Generalised Cantor sets and the dimension of products

In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of ‘equi-homogeneity’ of a set, which requires a uniformity in the cardinality of local covers at all length-scales and at all points, and we show that a large class of homogeneous Moran sets have this property. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set ‘attains’ these dimensions (analogous to ‘s-sets’ when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any α ∈ (0, 1) and any β, γ ∈ (0, 1) such that β + γ ≥ 1 we can construct two generalised Cantor sets C and D such that dimBC = αβ, dimBD = α γ, and dimAC = dimAD = dimA (C × D) = dimB (C × D) = α

Motor Learning Deficits in Parkinson\u27s Disease (PD) and Their Effect on Training Response in Gait and Balance: A Narrative Review

Parkinson\u27s disease (PD) is a neurological disorder traditionally associated with degeneration of the dopaminergic neurons within the substantia nigra, which results in bradykinesia, rigidity, tremor, and postural instability and gait disability (PIGD). The disorder has also been implicated in degradation of motor learning. While individuals with PD are able to learn, certain aspects of learning, especially automatic responses to feedback, are faulty, resulting in a reliance on feedforward systems of movement learning and control. Because of this, patients with PD may require more training to achieve and retain motor learning and may require additional sensory information or motor guidance in order to facilitate this learning. Furthermore, they may be unable to maintain these gains in environments and situations in which conscious effort is divided (such as dual-tasking). These shortcomings in motor learning could play a large part in degenerative gait and balance symptoms often seen in the disease, as patients are unable to adapt to gradual sensory and motor degradation. Research has shown that physical and exercise therapy can help patients with PD to adapt new feedforward strategies to partially counteract these symptoms. In particular, balance, treadmill, resistance, and repeated perturbation training therapies have been shown to improve motor patterns in PD. However, much research is still needed to determine which of these therapies best alleviates which symptoms of PIGD, the needed dose and intensity of these therapies, and long-term retention effects. The benefits of such technologies as augmented feedback, motorized perturbations, virtual reality, and weight-bearing assistance are also of interest. This narrative review will evaluate the effect of PD on motor learning and the effect of motor learning deficits on response to physical therapy and training programs, focusing specifically on features related to PIGD. Potential methods to strengthen therapeutic effects will be discussed

X-Ray Emission Produced in Charge-Exchange Collisions between Highly Charged Ions and Argon: Role of the Multiple Electron Capture

In this work we use the classical trajectory Monte Carlo method within an eight-electron scheme to theoretically study photonic spectra that follow charge-exchange processes between highly charged ions of charge states 10+, 17+, 18+, and 36+ with neutral argon. The energy range considered is 18 eV/amu to 4 keV/amu, covering typical electron beam ion traps and solar wind energies. The role played by multiple electron capture processes for the different collision systems under consideration is explicitly analyzed and its contribution separated as arising from radiative decay and autoionizing multiple capture. For the present collision systems we find that multiple electron capture is responsible for 50%-60% of the resulting x-ray spectra. The present results are of direct relevance to the astrophysical program

The Role of Multiple Electron Capture in the X-Ray Emission Process Following Charge Exchange Collisions with Neutral Targets

In this work we theoretically study photonic spectra that follow charge exchange processes between highly charged ions and neutral argon and CO targets. The range of collision energies studied is 5 eV/amu-10 keV/amu, covering typical EBIT-traps and Solar Wind energies. Our studies are based on multiple electrons schemes within the classical trajectory Monte Carlo method. Electrons are sorted with the sequential binding energies for the target under consideration. The role played by the multiple electron capture process for the different collision systems under consideration is explicitly analyzed and its contribution separated as arising from double radiative decay and autoionizing multiple capture. Present studies are stimulated by the upcoming launch of the Astro-H mission in 2015, which will provide high resolution spectra in the 0.3 keV-12keV band

Reply to Comment on \u27Classical Description of H(1s) and H* (n=2) for Cross-Section Calculations Relevant to Charge-Exchange Diagnostics\u27

In reply to the Comment of Jorge et al. [Phys. Rev. A 93, 066701 (2016)], we agree and reconfirm that the alternative classical trajectory Monte Carlo method (called hydrogenic-Z-CTMC) radial distributions for H*(n = 2) we recently published are not stable in time. However, we show that such lack of stability which is more noticeable for H(2s) than for H(2p) is due to the initialization procedure employed and not to the hydrogenic-Z-CTMC method itself. A new set of completely stable hydrogenic-Z-CTMC calculations for H*(n = 2) is introduced and found in very good agreement with standard microcanonical results reinforcing our previous findings. A second criticism of Jorge et al. concerning the number of components in hydrogenic-Z-CTMC with n \u3e 1 for H(1s) is shown not to have a significant impact on relative (n,l) populations in the final state

Prediction of Axial Pile Capacity Based on Case Histories

A data bank containing records from 1000 load tests on driven piles was set up. A computer program was developed to access the data bank and perform capacity analyses using a variety of methods. Analyses using six methods in clay and three in sand are reported here. For piles in clay, the capacities were predicted with tolerable accuracy by all methods, whereas the scatter was large for all methods for piles in sand. Generally, capacities were higher for tapered piles then indicated by the analyses. Tensile and compressive side shear capacities were essentially the same. The capacities of open and closed ended pipe piles were predicted with equal accuracy. Limits on side shear and tip stresses were helpful in reducing overpredictions

Imperfect identity

Questions of identity over time are often hard to answer. A long tradition has it that such questions are somehow soft: they have no unique, determinate answer, and disagreements about them are merely verbal. I argue that this claim is not the truism it is taken to be. Depending on how it is understood, it turns out either to be false or to presuppose a highly contentious metaphysical claim

A nullstellensatz for sequences over F_p

Let p be a prime and let A=(a_1,...,a_l) be a sequence of nonzero elements in F_p. In this paper, we study the set of all 0-1 solutions to the equation a_1 x_1 + ... + a_l x_l = 0. We prove that whenever l >= p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for l < p. The critical case l=p is of particular interest. In this context, we prove that whenever l=p and A is nonconstant, the above equation has at least p-1 minimal 0-1 solutions, thus refining a theorem of Olson. The subcritical case l=p-1 is studied in detail also. Our approach is algebraic in nature and relies on the Combinatorial Nullstellensatz as well as on a Vosper type theorem.Comment: 23 page