18 results
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2. Extremal energies of Laplacian operator: Different configurations for steady vortices.
- Author
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Mohammadi, Seyyed Abbas
- Subjects
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LAPLACIAN operator , *POISSON processes , *BOUNDARY value problems , *ALGORITHMS , *MATHEMATICAL optimization - Abstract
In this paper, we study a maximization and a minimization problem associated with a Poisson boundary value problem. Optimal solutions in a set of rearrangements of a given function define stationary and stable flows of an ideal fluid in two dimensions. The main contribution of this paper is to determine the optimal solutions. At first, we determine a nearly optimal solution which is an approximation of the optimal solution when the problems are in low contrast regime. Secondly, for the high contrast regime, two optimization algorithms are developed. For the minimization problem, we prove that our algorithm converges to the global minimizer regardless of the initializer. The maximization algorithm is capable of deriving all local maximizers including the global one. Numerical experiments lead us to a conjecture about the location of the maximizers in the set of rearrangements of a function. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
3. Existence and concentrating behavior of solutions for Kirchhoff type equations with steep potential well.
- Author
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Jia, Huifang and Luo, Xiao
- Subjects
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LAPLACIAN matrices , *POTENTIAL well , *LINEAR operators , *ALGORITHMS , *HILBERT space - Abstract
In this paper, we consider the following Kirchhoff type equations (0.1) { − ( a + b ∫ R 3 | ∇ u | 2 ) Δ u + λ V ( x ) u = q ( x ) f ( u ) in R 3 , u ∈ H 1 ( R 3 ) , where a , b , λ > 0 , V ∈ C ( R 3 , R ) is a potential well, q ( x ) is a positive bounded function, f ( s ) is either asymptotically linear or asymptotically 3-linear in s at infinity. Under some other suitable conditions on V , q and f , the existence, nonexistence and concentrating behavior of solutions to problem (0.1) are obtained by using variational methods. We mainly extend the results in J. Sun and T. Wu (2014) [26] , which dealt with Kirchhoff type equations with positive potential well, to Kirchhoff type equations with sign-changing potential well. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
4. Some sharp inequalities related to Trudinger–Moser inequality.
- Author
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de Souza, Manassés
- Subjects
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MATHEMATICAL inequalities , *NONLINEAR operators , *DENSITY functional theory , *PROBLEM solving , *ALGORITHMS - Abstract
This paper deals with improvements of the Trudinger–Moser inequality related to the operator Q V ( u ) : = − Δ n u + V ( x ) | u | n − 2 u , where n ≥ 2 and the potential V : R n → R belongs to a class of nonnegative and continuous functions. Precisely, under suitable assumptions on V we consider the subspace E : = { u ∈ W 1 , n ( R n ) : ∫ R n V ( x ) | u | n d x < ∞ } endowed with the norm ‖ u ‖ : = [ ∫ R n ( | ∇ u | n + V ( x ) | u | n ) d x ] 1 / n and we prove that if ( u k ) is a sequence in E such that ‖ u k ‖ = 1 , u k ⇀ u ≢ 0 in E and 0 < p < p n ( u ) : = β n ( 1 − ‖ u ‖ n ) − 1 / ( n − 1 ) , then ( ⁎ ) sup k ∫ R n Ψ ( p | u k | n / ( n − 1 ) ) d x < ∞ , where Ψ ( t ) : = e t − ∑ i = 0 n − 2 t i i ! , β n : = n ω n − 1 1 / ( n − 1 ) and ω n − 1 is the measure of the unit sphere in R n . Furthermore, p n ( u ) is sharp in the sense that there exists a sequence ( u k ) ⊂ E satisfying ‖ u k ‖ = 1 and u k ⇀ u ≢ 0 in E such that the supremum (⁎) is infinite for p ≥ p n ( u ) . As an application of the previous result we prove the following sharp form of the Trudinger–Moser inequality for the subspace E . Considering ℓ ( α ) : = sup { u ∈ E : ‖ u ‖ = 1 } ∫ R n Ψ ∘ ν α ( u ) d x , where ν α ( u ) : = β n ( 1 + α ‖ u ‖ n n ) 1 / ( n − 1 ) | u | n / ( n − 1 ) , assuming some conditions of symmetry on V it is established (1) for 0 ≤ α < λ 1 ( V ) we have ℓ ( α ) < ∞ , (2) for α ≥ λ 1 ( V ) , ℓ ( α ) = ∞ and (3) moreover, we prove that for 0 ≤ α < λ 1 ( V ) , an extremal function for ℓ ( α ) exists. Here λ 1 ( V ) denotes the first eigenvalue of Q V ( u ) . [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
5. Backward–forward algorithms for structured monotone inclusions in Hilbert spaces.
- Author
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Attouch, Hédy, Peypouquet, Juan, and Redont, Patrick
- Subjects
- *
ALGORITHMS , *MONOTONE operators , *HILBERT space , *CONVEX domains , *COMPUTATIONAL complexity , *DISCRETIZATION methods - Abstract
In this paper, we study the backward–forward algorithm as a splitting method to solve structured monotone inclusions, and convex minimization problems in Hilbert spaces. It has a natural link with the forward–backward algorithm and has the same computational complexity, since it involves the same basic blocks, but organized differently. Surprisingly enough, this kind of iteration arises when studying the time discretization of the regularized Newton method for maximally monotone operators. First, we show that these two methods enjoy remarkable involutive relations, which go far beyond the evident inversion of the order in which the forward and backward steps are applied. Next, we establish several convergence properties for both methods, some of which were unknown even for the forward–backward algorithm. This brings further insight into this well-known scheme. Finally, we specialize our results to structured convex minimization problems, the gradient-projection algorithms, and give a numerical illustration of theoretical interest. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
6. Noether's theorem of fractional Birkhoffian systems.
- Author
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Zhang, Hong-Bin and Chen, Hai-Bo
- Subjects
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NOETHER'S theorem , *FRACTIONAL calculus , *RIEMANN surfaces , *LIOUVILLE'S theorem , *ALGORITHMS - Abstract
In this paper, we study Noether type symmetry theorem to fractional Birkhoffian systems with Riemann–Liouville derivatives. This theorem provides an explicit algorithmic way to compute a constant for any Birkhoffian systems admitting a symmetry. Finally, we extend our Noether's theorem to fractional Birkhoffian systems base on Caputo or Riesz derivatives. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
7. Invariant measures for continued fraction algorithms with finitely many digits.
- Author
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Kraaikamp, Cor and Langeveld, Niels
- Subjects
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INVARIANT measures , *CONTINUED fractions , *ALGORITHMS , *MATHEMATICAL expansion , *STOCHASTIC convergence - Abstract
In this paper we consider continued fraction (CF) expansions on intervals different from [ 0 , 1 ] . For every x in such interval we find a CF expansion with a finite number of possible digits. Using the natural extension, the density of the invariant measure is obtained in a number of examples. In case this method does not work, a Gauss–Kuzmin–Lévy based approximation method is used. Convergence of this method follows from [32] but the speed of convergence remains unknown. For a lot of known densities the method gives a very good approximation in a low number of iterations. Finally, a subfamily of the N -expansions is studied. In particular, the entropy as a function of a parameter α is estimated for N = 2 and N = 36 . Interesting behavior can be observed from numerical results. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
8. On the abelian complexity of the Rudin–Shapiro sequence.
- Author
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Lü, Xiaotao, Chen, Jin, Wen, Zhixiong, and Wu, Wen
- Subjects
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ABELIAN groups , *COMPUTATIONAL complexity , *MATHEMATICAL sequences , *ALGORITHMS , *POINT mappings (Mathematics) - Abstract
In this paper, we study the abelian complexity of the Rudin–Shapiro sequence and a related sequence. We show that these two sequences share the same complexity function ρ ( n ) , which satisfies certain recurrence relations. As a consequence, the abelian complexity function is 2-regular. Further, we prove that the box dimension of the graph of the asymptotic function λ ( x ) is 3/2, where λ ( x ) = lim k → ∞ ρ ( 4 k x ) / 4 k x and ρ ( x ) = ρ ( ⌊ x ⌋ ) for every x > 0 . [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
9. Over relaxed hybrid proximal extragradient algorithm and its application to several operator splitting methods.
- Author
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Shen, Li
- Subjects
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LINEAR operators , *STOCHASTIC convergence , *ALGORITHMS , *MATHEMATICAL models , *METRIC system - Abstract
In this paper we propose a new over-relaxed variant of the hybrid proximal extragradient (HPE) algorithm, for the monotone inclusion problem, which uses a projection-free extragradient step with explicit over relaxed stepsize. Its global convergence as well as ergodic and nonergodic complexity rates are established. Moreover, local linear convergence rates are derived under some mild regularity condition. One benefit of the new over relaxed variant of the HPE is that it covers a large class of popular operator splitting methods and their over relaxed versions, thus providing a comprehensive insight on these operators splitting methods. In particular, forward Douglas Rachford splitting method, forward Spingarn's Partial Inverse method, forward Spingarn's partial inverse forward method and Davis–Yin's three operator splitting method are all included as special cases of the over relaxed HPE algorithm. Another benefit is that the interval of stepsize relaxation is easily estimated for these operator splitting methods under the presented framework. Additionally, the over relaxed Korpelevich's method and over relaxed forward–backward–forward method are formulated directly with convergence guarantee based on the proposed framework. The third benefit is that the local linear convergence for a large class of operator splitting methods is established effortlessly under metric subregularity condition. Moreover, this linear convergence condition is shown weaker than some existing ones that almost all require the strong monotonicity of the composite operators. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
10. Numerical computation of connecting orbits in planar piecewise smooth dynamical system.
- Author
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Zou, Yongkui, Zheng, Dan, and Chai, Shimin
- Subjects
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DYNAMICAL systems , *ALGORITHMS , *BIFURCATION theory , *MATHEMATICAL analysis , *EXPONENTIAL dichotomy - Abstract
In this paper, a numerical algorithm for computing the connecting orbits in piecewise smooth dynamical systems is derived and is analyzed. A nondegenerate condition for the connecting orbit with respect to its bifurcation parameter is presented to ensure the defining equation being well posed, which is a generalization of the Melnikov condition for smooth systems. The error caused by the truncation of time interval is also analyzed. Some numerical calculations are carried out to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
11. A 3D optimal control problem related to the urban heat islands.
- Author
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Fernández, F.J., Alvarez-Vázquez, L.J., Martínez, A., and Vázquez-Méndez, M.E.
- Subjects
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OPTIMAL control theory , *URBAN heat islands , *COMPUTER simulation , *PARTIAL differential equations , *INTERIOR-point methods , *ALGORITHMS - Abstract
Within the framework of numerical simulation and optimal control of partial differential equations, in this work we deal with the mathematical modelling and control of the processes related to the urban heat island effect. In particular, we are interested in finding the optimal locations of green zones inside metropolitan areas in order to mitigate the consequences of this harmful phenomenon. So, we consider a three-dimensional climate model and formulate a constrained optimal control problem, that is extensively analyzed in the first part of the paper. Then, we propose a complete numerical algorithm for its resolution, interfacing the interior point algorithm IPOPT with the FreeFem++ software package. Finally, we present several numerical tests for a simple realistic case, where the advantages of our approach are shown. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
12. Computation of local ISS Lyapunov functions for discrete-time systems via linear programming.
- Author
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Li, Huijuan and Grüne, Lars
- Subjects
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LYAPUNOV functions , *DISCRETE-time systems , *LINEAR programming , *ALGORITHMS , *MATHEMATICAL optimization , *TRIANGULATION - Abstract
This paper presents a numerical algorithm for computing ISS Lyapunov functions for discrete-time systems which are input-to-state stable (ISS) on compact subsets of the state space. The algorithm relies on solving a linear optimisation problem and delivers a continuous and piecewise affine ISS Lyapunov function on a suitable triangulation covering the given compact set excluding a small neighbourhood of the origin. The objective of the linear optimisation problem is to minimise the ISS gain. It is shown that for every ISS system there exists a suitable triangulation such that the proposed algorithm terminates successfully. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
13. Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements.
- Author
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Farhat, Aseel, Lunasin, Evelyn, and Titi, Edriss S.
- Subjects
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ALGORITHMS , *RAYLEIGH-Benard convection , *POROUS materials , *TEMPERATURE measurements , *EVOLUTION equations , *EXPONENTIAL functions - Abstract
In this paper we propose a continuous data assimilation (downscaling) algorithm for the Bénard convection in porous media using only discrete spatial-mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed (finite dimensional projection of) temperature data. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
14. Constructive analysis for coefficient regularization regression algorithms.
- Author
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Nie, Weilin and Wang, Cheng
- Subjects
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COEFFICIENTS (Statistics) , *MATHEMATICAL regularization , *REGRESSION analysis , *ALGORITHMS , *LEAST squares - Abstract
In this paper, we consider the least squares regression algorithm with a generalized coefficient regularization term. A novel error decomposition involving a constructive stepping-stone function is introduced. By choosing appropriate parameters for the constructive function we finally derive a satisfactory learning rate under some condition for the goal function and capacity of the hypothesis space. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
15. Identifying weak foci and centers in the Maxwell–Bloch system.
- Author
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Liu, Lingling, Aybar, O. Ozgur, Romanovski, Valery G., and Zhang, Weinian
- Subjects
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MATHEMATICAL decomposition , *ALGORITHMS , *COMMUTATIVE algebra , *MANIFOLDS (Mathematics) , *MATHEMATICAL singularities - Abstract
In this paper we identify weak foci and centers in the Maxwell–Bloch system, a three dimensional quadratic system whose three equilibria are all possible to be of center-focus type. Applying irreducible decomposition and the isolation of real roots in computation of algebraic varieties of Lyapunov quantities on an approximated center manifold, we prove that at most 6 limit cycles arise from Hopf bifurcations and give conditions for exact number of limit cycles near each weak focus. Further, applying algorithms of computational commutative algebra we find Darboux polynomials and give some center manifolds in closed form globally, on which we identify equilibria to be centers or singular centers by integrability and time-reversibility on a center manifold. We prove that those centers are of at most second order. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
16. Direct algorithm for multipolar sources reconstruction.
- Author
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Abdelaziz, Batoul, El Badia, Abdellatif, and El Hajj, Ahmad
- Subjects
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ALGORITHMS , *ELLIPTIC equations , *MATHEMATICAL programming , *PARTIAL differential equations , *HELMHOLTZ equation - Abstract
This paper proposes an identification algorithm for identifying multipolar sources F in the elliptic equation Δ u + μ u = F from boundary measurements. The reconstruction question of this class of sources appears naturally in Helmholtz equation ( μ > 0 ) and in biomedical phenomena particularly in EEG/MEG problems ( μ = 0 ) and bioluminescence tomography (BLT) applications ( μ < 0 ) . Previous works have treated the inverse multipolar source problems, only for equations with μ = 0 , using algebraic approaches depending on the complex calculation of determinants. Knowing that the novelty in our method concerns several points, the principal one is its simplicity where its proof is not founded on the determinants calculation and its ease in implementation. Moreover, this work involves the general form of equations considering μ ∈ R and at the same time considers a more general type of sources than former related works including sources having small compact support within a finite number of subdomains. Finally, some numerical results are shown to prove the robustness of our identification algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
17. New properties of the lemniscate function and its transformation.
- Author
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Nishimura, Ryo
- Subjects
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MATHEMATICAL functions , *MATHEMATICAL transformations , *INFINITY (Mathematics) , *ALGORITHMS , *GEOMETRIC analysis - Abstract
In this paper, we show several formulas for the lemniscate function which include an infinite product formula for the lemniscate sine. Furthermore, we show the relation between the product formula and Carlson's algorithm which is known as the variant of the arithmetic geometric mean of Gauss. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
18. When is the derivative of an eta quotient another eta quotient?
- Author
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Aygin, Zafer Selcuk and Toh, Pee Choon
- Abstract
In this paper we use techniques from the theory of modular forms to determine all eta quotients whose derivative is also an eta quotient of levels up to 36. In addition, we present an algorithm that determines all eta quotients in M 2 k (Γ 0 (N)). We also discuss some applications of these results. In particular, we evaluate a number of integrals in terms of algebraic constants. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
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