Necessary Conditions For Convergence Research Articles

Some Applications of Series Theory in Solving Economic Problems

The paper provides a formula for finding the present value of the payments stream, annuity. The application of the formula of the sum of an infinitely decreasing geometric progression to find the present value of infinite annuity is considered. The concepts of convergence, divergence of series, a necessary condition for convergence, and a sign of comparison are also given in this paper.

Convergence of deep ReLU networks

We explore convergence of deep neural networks with the popular ReLU activation function, as the depth of the networks tends to infinity. To this end, we introduce the notion of activation domains and activation matrices of a ReLU network. By replacing applications of the ReLU activation function by multiplications with activation matrices on activation domains, we obtain an explicit expression of the ReLU network. We then identify the convergence of the ReLU networks as convergence of a class of infinite products of matrices. Sufficient and necessary conditions for convergence of these infinite products of matrices are studied. As a result, we establish necessary conditions for ReLU networks to converge that the sequence of weight matrices converges to the identity matrix and the sequence of the bias vectors converges to zero as the depth of ReLU networks increases to infinity. Moreover, we obtain sufficient conditions in terms of the weight matrices and bias vectors at hidden layers for pointwise convergence of deep ReLU networks. These results provide mathematical insights to convergence of deep neural networks. Experiments are conducted to mathematically verify the results and to illustrate their potential usefulness in initialization of deep neural networks.

Demographic Globalization: Contribution to Global Economic Growth in Long-Term Perspective

Demography is one of the important factors of the Great Convergence. We estimate the contribution of the demographic factor to the prospective GDP dynamics for all countries of the world up to 2100. We use a long-term economic growth model that is based on estimates of labor productivity growth rates and incorporates the demographic forecast for both population and life expectancy. Globally, our model estimates long-term economic growth of world GDP at an average annual rate of 1.7 per cent, of which the demographic factor accounts for 0.54 percentage points, or about one-third (31.8 per cent) of the projected world economic growth in the twenty-first century. In our forecasts, we have relied on the logic of the previous research on convergence, which considered a sufficiently high level of the human capital development and a sufficient degree of economic openness as necessary conditions for convergence, which have now largely been achieved throughout the world precisely through the globalization processes (especially in the core and the semi-periphery of the World System). However, the inclusion of the demographic projections in our forecasts allows us to emphasize an important feature of the perspective convergence – it is not the semi-periphery, but rather the periphery of the World System, or the ‘bottom billion’ according to Collier, that is likely to converge particularly rapidly with both the core and the semi-periphery (such as the BRICS countries) thanks to favorable demographic structure. The paper has been prepared within the framework of RANEPA state assignment in 2023.

Extrapolated splitting methods for multilinear PageRank computations

Multilinear PageRank is a variant of the well-known PageRank model. With this model, web page ranking can be more accurate and efficient by taking into account higher-order connections between pages. The higher-order power method is commonly employed for computing the multilinear PageRank vector, since it is a natural extension of the traditional power method used in the PageRank algorithm. However, this method may not be efficient when the hyperlink tensor becomes large or the damping factor value fails to meet the necessary conditions for convergence. In this work, we propose a novel approach to efficiently computing the multilinear PageRank vector using tensor splitting and vector extrapolation methods.

Some properties of branching processes with random control functions and affected by viral infectivity in random environments

In this paper, a model of branching processes with random control functions and affected by viral infectivity in independent and identically distributed random environments is established, and the Markov property of the model and a sufficient condition for the model to be certainly extinct under some conditions are discussed. Then, the limit properties of the model are studied. Under the normalization factor {S_{n}:nin N}, the normalization processes {hat{W}_{n}:nin N} are studied, and the sufficient conditions of {hat{W}_{n}:nin N} a.s., L^{1} and L^{2} convergence are given; A sufficient condition and a necessary condition for convergence to a nondegenerate at zero random variable are obtained. Under the normalization factor {I_{n}:nin N}, the normalization processes {bar{W}_{n}:nin N} are studied, and the sufficient conditions of {bar{W}_{n}:nin N} a.s., and L^{1} convergence are obtained.

An Analytical Study on Two High-Order Hybrid Methods to Solve Systems of Nonlinear Equations

In order to solve systems of nonlinear equations, two novel iterative methods are presented. The successive over-relaxation method and the Chebyshev-like iterative methods to solve systems of nonlinear equations have combined to obtain the new algorithms. By this combination, two powerful hybrid methods are obtained. Necessary conditions for convergence of these methods are presented. Furthermore, the stability analysis of both algorithms is investigated. These algorithms are applied for solving two real stiff systems of ordinary differential equations. These systems arise from an HIV spreading model and an SIR model of an epidemic which formulates the spread of a nonfatal disease in a certain population. Numerical results show promising convergence and stability for both new hybrid methods.

Критерий сходимости квантовой относительной энтропии и его использование

Получены критерий и необходимое условие сходимости (локальной непрерывности) квантовой относительной энтропии. Рассмотрены некоторые приложения этих результатов. В частности, доказано сохранение локальной непрерывности квантовой относительной энтропии при действии вполне положительных линейных отображений. Библиография: 29 названий.

Phase-space structure of protohalos: Vlasov versus particle-mesh

The phase-space structure of primordial dark matter halos is revisited using cosmological simulations with three sine waves and cold dark matter (CDM) initial conditions. The simulations are performed with the tessellation based Vlasov solver ColDICE and a particle-mesh (PM) N-body code. The analyses include projected density, phase-space diagrams, radial density ρ(r), and pseudo-phase space density: Q(r) = ρ(r)/σv(r)3 with σv the local velocity dispersion. Particular attention is paid to force and mass resolution. Because the phase-space sheet complexity, estimated in terms of total volume and simplex (tetrahedron) count, increases very quickly, ColDICE can follow only the early violent relaxation phase of halo formation. During the violent relaxation phase, agreement between ColDICE and PM simulations having one particle per cell or more is excellent and halos have a power-law density profile, ρ(r) ∝ r−α, α ∈ [1.5, 1.8]. This slope, measured prior to any merger, is slightly larger than in the literature. The phase-space diagrams evidence complex but coherent patterns with clear signatures of self-similarity in the sine wave simulations, while the CDM halos are somewhat scribbly. After additional mass resolution tests, the PM simulations are used to follow the next stages of evolution. The power law progressively breaks down with a convergence of the density profile to the well-known Navarro–Frenk–White universal attractor, irrespective of initial conditions, that is even in the three-sine-wave simulations. This demonstrates again that mergers do not represent a necessary condition for convergence to the dynamical attractor. Not surprisingly, the measured pseudo phase-space density is a power law Q(r) ∝ r−αQ, with αQ close to the prediction of secondary spherical infall model, αQ ≃ 1.875. However this property is also verified during the early relaxation phase, which is non-trivial.

Variational assimilation of surface wave data for bathymetry reconstruction. Part I: algorithm and test cases

Accurate mapping of ocean bathymetry is needed for effective modelling of ocean dynamics, such as tsunami prediction. Available bathymetry data does not always provide the resolution to model such nonlinear waves accurately, and collection of accurate data is logistically challenging. As an alternative, in this study we develop and evaluate a variational data assimilation scheme for the one-dimensional nonlinear shallow water equations that estimates bathymetry using a finite set of observations of surface wave height. We demonstrate that convergence to exact bathymetry is improved by including more observation locations and by implementing a low-pass filter in the data assimilation algorithm to remove small-scale noise. A necessary condition for convergence of the bathymetry reconstruction is that the amplitude of the initial conditions is less than 1% of the bathymetry height. We use density-based global sensitivity analysis (GSA) to assess the sensitivity of the surface wave and reconstruction error to model parameters. By demonstrating low sensitivity of the surface wave to the reconstruction error, we show that reconstructing the bathymetry with a relative error of about 10% is sufficiently accurate for surface wave modelling in most cases. These results can be used to guide the development of similar assimilation schemes in higher dimensions and more realistic geometries.

Monte-Carlo for solving large linear systems of ordinary differential equations

Monte-Carlo approach towards solving Cauchy problem for large systems of linear differential equations is being proposed in this paper. Firstly, a quick overlook of previously obtained results from applying the approach towards Fredholm-type integral equations is being made. In the main part of the paper, a similar method is being applied towards a linear system of ODE. It is transformed into an equivalent system of Volterra-type integral equations, which relaxes certain limitations being present due to necessary conditions for convergence of majorant series. The following theorems are being stated. Theorem 1 provides necessary compliance conditions that need to be imposed upon initial and transition distributions of a required Markov chain, for which an equality between estimate’s expectation and a desirable vector product would hold. Theorem 2 formulates an equation that governs estimate’s variance, while theorem 3 states a form for Markov chain parameters that minimise the variance. Proofs are given, following the statements. A system of linear ODEs that describe a closed queue made up of ten virtual machines and seven virtual service hubs is then solved using the proposed approach. Solutions are being obtained both for a system with constant coefficients and time-variable coefficients, where breakdown intensity is dependent on t. Comparison is being made between Monte-Carlo and Rungge Kutta obtained solutions. The results can be found in corresponding tables.

Non‐linear estimation and observer‐based output feedback control

In this study, a new observer is proposed for a class of non-linear systems which is based on an output-dependent Riccati equation. Necessary conditions for convergence of the state estimation to the system state are investigated through a theorem. Then, based on the proposed observer, two techniques are developed to solve non-linear stabilisation and non-linear tracking problems. It is shown that the separation principle between the estimation and control holds. Indeed, just like a linear system, a decentralised observer-based state feedback controller can be designed for the non-linear system while the stability of the closed-loop system is guaranteed. For the tracking problem, it is proved that the closed-loop system states converge asymptotically to the states of the desired model. Numerical simulations are given to demonstrate the effectiveness of the proposed observer and the observer-based controllers.

Vehicle Lateral Dynamic Identification Method Based on Adaptive Algorithm

The development of advanced driver assistance systems relies on an accurate estimation of the tire-road friction coefficient and cornering stiffness of the vehicle, which are closely linked to internal and external driving conditions. In this paper, an identification algorithm capable of simultaneously estimate the friction coefficient and cornering stiffness of the front and rear tires is pursued. A nonlinear adaptive law is proposed for the estimation of vehicle parameters under certain excitation conditions. It is shown that, by exploring the lateral dynamic of the vehicle, the convergence of the parameters to their true values can be guaranteed. A comprehensive study has been carried out in order to reveal the necessary conditions for convergence and observability of the parameters. Simulation results with a high fidelity full order Carsim model show a good performance of the proposed identification method.

Convergence for Slow Discrete Dynamical Systems with Identity Linearization

In this work we give sufficient and necessary conditions for convergence for nonhyperbolic fixed points of dynamical systems of arbitrary dimension whose linearization around zero is the identity function. To achieve this goal, we first rewrite the dynamical system in terms of spherical polar coordinates and by approximation of the radial iteration function we discover a necessary condition depending on a remarkable angular function. Searching for conditions that are sufficient, we discover more angular functions that together with the first one gives a complete set that plays the role of the iteration derivative for unidimensional discrete systems.

Numerical solutions of compressible convection with an infinite Prandtl number: comparison of the anelastic and anelastic liquid models with the exact equations

We developed a numerical method for the set of equations governing fully compressible convection in the limit of infinite Prandtl numbers. Reduced models have also been analysed, such as the anelastic approximation and the anelastic liquid approximation. The tests of our numerical schemes against self-consistent criteria have shown that our numerical simulations are consistent from the point of view of energy dissipation, heat transfer and entropy budget. The equation of state of an ideal gas has been considered in this work. Specific effects arising because of the compressibility of the fluid are studied, like the scaling of viscous dissipation and the scaling of the heat flux contribution due to the mechanical power exerted by viscous forces. We analysed the solutions obtained with each model (fully compressible model, anelastic and anelastic liquid approximations) in a wide range of dimensionless parameters and determined the errors induced by each approximation with respect to the fully compressible solutions. Based on a rationale on the development of the thermal boundary layers, we can explain reasonably well the differences between the fully compressible and anelastic models, in terms of both the heat transfer and viscous dissipation dependence on compressibility. This could be mostly an effect of density variations on thermal diffusivity. Based on the different forms of entropy balance between exact and anelastic models, we find that a necessary condition for convergence of the anelastic results to the exact solutions is that the product $\unicode[STIX]{x1D716}q$ must be small compared to unity, where $\unicode[STIX]{x1D716}$ is the ratio of the superadiabatic temperature difference to the adiabatic difference, and $q$ is the ratio of the superadiabatic heat flux to the heat flux conducted along the adiabat. The same condition seems also to be associated with a convergence of the computed heat fluxes. Concerning the anelastic liquid approximation, we confirm previous estimates by Anufriev et al. (Phys. Earth Planet. Inter., vol. 152, 2005, pp. 163–190) and find that its results become generally close to those of the fully compressible model when $\unicode[STIX]{x1D6FC}T{\mathcal{D}}$ is small compared to unity, where $\unicode[STIX]{x1D6FC}$ is the isobaric thermal expansion coefficient, $T$ is the temperature (here $\unicode[STIX]{x1D6FC}T=1$ for an ideal gas) and ${\mathcal{D}}$ is the dissipation number.

Sufficient and necessary conditions of convergence for ρ͠ mixing random variables

Abstract In the present paper, the sufficient and necessary conditions of the complete convergence and complete moment convergence for ρ͠-mixing random variables are established, which extend some well-known results.

Convergence on Two Dimensional 1-Slow Discrete Dynamical Systems

In this work we give sufficient and necessary conditions for convergence for nonhyperbolic fixed points of two dimensional 1-slow discrete dynamical systems. To achieve this goal, we introduce two angular functions that play the role of the iteration derivatives for one dimensional discrete systems. We also discuss the convergence on different regions of the two dimensional phase space, including the quadrants. In this case, sufficient convergence conditions are given. Several two dimensional examples are presented to illustrate the use of the previous results.

A Simple Decision Market Model

Economic modeling of decision markets has mainly considered the market scoring rule setup. Literature has made reference to the alternative, joint elicitation type decision market, but no in depth analysis of it appears to have been published. This paper develops a simple decision market model of the joint elicitation type, that provides a specific decision market nomenclature on which to base future analysis. A generally accepted prediction market model is modified, by introducing two additional concepts: “proper information market” and “relevant information”. Our work then provides original contributions to the theoretical discourse on information markets, including finding the sufficient and necessary condition for convergence to the best possible prediction. It is shown in our new prediction market model that “all agents express relevant information” is a sufficient and necessary condition for convergence to the direct communication equilibrium in a proper information (prediction) market. Our new prediction market model is used to formulate a simple decision market model of the joint elicitation market type. It is shown that our decision market will select the best decision if a specific selection and payout rule is defined. Importantly, our decision market model does not need to delay payment of any contracts to the observation of the desired outcome. Therefore, when dealing with long-term outcome projects, our decision market does not need to be a long running market. Future work will test for the statistical significance of relevant information (identified as important in our idealized decision market model) in laboratory and real world settings.

Diffuse scattering on graphs

We formulate and analyze difference equations on graphs analogous to time-independent diffusion equations arising in the study of diffuse scattering in continuous media. Moreover, we show how to construct solutions in the presence of weak scatterers from the solution to the homogeneous (background problem) using Born series, providing necessary conditions for convergence and demonstrating the process through numerous examples. In addition, we outline a method for finding Green's functions for Cayley graphs for both abelian and non-abelian groups. Finally, we conclude with a discussion of the effects of sparsity on our method and results, outlining the simplifications that can be made provided that the scatterers are weak and well-separated.

Necessary conditions for convergence of difference schemes for fractional-derivative two-point boundary value problems

A fractional-derivative two-point boundary value problem of the form \({\tilde{D}}^\delta u=f\) on (0, 1) with Dirichlet boundary conditions is studied. Here \({\tilde{D}}^\delta \) is a Caputo or Riemann–Liouville fractional derivative operator of order \(\delta \in (1,2)\). The discretisation of this problem by an arbitrary difference scheme is examined in detail when u or f is a polynomial. For any convergent difference scheme, it is proved rigorously that the entries of the associated matrix must satisfy certain identities. It is shown that some of these identities are not satisfied by certain well-known schemes from the research literature; this clarifies the type of problem to which these schemes can be applied successfully. The effects of the special boundary condition \(u(0)=0\) and the special right-hand-side condition \(f(0)=0\) are also investigated. This leads, under certain circumstances, to a sharpening of a recently-published finite difference scheme convergence result of two of the authors.

A new convergence rate estimation of general artificial immune algorithm

Artificial immune algorithm has been used widely and successfully in many computational optimization areas, but the theoretical research exploring the convergence rate characteristics of artificial immune algorithm is yet inadequate. In this paper, instead of the traditional eigenvalue estimation of state transition matrix, stochastic processes theory is introduced to study the convergence rate of general artificial immune algorithm. The method begins by analyzing the necessary condition for convergence of artificial immune algorithm and takes it as the sufficient condition for a class of general artificial immune algorithm. Through the definition of Markov chain convergence rate, a probability strong convergence rate estimation method of general artificial immune algorithm is proposed. This method is judged by the final convergence of the best antibody, which overcomes the conservative defect of traditional estimation methods. The simulation results show the correctness of the proposed estimation method, and the estimation method can be used to judge the convergence and convergence rate of a class of artificial immune algorithms. This research has a certain theoretical reference value to optimize the convergence rate in the practical application of artificial immune algorithm.