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With increasing of data size and development of multi-core computers, asynchronous parallel stochastic optimization algorithms such as KroMagnon have gained significant attention. In this paper, we propose a new Sparse approximation and asynchronous parallel Stochastic Variance Reduced Gradient (SSVRG) method for sparse and high-dimensional machine learning problems. Unlike standard SVRG and its asynchronous parallel variant, KroMagnon, the snapshot point of SSVRG is set to the average of all the iterates in the previous epoch, which allows it to take much larger learning rates and makes it more robust to the choice of learning rates. In particular, we use the sparse approximation of the popular SVRG estimator to perform completely sparse updates. Therefore, SSVRG has a much lower per-iteration computational cost than its dense counterpart, SVRG++, and is very friendly to asynchronous parallel implementation. Moreover, we provide the convergence guarantees of SSVRG for both SC and non-SC problems, while existing asynchronous algorithms (e.g., KroMagnon) only have convergence guarantees for SC problems. Finally, we extend SSVRG to non-smooth and asynchronous parallel settings. Numerical results demonstrate that SSVRG converges significantly faster than the state-of-the-art asynchronous parallel methods, e.g., KroMagnon, and is usually more than three orders of magnitude faster than SVRG++.

In this paper, we present a new hybrid spectral-conjugate gradient (SCG) algorithm for finding approximate solutions to nonlinear monotone operator equations. The hybrid conjugate gradient parameter has the Polak–Ribiere–Polyak (PRP), Dai–Yuan (DY), Hestenes–Stiefel (HS) and Fletcher–Reeves (FR) as special cases. Moreover, the spectral parameter is selected such that the search direction has the descent property. Also, the search directions are bounded and the sequence of iterates generated by the new hybrid algorithm converge globally. Furthermore, numerical experiments were conducted on some benchmark nonlinear monotone operator equations to assess the efficiency of the proposed algorithm. Finally, the algorithm is shown to have the ability to recover disturbed signals.

Lisätään OA-artikkeli, kun julkaistu We address the problem of autonomous tracking and state estimation for marine vessels, autonomous vehicles, and other dynamic signals under a (structured) sparsity assumption. The aim is to improve the tracking and estimation accuracy with respect to the classical Bayesian filters and smoothers. We formulate the estimation problem as a dynamic generalized group Lasso problem and develop a class of smoothing-and-splitting methods to solve it. The Levenberg-Marquardt iterated extended Kalman smoother-based multiblock alternating direction method of multipliers (LM-IEKS-mADMMs) algorithms are based on the alternating direction method of multipliers (ADMMs) framework. This leads to minimization subproblems with an inherent structure to which three new augmented recursive smoothers are applied. Our methods can deal with large-scale problems without preprocessing for dimensionality reduction. Moreover, the methods allow one to solve nonsmooth nonconvex optimization problems. We then prove that under mild conditions, the proposed methods converge to a stationary point of the optimization problem. By simulated and real-data experiments, including multisensor range measurement problems, marine vessel tracking, autonomous vehicle tracking, and audio signal restoration, we show the practical effectiveness of the proposed methods.

The online social network has become an integral part of modern society and serves as an excellent platform for information diffusion, marketing, and so on. The minimum positive influence dominating set (MPIDS) problem aims to find a set of at least half neighbors of every individual with minimum cardinality and has lots of applications in the social network. Even though numerous algorithms have been proposed for this hard combinatorial problem, challenges such as poor quality and scalability have motivated the researchers for better solutions. In this article, we focus on designing a simple and effective method and propose an iterated carousel greedy (ICG) algorithm for the MPIDS problem. The destruction, carousel, and reconstruction (DCR) procedures are well-designed based on the problem-specific knowledge to improve the algorithm performance, while maintaining the simplicity of the ICG algorithm. The ICG algorithm uses a fast greedy algorithm to generate an initial solution with a certain quality. It then obtains a sequence of better solutions by iterating over DCR phases. A carefully designed experiment is carried out to determine the best parameter configuration. The experimental results on real-world networks indicate that the proposed ICG algorithm significantly outperforms the current state-of-the-art algorithms.

Fractional differential equations with constant coefficients can be readily handled by a number of methods, but those with variable coefficients are much more challenging. Recently, a method has appeared in the literature for solving fractional differential equations with variable coefficients, the solution being in the form of an infinite series of iterated fractional integrals. In the current work, we consider fractional differential equations with Atangana–Baleanu integro-differential operators and continuous variable coefficients, and establish analytical solutions for such equations. The representation of the solution is given by a uniformly convergent infinite series involving Atangana–Baleanu operators. To the best of our knowledge, this is the first time that explicit analytical solutions have been given for such general Atangana–Baleanu differential equations with variable coefficients. The corresponding results for fractional differential equations with constant coefficients are also given.

We propose a biobjective robust simulation-based optimization (BORSO) method to solve unconstrained problems involving implementation errors and parameter perturbations. We adopt the notion that a solution is robust efficient (RE) if the region that dominates its worst-case realizations of the biobjectives under uncertainty does not contain (all) the worst-case realizations of the biobjectives of any other solution under uncertainty. Our algorithm aims to efficiently find a set of RE solutions through a series of function evaluations or simulations. First, we design surrogate-model guided search mechanisms for the worst-case neighbors of the current iterate. Subsequently, we determine the iteration distance along an effective local move direction, which excludes the worst-case neighbors from the neighborhood of the new iterate. Depending on the practical need for solution diversity, multiple initial solutions can be specified in our algorithm, and the final iterates of these solutions generate a set of RE solutions. The test results of a synthetic biobjective nonconvex optimization problem show the effectiveness of the BORSO method and its superior performance against a sampling-based robust multiobjective optimization solver. Furthermore, when the proposed algorithm is applied to a real-world biobjective traffic signal timing problem, satisfactory solutions can be obtained under a limited computational budget. These results indicate that the proposed BORSO method can solve unconstrained biobjective simulation-based optimization problems under uncertainties.

Packing equal circles in a circle is a classic global optimization problem that has a rich research history and a number of relevant applications. The problem is computationally challenging due to the fact that the number of possible packing configurations grows exponentially with the number of circles. In this work, we propose a highly effective iterated dynamic thresholding search algorithm for solving this difficult problem. The algorithm integrates several features including a two-phase local optimization method, a dynamic thresholding search and a container adjustment procedure. Computational experiments on popular benchmark instances with up to N = 320 circles show that the algorithm outperforms significantly the state-of-the-art algorithms. In particular, it improves the best-known results for 136 instances, while matching the best-known results for other 175 instances.

Momentum technique has recently emerged as an effective strategy in accelerating convergence of gradient descent (GD) methods and exhibits improved performance in deep learning as well as regularized learning. Typical momentum examples include Nesterov's accelerated gradient (NAG) and heavy-ball (HB) methods. However, so far, almost all the acceleration analyses are only limited to NAG, and a few investigations about the acceleration of HB are reported. In this article, we address the convergence about the last iterate of HB in nonsmooth optimizations with constraints, which we name individual convergence. This question is significant in machine learning, where the constraints are required to impose on the learning structure and the individual output is needed to effectively guarantee this structure while keeping an optimal rate of convergence. Specifically, we prove that HB achieves an individual convergence rate of O(1/√t) , where t is the number of iterations. This indicates that both of the two momentum methods can accelerate the individual convergence of basic GD to be optimal. Even for the convergence of averaged iterates, our result avoids the disadvantages of the previous work in restricting the optimization problem to be unconstrained as well as limiting the performed number of iterations to be predefined. The novelty of convergence analysis presented in this article provides a clear understanding of how the HB momentum can accelerate the individual convergence and reveals more insights about the similarities and differences in getting the averaging and individual convergence rates. The derived optimal individual convergence is extended to regularized and stochastic settings, in which an individual solution can be produced by the projection-based operation. In contrast to the averaged output, the sparsity can be reduced remarkably without sacrificing the theoretical optimal rates. Several real experiments demonstrate the performance of HB momentum strategy.

In this paper, we propose two inertial projection methods to solve mixed variational inequalities without assuming monotonicity on the cost operator. Consequently, we show that the sequence of iterates generated by our proposed methods converge globally to a solution of the mixed variational inequality under the condition that the set of solutions of the dual mixed variational inequality is nonempty. Our convergence results are obtained without assuming any further stringent condition imposed in other related results in the literature. Moreover, our results improve on many important related results in the literature. Some numerical illustrations are given to show the efficiency of our proposed methods.

We propose a novel formalization of roles in social networks that unifies the most commonly used definitions of role equivalence. As one consequence, we obtain a single, straightforward proof that role equivalences form lattices. Our formalization focuses on the evolution of roles from arbitrary initial conditions and thereby generalizes notions of relative and iterated roles that have been suggested previously. In addition to the unified structure result this provides a potential micro-foundation for the emergence of roles. Considering the genesis of roles may explain, and help overcome, the problem that social networks rarely exhibit interesting role equivalences of the traditional kind. Finally, we discuss extensions of the role concept to multivariate networks with actor and tie attributes by example., Social Networks, 68

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/). Battery energy storage systems (BESSs) are expect‐ ed to play a crucial role in the operation and control of active distribution networks (ADNs). In this paper, a holistic state esti‐ mation framework is developed for ADNs with BESSs integrat‐ ed. A dynamic equivalent model of BESS is developed, and the state transition and measurement equations are derived. Based on the equivalence between the correction stage of the iterated extended Kalman filter (IEKF) and the weighted least squares (WLS) regression, a holistic state estimation framework is pro‐ posed to capture the static state variables of ADNs and the dy‐ namic state variables of BESSs, especially the state of charge (SOC). A bad data processing method is also presented. The simulation results show that the proposed holistic state estima‐ tion framework improves the accuracy of state estimation as well as the capability of bad data detection for both ADNs and BESSs, providing comprehensive situational awareness for the whole system.

We consider the problem of sensor and actuator (SaA) placement to minimize an infinite-horizon LQG cost for a discrete-time Gauss-Markov system. Due to financial, topology and bandwidth limitations, only a subset of SaAs can be selected for placement. Different from existing literature, which successively iterates partial selection to reach a suboptimal solution, this paper focuses on a joint placement and encounters fundamental difficulty in the sense that joint SaA placement introduces a term in the LQG cost that is difficult to be convexified. A branch-and-bound algorithm is introduced to search for solutions to an approximate problem obtained by relaxing the Boolean constraints. By deriving a compact search space where any optimal solution belongs to or resides, we derive lower and upper bounds to the optimal solution in each subregion of the search space, and subsequently refine the search space. A suboptimal solution to the original problem is obtained from integer rounding and the optimality gap is further analyzed. Numerical examples are provided to illustrate the effectiveness of the proposed algorithm. This algorithm has significant reduction of iteration numbers compared with the brute-force enumeration especially when the number of states is not large but the number of placement choices is large. In addition, an improvement of LQG cost is obtained compared with the successively iterating partial selection methods, a canonical algorithm for SaA placement in the literature.

This paper compiles several aspects of the dynamics of stochastic approximation algorithms with Markov iterate-dependent noise when the iterates are not known to be stable beforehand. We achieve the same by extending the lock-in probability (i.e. the probability of convergence of the iterates to a specific attractor of the limiting o.d.e. given that the iterates are in its domain of attraction after a sufficiently large number of iterations (say) n0) framework to such recursions. We use these results to prove almost sure convergence of the iterates to the specified attractor when the iterates satisfy an asymptotic tightness condition. The novelty of our approach is that if the state space of the Markov process is compact we prove almost sure convergence under much weaker assumptions compared to the work by Andrieu et al. which solves the general state space case under much restrictive assumptions. We also extend our single timescale results to the case where there are two separate recursions over two different timescales. This, in turn, is shown to be useful in analyzing the tracking ability of general adaptive algorithms.

In previous studies, the Huckel-London-Pople-McWeeny (HLPM) method was used to calculate the "topological" ring currents and bond currents that are associated with the mono-altans, and the doubly and triply iterated altans, of corannulene and coronene and their respective dianions. This gave rise to some intriguing patterns in these currents, which were partially rationalized by a detailed examination of the lone nonbonding orbitals that each of these structures possesses. In the present work, the same HLPM approach is used to investigate whether analogous changes in bond-current patterns are observed when going from the neutral species to the dianions of the iterated altans derived from some other, simpler, "parents" having molecular graphs with a nullity of zero. The altans and iterated altans of three of the simplest parents are studied: the neutral species (altan)n-benzene, (altan)n-naphthalene, and (altan)n-azulene, with n = 1, 2, and 3, together with their corresponding dianions. Analogous bond-current patterns are indeed encountered, some of which are likewise rationalized by a detailed consideration of the nonbonding orbitals.

Let $\alpha\in (0, 1]$, $\beta\in [0, n)$ and $T_{\Omega,\beta}$ be a singular or fractional integral operator with homogeneous kernel $\Omega$. In this article, a CMO type space ${\rm CMO}_\alpha(\mathbb R^n)$ is introduced and studied. In particular, the relationship between ${\rm CMO}_\alpha(\mathbb R^n)$ and the Lipchitz space $Lip_\alpha(\mathbb R^n)$ is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator $(T_{\Omega,\beta})^m_b$ on weighted Lebesgue spaces via functions in $Lip_\alpha(\mathbb R^n)$, and an equivalent characterization of the compactness for $(T_{\Omega,\beta})^m_b$ via functions in ${\rm CMO}_\alpha(\mathbb R^n)$ are obtained. Some results are new even in the unweighted setting for the first order commutators., Comment: arXiv admin note: text overlap with arXiv:1712.08292

In this paper, we prove the weak convergence of the iterates generated by the nonlinearly preconditioned forward-backward splitting method for the sum of a maximally hypermonotone operator A and a ...

In this article, we study the fundamental solution of the operator $$\left((\diamond+m^2)\left(\frac{\triangle^2+\boxdot^2}{2}\right)\right)^{k}$$ , iterated $k$ -times and is defined by ( \ref { odot }) ,where $m$ is a nonnegative real number, and $k$ is a nonnegative integer. After that, we study the Fourier transform of the operator $ \left((\diamond+m^2)\left(\frac{\triangle^2+\boxdot^2}{2}\right)\right)^{k}\delta$ .

In the Nash demand game (NDG) n players announce utility demands, the demands are implemented if they are jointly feasible, and otherwise no one gets anything. If the utilities set is the simplex, the game is called “divide-the-dollar”. Brams and Taylor (Theory Decis 37:211–231, 1994) studied variants of divide-the-dollar, on which they imposed reasonableness conditions. I explore the implications of these conditions on general NDGs. In any reasonable NDG, the egalitarian demand profile cannot be obtained via iterated elimination of weakly dominated strategies. Further, a reasonable NDG may fail to have a Nash equilibrium, even in mixed strategies. In the 2-person case, existence of pure strategy equilibrium is equivalent to the existence of a value, in the sense that each player can secure the egalitarian payoff level independent of his opponent’s play. This result does not extend to reasonable NDGs with more than two players. Interestingly, there are results for reasonable NDGs that hold for two and three players, but not for $$n\ge 4$$ players.

In the linguistics literature, the derivation of scalar implicatures has often been handled in a relatively modular way, using computations that are sensitive to logical relations among alternatives such as entailment but are blind to other notions such as the probabilities that participants in a conversation might associate with these alternatives (or with related propositions). In recent years, a family of models that we refer to as iterated rationality models (IRMs) have offered an interestingly different perspective on such alternative-sensitive processes in terms of multiple iterations of (typically probabilistic) reasoning. Our paper investigates what at first sight seems like a very interesting argument for the adequacy of IRMs as models of scalar implicatures, coming from the conjunctive interpretation of disjunctive sentences. We then outline challenges for the argument based on a theoretical comparison with the grammatical theory of scalar implicatures. The comparison focuses on the full distribution of conjunctive interpretation, on the question of sensitivity to priors, and on other results achieved within the grammatical theory that the IRM literature has yet to engage with.

The logic $${\mathbb{L}}_\theta ^1$$ was introduced in [She12]; it is the maximal logic below $${{\mathbb{L}}_{\theta, \theta}}$$ in which a well ordering is not definable. We investigate it for θ a compact cardinal. We prove that it satisfies several parallels of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are $${\mathbb{L}}_\theta ^1$$ -equivalent iff for some ω-sequence of θ-complete ultrafilters, the iterated ultrapowers by it of those two models are isomorphic. Also for strong limit λ>θ of cofinality $${\aleph _0}$$ , every complete $${\mathbb{L}}_\theta ^1$$ -theory has a so-called special model of cardinality λ, a parallel of saturated. For first order theory T and singular strong limit cardinal λ, T has a so-called special model of cardinality λ. Using “special” in our context is justified by: it is unique (fixing T and λ), all reducts of a special model are special too, so we have another proof of interpolation in this case.

The ill-posed problem of unknown source identification in the time-fractional radial heat conduction equation is studied. In order to overcome the ill-posedness of the problem, a stationary iterated weighted Tikhonov regularization method is proposed. The a-priori and the a-posteriori choice rules for regularization parameters are discussed and the corresponding convergence rates of both are obtained. The stationary iterated weighted Tikhonov regularization method goes beyond the saturation results of weighted Tikhonov regularization method. A numerical example shows the effectiveness of this method.

We consider a Urysohn integral operator $\mathcal{K}$ with kernel of the type of Green's function. For $r \geq 1$, a space of piecewise polynomials of degree $\leq r-1 $ with respect to a uniform partition is chosen to be the approximating space and the projection is chosen to be the orthogonal projection. Iterated Galerkin method is applied to the integral equation $x - \mathcal{K}(x) = f$. It is known that the order of convergence of the iterated Galerkin solution is $r+2$ and, at the above partition points it is $2r$. We obtain an asymptotic expansion of the iterated Galerkin solution at the partition points of the above Urysohn integral equation. Richardson extrapolation is used to improve the order of convergence. A numerical example is considered to illustrate our theoretical results.

The continuous iteration of an exponential-type function allows to describe a whole range of growths between logarithmical and exponential. This principle has been applied to a class of lifetime di...

We introduce reversed mean Lipschitzian mappings and find the bound for reversed mean Lipschitz constants of its iterates under certain conditions. Moreover, we prove that if a bounded closed and c...

Phase-only beamforming (POB) plays an important role in modern radar and communication systems. The NP-hard nature of POB makes it difficult to be solved for large-scale antenna arrays. This paper studies two fast and simple GP algorithms for the problem of POB. First we give a new interpretation of the GP method under the framework of sequential quadratic programming (SQP). Then we prove that the GP operation gives a strict global minimizer of the subproblem of SQP. Moreover, the larger the step size in GP operation, the smaller the minimum value of the subproblem. Based on this observation we design two variable step size GP (VSGP) algorithms that improve the convergence speed of the GP method significantly. Furthermore, the VSGP algorithms have a simple structure, making them easy to be implemented in chips. Finally, we derive a necessary and sufficient condition for the Karush-Kuhn-Tucker (KKT) points of the POB problem. We also prove that, for any initialization, all limit points of the iterates generated by the VSGP algorithms converge to the KKT points. Simulation results show that the proposed algorithms outperform the state-of-the-art algorithms in terms of convergence speed and computational cost.

In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations ( fie s) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems ( evp s) associated with the two-dimensional fie s. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional fie s in both L 2 and L ∞ norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in L 2 and L ∞ norms. The numerical illustrations are introduced for the error of these methods.

Euler [6] and Eisenstein [5] explored the concept of iterated exponents and their convergence. One particular iterated exponent of interest is i(i(i⋰)), which can be shown to converge to z⋆≈0.43828...

It is crucial to support students in better understanding water and sustainability issues because water plays a vital role in maintaining global ecosystems, including human life. A wide range of cu...

It is known that if a bivariate meanKis symmetric, continuous and strictly increasing in each variable, then for every meanMthere is a unique mean$$N\,$$Nsuch thatKis invariant with respect to the mean-type mapping$$\left( M,N\right) ,$$M,N,which means that$$K\circ \left( M,N\right) =K$$K∘M,N=KandNis called aK-complementary mean forM(Matkowski in Aequ Math 57(1):87–107, 1999). This paper extends this result for a large class of nonsymmetric means. As an application, the limits of the sequences of iterates of the related mean-type mappings are determined, which allows us to find all continuous solutions of some functional equations.

Given an undirected connected graph G = ( V , E ) with vertex set V and edge set E, the minimum conductance graph partitioning problem is to partition V into two disjoint subsets such that the conductance, i.e., the ratio of the number of cut edges to the smallest volume of two partition subsets is minimized. This problem has a number of practical applications in various areas such as community detection, bioinformatics , and computer vision . However, the problem is computationally challenging, especially for large problem instances. This work presents the first iterated multilevel simulated annealing algorithm for large-scale graph conductance minimization. The algorithm features a novel solution-guided coarsening method and an effective solution refinement procedure based on simulated annealing. Computational experiments demonstrate the high performance of the algorithm on 66 very large real-world sparse graphs with up to 23 million vertices. Additional experiments are presented to get insights into the influences of its algorithmic components . The source code of the proposed algorithm is publicly available, which can be used to solve various real world problems .

For the iteration of spherical average (A1)N and the Laplace operator Δ, we consider the boundedness of the operator Δ(A1)N on the α-modulation spaces $$M_{p,q}^{s,\alpha }$$ . The authors obtain some sufficient and necessary conditions to ensure the boundedness on the α-modulation spaces. The main theorems significantly improve some known results.

In this paper, we propose an accelerated variant of the proximal point algorithm for computing a zero of an arbitrary maximally monotone operator A. The method incorporates an inertial term (based upon the acceleration techniques introduced by Nesterov), relaxation factors and a correction term. In a general Hilbert space setting, we obtain the weak convergence of the iterates $$(x_n)$$ to equilibria, with the fast rate $$\Vert x_{n+1}-x_{n}\Vert =o(n^{-1})$$ (for the discrete velocity). In particular, when using constant proximal indexes, we establish the accuracy of $$(x_n)$$ to a zero of A with the worst-case rate $$\Vert A_{\lambda }(x_n)\Vert =o(n^{-1})$$ ( $$A_{\lambda }$$ being the Yosida regularization of A with some index $$\lambda $$ ). Furthermore, given any positive integer p and using an appropriate adjustment of increasing proximal indexes, we obtain the worst-case convergence rate $$\Vert A_{\lambda }(x_n)\Vert =o(n^{-(p+1)})$$ . This acceleration process can be applied to various proximal-type algorithms such as the augmented Lagrangian method, the alternating direction method of multipliers, operator splitting methods and so on. Our methodology relies on a Lyapunov analysis combined with a suitable reformulation of the considered algorithm. Numerical experiments are also performed.

In this paper, we investigate the growth and fixed points of solutions of higher-order linear differential equations in the unit disc. We extend the coefficient conditions to a type of one-constant-control coefficient comparison and obtain the same estimates of iterated order of solutions. We also obtain better estimates by providing a precise value of iterated order of solution instead of a range of that in the case of coefficient characteristic function comparison. Moreover, we utilize iteration to investigate and estimate the fixed points of solutions’ arbitrary-order derivatives with higher-order equations $f^{(k)}+A_{k-1}(z)f^{(k-1)}+{\cdots }+A_{1}(z)f'+A_{0}(z)f=0$ f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = 0 and provide a concise method to judge if the items generated by the iteration do not vanish identically and ensure the iteration proceeds. Our results are an improvement over those by B. Belaïdi, T. B. Cao, G. W. Zhang and A. Chen.

We examine a special case of an approximation of the joint spectral radius given by Blondel and Nesterov, which we call the outer spectral radius. The outer spectral radius is given by the square root of the ordinary spectral radius of the n2 by n2 matrix $$\sum \overline {{X_i}} \otimes {X_i}$$ . We give an analogue of the spectral radius formula for the outer spectral radius which can be used to quickly obtain the error bounds in methods based on the work of Blondel and Nesterov. The outer spectral radius is used to analyze the iterates of a completely positive map, including the special case of quantum channels. The average of the iterates of a completely positive map approach to a completely positive map where the Kraus operators span an ideal in the algebra generated by the Kraus operators of the original completely positive map. We also give an elementary treatment of Popescu’s theorems on similarity to row contractions in the matrix case, describe connections to the Parrilo-Jadbabaie relaxation, and give a detailed analysis of the maximal spectrum of a completely positive map.

The present work presents a new method of integration of non-elementary exponential functions where Fubinni's iterated integrals were used. In this research, some approximations were used in order to generalize the results obtained through mathematical series, in addition to integration methods and double integrals. In addition to the integration methods, the Taylor series was used, where the value found and compatible with the values ​​of the power series that are used to calculate the value of the exponential function demonstrated in the work was verified. In addition to the methods described, a comparison of the values ​​obtained by the series and the values ​​described in the method was improvised, where it was noticed that the higher the value of the variable, the closer the results show a stability for the variable greater than the value 4, described in table 01. The conclusions point to a great improvement, mainly for solving elliptic differential equations and statistical functions.

Funder: Institute of Science and Technology (IST Austria), We study the problem of recovering an unknown signal $${\varvec{x}}$$ x given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator $$\hat{\varvec{x}}^\mathrm{L}$$ x ^ L and a spectral estimator $$\hat{\varvec{x}}^\mathrm{s}$$ x ^ s . The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine $$\hat{\varvec{x}}^\mathrm{L}$$ x ^ L and $$\hat{\varvec{x}}^\mathrm{s}$$ x ^ s . At the heart of our analysis is the exact characterization of the empirical joint distribution of $$({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})$$ ( x , x ^ L , x ^ s ) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of $$\hat{\varvec{x}}^\mathrm{L}$$ x ^ L and $$\hat{\varvec{x}}^\mathrm{s}$$ x ^ s , given the limiting distribution of the signal $${\varvec{x}}$$ x . When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form $$\theta \hat{\varvec{x}}^\mathrm{L}+\hat{\varvec{x}}^\mathrm{s}$$ θ x ^ L + x ^ s and we derive the optimal combination coefficient. In order to establish the limiting distribution of $$({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})$$ ( x , x ^ L , x ^ s ) , we design and analyze an approximate message passing algorithm whose iterates give $$\hat{\varvec{x}}^\mathrm{L}$$ x ^ L and approach $$\hat{\varvec{x}}^\mathrm{s}$$ x ^ s . Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.

In this article we review a model of stochastic evolution under general noisy best-response protocols, allowing the probabilities of suboptimal choices to depend on their payoff consequences. We survey the methods developed by the authors which allow for a quantitative analysis of these stochastic evolutionary game dynamics. We start with a compact survey of techniques designed to study the long run behavior in the small noise double limit (SNDL). In this regime we let the noise level in agents' decision rules to approach zero, and then the population size is formally taken to infinity. This iterated limit strategy yields a family of deterministic optimal control problems which admit an explicit analysis in many instances. We then move in by describing the main steps to analyze stochastic evolutionary game dynamics in the large population double limit (LPDL). This regime refers to the iterated limit in which first the population size is taken to infinity and then the noise level in agents' decisions vanishes. The mathematical analysis of LPDL relies on a sample-path large deviations principle for a family of Markov chains on compact polyhedra. In this setting we formulate a set of conjectures and open problems which give a clear direction for future research activities.

We prove a Korovkin-type theorem for sequences of positive linear operators on C([0, 1]). This theorem allows us to discuss some general results concerning positive linear operators. In particular, we give simple and efficient convergence criteria for the iterates of positive linear operators preserving the affine functions. By way of examples, we consider the Bernstein, the Meyer-Konig and Zeller, the q-Bernstein operators and their iterates.