570 research outputs found
A Unifying Theory for Graph Transformation
The field of graph transformation studies the rule-based transformation of graphs. An important branch is the algebraic graph transformation tradition, in which approaches are defined and studied using the language of category theory. Most algebraic graph transformation approaches (such as DPO, SPO, SqPO, and AGREE) are opinionated about the local contexts that are allowed around matches for rules, and about how replacement in context should work exactly. The approaches also differ considerably in their underlying formal theories and their general expressiveness (e.g., not all frameworks allow duplication). This dissertation proposes an expressive algebraic graph transformation approach, called PBPO+, which is an adaptation of PBPO by Corradini et al. The central contribution is a proof that PBPO+ subsumes (under mild restrictions) DPO, SqPO, AGREE, and PBPO in the important categorical setting of quasitoposes. This result allows for a more unified study of graph transformation metatheory, methods, and tools. A concrete example of this is found in the second major contribution of this dissertation: a graph transformation termination method for PBPO+, based on decreasing interpretations, and defined for general categories. By applying the proposed encodings into PBPO+, this method can also be applied for DPO, SqPO, AGREE, and PBPO
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Measuring the impact of COVID-19 on hospital care pathways
Care pathways in hospitals around the world reported significant disruption during the recent COVID-19 pandemic but measuring the actual impact is more problematic. Process mining can be useful for hospital management to measure the conformance of real-life care to what might be considered normal operations. In this study, we aim to demonstrate that process mining can be used to investigate process changes associated with complex disruptive events. We studied perturbations to accident and emergency (A &E) and maternity pathways in a UK public hospital during the COVID-19 pandemic. Co-incidentally the hospital had implemented a Command Centre approach for patient-flow management affording an opportunity to study both the planned improvement and the disruption due to the pandemic. Our study proposes and demonstrates a method for measuring and investigating the impact of such planned and unplanned disruptions affecting hospital care pathways. We found that during the pandemic, both A &E and maternity pathways had measurable reductions in the mean length of stay and a measurable drop in the percentage of pathways conforming to normative models. There were no distinctive patterns of monthly mean values of length of stay nor conformance throughout the phases of the installation of the hospital’s new Command Centre approach. Due to a deficit in the available A &E data, the findings for A &E pathways could not be interpreted
On linear, fractional, and submodular optimization
In this thesis, we study four fundamental problems in the theory of optimization. 1. In fractional optimization, we are interested in minimizing a ratio of two functions over some domain. A well-known technique for solving this problem is the Newton– Dinkelbach method. We propose an accelerated version of this classical method and give a new analysis using the Bregman divergence. We show how it leads to improved or simplified results in three application areas. 2. The diameter of a polyhedron is the maximum length of a shortest path between any two vertices. The circuit diameter is a relaxation of this notion, whereby shortest paths are not restricted to edges of the polyhedron. For a polyhedron in standard equality form with constraint matrix A, we prove an upper bound on the circuit diameter that is quadratic in the rank of A and logarithmic in the circuit imbalance measure of A. We also give circuit augmentation algorithms for linear programming with similar iteration complexity. 3. The correlation gap of a set function is the ratio between its multilinear and concave extensions. We present improved lower bounds on the correlation gap of a matroid rank function, parametrized by the rank and girth of the matroid. We also prove that for a weighted matroid rank function, the worst correlation gap is achieved with uniform weights. Such improved lower bounds have direct applications in submodular maximization and mechanism design. 4. The last part of this thesis concerns parity games, a problem intimately related to linear programming. A parity game is an infinite-duration game between two players on a graph. The problem of deciding the winner lies in NP and co-NP, with no known polynomial algorithm to date. Many of the fastest (quasi-polynomial) algorithms have been unified via the concept of a universal tree. We propose a strategy iteration framework which can be applied on any universal tree
Left Regular Bands of Groups and the Mantaci--Reutenauer algebra
We develop the idempotent theory for algebras over a class of semigroups
called left regular bands of groups (LRBGs), which simultaneously generalize
group algebras of finite groups and left regular band (LRB) algebras. Our
techniques weave together the representation theory of finite groups and LRBs,
opening the door for a systematic study of LRBGs in an analogous way to LRBs.
We apply our results to construct complete systems of primitive orthogonal
idempotents in the Mantaci--Reutenauer algebra associated to
any finite group . When is abelian, we give closed form expressions for
these idempotents, and when is the cyclic group of order two, we prove that
these recover idempotents introduced by Vazirani
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Symmetries and intrinsic vs. extrinsic properties of
We consider the following question: How much of the combinatorial structure
determining properties of is ``intrinsic'' and
how much new information do we obtain from using properties specific to this
space? Our approach is to study the effect of the -action. Apart from
being a natural action to consider, it is known that this action does not
extend to other wonderful compactifications associated to the
hyperplane arrangement. We find the differences in intersection patterns of
faces on associahedra and permutohedra which characterize the failure to extend
to other compactifications and show that this is reflected by most terms of
degree of the cohomology/Chow ring.
Even from a combinatorial perspective, terms of degree 1 are more naturally
related to geometric properties. In particular, imposing -invariance
implies that many of the log concave sequences obtained from degree 1
Hodge--Riemann relations (and all of them for ) on the Chow ring of
can be restricted to those with a special
recursive structure. A conjectural result implies that this is true for all
. Elements of these sequences can be expressed as polynomials in quantum
Littlewood--Richardson coefficients multiplied by terms such as partition
components, factorials, and multinomial coefficients. After dividing by
binomial coefficients, polynomials with these numbers as coefficients can be
interepreted in terms of volumes or resultants. Finally, we find a connection
between the geometry of and higher degree
Hodge--Riemann relations of other rings via Toeplitz matrices.Comment: 19 pages, Comments welcome
Crystals for shifted key polynomials
This article continues our study of - and -key polynomials, which are
(non-symmetric) "partial" Schur - and -functions as well as "shifted"
versions of key polynomials. Our main results provide a crystal interpretation
of - and -key polynomials, namely, as the characters of certain connected
subcrystals of normal crystals associated to the queer Lie superalgebra
. In the -key case, the ambient normal crystals are the
-crystals studied by Grantcharov et al., while in the -key
case, these are replaced by the extended -crystals recently
introduced by the first author and Tong. Using these constructions, we propose
a crystal-theoretic lift of several conjectures about the decomposition of
involution Schubert polynomials into - and -key polynomials. We verify
these generalized conjectures in a few special cases. Along the way, we
establish some miscellaneous results about normal -crystals and
Demazure -crystals.Comment: 60 pages, 6 figure
Local infrared safety in time-ordered perturbation theory
We develop a general expression for weighted cross sections in leptonic
annihilation to hadrons based on time-ordered perturbation theory (TOPT). The
analytic behavior of the resulting integrals over spatial momenta can be
analyzed in the language of Landau equations and infrared (IR) power counting.
For any infrared-safe weight, the cancellation of infrared divergences is
implemented locally at the integrand level, and in principle can be evaluated
numerically in four dimensions. We go on to show that it is possible to
eliminate unphysical singularities that appear in time-ordered perturbation
theory for arbitrary amplitudes. This is done by reorganizing TOPT into an
equivalent form that combines classes of time orderings into a ``partially
time-ordered perturbation theory". Applying the formalism to leptonic
annihilation, we show how to derive diagrammatic expressions with only physical
unitarity cuts.Comment: 60 pages, 9 figure
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