Showing papers on "Constant (mathematics) published in 2014"

Communication-Efficient Distributed Optimization using an Approximate Newton-type Method

Abstract: We present a novel Newton-type method for distributed optimization, which is particularly well suited for stochastic optimization and learning problems. For quadratic objectives, the method enjoys a linear rate of convergence which provably improves with the data size, requiring an essentially constant number of iterations under reasonable assumptions. We provide theoretical and empirical evidence of the advantages of our method compared to other approaches, such as one-shot parameter averaging and ADMM.

Frictional velocity-weakening in landslides on Earth and on other planetary bodies

Abstract: One of the ultimate goals in landslide hazard assessment is to predict maximum landslide extension and velocity. Despite much work, the physical processes governing energy dissipation during these natural granular flows remain uncertain. Field observations show that large landslides travel over unexpectedly long distances, suggesting low dissipation. Numerical simulations of landslides require a small friction coefficient to reproduce the extension of their deposits. Here, based on analytical and numerical solutions for granular flows constrained by remote-sensing observations, we develop a consistent method to estimate the effective friction coefficient of landslides. This method uses a constant basal friction coefficient that reproduces the first-order landslide properties. We show that friction decreases with increasing volume or, more fundamentally, with increasing sliding velocity. Inspired by frictional weakening mechanisms thought to operate during earthquakes, we propose an empirical velocity-weakening friction law under a unifying phenomenological framework applicable to small and large landslides observed on Earth and beyond.

Neural antecedents of self-initiated actions in secondary motor cortex

Abstract: The neural origins of spontaneous or self-initiated actions are not well understood and their interpretation is controversial. To address these issues, we used a task in which rats decide when to abort waiting for a delayed tone. We recorded neurons in the secondary motor cortex (M2) and interpreted our findings in light of an integration-to-bound decision model. A first population of M2 neurons ramped to a constant threshold at rates proportional to waiting time, strongly resembling integrator output. A second population, which we propose provide input to the integrator, fired in sequences and showed trial-to-trial rate fluctuations correlated with waiting times. An integration model fit to these data also quantitatively predicted the observed inter-neuronal correlations. Together, these results reinforce the generality of the integration-to-bound model of decision-making. These models identify the initial intention to act as the moment of threshold crossing while explaining how antecedent subthreshold neural activity can influence an action without implying a decision.

Diffusion with resetting in arbitrary spatial dimension

Abstract: We consider diffusion in arbitrary spatial dimension d with the addition of a resetting process wherein the diffusive particle stochastically resets to a fixed position at a constant rate r. We compute the nonequilibrium stationary state which exhibits non-Gaussian behaviour. We then consider the presence of an absorbing target centred at the origin and compute the survival probability and mean time to absorption of the diffusive particle by the target. The mean absorption time is finite and has a minimum value at an optimal resetting rate r which depends on dimension. Finally we consider the problem of a finite density of diffusive particles, each resetting to its own initial position. While the typical survival probability of the target at the origin decays exponentially with time regardless of spatial dimension, the average survival probability decays asymptotically as exp ( − A(ln t)d) where A is a constant. We explain these findings using an interpretation as a renewal process and arguments invoking extreme value statistics.

Statistical mechanics for non-reciprocal forces

Abstract: A basic statistical mechanics analysis of many-body systems with non-reciprocal pair interactions is presented. Different non-reciprocity classes in two- and three-dimensional binary systems (relevant to real experimental situations) are investigated, where the action-reaction symmetry is broken for the interaction between different species. The asymmetry is characterized by a non-reciprocity parameter $\Delta$, which is the ratio of the non-reciprocal to reciprocal pair forces. It is shown that for the "constant" non-reciprocity (when $\Delta$ is independent of the interparticle distance $r$) one can construct a pseudo-Hamiltonian and such systems, being intrinsically non-equilibrium, can nevertheless be described in terms of equilibrium statistical mechanics and exhibit detailed balance with distinct temperatures for the different species. For a general case (when $\Delta$ is a function of $r$) the temperatures grow with time, approaching a universal power-law scaling, while their ratio is determined by an effective constant non-reciprocity which is uniquely defined for a given interaction.

Inflation with a constant rate of roll

Abstract: We consider an inflationary scenario where the rate of inflaton roll defined by $\ddot\phi/H\dot \phi$ remains constant. The rate of roll is small for slow-roll inflation, while a generic rate of roll leads to the interesting case of `constant-roll' inflation. We find a general exact solution for the inflaton potential required for such inflaton behaviour. In this model, due to non-slow evolution of background, the would-be decaying mode of linear scalar (curvature) perturbations may not be neglected. It can even grow for some values of the model parameter, while the other mode always remains constant. However, this always occurs for unstable solutions which are not attractors for the given potential. The most interesting particular cases of constant-roll inflation remaining viable with the most recent observational data are quadratic hilltop inflation (with cutoff) and natural inflation (with an additional negative cosmological constant). In these cases even-order slow-roll parameters approach non-negligible constants while the odd ones are asymptotically vanishing in the quasi-de Sitter regime.

High-dimensional estimation with geometric constraints

Abstract: Consider measuring an n-dimensional vector x through the inner product with several measurement vectors, a_1, a_2, ..., a_m. It is common in both signal processing and statistics to assume the linear response model y_i = + e_i, where e_i is a noise term. However, in practice the precise relationship between the signal x and the observations y_i may not follow the linear model, and in some cases it may not even be known. To address this challenge, in this paper we propose a general model where it is only assumed that each observation y_i may depend on a_i only through . We do not assume that the dependence is known. This is a form of the semiparametric single index model, and it includes the linear model as well as many forms of the generalized linear model as special cases. We further assume that the signal x has some structure, and we formulate this as a general assumption that x belongs to some known (but arbitrary) feasible set K. We carefully detail the benefit of using the signal structure to improve estimation. The theory is based on the mean width of K, a geometric parameter which can be used to understand its effective dimension in estimation problems. We determine a simple, efficient two-step procedure for estimating the signal based on this model -- a linear estimation followed by metric projection onto K. We give general conditions under which the estimator is minimax optimal up to a constant. This leads to the intriguing conclusion that in the high noise regime, an unknown non-linearity in the observations does not significantly reduce one's ability to determine the signal, even when the non-linearity may be non-invertible. Our results may be specialized to understand the effect of non-linearities in compressed sensing.

Null tests of the cosmological constant using supernovae

Abstract: The standard concordance model of the Universe is based on the cosmological constant as the driver of accelerating expansion. This concordance model is being subjected to a growing range of interlocking observations. In addition to using generic observational tests, one can also design tests that target the specific properties of the cosmological constant. These null tests do not rely on parametrizations of observables, but focus on quantities that are constant only if dark energy is a cosmological constant. We use supernova data in null tests that are based on the luminosity distance. In order to extract derivatives of the distance in a model-independent way, we use Gaussian processes. We find that the concordance model is compatible with the Union 2.1 data, but the error bars are fairly large. Simulated data sets are generated for the Dark Energy Survey supernova survey and we show that this survey will allow for a sharper null test of the cosmological constant if we assume the Universe is flat. Allowing for spatial curvature degrades the power of the null test. © 2014 American Physical Society.

A determination of Planck's constant using the NRC watt balance

Abstract: We have measured Planck's constant and have obtained a value of 6.626?070?34(12)???10?34?J?s. To our knowledge this measurement of h has the lowest uncertainty reported to date. This result has been obtained from measurements of four masses of different material and nominal values varying from 1?kg to 250?g. The experimental procedures and the measurement uncertainties are described in detail.

Recent development and application of constant pH molecular dynamics.

Abstract: Solution pH is a critical environmental factor for chemical and biological processes. Over the last decade, significant efforts have been made in the development of constant pH molecular dynamics (...

Estimating the value of von Kármán’s constant in turbulent pipe flow

Abstract: Five separate data sets on the mean velocity distributions in the Princeton University/ONR Superpipe are used to establish the best estimate for the value of von Karman’s constant for the flow in a fully developed, hydraulically smooth pipe. The profiles were taken using Pitot tubes, conventional hot wires and nanoscale thermal anemometry probes. The value of the constant was found to vary significantly due to measurement uncertainties in the mean velocity, friction velocity and the wall distance, and the number of data points included in the analysis. The best estimate for the von Karman constant in turbulent pipe flow is found to be . A more precise estimate will require improved instrumentation.

A Matlab Toolbox for Efficient Perfect Reconstruction Time-Frequency Transforms with Log-Frequency Resolution

Abstract: In this paper, we propose a time-frequency representation where the frequency bins are distributed uniformly in log-frequency and their Q-factors obey a linear function of the bin center frequencies. The latter allows for time-frequency representations where the bandwidths can be e.g. constant on the log-frequency scale (constant Q) or constant on the auditory critical-band scale (smoothly varying Q). The proposed techniques are published as a Matlab toolbox that extends [3]. Besides the features that stem from [3] – perfect reconstruction and computational efficiency – we propose here a technique for computing coefficient phases in a way that makes their interpretation more natural. Other extensions include flexible control of the Qvalues and more regular sampling of the time-frequency plane in order to simplify signal processing in the transform domain.

Trajectory Tracking Nonlinear Model Predictive Control for Autonomous Surface Craft

Abstract: This paper presents a solution to the problem of trajectory tracking control for autonomous surface craft (ASC) in the presence of ocean currents. The proposed solution is rooted in nonlinear model predictive control (NMPC) techniques and addresses explicitly state and input constraints. Whereas state saturation constraints are added to the underlying optimization cost functional as penalties, input saturation constraints are made intrinsic to the nonlinear model used in the optimization problem, thus reducing the computational burden of the resulting NMPC algorithm. Simulation results, obtained with a nonlinear dynamic model of a prototype ASC, show that the NMPC strategy adopted yields good performance in the presence of constant currents. Experimental results are also provided to validate the real-time implementation of the NMPC techniques for ASCs.

Determination of magneto-optical constant of Fe films with weak measurements

Abstract: In this letter, a detecting method for the magneto-optical constant is presented by using weak measurements. The photonic spin Hall effect (PSHE), which manifests itself as spin-dependent splitting, is introduced to characterize the magneto-optical constant, and a propagation model to describe the quantitative relation between the magneto-optical constant and the PSHE is established. According to the amplified shift of the PSHE detected by weak measurements, we determinate the magneto-optical constant of the Fe film sample. The Kerr rotation is measured via the standard polarimetry method to verify the rationality and feasibility of our method. These findings may provide possible applications in magnetic physics research.

Constraints from Conformal Symmetry on the Three Point Scalar Correlator in Inflation

Abstract: Using symmetry considerations, we derive Ward identities which relate the three point function of scalar perturbations produced during inflation to the scalar four point function, in a particular limit. The derivation assumes approximate conformal invariance, and the conditions for the slow roll approximation, but is otherwise model independent. The Ward identities allow us to deduce that the three point function must be suppressed in general, being of the same order of magnitude as in the slow roll model. They also fix the three point function in terms of the four point function, upto one constant which we argue is generically suppressed. Our approach is based on analyzing the wave function of the universe, and the Ward identities arise by imposing the requirements of spatial and time reparametrization invariance on it.

A two-stage DEA approach for environmental efficiency measurement

Abstract: The slacks-based measure (SBM) model based on the constant returns to scale has achieved some good results in addressing the undesirable outputs, such as waste water and water gas, in measuring environmental efficiency. However, the traditional SBM model cannot deal with the scenario in which desirable outputs are constant. Based on the axiomatic theory of productivity, this paper carries out a systematic research on the SBM model considering undesirable outputs, and further expands the SBM model from the perspective of network analysis. The new model can not only perform efficiency evaluation considering undesirable outputs, but also calculate desirable and undesirable outputs separately. The latter advantage successfully solves the "dependence" problem of outputs, that is, we can not increase the desirable outputs without producing any undesirable outputs. The following illustration shows that the efficiency values obtained by two-stage approach are smaller than those obtained by the traditional SBM model. Our approach provides a more profound analysis on how to improve environmental efficiency of the decision making units.

Conditions for Existence of Equilibria of Systems With Constant Power Loads

Abstract: In this paper we investigate the sine qua non condition of existence of equilibria for electrical systems with external (AC or DC) sources furnishing constant power to the loads, which is a scenario encountered in modern applications. Two general cases are considered, when the system is i) linear time-invariant or ii) nonlinear, with dynamic behavior described by a port-Hamiltonian model with constant dissipation and switching interconnection matrix. The latter class includes the practically important case of power converters. For both cases necessary and sufficient conditions for existence of equilibria are given, which give an upper bound on the power dissipated in steady-state that should exceed the extracted constant power. The existence of the equilibrium is ensured if and only if the inequality is satisfied.

Polynomiality for bin packing with a constant number of item types

Abstract: We consider the bin packing problem with d different item sizes si and item multiplicities ai, where all numbers are given in binary encoding. This problem formulation is also known as the 1-dimensional cutting stock problem.In this work, we provide an algorithm which, for constant d, solves bin packing in polynomial time. This was an open problem for all d ≥ 3.In fact, for constant d our algorithm solves the following problem in polynomial time: given two d-dimensional polytopes P and Q, find the smallest number of integer points in P whose sum lies in Q.Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant.

On hypercontractivity and a data processing inequality.

Abstract: In this paper we provide the correct tight constant to a data-processing inequality claimed by Erkip and Cover. The correct constant turns out to be a particular hypercontractivity parameter of (X,Y ), rather than their squared maximal correlation. We also provide alternate geometric characterizations for both maximal correlation as well as the hypercontractivity parameter that characterizes the data-processing inequality.

Kähler–Einstein Metrics and Stability

Abstract: This is a note to announce a solution of the well-known question: which Fano manifolds admit Kahler–Einstein metrics? The idea that the appropriate condition should be in terms of “algebro-geometric stability” was proposed about 20 years ago by Yau [21, 22] (partly by analogy with the “Kobayashi–Hitchin correspondence” in the case of holomorphic bundles). For more on the history of this problem please see our forthcoming full paper. Over the years various different notions of stability have been discussed in the literature, both in the Fano/Kahler–Einstein case and in the more general situation of constant scalar curvature Kahler metrics on polarized manifolds. But the condition we end up using is essentially as proposed by Tian [20] and which we now recall (see also [14] for the equivalence with a priori stronger definitions).

Modeling heterogeneous variance–Covariance components in two-level models

Abstract: Applications of multilevel models to continuous outcomes nearly always assume constant residual variance and constant random effects variances and covariances. However, modeling heterogeneity of va...

A space-time hp-interpolation-based certified reduced basis method for Burgers' equation

Abstract: We present a space-time interpolation-based certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval (0, T] parametrized with respect to the Peclet number. We first introduce a Petrov–Galerkin space-time finite element discretization which enjoys a favorable inf–sup constant that decreases slowly with Peclet number and final time T. We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi–Rappaz–Raviart a posteriori error bounds. We describe computational offline–online decomposition procedures for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf–sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L2-error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T, in marked contrast to the exponentially growing estimate of the classical formulation for high Peclet number cases.

Small-Signal Analysis and Optimal Design of External Ramp for Constant On-Time V $^{\bf 2}$ Control With Multilayer Ceramic Caps

Abstract: Recently, constant on-time V2 control, and its variety named constant on-time control, or constant on-time ripple-based control is more and more popular in industry products due to features of high light-load efficiency, simple implementation, and fast transient response. However, subharmonic oscillation occurs when using multilayer ceramic caps due to the lagging phase of the capacitor voltage relative to the inductor current. External ramp compensation is one simple solution to solve the instability issue. However, the characteristics of constant on-time V2 control with external ramp are not fully understood and no explicit design guidelines for the external ramp are provided. This paper investigates the small-signal characteristics of constant on-time V2 control with external ramp compensation by providing a factorized, easy-to-use small-signal model. The external ramp is a critical parameter because it directly affects the position and damping of two pairs of double poles. Based on this fact, design guidelines of the external ramp for optimal dynamic performance are provided. Moreover, the effect of duty cycle is investigated. Finally, the small-signal experimental results and load transient performance are presented to verify the small-signal analysis and proposed design guideline.

Notes on higher-spin algebras: minimal representations and structure constants

Abstract: The higher-spin (HS) algebras so far known can be interpreted as the symmetries of the minimal representation of the isometry algebra. After discussing this connection briefly, we generalize this concept to any classical Lie algebras and consider the corresponding HS algebras. For sp(2N) and so(N), the minimal representations are unique so we get unique HS algebras. For sl(N), the minimal representation has one-parameter family, so does the corresponding HS algebra. The so(N) HS algebra is what underlies the Vasiliev theory while the sl(2) one coincides with the 3D HS algebra hs[lambda]. Finally, we derive the explicit expression of the structure constant of these algebras --- more precisely, their bilinear and trilinear forms. Several consistency checks are carried out for our results.