Artículos con la etiqueta ‘matemáticas’

The changing concept of matter in H. Weyl’s thought, 1918 -1930

Por • 3 abr, 2014 • Category: Educacion

During the “long decade” of transformation of mathematical physics between 1915 and 1930, H. Weyl interacted with physics in two highly productive phases and contributed to it, among others, by his widely read book on Space – Time – Matter (Raum- Zeit – Materie), (1918-1923) and on Group Theory and Quantum Mechanics (Gruppentheorie und Quantenmechanik) (1928-1931). In this time Weyl’s understanding of the constitution of matter and its mathematical description changed considerably. At the beginning of the period he started from a «dynamistic» and geometrical conception of matter, following and extending the Mie-Hilbert approch, which he gave up during the year 1920. After transitional experiments with a singularity (and in this sense topological) approach in 1921/22, he developed an open perspective of what he called an “agency theory” of matter. The idea for it was formulated already before the advent of the “new” quantum mechanics in 1925/26.

Restructuring Logic

Por • 12 mar, 2014 • Category: Filosofía

The outline of a programme for restructuring mathematical logic. We explain what we mean by «restructuring» and carry out exemplary parts of the programme

Topos Semantics for Higher-Order Modal Logic

Por • 6 mar, 2014 • Category: Educacion

We define the notion of a model of higher-order modal logic in an arbitrary elementary topos E . In contrast to the well-known interpretation of (non-modal) higher-order logic, the type of propositions is not interpreted by the subobject classifier ΩE , but rather by a suitable complete Heyting algebra H . The canonical map relating H and ΩE both serves to interpret equality and provides a modal operator on H in the form of a comonad. Examples of such structures arise from surjective geometric morphisms f:F→E , where H=f∗ΩF . The logic differs from non-modal higher-order logic in that the principles of functional and propositional extensionality are no longer valid but may be replaced by modalized versions. The usual Kripke, neighborhood, and sheaf semantics for propositional and first-order modal logic are subsumed by this notion.

Trends in Computer Network Modeling Towards the Future Internet

Por • 6 mar, 2014 • Category: Ciencia y tecnología

This article provides a taxonomy of current and past network modeling efforts. In all these efforts over the last few years we see a trend towards not only describing the network, but connected devices as well. This is especially current given the many Future Internet projects, which are combining different models, and resources in order to provide complete virtual infrastructures to users. An important mechanism for managing complexity is the creation of an abstract model, a step which has been undertaken in computer networks too. The fact that more and more devices are network capable, coupled with increasing popularity of the Internet, has made computer networks an important focus area for modeling. The large number of connected devices creates an increasing complexity which must be harnessed to keep the networks functioning.

The axiomatic deduction of the quadratic Hencky strain energy by Heinrich Hencky

Por • 23 feb, 2014 • Category: Crítica

The introduction of the quadratic Hencky strain energy based on the logarithmic strain tensor log V is a milestone in the development of nonlinear elasticity theory in the first half of the 20th century. Since the original manuscripts are written in German, they are not easily accessible today. However, we believe that the deductive approach taken by Hencky deserves to be rediscovered today.

Creature forcing and five cardinal characteristics of the continuum

Por • 13 feb, 2014 • Category: Opinion

We use a (countable support) creature construction to show that consistently
d=ℵ 1 =cov(NULL)

Forgotten Motives: the Varieties of Scientific Experience

Por • 13 feb, 2014 • Category: Crítica

Personal recollections about Alexandre Grothendieck and early days of his theory of motives

Generalizations of the Kolmogorov-Barzdin embedding estimates

Por • 11 feb, 2014 • Category: Ciencia y tecnología

We consider several ways to measure the `geometric complexity’ of an embedding from a simplicial complex into Euclidean space. One of these is a version of `thickness’, based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved for graphs. We also consider the distortion of knots. We give an alternate proof of a theorem of Pardon that there are isotopy classes of knots requiring arbitrarily large distortion. This proof is based on the expander-like properties of arithmetic hyperbolic manifolds.

Chasing diagrams in cryptography

Por • 1 feb, 2014 • Category: Opinion

Cryptography is a theory of secret functions. Category theory is a general theory of functions. Cryptography has reached a stage where its structures often take several pages to define, and its formulas sometimes run from page to page. Category theory has some complicated definitions as well, but one of its specialties is taming the flood of structure. Cryptography seems to be in need of high level methods, whereas category theory always needs concrete applications. So why is there no categorical cryptography? One reason may be that the foundations of modern cryptography are built from probabilistic polynomial-time Turing machines, and category theory does not have a good handle on such things. On the other hand, such foundational problems might be the very reason why cryptographic constructions often resemble low level machine programming. I present some preliminary explorations towards categorical cryptography. It turns out that some of the main security concepts are easily characterized through the categorical technique of *diagram chasing*, which was first used Lambek’s seminal `Lecture Notes on Rings and Modules’.

A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture

Por • 30 ene, 2014 • Category: Educacion

This is an extension and background to a talk I gave on 9 October 2013 to the Brown Graduate Student Seminar, called `A friendly intro to sieves with a look towards recent progress on the twin primes conjecture.’ During the talk, I mention several sieves, some with a lot of detail and some with very little detail. I also discuss several results and built upon many sources. I’ll provide missing details and/or sources for additional reading here.