Complexity of Ising Polynomials

view Abstract Citations (753) References (12) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Does the missing mass problem signal the breakdown of Newtonian gravity? Bekenstein, J. ; Milgrom, M. Abstract A nonrelativistic potential theory for gravity is considered which differs from the Newtonian theory. The theory is built on the basic assumptions of the modified dynamics, which were shown earlier to reproduce dynamical properties of galaxies and galaxy aggregates without having to assume the existence of hidden mass. The theory involves a modification of the Poisson equation and can be derived from a Lagrangian. The total momentum, angular momentum, and (properly defined) energy of an isolated system are conserved. The center-of-mass acceleration of an arbitrary bound system in a constant external gravitational field is independent of any property of the system. In other words, all isolated objects fall in exactly the same way in a constant external gravitational field (the weak equivalence principle is satisfied). However, the internal dynamics of a system in a constant external field is different from that of the same system in the absence of the external field, in violation of the strong principle of equivalence. These two results are consistent with the phenomenological requirements of the modified dynamics. A toy relativistic theory is sketched which has a nonrelativistic limit satisfying the requirements of the modified dynamics. Publication: The Astrophysical Journal Pub Date: November 1984 DOI: 10.1086/162570 Bibcode: 1984ApJ...286....7B Keywords: Galactic Rotation; Gravitation Theory; Mass Distribution; Missing Mass (Astrophysics); Newton Theory; Nonrelativistic Mechanics; Potential Theory; Computational Astrophysics; Cosmology; Equivalence; Poisson Equation; Astrophysics full text sources ADS |

We find dispersion laws for the photon propagating in the presence of mutually orthogonal constant external electric and magnetic fields in the context of the $\theta $-expanded noncommutative QED. We show that there is no birefringence to the first order in the noncommutativity parameter $% \theta .$ By analyzing the group velocities of the photon eigenmodes we show that there occurs superluminal propagation for any direction. This phenomenon depends on the mutual orientation of the external electromagnetic fields and the noncommutativity vector. We argue that the propagation of signals with superluminal group velocity violates causality in spite of the fact that the noncommutative theory is not Lorentz-invariant and speculate about possible workarounds.

The exact solution for the chiral field superpropagator in a constant external Abelian field is given.

This work explores the conditions that allow the injection and pinning of different magnetic domain walls (DWs) in Permalloy nanostripes with notches of different shapes. The injection is done under a constant external field by applying a 10 ns current pulse through an adjacent current line. The type of DW is identified by its anisotropic magnetoresistance (AMR). We find that, while a quasi-static pinning (nucleating at zero field and propagating the DW to the notch by slowly increasing the external field) would allow to distinguish different types of DWs pinned at the notch, a dynamic pinning (nucleating, propagating, and pinning the DW under a constant non-zero magnetic field) makes the discrimination of different DWs very difficult. Micromagnetic simulations indicate that the AMR of the different types of DWs become quite similar to each other as the injection field increases. This might explain why at large injection fields, usually only one or two values of resistance are detected, depending on the shape of the notch. Therefore, caution should be taken when establishing a one-to-one relationship between a type of DW and a value of resistance, especially if the injection is done under a non-zero external magnetic field.

Quantum electrodynamics in an external constant field

Massive scalar and spin 1/2 fields are considered on a two-dimensional space-time with constant background curvature and in the presence of an external constant field. For each case, the Euler–Heisenberg effective action and the pair creation rate and their dependence on both the curvature and external field is determined exactly.

We study the problem of approximating the partition function of the ferromagnetic Ising model with both pairwise as well as higher order interactions (equivalently, in graphs as well as hypergraphs). Our approach is based on the classical Lee–Yang theory of phase transitions, along with a new Lee–Yang theorem for the Ising model with higher order interactions, and on an extension of ideas developed recently by Barvinok, and Patel and Regts that can be seen as an algorithmic realization of the Lee–Yang theory. Our first result is a deterministic polynomial time approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters \(\beta \) (the interaction) and \(\lambda \) (the external field), except for the case \(\left| \lambda \right| =1\) (the “zero-field” case). A polynomial time randomized approximation scheme (FPRAS) for all graphs and all \(\beta ,\lambda \), based on Markov chain Monte Carlo simulation, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the “decay of correlations” property, but, as pointed out above, on Lee–Yang theory. This approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this latter extension, we establish a tight version of the Lee–Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.

Article Free Access Share on Exact sampling and approximate counting techniques Author: Mark Huber School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NYView Profile Authors Info & Claims STOC '98: Proceedings of the thirtieth annual ACM symposium on Theory of computingMay 1998 Pages 31–40https://doi.org/10.1145/276698.276709Published:23 May 1998Publication History 25citation666DownloadsMetricsTotal Citations25Total Downloads666Last 12 Months66Last 6 weeks15 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

We consider optimization problems on weighted graphs where vertex and edge weights are polynomial functions of a parameter λ. We show that, if a problem satisfies certain regularity properties and the underlying graph has bounded tree-width, the number of changes in the optimum solution is polynomially bounded. We also show that the description of the sequence of optimum solutions can be constructed in polynomial time and that certain parametric search problems can be solved in O(n log n) time, where n is the number of vertices in the graph.KeywordsComposition OperatorSteiner TreeParse TreeLower EnvelopeOuterplanar GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Within the QCD sum-rule approach in an external field, we calculate the baryon matrix element of isovector-scalar current, $H_{\rm B}=\langle B|\overline{u}u- \overline{d}d|B\rangle/2M_{\rm B}$, for octet baryons, which appears in the response of the correlator of baryon interpolating fields to a constant isovector-scalar external field. The sum rules are obtained for a general baryon interpolating field with an appropriate form for the phenomenological ansatz of the spectral density. The key phenomenological input is the response of the quark condensates to the external field. To first order in the quark mass difference $\delta m=m_d-m_u$, the non-electromagnetic part of the baryon isospin mass splitting is given by the product of $\delta m$ and $H_{\rm B}$. Therefore, QCD sum-rule calculation of $H_{\rm B}$ leads to an estimate of the octet baryon isospin mass splittings. The resulting values are comparable to the experimental values; however, the sum-rule predictions for $H_{\rm B}$ are sensitive to the values of the response of the quark condensates to the external source, which are not well determined.

We study a problem of energy-efficiently connecting a symmetric wireless communication network: given an n-vertex graph with edge weights, find a connected spanning subgraph of minimum cost, where the cost is determined by each vertex paying the heaviest edge incident to it in the subgraph. The problem is known to be NP-hard. Strengthening this hardness result, we show that even o(log n)-approximating the difference d between the optimal solution cost and a natural lower bound is NP-hard. Moreover, we show that under the exponential time hypothesis, there are no exact algorithms running in 2o(n) time or in [Formula: see text] time for any computable function f. We also show that the special case of connecting c network components with minimum additional cost generally cannot be polynomial-time reduced to instances of size cO(1) unless the polynomial-time hierarchy collapses. On the positive side, we provide an algorithm that reconnects O(log n)-connected components with minimum additional cost in polynomial time. These algorithms are motivated by application scenarios of monitoring areas or where an existing sensor network may fall apart into several connected components because of sensor faults. In experiments, the algorithm outperforms CPLEX with known integer linear programming (ILP) formulations when n is sufficiently large compared with c. Summary of Contribution: Wireless sensor networks are used to monitor air pollution, water pollution, and machine health; in forest fire and landslide detection; and in natural disaster prevention. Sensors in wireless sensor networks are often battery-powered and disposable, so one may be interested in lowering the energy consumption of the sensors in order to achieve a long lifetime of the network. We study the min-power symmetric connectivity problem, which models the task of assigning transmission powers to sensors so as to achieve a connected communication network with minimum total power consumption. The problem is NP-hard. We provide perhaps the first parameterized complexity study of optimal and approximate solutions for the problem. Our algorithms work in polynomial time in the scenario where one has to reconnect a sensor network with n sensors and O(log n)-connected components by means of a minimum transmission power increase or if one can find transmission power lower bounds that already yield a network with O(log n)-connected components. In experiments, we show that, in this scenario, our algorithms outperform previously known exact algorithms based on ILP formulations.

Multi-way graph partition by stochastic probe

Graphical games have been proposed as a game-theoretic model of large-scale distributed networks of non-cooperative agents. When the number of players is large, and the underlying graph has low degree, they provide a concise way to represent the players' payoffs. It has recently been shown that the problem of finding Nash equilibria in a general degree-3 graphical game with two actions per player is complete for the complexity class PPAD, indicating that it is unlikely that there is any polynomial-time algorithm for this problem. In this paper, we study the complexity of graphical games with two actions per player on bounded-degree trees. This setting was first considered by Kearns, Littman and Singh, who proposed a dynamic programming-based algorithm that computes all Nash equilibria of such games. The running time of their algorithm is exponential, though approximate equilibria can be computed efficiently. Later, Littman, Kearns and Singh proposed a modification to this algorithm that can find a single Nash equilibrium in polynomial time. We show that this modified algorithm is incorrect-the output is not always a Nash equilibrium. We then propose a new algorithm that is based on the ideas of Kearns et al. and computes all Nash equilibria in quadratic time if the input graph is a path, and in polynomial time if it is an arbitrary graph of maximum degree 2. Moreover, our algorithm can be used to compute Nash equilibria of graphical games on arbitrary trees, but the running time can be exponential, even when the tree has bounded degree. We show that this is inevitable -- any algorithm of this type will take exponential time, even on bounded-degree trees with pathwidth 2. It is an open question whether our algorithm runs in polynomial time on graphs with pathwidth 1, but we show that finding a Nash equilibrium for a 2-action graphical game in which the underlying graph has maximum degree 3 and constant pathwidth is PPAD-complete (so is unlikely to be tractable).

Given a set W of k sequences (strings) and a tree structure T with k leaves, each of which is labeled with a unique sequence in W, a tree alignment is to label a sequence to each internal node of T. The weight of an edge of the tree alignment is the distance, such as Hamming distance, Levenshtein (edit) distance or reversal distance, between the two sequences labeled to the two ends of the edge. The bottleneck tree alignment problem is to find a tree alignment such that the weight of the largest edge is minimized. A lifted tree alignment is a tree alignment, where each internal node v is labeled one of the sequences that was labeled to the children of v. The bottleneck lifted tree alignment problem is to find a lifted tree alignment such that the weight of the largest edge is minimized. In this paper, we show that the bottleneck tree alignment problem is NP-complete even when the tree structure is the binary tree and the weight function is metric. For special cases, we present an exact algorithm to solve the bottleneck lifted tree alignment problem in polynomial time. If the weight function is ultrametric, we show that any lifted tree alignment is an optimal bottleneck tree alignment.KeywordsEdit distancebottleneck tree alignmentmetricultrametricNP-complete

The dynamical stability of disk galaxies is sensitive to whether their anomalous rotation curves are caused by dark matter halos or Milgromian dynamics (MOND). We investigate this by setting up a MOND model of M33. We first simulate it in isolation for 6 Gyr, starting from an initial good match to the rotation curve (RC). Too large a bar and bulge form when the gas is too hot, but this is avoided by reducing the gas temperature. A strong bar still forms in 1 Gyr, but rapidly weakens and becomes consistent with the observed weak bar. Previous work showed this to be challenging in Newtonian models with a live dark matter halo, which developed strong bars. The bar pattern speed implies a realistic corotation radius of 3 kpc. However, the RC still rises too steeply, and the central line-of-sight velocity dispersion (LOSVD) is too high. We then add a constant external acceleration field of 8.4 × 10−12 m s−2 at 30° to the disk as a first-order estimate for the gravity exerted by M31. This suppresses buildup of material at the center, causing the RC to rise more slowly and reducing the central LOSVD. Overall, this simulation bears good resemblance to several global properties of M33, and highlights the importance of including even a weak external field on the stability and evolution of disk galaxies. Further simulations with a time-varying external field, modeling the full orbit of M33, will be needed to confirm its resemblance to observations.