17,068 research outputs found
The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory
The Jacobian conjecture is an old unsolved problem in mathematics, which has
been unsuccessfully attacked from many different angles. We add here another
point of view pertaining to the so called formal inverse approach, that of
perturbative quantum field theory.Comment: 22 pages, 13 diagram
Solving Hard Control Problems in Voting Systems via Integer Programming
Voting problems are central in the area of social choice. In this article, we
investigate various voting systems and types of control of elections. We
present integer linear programming (ILP) formulations for a wide range of
NP-hard control problems. Our ILP formulations are flexible in the sense that
they can work with an arbitrary number of candidates and voters. Using the
off-the-shelf solver Cplex, we show that our approaches can manipulate
elections with a large number of voters and candidates efficiently
Mass problems and intuitionistic higher-order logic
In this paper we study a model of intuitionistic higher-order logic which we
call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the
category of sheaves of sets over the topological space consisting of the Turing
degrees, where the Turing cones form a base for the topology. We note that our
Muchnik topos interpretation of intuitionistic mathematics is an extension of
the well known Kolmogorov/Muchnik interpretation of intuitionistic
propositional calculus via Muchnik degrees, i.e., mass problems under weak
reducibility. We introduce a new sheaf representation of the intuitionistic
real numbers, \emph{the Muchnik reals}, which are different from the Cauchy
reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice
principle} and a \emph{bounding principle} where range over Muchnik
reals, ranges over functions from Muchnik reals to Muchnik reals, and
is a formula not containing or . For the convenience of the
reader, we explain all of the essential background material on intuitionism,
sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems,
Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an
English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page
Complexity vs Energy: Theory of Computation and Theoretical Physics
This paper is a survey dedicated to the analogy between the notions of {\it
complexity} in theoretical computer science and {\it energy} in physics. This
analogy is not metaphorical: I describe three precise mathematical contexts,
suggested recently, in which mathematics related to (un)computability is
inspired by and to a degree reproduces formalisms of statistical physics and
quantum field theory.Comment: 23 pages. Talk at the satellite conference to ECM 2012, "QQQ Algebra,
Geometry, Information", Tallinn, July 9-12, 201
Choreographies with Secure Boxes and Compromised Principals
We equip choreography-level session descriptions with a simple abstraction of
a security infrastructure. Message components may be enclosed within (possibly
nested) "boxes" annotated with the intended source and destination of those
components. The boxes are to be implemented with cryptography. Strand spaces
provide a semantics for these choreographies, in which some roles may be played
by compromised principals. A skeleton is a partially ordered structure
containing local behaviors (strands) executed by regular (non-compromised)
principals. A skeleton is realized if it contains enough regular strands so
that it could actually occur, in combination with any possible activity of
compromised principals. It is delivery guaranteed (DG) realized if, in
addition, every message transmitted to a regular participant is also delivered.
We define a novel transition system on skeletons, in which the steps add
regular strands. These steps solve tests, i.e. parts of the skeleton that could
not occur without additional regular behavior. We prove three main results
about the transition system. First, each minimal DG realized skeleton is
reachable, using the transition system, from any skeleton it embeds. Second, if
no step is possible from a skeleton A, then A is DG realized. Finally, if a DG
realized B is accessible from A, then B is minimal. Thus, the transition system
provides a systematic way to construct the possible behaviors of the
choreography, in the presence of compromised principals
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