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Below we derive necessary conditions of optimality for problems with mixed constraints (see Dmitruk in Control Cybern 38(4A):923–957, 2009) using the method of penalty functions similar to the one we previously used to solve optimization problems for control sweeping processes (see, e.g., De Pinho et al. in Optimization 71(11):3363–3381, 2022) and, more recently, to solve optimal control problems with pure state constraints (see De Pinho et al. in Syst Control Lett 188:105816, 2024). We intentionally consider a smooth case and the simplest boundary conditions; we consider global minimum and assume that the set of trajectories of the control system is compact. Based on our penalty functions approach we develop a numerical method admitting estimates for its parameters needed to achieve a given precision. [ABSTRACT FROM AUTHOR]

The numerical method developed in Nour and Zeidan (IEEE Control Syst. Lett. 6:1190-1195, 2022) via the exponential penalization technique for optimal control problems involving sweeping processes with sweeping set C generated by onesmooth function, is generalized in this paper to the case where C is nonsmooth. That is, C is the intersection of a finite number (greater than one) of sublevel sets of smooth functions. The change from one to greater than one generating smooth functions is quite challenging. Indeed, while in the latter case C could be reformulated as being generated by one function, however, this function is only Lipschitz, and hence, the method established for one generating smooth function is not applicable to this framework. Therefore, this general setting requires a new approach, which represents the novelty of this paper. [ABSTRACT FROM AUTHOR]

We consider a continuous-review inventory control problem for a hybrid manufacturing and remanufacturing system with product recovery and setup costs. A pull policy has been proposed in the literature, however, finding its optimal parameters requires an exhaustive search and a single cost function evaluation is itself complex. We propose a tractable approach to finding these parameters using an interim policy called double (r, Q) with parameters (r m , Q m , r r , Q r) . When the inventory position of the serviceable item reaches rr, a remanufacturing lot size Qr is setup if recoverable inventory is sufficient. Otherwise, we allow the inventory position to decrease further. As it drops to rm, a manufacturing lot size Qm is placed. Unlike the pull policy, this interim policy suspends the remanufacturing option when the inventory position is less than rr. This facilitates an efficient approximation of the recoverable inventory which decouples the double (r, Q) problem into two standard (r, Q) problems. It can then be efficiently solved using a modification of existing (r, Q) algorithms for two instances. Numerical studies show that our approach performs well relative to the optimal pull policy with parameters estimated from an extensive Simulation-Optimization method and other heuristics found in the literature. The approach is also extended to correlated demand and return arrivals. [ABSTRACT FROM AUTHOR]

Global approximations of scalar optimization problems in terms of variational convergence have been studied for several decades. However, there are very few results for vector models in the literature, which are limited to epi-convergence of vector functions, nothing has been obtained for vector bifunctions. This paper is the first attempt to consider approximations of vector optimization in terms of variational convergence of bifunctions. We first consider the vector quasi-equilibrium problem since it encompasses most optimization-related models. Then, we apply the obtained results to the vector Nash quasi-game (vector generalized non-cooperative game), which is one of the most important practical problems in applied mathematics. We approach to the vector problems in question via scalarization. Our scalarization tools are the Tammer and the Hiriart-Urruty scalarization functions. By scalarizing, we define types of vector variational convergence corresponding to some types of scalar variational convergence. We show that under these types of convergence of problems approximating the vector problems under consideration, approximate solutions of the former problems tend to solutions of the latter ones in the sense of set convergence. [ABSTRACT FROM AUTHOR]

Working with approximations is a common practice in physics. This paper presents an exploratory study of student understanding of some of the elementary aspects of approximations encountered in college physics. For this purpose, a questionnaire (a set of 14 multiple-choice questions) was developed to probe how students dealt with these aspects in a variety of contexts. The questionnaire was finalized after a trial of its pilot version on 7 students on the basis of their written scripts and interviews. After content validation by three external experts, the test was administered to a sample of 141 senior undergraduate physics students from a diverse set of colleges and national institutes. Data analysis reveals several significant student pitfalls in the topic related principally to (i) the identification of a dimensionless ratio to characterize an approximation, (ii) the notion of the order of approximation and consistent handling of a problem up to the required order, and (iii) the notion of the order of infinitesimal in calculus-based derivations. The findings are of direct pedagogic relevance. [ABSTRACT FROM AUTHOR]

In this article, we consider two- and three- dimensional stochastic convective Brinkman-Forchheimer extended Darcy (CBFeD) equations ∂ u ∂ t − μ Δ u + (u ⋅ ∇) u + α | u | q − 1 u + β | u | r − 1 u + ∇ p = f , ∇ ⋅ u = 0 , on a torus, where μ , β > 0 , α ∈ R , r ∈ [ 1 , ∞) and q ∈ [ 1 , r) . The goal is to show that the solutions of 2D and 3D stochastic CBFeD equations driven by Brownian motion can be approximated by 2D and 3D stochastic CBFeD equations forced by pure jump noise/random kicks on the state space D ([ 0 , T ] ; H) . For the cases d = 2 , r ∈ [ 1 , ∞) and d = 3 , r ∈ (3 , ∞) , by using minimal regularity assumptions on the noise coefficient, the results are established for any μ , β > 0 . For the case d = r = 3 , the same results are obtained for 2 β μ ≥ 1 . [ABSTRACT FROM AUTHOR]

This work is dedicated to advancing the approximation of initial value problems through the introduction of an innovative and superior method inspired by Taylor's approach. Specifically, we present an enhanced variant achieved by accelerating the expansion of the Obreschkoff formula. This results in a higher-order implicit corrected method that outperforms Taylor's method in terms of accuracy. We derive an error bound for the Obreschkoff higher-order method, showcasing its stability, convergence, and greater efficiency compared to the conventional Taylor method. To substantiate our claims, numerical experiments are provided, which highlight the exceptional efficacy of our proposed method over the traditional Taylor method. [ABSTRACT FROM AUTHOR]

Motivated by important applications to the analysis of complex noise-induced phenomena, we consider a problem of the constructive description of randomly forced equilibria for nonlinear systems with multiplicative noise. Using the apparatus of the first approximation systems, we construct an approximation of mean square deviations that explicitly takes into account the presence of multiplicative noises, depending on the current system state. A spectral criterion of existence and exponential stability of the stationary second moments for the solution of the first approximation system is presented. For mean square deviation, we derive an expansion in powers of the small parameter of noise intensity. Based on this theory, we derive a new, more accurate approximation of mean square deviations in a general nonlinear system with multiplicative noises. This approximation is compared with the widely used approximation based on the stochastic sensitivity technique. The general mathematical results are illustrated with examples of the model of climate dynamics and the van der Pol oscillator with hard excitement. [ABSTRACT FROM AUTHOR]

For numerical approximations to stochastic differential equations using the Euler-- Maruyama scheme, we propose incorporating approximate random variables computed using low precisions, such as single and half precision. We propose and justify a model for the rounding error incurred, and produce an average case error bound for two and four way differences, appropriate for regular and nested multilevel Monte Carlo estimations. Our rounding error model recovers and extends the statistical model by Arciniega and Allen [Stoch. Anal. Appl., 21 (2003), pp. 281-- 300], while bounding the size that systematic and biased rounding errors are permitted to be. By considering the variance structure of multilevel Monte Carlo correction terms in various precisions with and without a Kahan compensated summation, we compute the potential speed ups offered from the various precisions. We find single precision offers the potential for approximate speed improvements by a factor of 7 across a wide span of discretization levels. Half precision offers comparable improvements for several levels of coarse simulations, and even offers improvements by a factor of 10--12 for the very coarsest few levels, which are likely to dominate higher order methods such as the Milstein scheme. [ABSTRACT FROM AUTHOR]

In this article, we introduce and study a natural version of the directional compactness, which can be viewed as one of the effective tools for constructing sufficient conditions in nonsmooth vector equilibrium problems. We also provide the generalized Hadamard directional derivative notion which is closely related to a version of contingent derivative. The relation among the first-order approximations/the Clarke generalized Jacobian and the generalized Hadamard directional differentiability is formulated. Using the tool of approximations, a new version of the constraint qualification of the (CQ) type is proposed for establishing KT-type necessary nonsmooth optimality conditions via the generalized Hadamard directional derivatives for the weak/and strict efficiency of constrained nonsmooth vector equilibrium problems. As applications, we study a nonsmooth vector equilibrium problem with set, cone constraints using approximations and the constraint qualification (CQ). [ABSTRACT FROM AUTHOR]

Gaussian filtering, being a convolution with a Gaussian kernel, is a widespread technique in image analysis and computer vision applications. It is the traditional approach for noise reduction. In some cases, performing the exact convolution can be computationally expensive and time-consuming. To address this problem, approximations of the convolution are often used to achieve a balance between accuracy and computational efficiency, such as with running sums, Bell blur, Deriche approximation, etc. At the same time, modern computing devices support data parallelism (vectorization) via Single Instruction Multiple Data (SIMD) and can process integer numbers faster than floating-point approaches. In this paper, we describe several methods for approximating a Gaussian filter, implement the SIMD and quantized versions, and compare them in terms of speed and accuracy. The experiments were performed on central processing units with a x86_64 architecture using a family of SSE SIMD extensions and an ARMv8 architecture using the NEON SIMD extension. All the optimized approximations demonstrated 10–20× speedup while maintaining the accuracy in the range of 1 × 10 − 5 or higher. The fastest method is a trivial Stack blur with a relatively high error, so we recommend using the second-order Vliet–Young–Verbeek filter and quantized Bell blur and running sums as more accurate and still computationally efficient alternatives. [ABSTRACT FROM AUTHOR]

In this short paper, we review the Euler-Rodrigues formula for three-dimensional rotation via fractional powers of matrices. We derive the rotations by any angle through the spectral behavior of the fractional powers of the rotation matrix by p2 in R3 about some axis. [ABSTRACT FROM AUTHOR]

Data exchange, the problem of transferring data from a source schema to a target schema, has been studied for several years. The semantics of answering positive queries over the target schema has been defined in early work, but little attention has been paid to more general queries. A few proposals of semantics for more general queries exist but they either do not properly extend the standard semantics under positive queries, giving rise to counterintuitive answers, or they make query answering undecidable even for the most important data exchange settings, for example, with weakly-acyclic dependencies. The goal of this paper is to provide a new semantics for data exchange that is able to deal with general queries. At the same time, we want our semantics to coincide with the classical one when focusing on positive queries, and to not trade-off too much in terms of complexity of query answering. We show that query answering is undecidable in general under the new semantics, but it is $\text{co}\text{NP}\text{-complete}$ when the dependencies are weakly-acyclic. Moreover, in the latter case, we show that exact answers under our semantics can be computed by means of logic programs with choice, thus exploiting existing efficient systems. For more efficient computations, we also show that our semantics allows for the construction of a representative target instance, similar in spirit to a universal solution, that can be exploited for computing approximate answers in polynomial time. [ABSTRACT FROM AUTHOR]

In this paper, we propose a sufficient condition at which a neural network can approximate a set of optimization algorithm solutions; we establish under which conditions a neural network can replace an optimization algorithm to solve a problem with the objective of safely deploying that network in a system where online solutions are necessary to simplify the hardware or allowing the processor to solve the optimization problem on time. To that end, first, we define the family of optimization problems to be addressed; then, we construct a vector with the parameters on which the solution depends, in order to propose a function based on the first-order Karush–Kuhn–Tucker conditions to find conditions under which the inverse of the proposed function maps the problem minimizer with respect to the constructed vector, we provide the sufficiency proof of, both, existence and feasibility of approximation by a neural network regarding the inverse function. Two case studies are proposed, one numerical case showing how a neural network can solve an optimization problem faster than popular solvers to illustrate how it can be implemented in applications where the computation time is tight, and the other case is a Model Predictive Control implementation with the optimization problem solver replaced by a neural network which allows a hardware downgrade; both cases are presented with time statistics comparisons. [ABSTRACT FROM AUTHOR]

The central objective of this work is to make a comparative analysis of the fable “The Cigarra and the Form”, by Aesop, and “Sem barra”, by José Paulo Paes. In this process, analyze the similarities and distances between the works, highlighting the aspects of intertextuality, imitation, influences and marks of originality based on the theoretical perspectives discussed by Nitrini (2021), Carvalhal (2001), Ana Maria Mello (1996), Koch, Bentes, Carvalho (2008). The works bring criticism and teachings. While the fable explicitly brings its moralizing content, which aims to convey the message, merely capitalist, that “art is a waste of time” and the ants are shown as a representation of the “ideal” social model, which work to accumulate food in the summer to enjoy in the winter, Paes' poem, despite having the influence and intertextuality relative to the fable, distances itself in several aspects, such as the fact of describing the characters from another perspective, the cicada is no longer portrayed as “lazy”, but rather, as of fundamental importance for the ants’ daily lives. [ABSTRACT FROM AUTHOR]

A Near-Optimal Capacity and Scheduling Policy for Cloud Users In "Cloud Cost Optimization: Model, Bounds, and Asymptotics," Qu, Dawande, and Janakiraman study a long-term, dynamic resource-optimization problem for firms managing various cloud resources to process incoming computing tasks over time. Cloud resources vary in attributes, such as computing speed, memory, accelerators, and storage, tailored to different computing tasks, such as data warehousing, scientific computing, machine learning, and data processing. Firms can choose reserved resources for long-term commitments at lower costs or on-demand resources for flexibility at a higher price. Firms face three key trade-offs: the cost disparity between reserved and on-demand resources, the variation in resource attributes affecting performance and cost, and the challenge of balancing delay and resource costs. We propose an asymptotically optimal policy, demonstrating its effectiveness through a detailed numerical study based on Amazon Web Services data. Motivated by the rapid growth of the cloud cost management and optimization (CCMO) industry to support the exploding cloud-computing market, we study an infinite-horizon, stochastic optimization problem from the viewpoint of a firm that employs cloud resources to process incoming orders (or jobs) over time. We model the following vital practical features of CCMO in our problem. There are several types of resources that differ in their costs and performance attributes (e.g., processor speed, memory, storage). For each type of resource, capacity can either be reserved over the long term at a discounted price or be used on demand at a relatively higher price. Orders of several types arrive stochastically through time; orders differ in their completion-time deadlines and in their resource-specific processing-time distributions. Moreover, the progress of an order can be observed periodically, and if required, the order can be moved from one resource type to another. Penalty costs are incurred for orders not completed by their deadlines. The firm's goal is to minimize the long-run average expected cost per period, taking into account reserved-capacity costs, on-demand capacity costs, and order-delay costs. We derive a lower bound on the optimal cost by considering a set of decoupled problems, one for each order. The solutions of these problems are then used to construct a feasible policy for the original problem and derive an upper bound on that policy's optimality gap. Importantly, we show that our policy is asymptotically optimal; when the demand rates of the orders are scaled by a factor θ > 0 , the policy's optimality gap scales proportional to 1 / θ . We also report results of a comprehensive numerical study—on a test bed informed by capacity and pricing data from Amazon Web Services—to demonstrate the impressive performance of our policy. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.0362. [ABSTRACT FROM AUTHOR]

Various neuron approximations can be used to reduce the computational complexity of neural networks. One such approximation based on summation and maximum operations is a bipolar morphological neuron. This paper presents an improved structure of the bipolar morphological neuron that enhances its computational efficiency and a new approach to training based on continuous approximations of the maximum and knowledge distillation. Experiments were carried out on the MNIST dataset using a LeNet-like neural network architecture and on the CIFAR10 dataset using a ResNet-22 model architecture. The proposed training method achieves 99.45% classification accuracy on the LeNet-like model (the same accuracy as that provided by the classical network) and 86.69% accuracy on the ResNet-22 model compared with 86.43% accuracy of the classical model. The results show that the proposed method with log-sum-exp (LSE) approximation of the maximum and layer-by-layer knowledge distillation makes it possible to obtain a simplified bipolar morphological network that is not inferior to the classical networks. [ABSTRACT FROM AUTHOR]

We obtain an approximation result of the weak solutions to elliptic and parabolic equations with Dirichlet boundary conditions. We show that the weak solution can be obtained with a limit of approximations by regularizing the nonlinearities and approximating the domains.

Motivated by important applications to the analysis of complex noise-induced phenomena, we consider a problem of the constructive description of randomly forced equilibria for nonlinear systems with multiplicative noise. Using the apparatus of the first approximation systems, we construct an approximation of mean square deviations that explicitly takes into account the presence of multiplicative noises, depending on the current system state. A spectral criterion of existence and exponential stability of the stationary second moments for the solution of the first approximation system is presented. For mean square deviation, we derive an expansion in powers of the small parameter of noise intensity. Based on this theory, we derive a new, more accurate approximation of mean square deviations in a general nonlinear system with multiplicative noises. This approximation is compared with the widely used approximation based on the stochastic sensitivity technique. The general mathematical results are illustrated with examples of the model of climate dynamics and the van der Pol oscillator with hard excitement.

Let ρ be a general law-invariant convex risk measure, for instance, the average value at risk, and let X be a financial loss, that is, a real random variable. In practice, either the true distribution μ of X is unknown, or the numerical computation of ρ (μ) is not possible. In both cases, either relying on historical data or using a Monte Carlo approach, one can resort to an independent and identically distributed sample of μ to approximate ρ (μ) by the finite sample estimator ρ (μ N) (μN denotes the empirical measure of μ). In this article, we investigate convergence rates of ρ (μ N) to ρ (μ). We provide nonasymptotic convergence rates for both the deviation probability and the expectation of the estimation error. The sharpness of these convergence rates is analyzed. Our framework further allows for hedging, and the convergence rates we obtain depend on neither the dimension of the underlying assets nor the number of options available for trading. Funding: Daniel Bartl is grateful for financial support through the Vienna Science and Technology Fund [Grant MA16-021] and the Austrian Science Fund [Grants ESP-31 and P34743]. Ludovic Tangpi is supported by the National Science Foundation [Grant DMS-2005832] and CAREER award [Grant DMS-2143861]. [ABSTRACT FROM AUTHOR]

We generalize a maximum principle for optimal control problems involving sweeping systems previously derived in de Pinho et al. (Optimization 71(11):3363–3381, 2022, https://doi.org/10.1080/02331934.2022.2101111) to cover the case where the moving set may be nonsmooth. Noteworthy, we consider problems with constrained end point. A remarkable feature of our work is that we rely upon an ingenious smooth approximating family of standard differential equations in the vein of that used in de Pinho et al. (Set Valued Var Anal 27:523–548, 2019, https://doi.org/10.1007/s11228-018-0501-8). [ABSTRACT FROM AUTHOR]

For random variables produced through the inverse transform method, approximate random variables are introduced, which are produced using approximations to a distribution's inverse cumulative distribution function. These approximations are designed to be computationally inexpensive and much cheaper than library functions, which are exact to within machine precision and, thus, highly suitable for use in Monte Carlo simulations. The approximation errors they introduce can then be eliminated through use of the multilevel Monte Carlo method. Two approximations are presented for the Gaussian distribution: a piecewise constant on equally spaced intervals and a piecewise linear using geometrically decaying intervals. The errors of the approximations are bounded and the convergence demonstrated, and the computational savings are measured for C and C++ implementations. Implementations tailored for Intel and Arm hardware are inspected alongside hardware agnostic implementations built using OpenMP. The savings are incorporated into a nested multilevel Monte Carlo framework with the Euler-Maruyama scheme to exploit the speedups without losing accuracy, offering speed ups by a factor of 5–7. These ideas are empirically extended to the Milstein scheme and the non-central χ2 distribution for the Cox-Ingersoll-Ross process, offering speedups of a factor of 250 or more. [ABSTRACT FROM AUTHOR]

Emden–Fowler type equations are nonlinear differential equations that appear in many fields such as mathematical physics, astrophysics and chemistry. In this paper, we perform an asymptotic analysis of a specific Emden–Fowler type equation that emerges in a queuing theory context as an approximation of voltages under a well-known power flow model. Thus, we place Emden–Fowler type equations in the context of electrical engineering. We derive properties of the continuous solution of this specific Emden–Fowler type equation and study the asymptotic behavior of its discrete analog. We conclude that the discrete analog has the same asymptotic behavior as the classical continuous Emden–Fowler type equation that we consider. [ABSTRACT FROM AUTHOR]

Using geometric considerations, we provided a clear derivation of the integral representation for the error function, known as the Craig formula. We calculated the corresponding power series expansion and proved the convergence. The same geometric means finally assisted in systematically deriving useful formulas that approximated the inverse error function. Our approach could be used for applications in high-speed Monte Carlo simulations, where this function is used extensively.

Gaussian filtering, being a convolution with a Gaussian kernel, is a widespread technique in image analysis and computer vision applications. It is the traditional approach for noise reduction. In some cases, performing the exact convolution can be computationally expensive and time-consuming. To address this problem, approximations of the convolution are often used to achieve a balance between accuracy and computational efficiency, such as with running sums, Bell blur, Deriche approximation, etc. At the same time, modern computing devices support data parallelism (vectorization) via Single Instruction Multiple Data (SIMD) and can process integer numbers faster than floating-point approaches. In this paper, we describe several methods for approximating a Gaussian filter, implement the SIMD and quantized versions, and compare them in terms of speed and accuracy. The experiments were performed on central processing units with a x86_64 architecture using a family of SSE SIMD extensions and an ARMv8 architecture using the NEON SIMD extension. All the optimized approximations demonstrated 10–20× speedup while maintaining the accuracy in the range of 1 × 10−5 or higher. The fastest method is a trivial Stack blur with a relatively high error, so we recommend using the second-order Vliet–Young–Verbeek filter and quantized Bell blur and running sums as more accurate and still computationally efficient alternatives.

This work deals with the problem of semantic optimization of the central class of conjunctive queries (CQs). Since CQ evaluation is NP-complete, a long line of research has focussed on identifying fragments of CQs that can be efficiently evaluated. One of the most general restrictions corresponds to generalized hypetree width bounded by a fixed constant k ≥ 1; the associated fragment is denoted GHWk. A CQ is semantically in GHWk if it is equivalent to a CQ in GHWk. The problem of checking whether a CQ is semantically in GHWk has been studied in the constraint-free case, and it has been shown to be NP-complete. However, in case the database is subject to constraints such as tuple-generating dependencies (TGDs) that can express, e.g., inclusion dependencies, or equality-generating dependencies (EGDs) that capture, e.g., key dependencies, a CQ may turn out to be semantically in GHWk under the constraints, while not being semantically in GHWk without the constraints. This opens avenues to new query optimization techniques. In this article, we initiate and develop the theory of semantic optimization of CQs under constraints. More precisely, we study the following natural problem: Given a CQ and a set of constraints, is the query semantically in GHWk, for a fixed k ≥ 1, under the constraints, or, in other words, is the query equivalent to one that belongs to GHWk over all those databases that satisfy the constraints? We show that, contrary to what one might expect, decidability of CQ containment is a necessary but not a sufficient condition for the decidability of the problem in question. In particular, we show that checking whether a CQ is semantically in GHW1 is undecidable in the presence of full TGDs (i.e., Datalog rules) or EGDs. In view of the above negative results, we focus on the main classes of TGDs for which CQ containment is decidable and that do not capture the class of full TGDs, i.e., guarded, non-recursive, and sticky sets of TGDs, and show that the problem in question is decidable, while its complexity coincides with the complexity of CQ containment. We also consider key dependencies over unary and binary relations, and we show that the problem in question is decidable in elementary time. Furthermore, we investigate whether being semantically in GHWk alleviates the cost of query evaluation. Finally, in case a CQ is not semantically in GHWk, we discuss how it can be approximated via a CQ that falls in GHWk in an optimal way. Such approximations might help finding "quick" answers to the input query when exact evaluation is intractable. [ABSTRACT FROM AUTHOR]

The rational development of sustainable polymeric materials demands tunable properties using mixtures of polymers with chemical variations. At the same time, the sheer number of potential variations and combinations makes experimentally or numerically studying every new mixture impractical. A direct predictive tool quantifying how material properties change when molecular features change provides a less time‐ and resource‐consuming route to optimization. Numerically solving Flory–Huggins theory provides such a tool for mono‐disperse mixtures with a limited number of components, but for multi‐component systems the large number of equations makes numerical computations challenging. Approximate solutions to Flory–Huggins theory relating miscibility and solubility to molecular features are presented. The set of approximate relations show a wider range of accuracy compared to existing approximations. The combination of the analytical, lower‐order, and more accurate higher‐order approximations together contribute to a broader applicability and extensibility of Flory–Huggins theory. [ABSTRACT FROM AUTHOR]

Communications and information theory use the Gaussian $Q$ -function, a positive and decreasing function, across the literature. Its approximations were created to simplify mathematical study of the Gaussian $Q$ -function expressions. This is important since the $Q$ -function cannot be represented in closed-form terms of elementary functions. In a noise model with the Gaussian distribution function and various digital modulation schemes, closed-form approximations of the Gaussian $Q$ -function are used to predict a digital communications system’s symbol error probability (SEP) or bit error probability (BEP). Another significant scenario pertains to fading channels, whereby it is important to accurately determine, through a closed-form expression, the precise evaluations of complex integrals involved in the computations of SEP or BEP. In addition to the aforementioned scenarios, it is imperative for a communications system designer to ascertain the requisite operational signal-to-noise ratio for the specific application, based on the target SEP (or BEP). In this scenario, the crucial role of the explicit invertibility of the Gaussian $Q$ -function approximation is of significant importance in achieving this objective. In this paper we propose a survey of the approximations of the Gaussian $Q$ -function found in the literature, reviewing also the approximations originally given for the 4 classical special functions related to it, restricting the analysis to the explicitly invertible ones, and classifying them on the basis of their accuracy (on the significant range), simplicity, and easiness of inversion, also distinguishing the bounds from approximations. We also list the inverses of some of them, already published or newly found in this research.

Rough set theory is a mathematical approach that deals with the problems of uncertainty and ambiguity in knowledge. Neighborhood systems are the most effective instruments for researching rough set theory in general. Investigations on boundary regions and accuracy measures primarily rely on two approximations, namely lower and upper approximations, by using these systems. The concept of the ideal, which is one of the most successful and effective mathematical tools, is used to obtain a better accuracy measure and to decrease the boundary region. Recently, a generalization of Pawlak's rough set concept has been represented by neighborhood systems of graphs based on rough sets. In this research article, we propose a new method by using the concepts of the ideal and different neighborhoods from graph vertices. We examine important aspects of these techniques and produce accuracy measures that exceed those previously = reported in the literature. Finally, we show that our method yields better results than previous techniques utilized in chemistry. [ABSTRACT FROM AUTHOR]

The data we need to deal with are getting bigger and bigger in recent years, and the same happens to multi-granulation rough sets, so updated schemes have been proposed with the variation of attributes or attribute values in multi-granulation rough sets. This paper puts forward a dynamic mechanism to update the approximations in multi-granulation rough sets when increasing or decreasing objects. Firstly, the relationships between the original approximations and updated approximations are explored when adding or deleting objects and the dynamic processes of updating the lower and upper approximations in optimistic and pessimistic approximations are proposed. Secondly, two corresponding dynamic algorithms and their time complexity are given. Finally, the experimental evaluations show the effectiveness of the proposed dynamic updating algorithms compared with the static algorithm. [ABSTRACT FROM AUTHOR]

In this work, we address the problem of piecewise affine approximations, that is, to find piecewise affine functions that well-approximate a given signal. The proposed approach is optimal in the sense of L 2 norm and formulated in a compact and explicit way; no fitting stage is needed. Also, affine parameters are obtained as closed formulas, and affine approximation functions are optimal in their corresponding subdomains. In addition, we state and prove a recursive formula for approximation errors, which makes the approach optimal and nonlinear, links also the subdomains and helps derive an algorithm of complexity of order O (M N 2) , where M represents the number of piecewise affine approximants and N is the number of samples of the processed signal. Finally, obtained qualitative and quantitative results show that the presented method obtains good approximations and provides improvement over piecewise constant approximations. [ABSTRACT FROM AUTHOR]

Asymptotic approximation formulas for polynomials of the type Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi with integer order and real parameters are obtained via hyperbolic functions. The derivation of the formulas is done using the principle of saddle point and expansion of appropriate hyperbolic function about a saddle point. [ABSTRACT FROM AUTHOR]

Motivated by a routing problem faced by banks to enhance their encashment services in the city of Perm, Russia, we solve versions of the traveling salesman problem (TSP) with clustering. To minimize the risk of theft, suppliers seek to operate multiple vehicles and determine an efficient routing; and, a single vehicle serves a set of locations that forms a cluster. This need to form independent clusters—served by distinct vehicles—allows the use of the so-called cluster-first route-second approach. We are especially interested in the use of heuristics that are easily implementable and understandable by practitioners and require only the use of open-source solvers. To this end, we provide a short survey of 13 such heuristics for solving the TSP, five for clustering the set of locations, and three to determine an optimal number of clusters—all using data from Perm. To demonstrate the practicality and efficiency of the heuristics, we further compare our heuristic solutions against the optimal tours. We then provide statistical guarantees on the quality of our solution. All of our anonymized code is publicly available allowing extensions by practitioners, and serves as a decision-analytic framework for both clustering data and solving a TSP. [ABSTRACT FROM AUTHOR]

This work proposes an efficient analytical design procedure for elliptical and circular 2D digital filters with an adjustable response. The design relies on a 1D low-pass prototype filter with a specified bandwidth, to which a particular frequency mapping is applied. The prototype can be scaled in frequency, having adjustable selectivity. This analytical procedure yields 2D filters with accurate shape even near the frequency plane margins, very low distortions and steep transition at a relatively low order, thus being very efficient. The elliptical filters can be tuned to a desired orientation angle and can be used as directional filters. Also an uniform circular filter bank with a specified number of bands is designed. The frequency response results directly factored, and the corresponding filter matrices result as a convolution of small size matrices, which simplifies implementation and allows for a sequential filtering, in several steps. The proposed method is based entirely on accurate approximations and frequency mappings, without using any numerical optimization algorithms. Several examples of filtering on various test images are provided, to illustrate the capabilities of this class of filters. [ABSTRACT FROM AUTHOR]

Using geometric considerations, we provided a clear derivation of the integral representation for the error function, known as the Craig formula. We calculated the corresponding power series expansion and proved the convergence. The same geometric means finally assisted in systematically deriving useful formulas that approximated the inverse error function. Our approach could be used for applications in high-speed Monte Carlo simulations, where this function is used extensively. [ABSTRACT FROM AUTHOR]

This article investigates the long-wave anti-plane shear motion in a symmetric three-layered laminate composed of pre-stressed compressible elastic layers. The layers of the laminate are perfectly bonded, while traction-free and fixed boundary conditions are considered on the outer faces of the laminate. In both cases, the dispersion relation is obtained in terms of symmetric and anti-symmetric decompositions. Numerical results and an asymptotic long-wave analysis are presented, corresponding to the three possible vibration modes. It is revealed that a low-frequency mode only exists in respect of symmetric motion with free-faces, while all other cases pose a series of non-zero cut-off frequencies. Comparisons between the exact and approximate asymptotic results are presented, and excellent agreement is observed.

The normal or Gaussian distribution plays a prominent role in almost all fields of science. However, it is well known that the Gauss (or Euler–Poisson) integral over a finite boundary, as is necessary, for instance, for the error function or the cumulative distribution of the normal distribution, cannot be expressed by analytic functions. This is proven by the Risch algorithm. Regardless, there are proposals for approximate solutions. In this paper, we give a new solution in terms of normal distributions by applying a geometric procedure iteratively to the problem.

An optimization model for traffic light control in an urban network of intersections is derived. The model is based on store-and-forward analytic relations, which account for the length of the queue of waiting vehicles in front of the traffic light intersection. The model is complicated with probabilistic relations that formalize the requirements for maintaining short queues of vehicles. Probabilistic inequalities apply to each intersection of the city network. Approximations of probability inequalities are given in the article. Quadratic deterministic inequalities, which are part of the set of the traffic flow control optimization problem, are derived. Numerical simulations are performed, applying mean estimated data for real traffic in an urban area of Sofia. The model predictive approach is applied to traffic light optimization and control. Empirical results give advantages of the obtained model compared to the classical store-and-forward optimization model for the total number of vehicles waiting in the considered urban network. [ABSTRACT FROM AUTHOR]

In this work, we consider polynomial approximations of the sine map, which is known to be chaotic. The dynamics of the maps obtained for different orders of approximations are studied using classical tools. Comparison of the behavior is made through common bifurcation diagrams, 0–1 test results, Lyapunov exponent diagrams and approximate entropy diagrams. Furthermore, the differences between the Lyapunov exponents and approximate entropy are depicted, to provide a better grasp on the comparison. It is demonstrated that different orders of approximations lead to the definition of distinct chaotic systems, which in cases attain more complex chaotic behavior compared to the original sine map. Furthermore, the basins of stability for the proposed maps are thoroughly studied. Subsequently, considering the fact that the sine map is commonly utilized in the definition of more complex maps, the effects of substituting the sine map with its approximates is investigated. It is shown that in many occasions the dynamics of the map defined using the Taylor approximation is more complex compared to the original map. Finally, two of the proposed maps are used for the definition of two pseudorandom bit generators, whose randomness is verified via statistical tests. [ABSTRACT FROM AUTHOR]

Maximum flow is one of the fundamental problems in graph theory with several applications such as bipartite matchings, image segmentation, disjoint paths, network connectivity, etc. Goldberg-Tarjan's well-known Push Relabel (PR) Algorithm calculates the maximum s–t (source–target) flow on a directed weighted graph. PR algorithm has been effectively parallelized on GPUs. However, computing the maximum flow even using the GPU parallel PR algorithm continues to be time-consuming for large graphs. For the maximum flow algorithm's error-tolerant applications, it is sufficient to compute the approximate maximum flow value. In this work, we propose multiple techniques for improving the push-relabel algorithm's performance on the GPUs keeping its error-tolerant applications in mind. Our proposed techniques improve performance by carefully reducing the impact of the particular property that hampers the performance of the GPU parallel PR algorithm. These techniques provide tunable knobs to control the amount of approximation added and the respective performance achieved. In the end, we propose the Pull Relabel algorithm, which is the natural symmetric counterpart of the Push Relabel algorithm. Further, we combine both algorithms to construct a Pull-Push Relabel maxflow algorithm and analyze its effect on the dynamically changing graphs. We illustrate the effectiveness of our proposed algorithm and techniques using several real-world and synthetic graphs from the DIMACS Challenge, SNAP, Konect, and Network Repository, along with three maximum flow applications (Maximum Bipartite Matching, Team Elimination Problem, and Supply–Demand Problem). The proposals achieve 1.05 × to 94.83 × speedup over the exact GPU parallel push-relabel algorithm, and 14.29 × , 40.40 × and 32.41 × speed-up on the three applications. [ABSTRACT FROM AUTHOR]