Transformations of random walks on groups via Markov stopping times

We describe a new construction of a family of measures on a group with the same Poisson boundary. Our approach is based on applying Markov stopping times to an extension of the original random walk

Asymptotic entropy of transformed random walks

We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (resp., rate of escape) of the transformed random walks is equal to the asymptotic entropy (resp., rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov's formula from ergodic theory, its particular cases were established earlier by Kaimanovich [1983] and Hartman, Lima, Tamuz [2014]

Random walks of infinite moment on free semigroups

We consider random walks on finitely or countably generated free semigroups, and identify their Poisson boundaries for classes of measures which fail to meet the classical entropy criteria. In particular, we introduce the notion of w-logarithmic moment, and we show that if a random walk on a free semigroup has either finite entropy or finite w-logarithmic moment for some word w, then the space of infinite words with the resulting hitting measure is the Poisson boundary

Shannon's theorem for locally compact groups

We consider random walks on locally compact groups, extending the geometric criteria for the identification of their Poisson boundary previously known for discrete groups. First, we prove a version of the Shannon-McMillan-Breiman theorem, which we then use to generalize Kaimanovich's ray approximation and strip approximation criteria. We give several applications to identify the Poisson boundary of locally compact groups which act by isometries on nonpositively curved spaces, as well as on Diestel-Leader graphs and horocylic products.Comment: 34 pages, no figure

Comment on "robustness and regularization of support vector machines" by H. Xu, et al., (Journal of Machine Learning Research, vol. 10, pp. 1485-1510, 2009, arXiv:0803.3490)

This paper comments on the published work dealing with robustness and regularization of support vector machines (Journal of Machine Learning Research, vol. 10, pp. 1485-1510, 2009) [arXiv:0803.3490] by H. Xu, etc. They proposed a theorem to show that it is possible to relate robustness in the feature space and robustness in the sample space directly. In this paper, we propose a counter example that rejects their theorem.Comment: 2 pages. This paper has been accepted with minor revision in journal of machine learning research (JMLR

A note on "A simple algorithm to search for all MCs in networks"

Recently, Yeh [Yeh, WC. (2006). A simple algorithm to search for all MCs in networks. European Journal of Operational Research, 174, 1694{1705.] has proposed a simple algorithm to find all the Minimal Cuts in an undirected graph. However, the algorithm does not work properly. Here, using an example, a defect of this algorithm is illustrated, and then the corresponding result is shown to be incorrect. Moreover, a correct version of the algorithm is established.Comment: 11 pages, 1 figures, 2algorithm

Technical Notes on "A new approach to the d-MC problem"

System reliability is the probability of the maximum flow in a stochastic-flow network from the source node to the sink node being more than a demand level d. There are several approaches to compute system reliability using upper boundary points, called d-MinCuts (d-MCs). Search for all the d-MCs in a stochastic-flow network is an NP-hard problem. Here, a work proposed by Yeh [Yeh WC. A new approach to the d-MC problem. Reliab Eng and Syst Saf 2002; 77(2): 201-206.] for determining all the d-MCs is investigated. Two results (Lemma 3 and Theorem 5) are shown to be incorrect and their correct versions are established. Also, the complexity result (Theorem 6) is shown to be incorrect and the correct count is provided.Comment: 7 pages, 1 figur

Control of uniflagellar soft robots at low Reynolds number using buckling instability

In this paper, we analyze the inverse dynamics and control of a bacteria-inspired uniflagellar robot in a fluid medium at low Reynolds number. Inspired by the mechanism behind the locomotion of flagellated bacteria, we consider a robot comprised of a flagellum -- a flexible helical filament -- attached to a spherical head. The flagellum rotates about the head at a controlled angular velocity and generates a propulsive force that moves the robot forward. When the angular velocity exceeds a threshold value, the hydrodynamic force exerted by the fluid can cause the soft flagellum to buckle, characterized by a dramatic change in shape. In this computational study, a fluid-structure interaction model that combines Discrete Elastic Rods (DER) algorithm with Lighthill's Slender Body Theory (LSBT) is employed to simulate the locomotion and deformation of the robot. We demonstrate that the robot can follow a prescribed path in three dimensional space by exploiting buckling of the flagellum. The control scheme involves only a single (binary) scalar input -- the angular velocity of the flagellum. By triggering the buckling instability at the right moment, the robot can follow an arbitrary path in three dimensional space. We also show that the complexity of the dynamics of the helical filament can be captured using a deep neural network, from which we identify the input-output functional relationship between the control inputs and the trajectory of the robot. Furthermore, our study underscores the potential role of buckling in the locomotion of natural bacteria

Higher Dimensional Particle Model in Third-Order Lovelock Gravity

By using the formalism of thin-shells, we construct a geometrical model of a particle in third-order Lovelock gravity. This particular theory which is valid at least in 7 dimensions, provides enough degrees of freedom and grounds towards such a construction. The particle consists of a flat interior and a non-black hole exterior spacetimes whose mass, charge, and radius are determined from the junction conditions, in terms of the parameters of the theory.Comment: 7 pages, 1 figure (7 subfigures

The Stability of Asymmetric Cylindrical Thin-Shell Wormholes

In continuation of a preceding work on introducing asymmetric thin-shell wormholes as an emerging class of traversable wormholes within the context, this time cylindrically symmetric spacetimes are exploited to construct such wormholes. Having established a generic formulation, first the Linet-Tian metric generally, and then the cosmic string metric and a black string metric in greater details are studied as constructing blocks of cylindrical asymmetric thin-shell wormholes. The corresponding wormholes are investigated within the linearized stability analysis framework to firstly, demonstrate that they can exist from the mechanical stability point of view, and secondly, indicate the correlation between the stability and symmetry in each case, if there is any at all. From here, we have extracted a pattern for the way stability changes with the asymmetry degree for the two examples; however, it was observed that the symmetric state is not the most neither the less stable state. There are also some side results: It was learned that any cylindrical thin-shell wormhole made of two cosmic string universes cannot be supported by a barotropic equation of state. Furthermore, as another side outcome, it was perceived that the radius dependency of the so-called variable equation of state, which is used all over this article, has a great impact on the mechanical stability of the cylindrical asymmetric thin-shell wormholes studied in this brief.Comment: 11 pages, 4 figure