385 research outputs found
New Symbolic Tools for Differential Geometry, Gravitation, and Field Theory
DifferentialGeometry is a Maple software package which symbolically performs
fundamental operations of calculus on manifolds, differential geometry, tensor
calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the
variational calculus. These capabilities, combined with dramatic recent
improvements in symbolic approaches to solving algebraic and differential
equations, have allowed for development of powerful new tools for solving
research problems in gravitation and field theory. The purpose of this paper is
to describe some of these new tools and present some advanced applications
involving: Killing vector fields and isometry groups, Killing tensors and other
tensorial invariants, algebraic classification of curvature, and symmetry
reduction of field equations.Comment: 42 page
Associated consistency and values for TU games
In the framework of the solution theory for cooperative transferable utility games, Hamiache axiomatized the well-known Shapley value as the unique one-point solution verifying the inessential game property, continuity, and associated consistency. The purpose of this paper is to extend Hamiache's axiomatization to the class of efficient, symmetric, and linear values, of which the Shapley value is the most important representative. For this enlarged class of values, explicit relationships to the Shapley value are exploited in order to axiomatize such values with reference to a slightly adapted inessential game property, continuity, and a similar associated consistency. The latter axiom requires that the solutions of the initial game and its associated game (with the same player set, but a different characteristic function) coincide
Minimal Proof Search for Modal Logic K Model Checking
Most modal logics such as S5, LTL, or ATL are extensions of Modal Logic K.
While the model checking problems for LTL and to a lesser extent ATL have been
very active research areas for the past decades, the model checking problem for
the more basic Multi-agent Modal Logic K (MMLK) has important applications as a
formal framework for perfect information multi-player games on its own.
We present Minimal Proof Search (MPS), an effort number based algorithm
solving the model checking problem for MMLK. We prove two important properties
for MPS beyond its correctness. The (dis)proof exhibited by MPS is of minimal
cost for a general definition of cost, and MPS is an optimal algorithm for
finding (dis)proofs of minimal cost. Optimality means that any comparable
algorithm either needs to explore a bigger or equal state space than MPS, or is
not guaranteed to find a (dis)proof of minimal cost on every input.
As such, our work relates to A* and AO* in heuristic search, to Proof Number
Search and DFPN+ in two-player games, and to counterexample minimization in
software model checking.Comment: Extended version of the JELIA 2012 paper with the same titl
Automaton theory approach for solving modified PNS problems
In this paper a modified version of the Process Network Synthesis (PNS) problem is studied. By using an automaton theoretical approach, a procedure for finding an optimal solution of this modified PNS problem is presented
Theory of Initialization-Free Decoherence-Free Subspaces and Subsystems
We introduce a generalized theory of decoherence-free subspaces and
subsystems (DFSs), which do not require accurate initialization. We derive a
new set of conditions for the existence of DFSs within this generalized
framework. By relaxing the initialization requirement we show that a DFS can
tolerate arbitrarily large preparation errors. This has potentially significant
implications for experiments involving DFSs, in particular for the experimental
implementation, over DFSs, of the large class of quantum algorithms which can
function with arbitrary input states
How to assign volunteers to tasks compatibly ? A graph theoretic and parameterized approach
In this paper we study a resource allocation problem that encodes correlation
between items in terms of \conflict and maximizes the minimum utility of the
agents under a conflict free allocation. Admittedly, the problem is
computationally hard even under stringent restrictions because it encodes a
variant of the {\sc Maximum Weight Independent Set} problem which is one of the
canonical hard problems in both classical and parameterized complexity.
Recently, this subject was explored by Chiarelli et al.~[Algorithmica'22] from
the classical complexity perspective to draw the boundary between {\sf
NP}-hardness and tractability for a constant number of agents. The problem was
shown to be hard even for small constant number of agents and various other
restrictions on the underlying graph. Notwithstanding this computational
barrier, we notice that there are several parameters that are worth studying:
number of agents, number of items, combinatorial structure that defines the
conflict among the items, all of which could well be small under specific
circumstancs. Our search rules out several parameters (even when taken
together) and takes us towards a characterization of families of input
instances that are amenable to polynomial time algorithms when the parameters
are constant. In addition to this we give a superior 2^{m}|I|^{\Co{O}(1)}
algorithm for our problem where denotes the number of items that
significantly beats the exhaustive \Oh(m^{m}) algorithm by cleverly using
ideas from FFT based fast polynomial multiplication; and we identify simple
graph classes relevant to our problem's motivation that admit efficient
algorithms
- …