3,370 research outputs found
Nagata type statements
Nagata solved Hilbert's 14-th problem in 1958 in the negative. The solution
naturally lead him to a tantalizing conjecture that remains widely open after
more than half a century of intense efforts. Using Nagata's theorem as starting
point, and the conjecture, with its multiple variations, as motivation, we
explore the important questions of finite generation for invariant rings, for
support semigroups of multigraded algebras, and for Mori cones of divisors on
blown up surfaces, and the rationality of Waldschimdt constants. Finally we
suggest a connection between the Mori cone of the Zariski-Riemann space and the
continuity of the Waldschmidt constant as a function on the space of
valuations.Comment: 45 pages. These notes correspond to the course of the same title
given by the first author in the workshop "Asymptotic invariants attached to
linear series" held in the Pedagogical University of Cracow from May 16 to
20, 201
A topos for algebraic quantum theory
The aim of this paper is to relate algebraic quantum mechanics to topos
theory, so as to construct new foundations for quantum logic and quantum
spaces. Motivated by Bohr's idea that the empirical content of quantum physics
is accessible only through classical physics, we show how a C*-algebra of
observables A induces a topos T(A) in which the amalgamation of all of its
commutative subalgebras comprises a single commutative C*-algebra. According to
the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter
has an internal spectrum S(A) in T(A), which in our approach plays the role of
a quantum phase space of the system. Thus we associate a locale (which is the
topos-theoretical notion of a space and which intrinsically carries the
intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which
is the noncommutative notion of a space). In this setting, states on A become
probability measures (more precisely, valuations) on S(A), and self-adjoint
elements of A define continuous functions (more precisely, locale maps) from
S(A) to Scott's interval domain. Noting that open subsets of S(A) correspond to
propositions about the system, the pairing map that assigns a (generalized)
truth value to a state and a proposition assumes an extremely simple
categorical form. Formulated in this way, the quantum theory defined by A is
essentially turned into a classical theory, internal to the topos T(A).Comment: 52 pages, final version, to appear in Communications in Mathematical
Physic
Information Physics: The New Frontier
At this point in time, two major areas of physics, statistical mechanics and
quantum mechanics, rest on the foundations of probability and entropy. The last
century saw several significant fundamental advances in our understanding of
the process of inference, which make it clear that these are inferential
theories. That is, rather than being a description of the behavior of the
universe, these theories describe how observers can make optimal predictions
about the universe. In such a picture, information plays a critical role. What
is more is that little clues, such as the fact that black holes have entropy,
continue to suggest that information is fundamental to physics in general.
In the last decade, our fundamental understanding of probability theory has
led to a Bayesian revolution. In addition, we have come to recognize that the
foundations go far deeper and that Cox's approach of generalizing a Boolean
algebra to a probability calculus is the first specific example of the more
fundamental idea of assigning valuations to partially-ordered sets. By
considering this as a natural way to introduce quantification to the more
fundamental notion of ordering, one obtains an entirely new way of deriving
physical laws. I will introduce this new way of thinking by demonstrating how
one can quantify partially-ordered sets and, in the process, derive physical
laws. The implication is that physical law does not reflect the order in the
universe, instead it is derived from the order imposed by our description of
the universe. Information physics, which is based on understanding the ways in
which we both quantify and process information about the world around us, is a
fundamentally new approach to science.Comment: 17 pages, 6 figures. Knuth K.H. 2010. Information physics: The new
frontier. J.-F. Bercher, P. Bessi\`ere, and A. Mohammad-Djafari (eds.)
Bayesian Inference and Maximum Entropy Methods in Science and Engineering
(MaxEnt 2010), Chamonix, France, July 201
Ehrenfeucht-Fraisse Games on Omega-Terms
Fragments of first-order logic over words can often be characterized in terms
of finite monoids or finite semigroups. Usually these algebraic descriptions
yield decidability of the question whether a given regular language is
definable in a particular fragment. An effective algebraic characterization can
be obtained from identities of so-called omega-terms. In order to show that a
given fragment satisfies some identity of omega-terms, one can use
Ehrenfeucht-Fraisse games on word instances of the omega-terms. The resulting
proofs often require a significant amount of book-keeping with respect to the
constants involved. In this paper we introduce Ehrenfeucht-Fraisse games on
omega-terms. To this end we assign a labeled linear order to every omega-term.
Our main theorem shows that a given fragment satisfies some identity of
omega-terms if and only if Duplicator has a winning strategy for the game on
the resulting linear orders. This allows to avoid the book-keeping. As an
application of our main result, we show that one can decide in exponential time
whether all aperiodic monoids satisfy some given identity of omega-terms,
thereby improving a result of McCammond (Int. J. Algebra Comput., 2001)
Reified valuations and adic spectra
We revisit Huber's theory of continuous valuations, which give rise to the
adic spectra used in his theory of adic spaces. We instead consider valuations
which have been reified, i.e., whose value groups have been forced to contain
the real numbers. This yields reified adic spectra which provide a framework
for an analogue of Huber's theory compatible with Berkovich's construction of
nonarchimedean analytic spaces. As an example, we extend the theory of
perfectoid spaces to this setting.Comment: v5: refereed versio
Information-Based Physics: An Observer-Centric Foundation
It is generally believed that physical laws, reflecting an inherent order in
the universe, are ordained by nature. However, in modern physics the observer
plays a central role raising questions about how an observer-centric physics
can result in laws apparently worthy of a universal nature-centric physics.
Over the last decade, we have found that the consistent apt quantification of
algebraic and order-theoretic structures results in calculi that possess
constraint equations taking the form of what are often considered to be
physical laws. I review recent derivations of the formal relations among
relevant variables central to special relativity, probability theory and
quantum mechanics in this context by considering a problem where two observers
form consistent descriptions of and make optimal inferences about a free
particle that simply influences them. I show that this approach to describing
such a particle based only on available information leads to the mathematics of
relativistic quantum mechanics as well as a description of a free particle that
reproduces many of the basic properties of a fermion. The result is an approach
to foundational physics where laws derive from both consistent descriptions and
optimal information-based inferences made by embedded observers.Comment: To be published in Contemporary Physics. The manuscript consists of
43 pages and 9 Figure
Integrals and Valuations
We construct a homeomorphism between the compact regular locale of integrals
on a Riesz space and the locale of (valuations) on its spectrum. In fact, we
construct two geometric theories and show that they are biinterpretable. The
constructions are elementary and tightly connected to the Riesz space
structure.Comment: Submitted for publication 15/05/0
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