Approximation Problem Research Articles

Global classical solution of the fractional Nernst-Planck-Poisson-Navier- Stokes system in $ \mathbb{R}^{3} $

<abstract><p>In this paper, we consider a fractional Nernst-Planck-Poisson-Navier-Stokes system in $ \mathbb{R}^{3} $. First, we obtain a priori estimates by using energy estimates. Then, we construct an iterative solution sequence by solving the approximate problem and obtaining the local existence and uniqueness of the classical solution. Finally, combining the local existence with a priori estimates, the global existence and uniqueness of the classical solution with small initial data are obtained.</p></abstract>

Existence of the optimal controls for a controlled elliptic system with an $ L^0 $ term in the cost functional

Optimal control problems with an $ L^0 $ term in the cost functional are considered. The $ L^0 $ term measures the size of the support of the control, and brings difficulty for studying the optimal control problems. By studying the existence and necessary conditions of optimal controls for a series of approximation problems and corresponding relaxed problems, the existence of optimal controls for a controlled elliptic system with an $ L^0 $-cost is obtained. Necessary conditions and the $ L^\infty $-regularity of optimal controls are showed in the paper as well.

Numerical Study of the Rate of Convergence of Chernoff Approximations to Solutions of the Heat Equation

The article is devoted to construction of examples illustrating (using computer calculation) the convergence rate of Chernoff approximations to the solution of the Cauchy problem for the heat equation. Two Chernoff functions (of the first and second order of Chernoff tangency to the double differentiation operator) and several initial conditions of different smoothness are considered. As an illustration for the initial condition equal to the absolute value of the sine function to the power of five over two, a graph of the exact solution of the Cauchy problem and graphs of the tenth Chernoff approximations given by two different Chernoff functions are plotted. It is visually determined that the approximations are close to the solution. For each of the two Chernoff functions, for several initial conditions of different smoothness and for the approximation numbers up to 11, the error corresponding to each approximation is numerically found. This error is understood as the supremum of the absolute value of the difference between the exact solution and its approximating function. As it turned out, in all the cases studied, the error dependence on the approximation number nearly power-law form. This follows from the fact that the dependence of the error logarithm on the logarithm of the approximation number is close to linear. By finding the equation of the approximating line using linear regression, we find the exponent in the power dependence of the error on the approximation number and call it the order of convergence. These orders for all studied initial conditions are compiled in a table. The empirical dependence of the convergence order on the smoothness class of the initial condition is found on the considered family of initial conditions.

Теоретические основы искусственных нейронных сетей для решения задачи аппроксимации и интерполяции

в статье проведено исследование теоретической базы искусственных нейронных сетей. Изучалось теоретическое обоснование возможности аппроксимации функций многих переменных суперпозицией функций одной переменной. Рассмотрены основные универсальные теоремы аппроксимации, представленные и доказанные к настоящему моменту зарубежными и отечественными авторами. Рассмотрены теоремы аппроксимации, в которых представлено необходимое количество нейронов в слое — ограничение по ширине; теоремы, в которых показано необходимое количество слоев в нейронной сети — ограничение по глубине; теоремы, в которых авторы доказывают минимальные границы одновременно для количества слоев в сети и количества нейронов на слое — ограничения по глубине и ширине we studied the theoretical foundations of artificial neural networks as applied to the possibility of approximating functions of many variables by superposition of functions of one variable. We considered the most important universal approximation theorems. We also studied the approximation theorems with the required number of neurons in a layer (width constraint) or the number of layers in a neural network (depth constraint), and the theorems in which their authors prove the existence of min bounds both for the number of layers and for the number of neurons per layer

An algorithm for stochastic convex-concave fractional programs with applications to production efficiency and equitable resource allocation

We propose an algorithm to solve convex and concave fractional programs and their stochastic counterparts in a common framework. Our approach is based on a novel reformulation that involves differences of square terms in the constraints, and subsequent employment of piecewise-linear approximations of the concave terms. Using the branch-and-bound (B&B) framework, our algorithm adaptively refines the piecewise-linear approximations and iteratively solves convex approximation problems. The convergence analysis provides a bound on the optimality gap as a function of approximation errors. Based on this bound, we prove that the proposed B&B algorithm terminates in a finite number of iterations and the worst-case bound to obtain an ϵ-optimal solution reciprocally depends on the square root of ϵ. Numerical experiments on Cobb–Douglas production efficiency and equitable resource allocation problems support that the algorithm efficiently finds a highly accurate solution while significantly outperforming the benchmark algorithms for all the small size problem instances solved. A modified branching strategy that takes the advantage of non-linearity in convex functions further improves the performance. Results are also discussed when solving a dual reformulation and using a cutting surface algorithm to solve distributionally robust counterpart of the Cobb–Douglas example models.

Behavioral Modeling of Memristors under Harmonic Excitation.

Memristors are devices built on the basis of fourth passive electrical elements in nanosystems. Because of the multitude of technologies used for memristor implementation, it is not always possible to obtain analytical models of memristors. This difficulty can be overcome using behavioral modeling, which is when mathematical models are constructed according to the input-output relationships on the input and output signals. For memristor modeling, piecewise neural and polynomial models with split signals are proposed. At harmonic input signals of memristors, this study suggests that split signals should be formed using a delay line. This method produces the minimum number of split signals and, as a result, simplifies behavioral models. Simplicity helps reduce the dimension of the nonlinear approximation problem solved in behavioral modeling. Based on the proposed method, the piecewise neural and polynomial models with harmonic input signals were constructed to approximate the transfer characteristic of the memristor, in which the current dynamics are described using the Bernoulli differential equation. It is shown that the piecewise neural model based on the feedforward network ensures higher modeling accuracy at almost the same complexity as the piecewise polynomial model.

Tidal Modeling Analysis For Potensial Tidal Power Generating In The District West Siberut

Numerical modeling is a combination of mathematical representation of the problem in physical form and numerical approximation of mathematical equations. Oceanographic numerical modeling is a system for studying ocean dynamic phenomena into discrete/numerical equations. Tidal power plants are renewable tidal energy. The working principle is the same as a hydroelectric power plant, where water is used to rotate a turbine and produce electrical energy. This research uses descriptive quantitative methods, data processing with MIKE 21 software which results in predicted data on tides and current speeds, ,Results from flow and pass modelingg low tide for potential electricity generation where the highest tide in 2024 in May was 1.53m and the lowest tide in April was -0.92m. In 2025, the highest tide in March will be 2.07m and the lowest tide in April will be -0.56m. for electrical energy potential in 2024 the average is 5.7kWh, and in 2025 the average is 9.07kWh.

Extreme problems of weight approximation on the real axis

The exact value was calculated and the polynomial for the best weighted polynomial approximation was found for the kernels of the form $\frac {A+Bx}{(x^2+\lambda^2)^2}$, where $A,B\in {\mathbb R}$ and $\lambda>0$, in the root-mean-square metric.

On a general approach to some problems of approximation of operators

In the article we propose a general scheme for solutions of some approximation problems under a rather general setting. We illustrate the application of the proposed scheme by a series of examples, in particular we show that many results in the area of Ostrowski type inequalities can be obtained by standard arguments.

Approximate sorting and its applications in I/O model

Sorting is a fundamental component in many different applications including web indexing engines, geographic information systems, data mining and database systems. However, the existing optimal algorithms still cause huge costs when the input data becomes very large. Thus, the approximate sorting for big data is considered in this paper. The goal of approximate sorting for big data is to generate an approximate sorted result, but using less CPU and I/O cost. For big data, we consider the approximate sorting and applications for approximate sorted results in I/O model.The quality of approximate sorting results is usually measured by the distance metrics on permutation space. However, the existing metrics on permutation space are not available for external approximate sorting algorithms. Thus, we propose a new kind of metric named External metric, which ignores the errors and dislocations that happened in each I/O block. The External metric of Spearman's footrule metric is named as External Spearman's footrule metric (short as ESP metric), which is analyzed in this paper.Furthermore, to facilitate a better evaluation of the approximate sorted result, we propose a new metric, named as errors, which directly states the number of dislocation of the elements. The External errors are also considered in this paper.Then, according to the rate-distortion relationship under these two metrics, the lower bound of these two metrics on external approximate sorting problem with t I/O operations is proved. We propose a k-passes external approximate sorting algorithm, named as EASORT, and prove that EASORT is asymptotically optimal under ESP metric.Finally, we consider the applications on approximate sorting results. An index for the approximate sorted result is proposed. The single and range query on approximate sorted result using this index are also analyzed. Further, the sort-merge join on two relations, where one of the relations is approximate sorted or both relations are approximate sorted, are all discussed in this paper.

Arithmetics of homogeneous spaces over p$p$‐adic function fields

AbstractLet be the function field of a smooth projective geometrically integral curve over a finite extension of . Following the works of Harari, Scheiderer, Szamuely, Izquierdo, and Tian, we study the local–global and weak approximation problems for homogeneous spaces of with geometric stabilizers extension of a group of multiplicative type by a unipotent group. The tools used are arithmetic (local and global) duality theorems in Galois cohomology, in combination with techniques similar to those used by Harari, Szamuely, Colliot‐Thélène, Sansuc, and Skorobogatov. As a consequence, we show that any finite abelian group is a Galois group over , rediscovering the positive answer to the abelian case of the inverse Galois problem over . In the case where the curve is defined over a higher dimensional local field instead of a finite extension of , coarser results are also given.

AIOL: An Improved Orthogonal Lattice Algorithm for the General Approximate Common Divisor Problem

The security of several fully homomorphic encryption (FHE) schemes depends on the intractability assumption of the approximate common divisor (ACD) problem over integers. Subsequent efforts to solve the ACD problem as well as its variants were also developed during the past decade. In this paper, an improved orthogonal lattice (OL)-based algorithm, AIOL, is proposed to solve the general approximate common divisor (GACD) problem. The conditions for ensuring the feasibility of AIOL are also presented. Compared to the Ding–Tao OL algorithm, the well-known LLL reduction method is used only once in AIOL, and when the error vector r is recovered in AIOL, the possible difference between the restored and the true value of p is given. Experimental comparisons between the Ding-Tao algorithm and ours are also provided to validate our improvements.

Shorter quantum circuits via single-qubit gate approximation

We give a novel procedure for approximating general single-qubit unitaries from a finite universal gate set by reducing the problem to a novel magnitude approximation problem, achieving an immediate improvement in sequence length by a factor of 7/9. Extending the works \cite{Hastings2017} and \cite{Campbell2017}, we show that taking probabilistic mixtures of channels to solve fallback \cite{BRS2015} and magnitude approximation problems saves factor of two in approximation costs. In particular, over the Clifford+T gate set we achieve an average non-Clifford gate count of 0.23log2⁡(1/ε)+2.13 and T-count 0.56log2⁡(1/ε)+5.3 with mixed fallback approximations for diamond norm accuracy ε.This paper provides a holistic overview of gate approximation, in addition to these new insights. We give an end-to-end procedure for gate approximation for general gate sets related to some quaternion algebras, providing pedagogical examples using common fault-tolerant gate sets (V, Clifford+T and Clifford+T). We also provide detailed numerical results for Clifford+T and Clifford+T gate sets. In an effort to keep the paper self-contained, we include an overview of the relevant algorithms for integer point enumeration and relative norm equation solving. We provide a number of further applications of the magnitude approximation problems, as well as improved algorithms for exact synthesis, in the Appendices.

Mathematical analysis of a fourth-order degenerate equation with a nonlinear second-order term

A fourth-order partial differential model in this paper is called the thin-film equation in the field of fluid theory. The existence of the weak solution is obtained by solving two approximation problems. In order to perform the limit for small parameters in the approximation problems, the Galerkin method and the entropy functional method are both used. By means of some classical compactness results we can give the existence of nonnegative weak solutions. Finally, by redefining the entropy functional, the long-time exponential decay is given in the sense of L1-norm.

Approximation of Dirichlet-to-Neumann operator for a planar thin layer and stabilization in the framework of couple stress elasticity with voids

In this paper, we start from a two dimensional transmission model problem in the framework of couple stress elasticity with voids which is defined in a fixed domain Ω − juxtaposed with a planar thin layer Ω + δ . We first derive a first approximation of Dirichlet-to-Neumann operator for the thin layer Ω + δ by using the techniques of asymptotic expansion with scaling, which allows us to approximate the transmission problem by a boundary value problem doesn’t take into account any more the thin layer Ω + δ , called approximate impedance problem; after that, we prove an error estimate between the solution of the transmission problem and the solution of the approximate impedance problem.

On the distribution of Born transmission eigenvalues in the complex plane

We analyze an approximate interior transmission eigenvalue problem in R d for d = 2 or d = 3, motivated by the transmission problem of a transformation optics-based cloaking scheme and obtained by replacing the refractive index with its first order approximation, which is an unbounded function. Using the radial symmetry we show the existence of (infinitely many) complex transmission eigenvalues and prove their discreteness. Moreover, it is shown that there exists a horizontal strip in the complex plane around the real axis, that does not contain any transmission eigenvalues.

Approximation and interpolation with neural network

In this paper we show that multilayer feedforward networks with one single hidden layer.and certain types of activation functions can approximate univariant continuous functions defined on a compact set. arbitrarily well. In particular, our results contain some usual activation functions such as sigmoidal functions, RELU functions and threshold functions. Besides, since interpolation problems are highly related to approximation problem, we demonstrate that a wide range of functions have the ability to interpolate and generalize our results to functions which are not polynomial on R. Compared to existing results by numerous work, our methods are more intuitive and less technical. Lastly, the paper discusses the possibility of combining interpolation property and approximating property together, and demonstrates that given any Riemann integrable functions on a compact set in R, with several points on its graph, the finite combination of monotone sigmoidal functions can pass through these points and approximate the given function arbitrarily well with respect to L^1 (dx) (in the sense of Riemann integral) when the number of points getting large.

Local well-posedness of the free-boundary incompressible magnetohydrodynamics with surface tension

We prove the local well-posedness of the 3D free-boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, which describe the motion of a perfect conducting fluid in an electromagnetic field. We adapt the ideas developed in the remarkable paper [11] by Coutand and Shkoller to generate an approximate problem with artificial viscosity indexed by κ>0 whose solution converges to that of the MHD equations as κ→0. However, the local well-posedness of the MHD equations is no easy consequence of Euler equations thanks to the strong coupling between the velocity and magnetic fields. This paper is the continuation of the second and third authors' previous work [38] in which the a priori energy estimate for incompressible free-boundary MHD with surface tension is established. But the existence is not a trivial consequence of the a priori estimate as it cannot be adapted directly to the approximate problem due to the loss of the symmetric structure.

Екстремальні значення найкращих наближень лінійних комбінацій гармонічних функцій

Extreme problems and their practical applications have been under the scrutiny of mathematicians since ancient times. An important step in the development of extreme problems was made by P. L. Chebyshev, who in the 50s of the 19th century laid the foundations of a section of destructive function theory – the theory of approximation. A significant role of the formation of the theory of approximation of functions was played by Carl Weierstrass’s theorem on the convergence to zero of best approximations by polynomials of a continuous function. As is well known, Weierstrass’s theorem is not constructive – it does not contained estimates of the approach speed. Thanks to the work of D. Jackson, S. N. Bernstein, Vallee-Poussin and others, such estimates began to appear in works on approximation theory. At the same time, at the first stages of the development of the theory of approximation, approximations of individual functions were studied. That beginning of a new period, a dipper study of the deviation values of functions from their approximating polynomials, dates back to the 30s and 40s of the 20th century and is associated with the names of A. M. Kolmogorov, S. M. Nikolsky, J. Favard, N. I. Achieser, M. G. Crane and B. Nagy. Thanks to their works, the main emphasis in the theory of approximations is shifted to the study of the best approximations or other approximation characteristics of functions that have certain differential-difference or smoothness properties. In particular, in 1936, J. Favard calculated the exact values of the best uniform approximations by trigonometric polynomials of order no higher than n – 1 on classes of differentiable 2π-periodic functions, whose r-th (r – natural) derivatives are in a unit sphere of the space of essentially bounded functions. The problem of obtaining exact values of the best approximations in uniform and integral metrics for various functional compacts was in sight of many prominent mathematicians of the XX century. General issues related to the study of the best approximation functional: the existence of a polynomial of the best approximation, its characteristic properties, are destribed in detail in many monographs, in particular, for example, in the book by M. P. Korneichuk [1]. In the 80s and 90s of the XX century, O. I. Stepanets (see, [2, section III]) developed a new approach to the classification of periodic functions, which allowed for a fairly fine classification of extremely wide sets of periodic functions. At the same time, the results obtained for these classes are, on the one hand, general, and on the other hand, they give a number of new, hitherto unknown results that were impossible to obtain on previously known classes. Following the approaches to the requirements of function classification, we can consider a linear combination of function classes of a more complex nature. And then the problem of finding the exact values of the upper bounds of the best joint approximations will be reduced to the problem of the best approximation of this composite class corresponding to convolutions with the composite kernel.

Approximate optimality conditions and duality results for non-smooth semi-infinite programming problems

This paper concentrates on investigating a semi-infinite optimization problem with cone constraints. Under the attainment of Abadie constraint qualification, an approximate necessary optimality condition by employing Clarke subdifferential for the semi-infinite optimization problem having cone constraints is developed. A new class of functions namely Q-quasiconvex functions is introduced and in the light of the approximate pseudoconvexity and Q-quasiconvexity assumptions, an approximate sufficient optimality condition is investigated using locally Lipschitz functions. Additionally, we formulate the approximate Wolfe's and approximate Mond-Weir dual problems for the non-smooth semi-infinite optimization problem. Subsequently, duality relations in terms of weak, strong and converse results between the semi-infinite optimization model and the aforementioned dual problems are established under the approximate pseudoconvexity and Q-quasiconvexity assumptions. Moreover, to justify the main results of the paper, numerical examples have been shown at suitable places.