Solvable Cases Research Articles

Frictionless contact problems for elastic hemitropic solids: Boundary variational inequality approach

The frictionless contact problems for two interacting hemitropic solids with different elastic properties is investigated under the condition of natural impenetrability of one medium into the other. We consider two cases, the so-called coercive case (when elastic media are fixed along some parts of their boundaries), and the semicoercive case (the boundaries of the interacting elastic media are not fixed). Using the potential theory we reduce the problems to the boundary variational inequalities and analyse the existence and uniqueness of weak solutions. In the semicoercive case, the necessary and sufficient conditions of solvability of the corresponding contact problems are written out explicitly.

Deductive inference for the interiors and exteriors of horn theories

In this article, we investigate deductive inference for interiors and exteriors of Horn knowledge bases, where interiors and exteriors were introduced by Makino and Ibaraki [1996] to study stability properties of knowledge bases. We present a linear time algorithm for deduction for interiors and show that deduction is coNP-complete for exteriors. Under model-based representation, we show that the deduction problem for interiors is NP-complete while the one for exteriors is coNP-complete. As for Horn envelopes of exteriors, we show that it is linearly solvable under model-based representation, while it is coNP-complete under formula-based representation. We also discuss polynomially solvable cases for all the intractable problems.

Computing vertex-surjective homomorphisms to partially reflexive trees

A homomorphism from a graph G to a graph H is a vertex mapping f:VG→VH such that f(u) and f(v) form an edge in H whenever u and v form an edge in G. The H-Coloring problem is that of testing whether a graph G allows a homomorphism to a given graph H. A well-known result of Hell and Nešetřil determines the computational complexity of this problem for any fixed graph H. We study a natural variant of this problem, namely the SurjectiveH-Coloring problem, which is that of testing whether a graph G allows a homomorphism to a graph H that is (vertex-)surjective. We classify the computational complexity of this problem for when H is any fixed partially reflexive tree. Thus we identify the first class of target graphs H for which the computational complexity of SurjectiveH-Coloring can be determined. For the polynomial-time solvable cases we show a number of parameterized complexity results, including in particular ones on graph classes with (locally) bounded expansion.

Quasi-exactly Solvable Cases of the N-Dimensional Symmetric Quartic Anharmonic Oscillator

The O(N) invariant quartic anharmonic oscillator is shown to be exactly solvable if the interaction parameter satisfies special conditions. The problem is directly related to that of a quantum double well anharmonic oscillator in an external field. A finite dimensional matrix equation for the problem is constructed explicitly, along with analytical expressions for some excited states in the system. The corresponding Niven equations for determining the polynomial solutions for the problem are given.

Following states in temperature in the spherical s + p-spin glass model

In many mean-field glassy systems, the low-temperature Gibbs measure is dominated by exponentially many metastable states. We analyze the evolution of the metastable states as temperature changes adiabatically in the solvable case of the spherical s + p-spin glass model, extending the work of Barrat et al (1997 J. Phys. A: Math. Gen. 30 5593). We confirm the presence of level crossings, bifurcations, and temperature chaos. For the states that are at equilibrium close to the so-called dynamical temperature Td, we find, however, that the following state method (and the dynamical solution of the model as well) is intrinsically limited by the vanishing of solutions with non-zero overlap at low temperature.

On the routing open shop problem with two machines on a two-vertex network

The open shop problem is under study with two machines and routing on a two-vertex network. This problem is NP-hard. We introduce an exact pseudopolynomial algorithm, propose a fully polynomial time approximation scheme for solving this problem, and consider some particular polynomially solvable cases.

Entropy production and fluctuation theorems under feedback control: the molecularrefrigerator model revisited

We revisit the model of a Brownian particle in a heat bath submitted to an activelycontrolled force proportional to the velocity that leads to thermal noise reduction (colddamping). We investigate the influence of the continuous feedback on the fluctuations ofthe total entropy production and show that the explicit expression of the detailedfluctuation theorem involves different dynamics and observables in the forward andbackward processes. As an illustration, we study the analytically solvable case of aharmonic oscillator and calculate the characteristic function of the entropy production in anonequilibrium steady state. We then determine the corresponding large deviationfunction which results from an unusual interplay between ‘boundary’ and ‘bulk’contributions.

A polynomially solvable case of a single machine scheduling problem when the maximal job processing time is a constant

We consider the problem of scheduling jobs with release times and due dates on a single machine to minimize the maximal job lateness. This problem is NP-hard, and its version when the job processing times are restricted to p, 2 p, 3 p, 4 p, …, for an integer p, is also NP-hard. We consider the case when the maximal job processing time is kp, for any constant k, and propose its polynomial-time solution. We easily establish that the version of this problem with unrestricted k is NP-hard. Moreover, it is strongly NP-hard if p has no exponential-time dependence on the maximal job due date. From a practical point of view, this is a realistic assumption.

On the complexity of the Eulerian closed walk with precedence path constraints problem

The Eulerian closed walk problem in a digraph is a well-known polynomial-time solvable problem. In this paper, we show that if we impose the feasible solutions to fulfill some precedence constraints specified by paths of the digraph, then the problem becomes NP-complete. We also present a polynomial-time algorithm to solve this variant of the Eulerian closed walk problem when the set of paths does not contain some forbidden structure. This allows us to give necessary and sufficient conditions for the existence of feasible solutions in this polynomial-time solvable case.

On scheduling credited projects

Consideration was given to scheduling investment projects with regard for possible use of credits. The net reduced profit was used as the optimization criterion. The problem was studied, a mathematical model constructed, an algorithm to solve it was suggested, and a polynomially solvable case was specified.

Information capacity of a quantum observable

In this paper we consider the classical capacities of quantum-classical channels corresponding to measurement of observables. Special attention is paid to the case of continuous observables. We give the formulas for unassisted and entanglement-assisted classical capacities $C,C_{ea}$ and consider some explicitly solvable cases which give simple examples of entanglement-breaking channels with $C<C_{ea}.$ We also elaborate on the ensemble-observable duality to show that $C_{ea}$ for the measurement channel is related to the $\chi$-quantity for the dual ensemble in the same way as $C$ is related to the accessible information. This provides both accessible information and the $\chi$-quantity for the quantum ensembles dual to our examples.

Polynomial-time solvable cases of the capacitated multi-echelon shipping network scheduling problem with delivery deadlines

Abstract We consider the problem of operations scheduling for a capacitated multi-echelon shipping network with delivery deadlines. Over the network, semi-finished goods are shipped from origins to many demand points through a capacitated network consisting of shipping links and capacitated processing centers. The shipping operations are performed by a fleet of transporters which require time to travel from one location to another. Each demand point has a specified shipment quantity and a deadline for delivery. The problem is to find a feasible operation schedule to minimize the shipping and penalty cost. This problem is a computationally difficult one because of its inherent combinatorial nature. We report three polynomial-time solvable cases of this problem with (a) identical order quantities; (b) designated suppliers; and (c) divisible customer order sizes. These results reveal some interesting properties of the problem, and can be used to facilitate the design of fast heuristics for operations scheduling of capacitated shipping networks.

Universality of Phases in QCD and QCD-like Theories

We argue that the whole or the part of the phase diagrams of QCD and QCD-like theories should be universal in the large-N_c limit through the orbifold equivalence. The whole phase diagrams, including the chiral phase transitions and the BEC-BCS crossover regions, are identical between SU(N_c) QCD at finite isospin chemical potential and SO(2N_c) and Sp(2N_c) gauge theories at finite baryon chemical potential. Outside the BEC-BCS crossover region in these theories, the phase diagrams are also identical to that of SU(N_c) QCD at finite baryon chemical potential. We give examples of the universality in some solvable cases: (i) QCD and QCD-like theories at asymptotically high density where the controlled weak-coupling calculations are possible, (ii) chiral random matrix theories of different universality classes, which are solvable large-N (large volume) matrix models of QCD. Our results strongly suggest that the chiral phase transition and the QCD critical point at finite baryon chemical potential can be studied using sign-free theories, such as QCD at finite isospin chemical potential, in lattice simulations.

Resilience and optimization of identifiable bipartite graphs

A bipartite graph G=(L,R;E) with at least one edge is said to be identifiable if for every vertex v∈L, the subgraph induced by its non-neighbors has a matching of cardinality |L|−1. This definition arises in the context of low-rank matrix factorization and is motivated by signal processing applications.In this paper, we study the resilience of identifiability with respect to edge additions, edge deletions and edge modifications. These can all be seen as measures of evaluating how strongly a bipartite graph possesses the identifiability property. On the one hand, we show that computing the resilience of this non-monotone property can be done in polynomial time for edge additions or edge modifications. On the other hand, for edge deletions this is an NP-complete problem. Our polynomial results are based on polynomial algorithms for computing the surplus of a bipartite graph G and finding a tight set in G, which might be of independent interest.We also deal with some complexity results for the optimization problem related to the isolation of a smallest set J⊆L that, together with all vertices with neighbors only in J, induces an identifiable subgraph. We obtain an APX-hardness result for the problem and identify some polynomially solvable cases.

The Arboreal Jump Number of an Order

Let \({\mathcal P}\) be a partial order and \({\mathcal A}\) an arboreal extension of it (i.e. the Hasse diagram of \({\mathcal A}\) is a rooted tree with a unique minimal element). A jump of \({\mathcal A}\) is a relation contained in the Hasse diagram of \({\mathcal A}\), but not in the order \({\mathcal P}\). The arboreal jump number of \({\mathcal A}\) is the number of jumps contained in it. We study the problem of finding the arboreal extension of \({\mathcal P}\) having minimum arboreal jump number—a problem related to the well-known (linear) jump number problem. We describe several results for this problem, including NP-completeness, polynomial time solvable cases and bounds. We also discuss the concept of a minimal arboreal extension, namely an arboreal extension whose removal of one jump makes it no longer arboreal.

Swimming holonomy principles, exemplified with a Euler fluid in two dimensions

The idealized problem of swimming—the self-propulsion phenomenon whereby a cyclic change of shape of a ‘swimmer’ produces a net movement—is well studied for the case of a very viscous incompressible liquid. The opposite limit of zero viscosity, the ideal or ‘Euler’ fluid, has also received some attention. There remain to be articulated and explored some points of principle, set here in the context of the Euler fluid in two dimensions, though partly common to both limits and to both two and three dimensions. (i) Perhaps surprisingly, both limits are purely geometric effects, ‘holonomies’, not dependent on any timings or rates, but only on the sequence of shapes adopted by the swimmer. (ii) A principle fully determining swimming in a Euler fluid is simply stated: the fluid moves at every moment so as to minimize the sum of its and the swimmer's kinetic energy. (iii) Euler swimming would be solvable explicitly were it not for the standard impasse of potential theory: to find the boundary normal derivative of a function obeying Laplace's equation given its value around the boundary (or vice versa). As usual more analytical progress is possible in two dimensions (by complexifying) than three, but full tractability still requires the extreme of slight, rapid swimming strokes, and a simple example is given. In both limits, for a non-symmetrical swimming stroke, a rotation or orientation holonomy accompanies the translational one—the swimmer has turned somewhat as well as translated. The whole holonomy is non-Abelian (the order of the shape sequence matters), but (iv) for two dimensions the rotation part is Abelian. A benefit (albeit cosmetic) is that the one-stroke displacement and turning can be written down as a complex line integral. (v) Another benefit is that while Stokes's theorem (in shape space) is normally sacrificed in non-Abelian holonomies, a partial recovery of the theorem is possible in two-dimensional swimming. To illustrate this last principle, a completely solvable case is analyzed: that of a Euler fluid where the swimmer's shape is fixed but its internal mass distribution varies cyclically.

Maximal Entropy Random Walk: solvable cases of dynamics

We focus on the study of dynamics of two kinds of random walk: generic random walk (GRW) and maximal entropy random walk (MERW) on two model networks: Cayley trees and ladder graphs. The stationary probability distribution for MERW is given by the squared components of the eigenvector associated with the largest eigenvalue \lambda_0 of the adjacency matrix of a graph, while the dynamics of the probability distribution approaching to the stationary state depends on the second largest eigenvalue \lambda_1. Firstly, we give analytic solutions for Cayley trees with arbitrary branching number, root degree, and number of generations. We determine three regimes of a tree structure that result in different statics and dynamics of MERW, which are due to strongly, critically, and weakly branched roots. We show how the relaxation times, generically shorter for MERW than for GRW, scale with the graph size. Secondly, we give numerical results for ladder graphs with symmetric defects. MERW shows a clear exponential growth of the relaxation time with the size of defective regions, which indicates trapping of a particle within highly entropic intact region and its escaping that resembles quantum tunneling through a potential barrier. GRW shows standard diffusive dependence irrespective of the defects.

The strong NP-hardness of the maximum lateness minimization scheduling problem with the processing-time based aging effect

In this paper, we analyse the single machine maximum lateness minimization scheduling problem with the processing time based aging effect, where the processing time of each job is described by a non-decreasing function dependent on the sum of the normal processing times of preceded jobs. The computational complexity of this problem was not determined. However, we show it is strongly NP-hard by proving the strong NP-hardness of the single machine maximum completion time minimization problem with this aging model and job deadlines. Furthermore, we determine the boundary between polynomially solvable and NP-hard cases.

Numerical Estimation of the Current Large Deviation Function in the Asymmetric Simple Exclusion Process with Open Boundary Conditions

We numerically study the large deviation function of the total current, which is the sum of local currents over all bonds, for the symmetric and asymmetric simple exclusion processes with open boundary conditions. We estimate the generating function by calculating the largest eigenvalue of the modified transition matrix and by population Monte Carlo simulation. As a result, we find a number of interesting behaviors not observed in the exactly solvable cases studied previously as follows. The even and odd parts of the generating function show different system-size dependences. Different definitions of the current lead to the same generating function in small systems. The use of the total current is important in the Monte Carlo estimation. Moreover, a cusp appears in the large deviation function for the asymmetric simple exclusion process. We also discuss the convergence property of the population Monte Carlo simulation and find that in a certain parameter region, the convergence is very slow and the gap between the largest and second largest eigenvalues of the modified transition matrix rapidly tends to vanish with the system size.

Master equation approach to the central spin decoherence problem: Uniform coupling model and role of projection operators

The generalized Master equation of the Nakajima-Zwanzig (NZ) type has been used extensively to investigate the coherence dynamics of the central spin model with the nuclear bath in a narrowed state characterized by a well defined value of the Overhauser field. We revisit the perturbative NZ approach and apply it to the exactly solvable case of a system with uniform hyperfine couplings. This is motivated by the fact that the effective Hamiltonian-based theory suggests that the dynamics of the realistic system at low magnetic fields and short times can be mapped onto the uniform coupling model. We show that the standard NZ approach fails to reproduce the exact solution of this model beyond very short times, while the effective Hamiltonian calculation agrees very well with the exact result on timescales during which most of the coherence is lost. Our key finding is that in order to extend the timescale of applicability of the NZ approach in this case, instead of using a single projection operator one has to use a set of correlated projection operators which properly reflect the symmetries of the problem and greatly improve the convergence of the theory. This suggests that the correlated projection operators are crucial for a proper description of narrowed state free induction decay at short times and low magnetic fields. Our results thus provide important insights toward the development of a more complete theory of central spin decoherence applicable in a broader regime of timescales and magnetic fields.