On the dual graph of Cohen-Macaulay algebras

Given a projective algebraic set X, its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshorne's connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshorne's result: 1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo-Mumford regularity of X. (The bound is best possible; for coordinate arrangements, it yields an algebraic extension of Balinski's theorem for simplicial polytopes.) 2) If X is a canonically embedded arrangement of lines no three of which meet in the same point, then the diameter of the graph G(X) is not larger than the codimension of X. (The bound is sharp; for coordinate arrangements, it yields an algebraic expansion on the recent combinatorial result that the Hirsch conjecture holds for flag normal simplicial complexes.)Comment: Minor changes throughout, Remark 4.1 expanded, to appear in IMR

Conversion of Artificial Recurrent Neural Networks to Spiking Neural Networks for Low-power Neuromorphic Hardware

In recent years the field of neuromorphic low-power systems that consume orders of magnitude less power gained significant momentum. However, their wider use is still hindered by the lack of algorithms that can harness the strengths of such architectures. While neuromorphic adaptations of representation learning algorithms are now emerging, efficient processing of temporal sequences or variable length-inputs remain difficult. Recurrent neural networks (RNN) are widely used in machine learning to solve a variety of sequence learning tasks. In this work we present a train-and-constrain methodology that enables the mapping of machine learned (Elman) RNNs on a substrate of spiking neurons, while being compatible with the capabilities of current and near-future neuromorphic systems. This "train-and-constrain" method consists of first training RNNs using backpropagation through time, then discretizing the weights and finally converting them to spiking RNNs by matching the responses of artificial neurons with those of the spiking neurons. We demonstrate our approach by mapping a natural language processing task (question classification), where we demonstrate the entire mapping process of the recurrent layer of the network on IBM's Neurosynaptic System "TrueNorth", a spike-based digital neuromorphic hardware architecture. TrueNorth imposes specific constraints on connectivity, neural and synaptic parameters. To satisfy these constraints, it was necessary to discretize the synaptic weights and neural activities to 16 levels, and to limit fan-in to 64 inputs. We find that short synaptic delays are sufficient to implement the dynamical (temporal) aspect of the RNN in the question classification task. The hardware-constrained model achieved 74% accuracy in question classification while using less than 0.025% of the cores on one TrueNorth chip, resulting in an estimated power consumption of ~17 uW

First principles studies of modulated Co/Cu superlattices with strongly and weakly exchange biased Co-monolayers

First-principles calculations have been performed in order to determine effective exchange integrals between {\it strongly} and {\it weakly} exchange-coupled Co monolayers in certain modulated periodic $CoCu_2/CoCu_n$-type superlattices with three non-equivalent Co planes, which have not yet been studied hitherto. For $3\le n\le 6$ we find that the two non-equivalent exchange integrals have opposite signs, i.e.~the strong coupling is antiferromagnetic and the weak coupling ferromagnetic, and differ for $n\ne 4$ from each other by one order of magnitude. It is shown that the results depend on the system as a whole and could not be obtained from separate parts. Finally we suggest that ''spin valve'' systems of such kind should be considered when trying to obtain good magneto-resistance together with low switching-fields.Comment: LaTex, 9 pages, including two .eps-figure

Persistence of small-scale anisotropy of magnetic turbulence as observed in the solar wind

The anisotropy of magnetophydrodynamic turbulence is investigated by using solar wind data from the Helios 2 spacecraft. We investigate the behaviour of the complete high-order moment tensors of magnetic field increments and we compare the usual longitudinal structure functions which have both isotropic and anisotropic contributions, to the fully anisotropic contribution. Scaling exponents have been extracted by an interpolation scaling function. Unlike the usual turbulence in fluid flows, small-scale magnetic fluctuations remain anisotropic. We discuss the radial dependence of both anisotropy and intermittency and their relationship.Comment: 7 pages, 2 figures, in press on Europhys. Let

Oscillatory Thickness Dependence of Magnetic Moments and interface-induced Changes of the Exchange Coupling in Co/Cu and Co-Ni/Cu Multilayers

We perform first-principles calculations for the three multilayer systems (100)-Co_1/Cu_n, -NiCo_2Ni/Cu_n and -Co_4/Cu_n, and find from a comparison x of the results for system 2 and 3 that amplitude and phase of the exchange coupling are sensitive to the magnetic-slab/nonmagnetic-spacer interface. Moreover, we observe that for the system 1 and 2 the averaged magnetic moment of the magnetic slab oscillates with the spacer thickness similarly as the exchange coupling.Comment: 5 pages (Latex, to be applied 2 times) + 2 figures (.ps-files

Stochastic Synapses Enable Efficient Brain-Inspired Learning Machines

Recent studies have shown that synaptic unreliability is a robust and sufficient mechanism for inducing the stochasticity observed in cortex. Here, we introduce Synaptic Sampling Machines, a class of neural network models that uses synaptic stochasticity as a means to Monte Carlo sampling and unsupervised learning. Similar to the original formulation of Boltzmann machines, these models can be viewed as a stochastic counterpart of Hopfield networks, but where stochasticity is induced by a random mask over the connections. Synaptic stochasticity plays the dual role of an efficient mechanism for sampling, and a regularizer during learning akin to DropConnect. A local synaptic plasticity rule implementing an event-driven form of contrastive divergence enables the learning of generative models in an on-line fashion. Synaptic sampling machines perform equally well using discrete-timed artificial units (as in Hopfield networks) or continuous-timed leaky integrate & fire neurons. The learned representations are remarkably sparse and robust to reductions in bit precision and synapse pruning: removal of more than 75% of the weakest connections followed by cursory re-learning causes a negligible performance loss on benchmark classification tasks. The spiking neuron-based synaptic sampling machines outperform existing spike-based unsupervised learners, while potentially offering substantial advantages in terms of power and complexity, and are thus promising models for on-line learning in brain-inspired hardware

Personality Traits and Socio-Demographic Variables as Correlates of Counselling Effectiveness of Counsellors in Enugu State, Nigeria

Quality personality traits and socio-demographic variables are essential elements of effective counselling. This correlational study investigated personality traits and socio-demographic variables as predictors of counselling effectiveness of counsellors in Enugu State. The instruments for data collection were Personality Traits Assessment Scale and Counselling Effectiveness Inventory. These instruments were validated by three experts in Counselling Psychology. The reliability coefficients of Personality Traits Assessment Scale and Counselling Effectiveness Inventory as established by test-retest method and Pearson product moment correlation technique were 0.78 and 0.83 respectively. A sample of 309 registered and practising counsellors participated in the study. Four research questions and four hypotheses were formulated to guide the study. Research questions 1 and 3 were answered using the coefficient of regression (r), while research questions 2 and 4 were answered using the beta values associated with multiple regression analysis. Hypotheses 1 and 3 were tested at 0.05 probability level using analysis of variance associated with multiple regression analysis while hypotheses 2 and 4 were tested at 0.05 probability level using independent sample t-test associated with multiple regression analysis. Some major findings of this study revealed that there was a positive and significant joint relationship between personality traits and counselling effectiveness; extraversion, openness to experience, agreeableness and conscientiousness each had a positive relationship with counselling effectiveness of counsellors while neurotism had a negative relationship with the counselling effectiveness; socio-demographic variables had a positive and significant joint relationship with counselling effectiveness. It was therefore recommended that for effective counselling to take place, the counsellors should endeavour to manifest only the personality traits that relate positively with counselling effectiveness so as to produce effective counselling. Keywords: Personality traits, conscientiousness, agreeableness, neuroticism, openness to experience, extraversion, socio-demographic variables and counselling effectiveness

Magnetic Discontinuities in Magnetohydrodynamic Turbulence and in the Solar Wind

Recent measurements of solar wind turbulence report the presence of intermittent, exponentially distributed angular discontinuities in the magnetic field. In this Letter, we study whether such discontinuities can be produced by magnetohydrodynamic (MHD) turbulence. We detect the discontinuities by measuring the fluctuations of the magnetic field direction, Delta theta, across fixed spatial increments Delta x in direct numerical simulations of MHD turbulence with an imposed uniform guide field B_0. A large region of the probability density function (pdf) for Delta theta is found to follow an exponential decay, proportional to exp(-Delta theta/theta_*), with characteristic angle theta_* ~ (14 deg) (b_rms/B_0)^0.65 for a broad range of guide-field strengths. We find that discontinuities observed in the solar wind can be reproduced by MHD turbulence with reasonable ratios of b_rms/B_0. We also observe an excess of small angular discontinuities when Delta x becomes small, possibly indicating an increasing statistical significance of dissipation-scale structures. The structure of the pdf in this case closely resembles the two-population pdf seen in the solar wind. We thus propose that strong discontinuities are associated with inertial-range MHD turbulence, while weak discontinuities emerge from near-dissipation-range turbulence. In addition, we find that the structure functions of the magnetic field direction exhibit anomalous scaling exponents, which indicates the existence of intermittent structures.Comment: To appear in Physical Review Letter