Showing papers on "Constant (mathematics) published in 2013"
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TL;DR: A new way of defining the effect transmitted through a restricted set of paths, without controlling variables on the remaining paths is presented, which permits the assessment of a more natural type of direct and indirect effects.
Abstract: The direct effect of one eventon another can be defined and measured byholding constant all intermediate variables between the two.Indirect effects present conceptual andpractical difficulties (in nonlinear models), because they cannot be isolated by holding certain variablesconstant. This paper shows a way of defining any path-specific effectthat does not invoke blocking the remainingpaths.This permits the assessment of a more naturaltype of direct and indirect effects, one thatis applicable in both linear and nonlinear models. The paper establishesconditions under which such assessments can be estimated consistentlyfrom experimental and nonexperimental data,and thus extends path-analytic techniques tononlinear and nonparametric models.
1,286 citations
TL;DR: An implementation of a Gaussian-based approach to deliver the dielectric constant distribution throughout the protein and surrounding water phase by utilizing the 3D structure of the corresponding macromolecule is reported.
Abstract: Implicit methods for modeling protein electrostatics require dielectric properties of the system to be known, in particular, the value of the dielectric constant of protein. While numerous values of the internal protein dielectric constant were reported in the literature, still there is no consensus of what the optimal value is. Perhaps this is due to the fact that the protein dielectric constant is not a "constant" but is a complex function reflecting the properties of the protein's structure and sequence. Here, we report an implementation of a Gaussian-based approach to deliver the dielectric constant distribution throughout the protein and surrounding water phase by utilizing the 3D structure of the corresponding macromolecule. In contrast to previous reports, we construct a smooth dielectric function throughout the space of the system to be modeled rather than just constructing a "Gaussian surface" or smoothing molecule-water boundary. Analysis on a large set of proteins shows that (a) the average dielectric constant inside the protein is relatively low, about 6-7, and reaches a value of about 20-30 at the protein's surface, and (b) high average local dielectric constant values are associated with charged residues while low dielectric constant values are automatically assigned to the regions occupied by hydrophobic residues. In terms of energetics, a benchmarking test was carried out against the experimental pKa's of 89 residues in staphylococcal nuclease (SNase) and showed that it results in a much better RMSD (= 1.77 pK) than the corresponding calculations done with a homogeneous high dielectric constant with an optimal value of 10 (RMSD = 2.43 pK).
429 citations
TL;DR: It is shown that upon application of a constant applied potential difference, the increase in the temperature, due to the Joule effect, associated with the creation of an electric current across the cell follows Ohm's law, while unphysically high temperatures are rapidly observed when constant charges are assigned to each carbon atom.
Abstract: Supercapacitors based on an ionic liquid electrolyte and graphite or nanoporous carbon electrodes are simulated using molecular dynamics. We compare a simplified electrode model in which a constant, uniform charge is assigned to each carbon atom with a realistic model in which a constant potential is applied between the electrodes (the carbon charges are allowed to fluctuate). We show that the simulations performed with the simplified model do not provide a correct description of the properties of the system. First, the structure of the adsorbed electrolyte is partly modified. Second, dramatic differences are observed for the dynamics of the system during transient regimes. In particular, upon application of a constant applied potential difference, the increase in the temperature, due to the Joule effect, associated with the creation of an electric current across the cell follows Ohm’s law, while unphysically high temperatures are rapidly observed when constant charges are assigned to each carbon atom.
215 citations
TL;DR: The three-loop remainder function as discussed by the authors describes the scattering of six gluons in the maximally-helicity-violating configuration in planar N = 4 super- Yang-Mills theory, as a function of the three dual conformal cross ratios.
Abstract: We present the three-loop remainder function, which describes the scattering of six gluons in the maximally-helicity-violating configuration in planar N = 4 super- Yang-Mills theory, as a function of the three dual conformal cross ratios. The result can be expressed in terms of multiple Goncharov polylogarithms. We also employ a more restricted class of hexagon functions which have the correct branch cuts and certain other restrictions on their symbols. We classify all the hexagon functions through transcendental weight five, using the coproduct for their Hopf algebra iteratively, which amounts to a set of first-order differential equations. The three-loop remainder function is a particular weight-six hexagon function, whose symbol was determined previously. The differential equations can be integrated numerically for generic values of the cross ratios, or analytically in certain kinematic limits, including the near-collinear and multi-Regge limits. These limits allow us to impose constraints from the operator product expansion and multi- Regge factorization directly at the function level, and thereby to fix uniquely a set of Riemann ζ valued constants that could not be fixed at the level of the symbol. The near- collinear limits agree precisely with recent predictions by Basso, Sever and Vieira based on integrability. The multi-Regge limits agree with the factorization formula of Fadin and Lipatov, and determine three constants entering the impact factor at this order. We plot the three-loop remainder function for various slices of the Euclidean region of positive cross ratios, and compare it to the two-loop one. For large ranges of the cross ratios, the ratio of the three-loop to the two-loop remainder function is relatively constant, and close to −7.
212 citations
TL;DR: This work predicts that the effective dielectric constant (ε) of few-layer MoS2 is tunable by an external electric field (Eext) and points to a promising way of understanding and controlling the screening properties ofMoS2 through external electric fields.
Abstract: The properties of two-dimensional materials, such as molybdenum disulfide, will play an important role in the design of the next generation of electronic devices. Many of those properties are determined by the dielectric constant which is one of the fundamental quantities used to characterize conductivity, refractive index, charge screening, and capacitance. We predict that the effective dielectric constant (e) of few-layer MoS2 is tunable by an external electric field (Eext). Through first-principles electronic structure calculations, including van der Waals interactions, we show that at low fields (Eext < 0.01 V/A) e assumes a nearly constant value ∼4 but increases at higher fields to values that depend on the layer thickness. The thicker the structure, the stronger the modulation of e with the electric field. Increasing of the external field perpendicular to the dichalcogenide layers beyond a critical value can drive the system to an unstable state where the layers are weakly coupled and can be easily ...
182 citations
TL;DR: In this article, an asymptotically equivalent formula for the finite-time ruin probability of a nonstandard risk model with a constant interest rate, in which both claim sizes and inter-arrival times follow a certain dependence structure, was given.
Abstract: This paper gives an asymptotically equivalent formula for the finite-time ruin probability of a nonstandard risk model with a constant interest rate, in which both claim sizes and inter-arrival times follow a certain dependence structure. This new dependence structure allows the underlying random variables to be either positively or negatively dependent. The obtained asymptotics hold uniformly in a finite time interval. Especially, in the renewal risk model the uniform asymptotics of the finite-time ruin probability for all times have been given. The obtained results have extended and improved some corresponding results.
173 citations
TL;DR: This work proves an L2(R) maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface, and takes advantage of the fact that the bound ‖∂xf0‖L∞ < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.
Abstract: . The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L2(R) maximum principle, in the form of a new “log” conservation law (3) which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance ‖f ‖1 ≤ 1/5. Previous results of this sort used a small constant 1 which was not explicit [7, 19, 9, 14]. Lastly, we prove a global existence result for Lipschitz continuous solutions with initial data that satisfy ‖f0‖L∞ < ∞ and ‖∂xf0‖L∞ < 1. We take advantage of the fact that the bound ‖∂xf0‖L∞ < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.
167 citations
TL;DR: A survey of Euler's work on the constant gamma = 0.57721 can be found in this paper, together with some of his related work on gamma function, values of the zeta function and divergent series.
Abstract: This paper has two parts. The first part surveys Euler's work on the constant gamma=0.57721... bearing his name, together with some of his related work on the gamma function, values of the zeta function and divergent series. The second part describes various mathematical developments involving Euler's constant, as well as another constant, the Euler-Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations and random matrix products. It includes recent results on Diophantine approximation and transcendence related to Euler's constant.
136 citations
TL;DR: This work improves the bound given by Ye and shows that the same bound applies to the number of iterations performed by the strategy iteration algorithm, a generalization of Howard’s policy iteration algorithm used for solving 2-player turn-based stochastic games with discounted zero-sum rewards.
Abstract: Ye [2011] showed recently that the simplex method with Dantzig’s pivoting rule, as well as Howard’s policy iteration algorithm, solve discounted Markov decision processes (MDPs), with a constant discount factor, in strongly polynomial time. More precisely, Ye showed that both algorithms terminate after at most O(mn1−γ log n1−γ) iterations, where n is the number of states, m is the total number of actions in the MDP, and 0
134 citations
TL;DR: The first part surveys Euler's work on the constant γ = 0.57721 · · · bearing his name, together with some of his related work on gamma function, values of the zeta function, and divergent series as mentioned in this paper.
Abstract: This paper has two parts. The first part surveys Euler’s work on the constant γ = 0.57721 · · · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant.
121 citations
TL;DR: In this article, the conjecture of Korevaar and Meyers that for each N cdt d, there exists a spherical t-design in the sphere S d consisting of n points, where cd is a constant depending only on d.
Abstract: In this paper we prove the conjecture of Korevaar and Meyers: for each N cdt d , there exists a spherical t-design in the sphere S d consisting of N points, where cd is a constant depending only on d.
TL;DR: The conservative part of the Lagrangian and the energy of a two-body system at fourth post-Newtonian order up to terms quadratic in the Newton constant were derived in this article.
Abstract: We derive the conservative part of the Lagrangian and the energy of a gravitationally bound two-body system at fourth post-Newtonian order, up to terms quadratic in the Newton constant. We also show that such terms are compatible with Lorentz invariance and we write an ansatz for the center-of-mass position. The remaining terms carrying higher powers of the Newton constant are currently under investigation.
01 Jan 2013
TL;DR: This work shows how to simulate any seeded system with a two- handed system that is essentially just a constant factor larger, and shows that verifying whether a given system uniquely assembles a desired supertile is co-NP-complete in the two-handed model, while it was known to be polynomially solvable in the seeded model.
Abstract: We study the dierence between the standard seeded model (aTAM) of tile self-assembly, and the “seedless” two-handed model of tile self-assembly (2HAM). Most of our results suggest that the two-handed model is more powerful. In particular, we show how to simulate any seeded system with a two-handed system that is essentially just a constant factor larger. We exhibit finite shapes with a busy-beaver separation in the number of distinct tiles required by seeded versus two-handed, and exhibit an infinite shape that can be constructed two-handed but not seeded. Finally, we show that verifying whether a given system uniquely assembles a desired supertile is co-NP-complete in the two-handed model, while it was known to be polynomially solvable in the seeded model. 1998 ACM Subject Classification F.1.2
TL;DR: A hybrid feedback control method is designed to achieve function projective synchronization for complex dynamical networks, one with constant time delay and one with time-varying coupling delay.
Abstract: This paper investigates the problem of function projective synchronization for general complex dynamical networks with time delay. A hybrid feedback control method is designed to achieve function projective synchronization for complex dynamical networks, one with constant time delay and one with time-varying coupling delay. Numerical examples are provided to show the effectiveness of the proposed method.
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TL;DR: Under mild assumptions on F, the error rate of ERM remains optimal even if the procedure is allowed to perform with constant probability and the bound is sharp in the minimax sense if F is convex.
Abstract: We obtain sharp oracle inequalities for the empirical risk minimization procedure in the regression model under the assumption that the target Y and the model F are subgaussian. The bound we obtain is sharp in the minimax sense if F is convex. Moreover, under mild assumptions on F, the error rate of ERM remains optimal even if the procedure is allowed to perform with constant probability. A part of our analysis is a new proof of minimax results for the gaussian regression model.
TL;DR: An application of the Euler-Maclaurin summation formula with the Bernoulli function is provided for the proof of a strengthened version of the half-discrete Hilbert inequality with the best constant factor in terms of theEuler-Mascheroni constant.
TL;DR: This is the first time that a constant factor change of the mutation probability changes the runtime by more than a constant factors, and shows that if c<1, then the (1+1) EA finds the optimum of every such function in iterations.
Abstract: Extending previous analyses on function classes like linear functions, we analyze how the simple 1+1 evolutionary algorithm optimizes pseudo-Boolean functions that are strictly monotonic. These functions have the property that whenever only 0-bits are changed to 1, then the objective value strictly increases. Contrary to what one would expect, not all of these functions are easy to optimize. The choice of the constant c in the mutation probability pn=c/n can make a decisive difference.
We show that if c iterations. For c=1, we can still prove an upper bound of On3/2. However, for , we present a strictly monotonic function such that the 1+1 EA with overwhelming probability needs iterations to find the optimum. This is the first time that we observe that a constant factor change of the mutation probability changes the runtime by more than a constant factor.
TL;DR: In this article, it was shown that small smooth perturbations of a constant background exist for all time and remain smooth (never develop shocks) in the Euler-Poisson system.
Abstract: We consider the (repulsive) Euler-Poisson system for the electrons in two dimensions and prove that small smooth perturbations of a constant background exist for all time and remain smooth (never develop shocks). This extends to 2D the work of Guo [6].
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TL;DR: In particular, the authors showed that polynomial-sized linear programs are exactly as powerful as programs arising from a constant number of rounds of the Sherali-Adams hierarchy for approximation versions of constraint satisfaction problems.
Abstract: We prove super-polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems We show that for these problems, polynomial-sized linear programs are exactly as powerful as programs arising from a constant number of rounds of the Sherali-Adams hierarchy
In particular, any polynomial-sized linear program for Max Cut has an integrality gap of 1/2 and any such linear program for Max 3-Sat has an integrality gap of 7/8
TL;DR: In this paper, the authors studied the approximate controllability of nonlinear fractional control systems with state-dependent delays and resolvent operators and established sufficient conditions to obtain the required result by employing semigroup theory, fixed point technique and fractional calculus.
Abstract: The systems governed by delay differential equations come up in different fields of science and engineering but often demand the use of non-constant or state-dependent delays The corresponding model equation is a delay differential equation with state-dependent delay as opposed to the standard models with constant delay The concept of controllability plays an important role in physics and mathematics In this paper, first we study the approximate controllability for a class of nonlinear fractional differential equations with state-dependent delays Then, the result is extended to study the approximate controllability fractional systems with state-dependent delays and resolvent operators A set of sufficient conditions are established to obtain the required result by employing semigroup theory, fixed point technique and fractional calculus In particular, the approximate controllability of nonlinear fractional control systems is established under the assumption that the corresponding linear control system is approximately controllable Also, an example is presented to illustrate the applicability of the obtained theory
06 Jul 2013
TL;DR: Although the constant optimization involves an overhead regarding the execution time, the achieved accuracy increases significantly as well as the ability of genetic programming to learn from provided data.
Abstract: In this publication a constant optimization approach for symbolic regression is introduced to separate the task of finding the correct model structure from the necessity to evolve the correct numerical constants. A gradient-based nonlinear least squares optimization algorithm, the Levenberg-Marquardt (LM) algorithm, is used for adjusting constant values in symbolic expression trees during their evolution. The LM algorithm depends on gradient information consisting of partial derivations of the trees, which are obtained by automatic differentiation. The presented constant optimization approach is tested on several benchmark problems and compared to a standard genetic programming algorithm to show its effectiveness. Although the constant optimization involves an overhead regarding the execution time, the achieved accuracy increases significantly as well as the ability of genetic programming to learn from provided data. As an example, the Pagie-1 problem could be solved in 37 out of 50 test runs, whereas without constant optimization it was solved in only 10 runs. Furthermore, different configurations of the constant optimization approach (number of iterations, probability of applying constant optimization) are evaluated and their impact is detailed in the results section.
TL;DR: In this article, the authors localize a condition satisfied by such stationary points to smooth bounded domains, and apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved.
Abstract: The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. In this paper, we localize a condition satisfied by such stationary points to smooth bounded domains. The condition involves a generalization of the static equations, and we interpret solutions (and their boundary values) of this equation variationally. On domains carrying a metric that does not satisfy the condition, we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric. We apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved. Finally, we note that starting from a counterexample of Min-Oo’s conjecture such as that of Brendle–Marques–Neves, counterexamples of arbitrarily large volume and different topological types can be constructed.
TL;DR: In this paper, a Lipschitz-type stability is established assuming a priori that the potential is piecewise constant with a bounded known number of unknown values, which is known as the Dirichlet-to-Neumann map.
Abstract: In this paper we study the inverse boundary value problem of determining the potential in the Schrodinger equation from the knowledge of the Dirichlet-to-Neumann map, which is commonly accepted as an ill-posed problem in the sense that, under general settings, the optimal stability estimate is of logarithmic type. In this work, a Lipschitz-type stability is established assuming a priori that the potential is piecewise constant with a bounded known number of unknown values.
TL;DR: In this paper, for the first time, a variable-step-length approach is presented that is applied in combination with the matrix approach for each ‘large step’, and opens the way to development of variable- and adaptive- step-length techniques for fractional- and distributed-order differential equations.
Abstract: In this paper, we further develop Podlubny's matrix approach to discretization of integrals and derivatives of non-integer order. Numerical integration and differentiation on non-equidistant grids is introduced and illustrated by several examples of numerical solution of differential equations with fractional derivatives of constant orders and with distributed-order derivatives. In this paper, for the first time, we present a variable-step-length approach that we call 'the method of large steps', because it is applied in combination with the matrix approach for each 'large step'. This new method is also illustrated by an easy-to-follow example. The presented approach allows fractional-order and distributed-order differentiation and integration of non-uniformly sampled signals, and opens the way to development of variable- and adaptive-step-length techniques for fractional- and distributed-order differential equations.
TL;DR: It is shown that, for such systems, the exponential stability with given decay rate is closely related to the bound of the delay, and sufficient conditions for the existence of static output feedback controllers are established in terms of linear programming (LP) problems.
Abstract: This paper investigates the problem of exponential stability analysis and static output feedback stabilization for discrete-time and continuous-time positive systems with bounded time-varying delays. Based on the relationship between the solution to the system with time-varying delay and that to the corresponding system with constant delay under specific conditions, the equivalence between the α - exponential stability of such two types of systems is established. Then some necessary conditions and sufficient conditions are provided for α - exponential stability of positive systems with bounded time-varying delays. It is shown that, for such systems, the exponential stability with given decay rate is closely related to the bound of the delay. Then by using the singular value decomposition approach, sufficient conditions for the existence of static output feedback controllers are established in terms of linear programming (LP) problems. Some illustrative examples are given to show the correctness of the obtained theoretical results.
TL;DR: A new 2-D random-switching pulsewidth modulation (PWM) technique is proposed to reduce the dominant harmonic clusters while retaining constant average inductor current and constant sampling frequency and the controller parameters of a digitally controlled power converter are not required to change.
Abstract: A new 2-D random-switching pulsewidth modulation (PWM) technique is proposed to reduce the dominant harmonic clusters while retaining constant average inductor current and constant sampling frequency. The special feature of constant average inductor current can reduce the output voltage ripple. Moreover, the controller parameters of a digitally controlled power converter are not required to change, which is quite essential to digitally controlled systems. Current random PWM methods are discussed and compared with the proposed method in this paper. It will be shown that the merits of the presented method include random-switching frequency, constant sampling frequency, and constant average inductor current. An field-programmable-gate-array-based digitally controlled buck converter experimental system has been set up. The specifications of the converter include input voltage = 5 V, output voltage = 1.5 V, and switching frequency = 200 kHz. The proposed random-switching pattern is implemented by software. Experimental results demonstrate the effectiveness of the proposed random-switching pattern.
TL;DR: In this article, the Lagrangian formulation for describing the Navier-Stokes equations with variable density was used to prove existence and uniqueness results in the case of discontinuous initial density, assuming only that the initial density is bounded and bounded away from zero.
Abstract: We investigate the incompressible Navier–Stokes equations with variable density. The aim is to prove existence and uniqueness results in the case of discontinuous initial density. In dimension n = 2,3, assuming only that the initial density is bounded and bounded away from zero, and that the initial velocity is smooth enough, we get the local-in-time existence of unique solutions. Uniqueness holds in any dimension and for a wider class of velocity fields. In particular, all those results are true for piecewise constant densities with arbitrarily large jumps. Global results are established in dimension two if the density is close enough to a positive constant, and in n dimensions if, in addition, the initial velocity is small. The Lagrangian formulation for describing the flow plays a key role in the analysis that is proposed in the present paper.
TL;DR: In this paper, the authors prove a stochastic version of the martingale inequality for non-negative continuous processes, where the pth moment of the supremum of a process is bounded by a constant κp (which does not depend on M) times the moment of H. They also provide an explicit numerical value for the constant cp appearing in the inequality which is at most four times as large as the optimal constant.
Abstract: We prove a stochastic Gronwall lemma of the following type: if Z is an adapted non-negative continuous process which satisfies a linear integral inequality with an added continuous local martingale M and a process H on the right-hand side, then for any p ∈ (0, 1) the pth moment of the supremum of Z is bounded by a constant κp (which does not depend on M) times the pth moment of the supremum of H. Our main tool is a martingale inequality which is due to D. Burkholder. We provide an alternative simple proof of the martingale inequality which provides an explicit numerical value for the constant cp appearing in the inequality which is at most four times as large as the optimal constant.
TL;DR: In this article, a continuation principle or uncertainty relation valid for Schrodinger operator eigenfunctions, or more generally solutions of a Schroding inequality, on cubes of side L is established.
Abstract: We prove a unique continuation principle or uncertainty relation valid for Schrodinger operator eigenfunctions, or more generally solutions of a Schrodinger inequality, on cubes of side \({L \in 2\mathbb{N} + 1}\) . It establishes an equi-distribution property of the eigenfunction over the box: the total L2-mass in the box of side L is estimated from above by a constant times the sum of the L2-masses on small balls of a fixed radius δ > 0 evenly distributed throughout the box. The dependence of the constant on the various parameters entering the problem is given explicitly. Most importantly, there is no L-dependence.
TL;DR: In this article, the authors consider the homogeneous one-dimensional wave equation on [0,π] with Dirichlet boundary conditions, and observe its solutions on a subset ω of [0π] of Lebesgue measure Lπ.
Abstract: In this paper, we consider the homogeneous one-dimensional wave equation on [0,π] with Dirichlet boundary conditions, and observe its solutions on a subset ω of [0,π]. Let L∈(0,1). We investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets ω of [0,π] of Lebesgue measure Lπ. We solve this problem by means of Fourier series considerations, give the precise optimal value and prove that there does not exist any optimal set except for L=1/2. When L≠1/2 we prove the existence of solutions of a relaxed minimization problem, proving a no gap result. Following Hebrard and Henrot (Syst. Control Lett., 48:199–209, 2003; SIAM J. Control Optim., 44:349–366, 2005), we then provide and solve a modal approximation of this problem, show the oscillatory character of the optimal sets, the so called spillover phenomenon, which explains the lack of existence of classical solutions for the original problem.