24,794 research outputs found
Satisfaction, Restriction and Amalgamation of Constraints in the Framework of M-Adhesive Categories
Application conditions for rules and constraints for graphs are well-known in
the theory of graph transformation and have been extended already to M-adhesive
transformation systems. According to the literature we distinguish between two
kinds of satisfaction for constraints, called general and initial satisfaction
of constraints, where initial satisfaction is defined for constraints over an
initial object of the base category. Unfortunately, the standard definition of
general satisfaction is not compatible with negation in contrast to initial
satisfaction.
Based on the well-known restriction of objects along type morphisms, we study
in this paper restriction and amalgamation of application conditions and
constraints together with their solutions. In our main result, we show
compatibility of initial satisfaction for positive constraints with restriction
and amalgamation, while general satisfaction fails in general.
Our main result is based on the compatibility of composition via pushouts
with restriction, which is ensured by the horizontal van Kampen property in
addition to the vertical one that is generally satisfied in M-adhesive
categories.Comment: In Proceedings ACCAT 2012, arXiv:1208.430
Shape-Driven Nested Markov Tessellations
A new and rather broad class of stationary (i.e. stochastically translation
invariant) random tessellations of the -dimensional Euclidean space is
introduced, which are called shape-driven nested Markov tessellations. Locally,
these tessellations are constructed by means of a spatio-temporal random
recursive split dynamics governed by a family of Markovian split kernel,
generalizing thereby the -- by now classical -- construction of iteration
stable random tessellations. By providing an explicit global construction of
the tessellations, it is shown that under suitable assumptions on the split
kernels (shape-driven), there exists a unique time-consistent whole-space
tessellation-valued Markov process of stationary random tessellations
compatible with the given split kernels. Beside the existence and uniqueness
result, the typical cell and some aspects of the first-order geometry of these
tessellations are in the focus of our discussion
Towers of solutions of qKZ equations and their applications to loop models
Cherednik's type A quantum affine Knizhnik-Zamolodchikov (qKZ) equations form
a consistent system of linear -difference equations for -valued
meromorphic functions on a complex -torus, with a module over the
GL-type extended affine Hecke algebra . The family
of extended affine Hecke algebras forms a tower of
algebras, with the associated algebra morphisms
the Hecke algebra descends of arc
insertion at the affine braid group level. In this paper we consider qKZ towers
of solutions, which consist of twisted-symmetric
polynomial solutions () of the qKZ equations that are
compatible with the tower structure on . The
compatibility is encoded by so-called braid recursion relations:
is required to coincide up to a quasi-constant
factor with the push-forward of by an intertwiner
of -modules, where
is considered as an -module through the tower
structure on .
We associate to the dense loop model on the half-infinite cylinder with
nonzero loop weights a qKZ tower of solutions. The
solutions are constructed from specialised dual non-symmetric
Macdonald polynomials with specialised parameters using the Cherednik-Matsuo
correspondence. In the special case that the extended affine Hecke algebra
parameter is a third root of unity, coincides with the (suitably
normalized) ground state of the inhomogeneous dense loop model on the
half-infinite cylinder with circumference .Comment: 45 pages. v2: main theorem (Thm. 4.7) strengthened. v3: minor typos
corrected. To appear in Ann. Henri Poincar
From duality to determinants for q-TASEP and ASEP
We prove duality relations for two interacting particle systems: the
-deformed totally asymmetric simple exclusion process (-TASEP) and the
asymmetric simple exclusion process (ASEP). Expectations of the duality
functionals correspond to certain joint moments of particle locations or
integrated currents, respectively. Duality implies that they solve systems of
ODEs. These systems are integrable and for particular step and half-stationary
initial data we use a nested contour integral ansatz to provide explicit
formulas for the systems' solutions, and hence also the moments. We form
Laplace transform-like generating functions of these moments and via residue
calculus we compute two different types of Fredholm determinant formulas for
such generating functions. For ASEP, the first type of formula is new and
readily lends itself to asymptotic analysis (as necessary to reprove GUE
Tracy--Widom distribution fluctuations for ASEP), while the second type of
formula is recognizable as closely related to Tracy and Widom's ASEP formula
[Comm. Math. Phys. 279 (2008) 815--844, J. Stat. Phys. 132 (2008) 291--300,
Comm. Math. Phys. 290 (2009) 129--154, J. Stat. Phys. 140 (2010) 619--634]. For
-TASEP, both formulas coincide with those computed via Borodin and Corwin's
Macdonald processes [Probab. Theory Related Fields (2014) 158 225--400]. Both
-TASEP and ASEP have limit transitions to the free energy of the continuum
directed polymer, the logarithm of the solution of the stochastic heat equation
or the Hopf--Cole solution to the Kardar--Parisi--Zhang equation. Thus,
-TASEP and ASEP are integrable discretizations of these continuum objects;
the systems of ODEs associated to their dualities are deformed discrete quantum
delta Bose gases; and the procedure through which we pass from expectations of
their duality functionals to characterizing generating functions is a rigorous
version of the replica trick in physics.Comment: Published in at http://dx.doi.org/10.1214/13-AOP868 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
First steps in synthetic guarded domain theory: step-indexing in the topos of trees
We present the topos S of trees as a model of guarded recursion. We study the
internal dependently-typed higher-order logic of S and show that S models two
modal operators, on predicates and types, which serve as guards in recursive
definitions of terms, predicates, and types. In particular, we show how to
solve recursive type equations involving dependent types. We propose that the
internal logic of S provides the right setting for the synthetic construction
of abstract versions of step-indexed models of programming languages and
program logics. As an example, we show how to construct a model of a
programming language with higher-order store and recursive types entirely
inside the internal logic of S. Moreover, we give an axiomatic categorical
treatment of models of synthetic guarded domain theory and prove that, for any
complete Heyting algebra A with a well-founded basis, the topos of sheaves over
A forms a model of synthetic guarded domain theory, generalizing the results
for S
RG stability of integrable fishnet models
We address the question of perturbative consistency in the scalar fishnet
models presented by Caetano, Gurdogan and Kazakov\cite{Gurdogan:2015csr,
Caetano:2016ydc}. We argue that their 3-dimensional fishnet model
becomes perturbatively stable under renormalization in the large limit, in
contrast to what happens in their 4-dimensional fishnet model, in
which double trace terms are known to be generated by the RG flow. We point out
that there is a direct way to modify this second theory that protects it from
such corrections. Additionally, we observe that the 6-dimensional
Lagrangian that spans an hexagonal integrable scalar fishnet is consistent at
the perturbative level as well. The nontriviality and simplicity of this last
model is illustrated by computing the anomalous dimensions of its
operators to all perturbative orders.Comment: 25 pages, 15 figures. V2: Added references, minor correction
Linear nested Artin approximation theorem for algebraic power series
We give a new and elementary proof of the nested Artin approximation Theorem
for linear equations with algebraic power series coefficients. Moreover, for
any Noetherian local subring of the ring of formal power series, we clarify the
relationship between this theorem and the problem of the com-mutation of two
operations for ideals: the operation of replacing an ideal by its completion
and the operation of replacing an ideal by one of its elimination ideals.Comment: Last version. To appear in Manuscripta Mat
Generating Bijections between HOAS and the Natural Numbers
A provably correct bijection between higher-order abstract syntax (HOAS) and
the natural numbers enables one to define a "not equals" relationship between
terms and also to have an adequate encoding of sets of terms, and maps from one
term family to another. Sets and maps are useful in many situations and are
preferably provided in a library of some sort. I have released a map and set
library for use with Twelf which can be used with any type for which a
bijection to the natural numbers exists.
Since creating such bijections is tedious and error-prone, I have created a
"bijection generator" that generates such bijections automatically together
with proofs of correctness, all in the context of Twelf.Comment: In Proceedings LFMTP 2010, arXiv:1009.218
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