Nonnegative Terms Research Articles

Two Convergence Theorems and an Extension of the Ratio Test for a Series

SummaryGiven a decreasing sequence of nonnegative terms, we prove that there exist subsequences of {n} such that and have the same convergence. In particular, converges if and only if converges for any natural number q > 1. Using this result, the ratio test is also extended. One example further shows that the new extended ratio test is applicable to discussing the convergence of a p-series for which the ratio test fails.

A Krasnoselskii-Mann Algorithm With an Improved EM Preconditioner for PET Image Reconstruction.

This paper presents a preconditioned Krasnoselskii-Mann (KM) algorithm with an improved EM preconditioner (IEM-PKMA) for higher-order total variation (HOTV) regularized positron emission tomography (PET) image reconstruction. The PET reconstruction problem can be formulated as a three-term convex optimization model consisting of the Kullback-Leibler (KL) fidelity term, a nonsmooth penalty term, and a nonnegative constraint term which is also nonsmooth. We develop an efficient KM algorithm for solving this optimization problem based on a fixed-point characterization of its solution, with a preconditioner and a momentum technique for accelerating convergence. By combining the EM precondtioner, a thresholding, and a good inexpensive estimate of the solution, we propose an improved EM preconditioner that can not only accelerate convergence but also avoid the reconstructed image being "stuck at zero." Numerical results in this paper show that the proposed IEM-PKMA outperforms existing state-of-the-art algorithms including, the optimization transfer descent algorithm and the preconditioned L-BFGS-B algorithm for the differentiable smoothed anisotropic total variation regularized model, the preconditioned alternating projection algorithm, and the alternating direction method of multipliers for the nondifferentiable HOTV regularized model. Encouraging initial experiments using clinical data are presented.

带有非负局部项的p-Laplacian问题的径向正解

带有非负局部项的p-Laplacian问题的径向正解

On the Grace–Danielsson inequality for tetrahedra

The difference between the (squared) sides of the Grace–Danielsson inequality for tetrahedra will be represented as a sum of two nonnegative terms. This gives another proof of the inequality. Examining the denominator allows us to characterize the infinite triangular prism as a degenerate tetrahedron. We give conditions for equality (for a zero gap) as well, and some examples are included.

The Identity of Information: How Deterministic Dependencies Constrain Information Synergy and Redundancy.

Understanding how different information sources together transmit information is crucial in many domains. For example, understanding the neural code requires characterizing how different neurons contribute unique, redundant, or synergistic pieces of information about sensory or behavioral variables. Williams and Beer (2010) proposed a partial information decomposition (PID) that separates the mutual information that a set of sources contains about a set of targets into nonnegative terms interpretable as these pieces. Quantifying redundancy requires assigning an identity to different information pieces, to assess when information is common across sources. Harder et al. (2013) proposed an identity axiom that imposes necessary conditions to quantify qualitatively common information. However, Bertschinger et al. (2012) showed that, in a counterexample with deterministic target-source dependencies, the identity axiom is incompatible with ensuring PID nonnegativity. Here, we study systematically the consequences of information identity criteria that assign identity based on associations between target and source variables resulting from deterministic dependencies. We show how these criteria are related to the identity axiom and to previously proposed redundancy measures, and we characterize how they lead to negative PID terms. This constitutes a further step to more explicitly address the role of information identity in the quantification of redundancy. The implications for studying neural coding are discussed.

Positive bounded solutions for nonlinear polyharmonic problems in the unit ball

Abstract In this paper, we study the existence of positive solutions for the following nonlinear polyharmonic equation (-∆)mu+λf(x, u) = 0 in B; subject to some boundary conditions, where m is a positive integer, λ is a nonnegative constant and B is the unit ball of ℝn (n ≥ 2). Under some appropriate assumptions on the nonnegative nonlinearity term f(x, u) and by using the Schäuder fixed point theorem, the existence of positive solutions is obtained. At last, examples are given for illustration.

Monotone decomposition of 2-additive Generalized Additive Independence models

The GAI (Generalized Additive Independence) model proposed by Fishburn is a generalization of the additive value function model, which need not satisfy preferential independence. Its great generality makes however its application and study difficult. We consider a significant subclass of GAI models, namely the discrete 2-additive GAI models, and provide for this class a decomposition into nonnegative monotone terms. This decomposition allows a reduction from exponential to quadratic complexity in any optimization problem involving discrete 2-additive models, making them usable in practice.

Pinning synchronization of complex directed dynamical networks under decentralized adaptive strategy for aperiodically intermittent control

In this paper, the problem of global synchronization for complex directed dynamical networks via adaptive aperiodically intermittent pinning control is studied. By constructing a piecewise Lyapunov function, some sufficient conditions to guarantee global synchronization are derived based on the analytical technique and theory of series with nonnegative terms. Different from previous works, the adaptive intermittent pinning control is aperiodic with non-fixed both control period and control width, and moreover, the adaptive approach is decentralized relying only on the state information of the controlled node. Hence, the adaptive intermittent pinning control strategy proposed in this paper is more practically applicable than those in previous works. Additionally, it is noted that the derived synchronization criteria are dependent on the control rates, but not the control widths or the control periods, which makes the theoretical results are less conservative than the corresponding results given in the existing works. A numerical example is finally provided to illustrate the validity of our theoretical results.

Entropy-based mixed three-moment description of fermionic radiation transport in slab and spherical geometries

The mixed three-moment hydrodynamic description of fermionic radiation transport based on the Boltzmann entropy optimization procedure is considered for the case of one-dimensional flows. The conditions for realizability of the mixed three moments chosen as the energy density and two partial heat fluxes are established. The domain of admissible values of those moments is determined and the existence of the solution to the optimization problem is proved. Here, the standard approaches related to either the truncated Hausdorff or Markov moment problems do not apply because the non-negative fermionic distribution function, denoted \begin{document}$f$\end{document} , must satisfy the inequality \begin{document}$f≤q 1$\end{document} and, at the same time, there are three different intervals of integration in the integral formulae defining the mixed moments. The hydrodynamic equations are obtained in the form of the symmetric hyperbolic system for the Lagrange multipliers of the optimization problem with constraints. The potentials generating this system are explicitly determined as dilogarithm and trilogarithm functions of the Lagrange multipliers. The invertibility of the relation between moments and Lagrange multipliers is proved. However, the inverse relation cannot be determined in a closed analytic form. Using the \begin{document}$H$\end{document} -theorem for the radiative transfer equation, it is shown that the derived system of hydrodynamic radiation equations has as a consequence an additional balance law with a non-negative source term.

Novel dissipative properties of the master equation

Recent studies have shown that the entropy production rate for the master equation consists of two non-negative terms: the adiabatic and non-adiabatic parts, where the non-adiabatic part is also known as the free energy dissipation rate. In this paper, we present some nonzero lower bounds for the free energy, the entropy production rate, and its adiabatic and non-adiabatic parts. These nonzero lower bounds not only reveal some novel dissipative properties for nonequilibrium dynamics which are much stronger than the second law of thermodynamics, but also impose some new constraints on thermodynamic constitutive relations. Moreover, we also give a mathematical application of the nonzero lower bounds by studying the long-time behavior of the master equation. Extensions to the Tsallis statistics are also discussed, including the nonzero lower bounds for the Tsallis-type free energy and its dissipation rate.

Optimization of Markov decision processes under the variance criterion

In this paper, we study a variance minimization problem in an infinite stage discrete time Markov decision process (MDP), regardless of the mean performance. For the Markov chain under the variance criterion, since the value of the cost function at the current stage will be affected by future actions, this problem is not a standard MDP and the traditional MDP theory is not applicable. In this paper, we convert the variance minimization problem into a standard MDP by introducing a concept called pseudo variance. Then we derive a variance difference formula that quantifies the difference of variances of Markov systems under any two policies. With the difference formula, the correlation of the variance cost function at different stages can be decoupled through a nonnegative term. A necessary condition of the optimal policy is obtained. It is also proved that the optimal policy with the minimal variance can be found in the deterministic policy space. Furthermore, we propose an efficient iterative algorithm to reduce the variance of Markov systems. We prove that this algorithm can converge to a local optimum. Finally, a numerical experiment is conducted to demonstrate the efficiency of our algorithm compared with the gradient-based method widely adopted in the literature.

On series of fuzzy numbers

The main purpose of the present paper is to derive some new results, by using the concept of the level set, related to the convergence of the series of fuzzy numbers. As a consequence of this goal, we give the fuzzy analogues of certain convergence tests and results concerning the series of nonnegative or arbitrary real terms.

A Limit Comparison Test for General Series

The well-known limit comparison test is only applicable for series with nonnegative terms. Thus, it can be used only for proving or disproving the absolute convergence of a series. In this note we formulate and justify a modified version of the limit comparison test for general series. The test can be used to prove the convergence of conditionally convergent series.

Solutions to a class of parabolic inhomogeneous normalized p-laplace equations

In this article, we prove that viscosity solutions of the parabolic inhomogeneous equations n+pput−ΔpNu=f(x,t)can be characterized using asymptotic mean value properties for all p ≥ 1, including p = 1 and p = ∞. We also obtain a comparison principle for the non-negative or non-positive inhomogeneous term f for the corresponding initial-boundary value problem and this in turn implies the uniqueness of solutions to such a problem.

Debating Multicultural Korea: Media Discourse on Migrants and Minorities in South Korea

Since the early 1990s, South Korea has experienced growing, public debate about migrants, minorities, and related government policies. Much of this ‘multicultural’ discourse occurs in the internet-based, cyber-media, which offer space for various producers, including mainstream media professionals, netizens, and resident foreigners. A prevailing discourse of victimhood is associated with small, ideologically homogeneous groups of media producers (mainstream media and nativist websites); in contrast, mixed and nuanced discourses are more common in large and heterogeneous, online communities, where users interact with a variety of viewpoints. Moreover, actors who are framed in non-negative terms — as neither xenophobes nor corrupt elites — serve as more effective brokers and bridges across various media. Most foreign-origin netizens are passive consumers of media discourse; but a few, active foreigners communicate effectively with both mainstream media and netizens and help to diversify the prevailing discourse. We find supportive evidence from a content analysis of 15 websites, supplemented with informant interviews and observations.

Debating Multicultural Korea: Media Discourse on Migrants and Minorities in South Korea

Since the early 1990s, South Korea has experienced growing public debate about migrants, minorities and related government policies. Much of this ‘multicultural’ discourse occurs in the internet-based cyber-media, which offer space for various producers, including mainstream media professionals, netizens and resident foreigners. A prevailing discourse of victimhood is associated with small, ideologically homogeneous groups of media producers (mainstream media and nativist websites); in contrast, mixed and nuanced discourses are more common in large and heterogeneous online communities, where users interact with a variety of viewpoints. Moreover, actors who are framed in non-negative terms—as neither xenophobes nor corrupt elites—serve as more effective brokers and bridges across various media. Most foreign-origin netizens are passive consumers of media discourse; but a few, active foreigners communicate effectively with both mainstream media and netizens and help to diversify the prevailing discourse. We find supportive evidence from a content analysis of 15 websites, supplemented with informant interviews and observations.

Double Series and Sums

Summary In this paper the author constructs several properties for double series and its convergence. The notions of convergence of double sequence have already been introduced in our previous paper [18]. In section 1 we introduce double series and their convergence. Then we show the relationship between Pringsheim-type convergence and iterated convergence. In section 2 we study double series having non-negative terms. As a result, we have equality of three type sums of non-negative double sequence. In section 3 we show that if a non-negative sequence is summable, then the sequence of rearrangement of terms is summable and it has the same sums. In the last section two basic relations between double sequences and matrices are introduced.

Time reversibility and nonequilibrium thermodynamics of second-order stochastic processes.

Nonequilibrium thermodynamics of a general second-order stochastic system is investigated. We prove that at steady state, under inversion of velocities, the condition of time reversibility over the phase space is equivalent to the antisymmetry of spatial flux and the symmetry of velocity flux. Then we show that the condition of time reversibility alone cannot always guarantee the Maxwell-Boltzmann distribution. Comparing the two conditions together, we find that the frictional force naturally emerges as the unique odd term of the total force at thermodynamic equilibrium, and is followed by the Einstein relation. The two conditions respectively correspond to two previously reported different entropy production rates. In the case where the external force is only position dependent, the two entropy production rates become one. We prove that such an entropy production rate can be decomposed into two non-negative terms, expressed respectively by the conditional mean and variance of the thermodynamic force associated with the irreversible velocity flux at any given spatial coordinate. In the small inertia limit, the former term becomes the entropy production rate of the corresponding overdamped dynamics, while the anomalous entropy production rate originates from the latter term. Furthermore, regarding the connection between the first law and second law, we find that in the steady state of such a limit, the anomalous entropy production rate is also the leading order of the Boltzmann-factor weighted difference between the spatial heat dissipation densities of the underdamped and overdamped dynamics, while their unweighted difference always tends to vanish.

A magnetic contribution to the Hardy inequality

We study the quadratic form associated to the kinetic energy operator in the presence of an external magnetic field in d = 3. We show that if the radial component of the magnetic field does not vanish identically, then the classical lower bound given by Hardy is improved by a non-negative potential term depending on properties of the magnetic field.

Positive bounded solutions for semilinear elliptic equations in smooth domains

We are concerned with the following semilinear elliptic equation $\Delta u=\lambda f\left( x,u\right) $ in $D,$ subject to some Dirichlet conditions, where $\lambda \geq 0$ is a parameter and $D$ is a smooth domain in $\mathbb{R}^{n}\left( n\geq 3\right) $. Under some appropriate assumptions on the nonnegative nonlinearity term $f\left( x,u\right) ,$ we show the existence of a positive bounded solution for the above semilinear elliptic equation. Our approach is based on Schauder's fixed point Theorem.