Can one hear the shape of a group?

Let Γ be a finitely generated group, and letS be a finite, non-necessarily symmetric, generating subset of Γ. Leth be the transition operator of the directed Cayley graphG(Γ,S), acting onl ² (Γ). Staring with Kesten’s seminal results, we give a survey of results linking group-theoretic properties of the pair (Γ,S) with spectral properties ofh.     

Can a ground-based vehicle hear the shape of a room?

Assume that a ground-based vehicle moves in a room with walls or other planar surfaces. Can the vehicle reconstruct the positions of the walls from the echoes of a single sound event? We assume that the vehicle carries some microphones and that a loudspeaker is either also mounted on the vehicle or placed at a fixed location in the room. We prove that the reconstruction is almost always possible if (1) no echoes are received from floors, ceilings, or sloping walls and the vehicle carries at least three noncollinear microphones, or if (2) walls of any inclination may occur, the loudspeaker is fixed in the room and there are four noncoplanar microphones. The difficulty lies in the echo-matching problem: How to determine which echoes come from the same wall. We solve this by using a Cayley–Menger determinant. Our proofs use methods from computational commutative algebra.     

Can you cover your shadows?

A subsetX of thed-dimensional Euclidean space ^d can cover its shadows inR ^d , if every orthogonal projection ofX onto a (d–1)-dimensional linear subspace of ^d is contained in some congruent copy ofX. Whereas every two-dimensional convex discC R ^d has this property, no (d–1)-polytope does, provided thatd>-4.     

Can Galileons support Lorentzian wormholes?

We discuss the possibility of constructing stable, static, spherically symmetric, asymptotically flat Lorentzian wormhole solutions in general relativity coupled to a generalized Galileon field π. Assuming that the Minkowski space–time is obtained at ?_π = 0, we find that there is tension between the properties of the energy–momentum tensor required to support a wormhole (violation of the average null energy conditions) and stability of the Galileon perturbations about the putative solution (absence of ghosts and gradient instabilities). In three-dimensional space–time, this tension is strong enough to rule out wormholes with the above properties. In higher dimensions, including the most physically interesting case of four-dimensional space–time, wormholes, if any, must have fairly contrived shapes.     

Can topology reshape segregation patterns?

We consider a metapopulation version of the Schelling model of segregation over several complex networks and lattices. We show that the segregation process is topology independent and hence it is intrinsic to the individual tolerance. The role of the topology is to fix the places where the segregation patterns emerge. In addition we address the question of the time evolution of the segregation clusters, resulting from different dynamical regimes of a coarsening process, as a function of the tolerance parameter. We show that the underlying topology may alter the early stage of the coarsening process, once large values of the tolerance are used, while for lower ones a different mechanism is at work and it results to be topology independent.     

When does gradient Ricci soliton have one end?

Annals of Global Analysis and Geometry - In this paper, we study a problem of finding compact hypersurfaces of prescribed mean curvature with free boundary in a ball. We transform the problem to an...     

Can Data Recognize Its Parent Distribution?

This study is concerned with model selection of lifetime and survival distributions arising in engineering reliability or in the medical sciences. We compare various distributions—including the gamma, Weibull, and lognormal—with a new distribution called geometric extreme exponential. Except for the lognormal distribution, the other three distributions all have the exponential distribution as special cases. A Monte Carlo simulation was performed to determine sample sizes for which survival distributions can distinguish data generated by their own families. Two methods for decision are by maximum likelihood and by Kolmogorov distance. Neither method is uniformly best. The probability of correct selection with more than one alternative shows some surprising results when the choices are close to the exponential distribution.     

Can time-homogeneous diffusions produce any distribution?

Given a centred distribution, can one find a time-homogeneous martingale diffusion starting at zero which has the given law at time 1? We answer the question affirmatively if generalized diffusions are allowed.     

Can international environmental cooperation be bought?

In this paper a two-stage game of international environmental agreement formation with asymmetric countries is analytically solved. The equilibrium of the game makes it possible to determine the size and composition of a stable agreement. Two cases are studied. In the first case, countries differ only in abatement costs, while in the second case, they differ in environmental damages. In both cases, two different institutional settings, one without transfers and another with transfers, are considered. The results establish that the asymmetry assumption has no important effects on the scope of cooperation in comparison with the symmetric case if transfers are not used or abatement costs represent the only difference among countries. However, when the only difference is in environmental damages, the level of cooperation that can be bought through a self-financed transfer scheme increases with the degree of asymmetry.     

Can adaption help on the average?

Summary We study adaptive information for approximation of linear problems in a separable Hilbert space equipped with a probability measure . It is known that adaption does not help in the worst case for linear problems. We prove that adaption also doesnot help on the average. That is, there exists nonadaptive information which is as powerful as adaptive information. This result holds for orthogonally invariant measures. We provide necessary and sufficient conditions for a measure to be orthogonally invariant. Examples of orthogonally invariant measures include Gaussian measures and, in the finite dimensional case, weighted Lebesgue measures.This research was supported in part by the National Science Foundation under Grant MCS-7823676     

Creativity and interdisciplinarity: one creativity or many creativities?

Psychologists and educators frequently debate whether creativity and problem solving are domain-general—applicable to all disciplines and tasks—or domain-specific—tailored to specific disciplines and tasks. In this paper, we briefly review the major arguments for both positions, identify conceptual and empirical weaknesses of both perspectives, and describe two relatively new hybrid models that attempt to address ways in which creativity and innovation are both domain-general and domain-specific.     

How far one can go with the Exp-function method?

A criterion determining if an exact solution of a differential equation can be expressed in a form comprising a finite number of exponential functions is constructed in this paper. This criterion is based on the concept of ranks of Hankel matrixes constructed from sequences of coefficients produced by symbolic multiplicative operator techniques. The employment of this criterion also gives an answer on the structure of the solution. Several examples are used to illustrate this concept.     

Can Monte-Carlo Tree Search learn to sacrifice?

One of the most basic activities performed by an intelligent agent is deciding what to do next. The decision is usually about selecting the move with the highest expectation, or exploring new scenarios. Monte-Carlo Tree Search (MCTS), which was developed as a game playing agent, deals with this exploration–exploitation ‘dilemma’ using a multi-armed bandits strategy. The success of MCTS in a wide range of problems, such as combinatorial optimisation, reinforcement learning, and games, is due to its ability to rapidly evaluate problem states without requiring domain-specific knowledge. However, it has been acknowledged that the trade-off between exploration and exploitation is crucial for the performance of the algorithm, and affects the efficiency of the agent in learning deceptive states. One type of deception is states that give immediate rewards, but lead to a suboptimal solution in the long run. These states are known as trap states, and have been thoroughly investigated in previous research. In this work, we study the opposite of trap states, known as sacrifice states, which are deceptive moves that result in a local loss but are globally optimal, and investigate the efficiency of MCTS enhancements in identifying this type of moves.     

Can We Compute the Similarity between Surfaces?

A suitable measure for the similarity of shapes represented by parameterized curves or surfaces is the Fréchet distance. Whereas efficient algorithms are known for computing the Fréchet distance of polygonal curves, the same problem for triangulated surfaces is NP-hard. Furthermore, it remained open whether it is computable at all. Using a discrete approximation, we show that it is upper semi-computable, i.e., there is a non-halting Turing machine which produces a decreasing sequence of rationals converging to the Fréchet distance. It follows that the decision problem, whether the Fréchet distance of two given surfaces lies below a specified value, is recursively enumerable.     

How Many Conjectures Can You Stand? A Survey

We survey results and open problems in hamiltonian graph theory centered around two conjectures of the 1980s that are still open: every 4-connected claw-free graph (line graph) is hamiltonian. These conjectures have lead to a wealth of interesting concepts, techniques, results and equivalent conjectures.