Approximate Point Research Articles

Iterative Approximations of Attractive Point of A New Generalized Bregman Nonspreading Mapping in Banach Spaces

In this paper, a generic 2-generalized Bregman nonspreading mapping is introduced. Also, a Halpern-type iterative scheme for the approximation of attractive point of such mapping is constructed in the setting of Banach space. The result established generalized some recently announced results in the literature.

Statistics of heat transport across a capacitively coupled double quantum dot circuit

We study heat current and the full statistics of heat fluctuations in a capacitively-coupled double quantum dot system. This work is motivated by recent theoretical studies and experimental works on heat currents in quantum dot circuits. As expected intuitively, within the (static) mean-field approximation, the system at steady-state decouples into two single-dot equilibrium systems with renormalized dot energies, leading to zero average heat flux and fluctuations. This reveals that dynamic correlations induced between electrons on the dots is solely responsible for the heat transport between the two reservoirs. To study heat current fluctuations, we compute steady-state cumulant generating function for heat exchanged between reservoirs using two approaches : Lindblad quantum master equation approach, which is valid for arbitrary coulomb interaction strength but weak system-reservoir coupling strength, and the saddle point approximation for Schwinger-Keldysh coherent state path integral, which is valid for arbitrary system-reservoir coupling strength but weak coulomb interaction strength. Using thus obtained generating functions, we verify steady-state fluctuation theorem for stochastic heat flux and study the average heat current and its fluctuations. We find that the heat current and its fluctuations change non-monotonically with the coulomb interaction strength ($U$) and system-reservoir coupling strength ($\Gamma$) and are suppressed for large values of $U$ and $\Gamma$.

Stack and register complexity of radix conversions

We investigate the question of computational resources (such as stacks and counters) necessary to perform radix conversions. To this end it is shown that no PDA can compute the significand of the best n-digit floating point approximation of a power of an incommensurable radix. This extends the results of W. Clinger. We also prove that a two counter machine with input is capable of such conversions. On the other hand we note a curious asymmetry with respect to the order in which the digits are input by showing that a two counter machine can decode its input online if the digits are presented in the most-to-least significant order while no such machine can decode its input in this manner if the digits are presented in the least-to-most significant order. Some structural results about two counter machines (with input) are also established.

Magnetic Order in Heisenberg Models on Non-Bravais Lattice: Popov-Fedotov Functional Method

We study magnetic properties of ordered phase in Heisenberg model on a non-Bravais lattice by means of Popov - Fedotov trick, which takes into account a rigorous constraint of a single occupancy. We derive magnetization and free energy using sadle point approximation in the functional integral formalism. We illustrate the application of the Popov -- Fedotov approach to the Heisenberg antiferromagnet on a honeycomb lattice.

Iterative approximation of common attractive points of $$(\alpha ,\beta )$$ ( α , β ) -generalized hybrid set-valued mappings

In this paper, we introduce the concepts of the set of all the common attractive points (CAP(S, T)) and the set of all the common strongly attractive points (CSAP(S, T)) for the set-valued mappings S and T in a Hilbert space. Moreover, some fundamental properties related to the sets A(T), F(T) and CAP(S, T) are given. Furthermore, we generalize the Agarwal iteration for the case of two set-valued mappings and obtain a weak convergence theorem for two $$(\alpha ,\beta )$$ -generalized hybrid set-valued mappings in a Hilbert space. Moreover, an example of two $$(\alpha ,\beta )$$ -generalized hybrid set-valued mappings which have a common strongly attractive point is shown. Then we finish the work using the proposed algorithm to find a common element of the set of solutions of an equilibrium problem and the sets of fixed points of two $$(\alpha ,\beta )$$ -generalized hybrid set-valued mappings.

Penalty variable sample size method for solving optimization problems with equality constraints in a form of mathematical expectation

Equality-constrained optimization problems with deterministic objective function and constraints in the form of mathematical expectation are considered. The constraints are approximated by employing the sample average where the sample size varies throughout the iterations in an adaptive manner. The proposed method incorporates variable sample size scheme with cumulative and unbounded sample into the well- known quadratic penalty iterative procedure. Line search is used for globalization and the sample size is updated in a such way to preserve the balance between two types of errors—errors coming from the sample average approximation and the approximation of the optimal point. Moreover, the penalty parameter is also updated in an adaptive way. We prove that the proposed algorithm pushes the sample size and the penalty parameter to infinity which further allows us to prove the almost sure convergence towards a Karush-Kuhn-Tucker optimal point of the original problem under the rather standard assumptions. Numerical comparison on a set of relevant problems shows the advantage of the proposed adaptive scheme over the heuristic (predetermined) sample scheduling in terms of number of function evaluations as a measure of the optimization cost.

Relative uniform convergence of a sequence of functions at a point and Korovkin-type approximation theorems

We prove a Korovkin-type approximation theorem using the relative uniform convergence of a sequence of functions at a point, which is a method stronger than the classical ones. We give some examples on this new convergence method and we study also rates of convergence.

The empirical optimal envelope and its application to local mean decomposition

Envelopes, which represent the overall information of the signal amplitude variation, are the necessary mediums in many signal decomposition methods. The phenomena of undershoot and overshoot result in the error of envelope estimation and unexpected signal decomposition components. In this paper, an accurate envelope estimation method, called the empirical optimal envelope (EOE), is proposed and applied to the local mean decomposition (LMD). First, an indicator of envelope distance is defined to describe the features of the ideal envelope. Utilizing the indicator, an iterative algorithm for the approximation of tangency points is designed. The tangency points, instead of the extreme points, are interpolated to realize the EOE. Then, the EOE is integrated with the LMD, and two interpolation functions, the cubic spline and the piecewise cubic Hermite interpolating polynomial, are combined to improve the efficiency and convergence of signal decomposition. Finally, the proposed method is verified by simulated signals and actual signals.

On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems.

Mathematical analysis with the numerical simulation of the newly formulated fractional version of the Adams-Bashforth method using the Atangana-Baleanu operator which has both nonlocal and nonsingular properties is considered in this paper. We adopt the fixed point theory and approximation method to prove the existence and uniqueness of the solution via general two-component time fractional differential equations. The method is tested with three nonlinear chaotic dynamical systems in which the integer-order derivative is modeled with the proposed fractional-order case. The simulation result for different α values in (0,1] is presented.

Iterative approximation of attractive points of further generalized hybrid mappings in Hadamard spaces

In this paper, we study the class of further generalized hybrid mappings due to Khan (Fixed Point Theory Appl. 2018:8, 2018) in the setting of Hadamard spaces. We prove a demiclosed principle for such mappings in Hadamard spaces. Furthermore, we also prove the Δ-convergence of the sequence generated by the S-iteration process for finding attractive points of further generalized hybrid mappings in Hadamard spaces satisfying the (mathbb{S}) property and the (overline{Q_{4}}) condition. Moreover, we provide a numerical example to illustrate the convergence behavior of the studied iteration and numerically compare the convergence of the studied iteration scheme with the existing schemes. Our results extend some known results which appeared in the literature.

Approximation diophantienne et distribution locale sur une surface torique

In this article we study Diophantine approximation and local distribution of a rational point on a toric surface obtained as a blow-up of $\mathbb{P}^1\times\mathbb{P}^1$. It turns out that outside a Zariski closed subset the best approximations are achieved through a family of nodal curves. Hence the investigation is reduced to the question of local distribution of a quadratic point on the projective line.

A Structure Topology Optimization With the First-Order Saddlepoint Approximation

In practical engineering, uncertain loads usually cause the variations of structure stiffness to affect the security of the structure. To enhance the reliability and find the optimum structure in engineering, a topology optimization method is developed for the optimization of structural stiffness in this paper. The first-order saddle point approximation method is introduced into the topology optimization process. A multi-constrained stiffness optimization model considering uncertain loads is proposed. The corresponding sensitivity mathematical formula for stiffness is also given. The complete optimization iteration process is given. The structural stiffness optimization problems with the joint constraints of volume and displacement are solved here to illustrate the application of the proposed method. In the examples, a U-shape groove in different working situations is considered which includes the single node displacement constraint and the multi-node displacement constraint. The corresponding optimization results of different working situations are compared, and the results accord with the law of engineering.

Generalized Gibbs Ensemble of 2d CFTs at large central charge in the thermodynamic limit

We discuss partition function of 2d CFTs decorated by higher qKdV charges in the thermodynamic limit when the size of the spatial circle goes to infinity. In this limit the saddle point approximation is exact and at infinite central charge generalized partition function can be calculated explicitly. We show that leading 1/c corrections to free energy can be reformulated as a sum over Young tableaux which we calculate for the first two qKdV charges. Next, we compare generalized ensemble with the “eigenstate ensemble” that consists of a single primary state. At infinite central charge the ensembles match at the level of expectation values of local operators for any values of qKdV fugacities. When the central charge is large but finite, for any values of the fugacities the aforementioned ensembles are distinguishable.

Aspects of massive gauge theories on three sphere in infinite mass limit

We study the S3 partition function of three-dimensional supersymmetric mathcal{N}=4 U(N) SQCD with massive matter multiplets in the infinite mass limit with the so-called Coulomb branch localization. We show that in the infinite mass limit a specific point of the Coulomb branch is selected and contributes dominantly to the partition function. Therefore, we can argue whether each multiplet included in the theory is effectively massless in this limit, even on S3, and conclude that the partition function becomes that of the effective theory on the specific point of the Coulomb branch in the infinite mass limit. In order to investigate which point of the Coulomb branch is dominant, we use the saddle point approximation in the large N limit because the solution of the saddle point equation can be regarded as a specific point of the Coulomb branch. Then, we calculate the partition functions for small rank N and confirm that their behaviors in the infinite mass limit are consistent with the conjecture from the results in the large N limit. Our result suggests that the partition function in the infinite mass limit corresponds to that of an interacting superconformal field theory.

Analysis of the accuracy of the gradient method for determining the mean integral refractive index of air

The article is devoted to the problem of improving the accuracy of taking into account the influence of the Earth’s atmosphere on the results of distance measurements carried out using electromagnetic waves. The main influencing factors in such measurements are the difference between the speed of propagation of an electromagnetic signal in the atmosphere and the speed of light in a vacuum, as well as the refractive distortion of the trajectory along which the signal propagates.To eliminate the influence of the atmosphere, special corrections are used, which are applied into the measurement results in order to compensate for the above influencing factors. The most important among them is the correction, which takes into account the mean integral refractive index of air along the trace being measured.At present, as a rule, model methods for determining the correction are used, taking into account the mean integral refractive index of air. These methods use a point approximation of continuous functions describing the spatial distribution of temperature, pressure and air humidity on the trace being measured, as well as the representation of the finite sum of a certain integral specifying the exact correlation for the correction. Due to these limitations in the framework of the commonly used model methods, it is not always possible to achieve the required accuracy of the results of distance measurements. As for the well-known hardware methods, which are potentially more accurate, they are still under development. In this regard, new possibilities of improving the accuracy of model methods, in particular, the recently proposed gradient method, are considered.Theoretical studies of the accuracy of the gradient method for determining the mean integral refractive index of air in distance measurements on near-Earth traces are carried out. It is shown that it is more accurate than the well-known trapezium method. The equations of measurement of the gradient method obtained with the use of Hermite interpolation polynomials, which are valid for non-uniform partition of the measured trace by points in which the local values of the refractive index are determined, are considered. The requirements for the procedure and accuracy of measurements of the parameters necessary for determining the gradients of the refractive index of air at the end points of the trace in the framework of the gradient method are substantiated.The results of the studies provide a rigorous justification for new methods for determining the mean integral refractive index of air on near-Earth traces with a non-uniform profile of the underlying surface in the presence of significant elevation differences.

Weak Approximation for Points with Coordinates in Rank-one Subgroups of Global Function Fields

AbstractFor every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field satisfies the required property of weak approximation for finite sets of places of this function field avoiding arbitrarily given finitely many places.

On the effect of electrode finiteness in small-scale electrical resistivity imaging

Electrical resistivity tomography (ERT) profiles including finite steel-rod electrodes have been widely used in assuming a surface-node current injection for inversion. This hypothesis was shown by others to be safe for ratios of electrode embedment to electrode spacing smaller than 20%. Relying on the conductive cell model (CCM), we took into account the complete electrodes in the DC forward problem. We found that an electrode effect is included in resistivity sections inverted with a surface point electrode model. Several synthetic examples indicated that this unwanted effect is particularly developed when the electrode spacing does not meet a double constraint from the characteristic size of a shallow heterogeneity and from the electrode embedment. This effect deserved correction. A point approximation for a finite electrode referred to as the equivalent electrode point (EEP) was sought by placing a point-source current in the ground along the electrode length. The appropriate EEP depth was the one for which the CCM and a buried point source minimized a systematic geometric error; i.e., the relative change of the geometric factors obtained with the CCM and with an EEP. An EEP placed at 73% of the electrode length was declared as a suitable point approximation for an electrode. Use of this point assumption for inversion remedied efficiently the electrode effect subject to conditions. More precisely, the electrode spacing should stay within a lower bound equal to twice the electrode embedment and an upper bound equal to the shallow heterogeneity characteristic size divided by 0.75. The interest of such a metrological appraisal of the suitable acquisition layout to be used on the field was illustrated by a small-scale ERT field survey. This case study permitted us to understand more reliably the impact from fires upon a centimetric shallow layer in a calcareous wall.

Approximation of Information Divergences for Statistical Learning with Applications

Abstract In this paper we give a partial response to one of the most important statistical questions, namely, what optimal statistical decisions are and how they are related to (statistical) information theory. We exemplify the necessity of understanding the structure of information divergences and their approximations, which may in particular be understood through deconvolution. Deconvolution of information divergences is illustrated in the exponential family of distributions, leading to the optimal tests in the Bahadur sense. We provide a new approximation of I-divergences using the Fourier transformation, saddle point approximation, and uniform convergence of the Euler polygons. Uniform approximation of deconvoluted parts of I-divergences is also discussed. Our approach is illustrated on a real data example.

Heat Out of Equilibrium Driven by Potential Pulling Beyond the Overdamped Limit

We investigate heat accumulated for a long period time in nonequilibrium motion of a colloid under a potential with center pulled by a constant velocity. Heat and work have the same average value in steady state, but the distribution functions for thermal fluctuations are different. Difference comes from everlasting memory of initial high energy on total amount of accumulated heat. As a result, the positive tail of the heat distribution exhibits a distinct correction to the large deviation function, compared to a usual logarithmic correction. Mathematically, difference originates from the singularity in the generating function, the Fourier transform, of the distribution function and makes usual saddle point approximation not valid. We use the modified saddle point approximation technique to compute the heat distribution function. We find the heat distribution for the underdamped case and find the correction to large deviation function more predominant than for the overdamped case.

Nonparametric Probabilistic Load Flow With Saddle Point Approximation

Because of uncertainties of renewable generation resources and load, efficient tools are required for load flow analysis in power systems. Many of the existing papers assume a set of given probability density functions (PDFs) to model uncertainties and develop parametric probabilistic load flow tools. However, the uncertainties might not fall in any standard class of PDFs. Thus, nonparametric tools are needed. This paper presents a method for nonparametric probabilistic load flow analysis to determine PDFs of the load flow outputs. The proposed method is based on the mean value first order saddle point approximation. For a system with $\boldsymbol {n}$ random variables, $\boldsymbol {(n+1)}$ load flow calculations are utilized to establish first order Taylor series expansion and then saddle point approximation is adopted to determine the probabilistic characteristic of desired output variables. The proposed nonparametric estimator provides accurate results while needing a reasonable computational effort. The estimator needs only a limited available data set of random variables without requiring information whether the data set belongs to a certain class of parametric distribution functions. Furthermore, both probability and cumulative distribution functions of load flow outputs are directly established without using integral or differential operators. The proposed method is tested on the IEEE 14-bus and the IEEE 118-bus test systems and promising results are obtained.