. When it is absolutely summable the subsum set is one of the following: a finite union of (nontrivial) compact intervals, a Cantor set, or a "symmetric Cantorval".
In version v1 (under a different title) I was trying to give a new proof of Wedderburn's Little Theorem (WLT), stating that a finite dision ring is commutative, but I failed. So I had to withdraw the paper (version v2). Firstly I became aware of a new theorem. So in the mean time in version v3 under a new title (see top of this page) I proved a useful property of the finite nonabelian groups stating that one of its maximally abelian subgroups is not an eigenheimer. In the present version v4 I prove finally WLT in a new section 2 entitled: A New Proof of Wedderburn's Little Theorem: A Finite Division Ring is Commutative.
This is an introduction (in German) to projective geometry by the late Heinz Lueneburg. Projective spaces are treated as lattices with particular properties, and finite geometries receive special attention. The final chapters deal with Segre and Grassmann manifolds.
Kramers' law describes the mean transition time of an overdamped Brownian particle between local minima in a potential landscape. We review different approaches that have been followed to obtain a mathematically rigorous proof of this formula. We also discuss some generalisations, and a case in which Kramers' law is not valid. This review is written for both mathematicians and theoretical physicists, and endeavours to link concepts and terminology from both fields.
The basic Monty Hall problem is explored to introduce into the fundamental concepts of the game theory and to give a complete Bayesian and a (noncooperative) game-theoretic analysis of the situation. Simple combinatorial arguments are used to exclude the holding action and to find minimax solutions.
In the famous Three-Door-Game Monte cannot help to win all the time by signaling location of the prize, using only the freedom he allowed to use. No matter which strategies played, there is always at least one door where the prize will not be found. However, already in the game with four doors cooperative Monte can reveal two useless doors in sequence (leaving two doors unrevealed), to inform Conie about location of the prize, so enabling her to beat the only-switching strategies and win all the time.
The cumulative hierarchy conception of set, which is based on the conception that sets are inductively generated from "former" sets, is generally considered a good way to create a set conception that seems safe from contradictions. This imposes two restrictions on sets. One is a "limitation of size," and the other is the rejection of non-well-founded sets. Quine's NF system of axioms, does not have any of the two restrictions, but it has a formal restriction on allowed formulas in its comprehension axiom schema, which reflects a similar notion of elements being prior to sets. Here we suggest that a possible reason for set antinomies is the tension between our perception of sets as entities formed from elements by an imaginary aggregation operator, and our wish to regard sets as existing "at once." A new approach to sets as totalities is presented based on a notion of "concurrent aggregation," which instead of avoiding "viscous circles," acknowledges the inherent circularities of some predicates, and provides a way to characterize and investigate these circularities.
The complete list of pairs of non-isomorphic finite simple groups having the same order is well-known. In particular for p>3, PSL_2(Z/p) is the "only" simple group of order (p^3-p)/2. It's less well-known that Frobenius proved this uniqueness result in 1902. This note presents a version of Frobenius' argument that might be used in an undergraduate honors algebra course. It also includes a short modern proof, aimed at the same audience, of the much earlier result that PSL_2(Z/p) is simple for p>3; a result stated by Galois in 1832.
Sparsity, Structure, Scaling and Stability in Computational Linear Algebra - Textbook from the IX School of Computer Science, held on July 24-31 of 1994 at Recife, Brazil. Esparsidade, Estrutura, Escalamento e Estabilidade em Algebra Linear Computacional - Livro texto da IX Escola de Computacao, realizada nos dias 24 a 31 de Julho de 1994 em Recife, Brasil. This textbook is written in Portuguese Language.
I have recently proposed that an object, found in an Egyptian tomb and exposed at the Egyptian Museum of Torino, could be a protractor. The tomb was that of architect Kha, supervisor at Deir El-Medina during the 18th Dynasty, and his wife Merit. Considering then the object as a protractor, the architect could have used it to measure the angle of the inclined planes.