Symmetric duality with (p, r) − ρ − (η, θ)-invexity

In this paper we introduce a new type of generalized invex function, called (p, r) − ρ − (η, θ)-invex function and study symmetric duality results under these assumptions. In our study the nonnegative orthants for the constraints are replaced by closed convex cones and their polars. We establish weak and strong duality theorems under (p, r) − ρ − (η, θ)-invexity assumptions for the symmetric dual problems. We also give many examples to justify our results.     

Approximation of signals of class Lip(α, p) by linear operators

Mittal, Rhoades [5], [6], [7] and [8] and Mittal et al. [9] and [10] have initiated a study of error estimates E_n(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4], where he has weakened the conditions on {p_n} given by Chandra [2], to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15].     

On Tauberian theorems for (A)(C, α) summability method

In this paper we introduce some Tauberian conditions for the (A)(C, α) summability method. These results extend and generalize some of the classical Tauberian theorems for the Abel summability method.     

New approach to the η-proximal point algorithm and nonlinear variational inclusion problems

A new approach using the over-relaxed proximal point algorithm in the context of solving a class of inclusion problems based on the notion of maximal (η)-monotonicity is developed and examined. Convergence analysis seems to be reasonable, and finally, some specializations are included. Furthermore, the model developed in this communication seems to be appropriate to the Yosida approximation in the sense that it can be applied to first-order evolution equations/inclusions as well.     

Nonsmooth minimax programming under locally Lipschitz (Φ, ρ)-invexity

Minimax programming problems involving locally Lipschitz (Φ, ρ)-invex functions are considered. The parametric and non-parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable (Φ, ρ)-invexity assumption imposed on objective and constraint functions. When the sufficient conditions are utilized, parametric and non-parametric dual problems in the sense of Mond-Weir and Wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem. With the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems.     

On simultaneous approximation to (α,α) with α+kα−1=0

We show that the modified Jacobi-Perron algorithm gives the best simultaneous approximation to (α,α²) with α³+kα−1=0. We claim the following facts:

(1)

the limit set of become an ellipse, where (p_n,q_n,r_n) is the nth convergent (p_n/q_n,r_n/q_n) of (α,α²) by the modified Jacobi-Perron algorithm,

(2)

the limit set of belongs to outside of the ellipse in (1).

     

Semiconductor final test scheduling with Sarsa(λ, k) algorithm

Semiconductor test scheduling problem is a variation of reentrant unrelated parallel machine problems considering multiple resource constraints, intricate {product, tester, kit, enabler assembly} eligibility constraints, sequence-dependant setup times, etc. A multi-step reinforcement learning (RL) algorithm called Sarsa(λ, k) is proposed and applied to deal with the scheduling problem with throughput related objective. Allowing enabler reconfiguration, the production capacity of the test facility is expanded and scheduling optimization is performed at the bottom level. Two forms of Sarsa(λ, k), i.e. forward view Sarsa(λ, k) and backward view Sarsa(λ, k), are constructed and proved equivalent in off-line updating. The upper bound of the error of the action-value function in tabular Sarsa(λ, k) is provided when solving deterministic problems. In order to apply Sarsa(λ, k), the scheduling problem is transformed into an RL problem by representing states, constructing actions, the reward function and the function approximator. Sarsa(λ, k) achieves smaller mean scheduling objective value than the Industrial Method (IM) by 68.59% and 76.89%, respectively for real industrial problems and randomly generated test problems. Computational experiments show that Sarsa(λ, k) outperforms IM and any individual action constructed with the heuristics derived from the existing heuristics or scheduling rules.     

Computational study of decaying annular vortex flow using the Rε/k − ε turbulence model

In this article, we present three dimensional CFD study of turbulent vortex flow in an annular passage using OpenFOAM 1.6. The vortex flow is generated by introducing the flow through a tangential entry to the passage. For the analysis presented in this article, turbulence was modeled using the R_ε/k − ε model, in addition, a comparison between such model with the standard k − ε model was conducted and discussed. The main characteristics of the flow such as vortex structure and recirculation zone were investigated. It was found that flow is subjected to Rankine vortex structure with three forced vortex regimes and a free vortex region near to the outer wall. The phenomenon of vortex decay was investigated by depicting the swirl number trend along the axial direction of the flow domain. It was found that the vortex decay is subjected to an exponential decay behavior. New coefficients for the exponential decay correlation were derived based on local values of velocity components in different radial planes.     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

Embeddability of L1(μ) in dual spaces, geometry of cones and a characterization of c0

In this article we suppose that (Ω,Σ,μ) is a measure space and T an one-to-one, linear, continuous operator of L₁(μ) into the dual E′ of a Banach space E. For any measurable set A consider the image T(L⁺₁(μ_A)) of the positive cone of the space L₁(μ_A) in E′, where μ_A is the restriction of the measure μ on A. We provide geometrical conditions on the cones T(L⁺₁(μ_A)) which yield that the measure μ is atomic, i.e., that L₁(μ) is lattice isometric to , where denotes the set of atoms of μ. This result yields also a new characterization of c₀(Γ).     

Improved T − ψ nodal finite element schemes for eddy current problems

The aim of this paper is to propose improved T − ψ finite element schemes for eddy current problems in the three-dimensional bounded domain with a simply-connected conductor. In order to utilize nodal finite elements in space discretization, we decompose the magnetic field into summation of a vector potential and the gradient of a scalar potential in the conductor; while in the nonconducting domain, we only deal with the gradient of the scalar potential. As distinguished from the traditional coupled scheme with both vector and scalar potentials solved in a discretizing equation system, the proposed decoupled scheme is presented to solve them in two separate equation systems, which avoids solving a saddle-point equation system like the traditional coupled scheme and leads to an important saving in computational effort. The simulation results and the data comparison of TEAM Workshop Benchmark Problem 7 between the coupled and decoupled schemes show the validity and efficiency of the decoupled one.     

Heat kernel bounds, ancient κ solutions and the Poincaré conjecture

We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3-dimensional ancient κ solutions to the Ricci flow. As an application, using the W entropy associated with the heat kernel, we give a different and much shorter proof of Perelman's classification of backward limits of these ancient solutions. The method is partly motivated by Cao (2007) [1] and Sesum (2006) [27]. The current paper or Chow and Lu (2004) [6] combined with Chen and Zhu (2006) [4] and Zhang (2009) [31] lead to a simplified proof of the Poincaré conjecture without using reduced distance and reduced volume.     

Effective computation of traces,determinants, and ζ-functions for Sturm–Liouville operators

The principal aim in this paper is to develop an effective and unified approach to the computation of traces of resolvents (and resolvent differences), Fredholm determinants, ζ- functions, and ζ-function regularized determinants associated with linear operators in a Hilbert space. In particular, we detail the connection between Fredholm and ζ-function regularized determinants.Concrete applications of our formalism to general (i.e., three-coefficient) regular Sturm–Liouville operators on bounded intervals with various (separated and coupled) boundary conditions, and Schrödinger operators on a half-line, are provided and further illustrated with an array of examples.     

Existence and algorithm of solutions for general set-valued Noor variational inequalities with relaxed (μ,ν)-cocoercive operators in Hilbert spaces

The purpose of this paper is to suggest and analyze a number of iterative algorithms for solving the generalized set-valued variational inequalities in the sense of Noor in Hilbert spaces. Moreover, we show some relationships between the generalized set-valued variational inequality problem in the sense of Noor and the generalized set-valued Wiener-Hopf equations involving continuous operator. Consequently, by using the equivalence, we also establish some methods for finding the solutions of generalized set-valued Wiener-Hopf equations involving continuous operator. Our results can be viewed as a refinement and improvement of the previously known results for variational inequality theory.     

A new type of approximation for the radical quintic functional equation in non-Archimedean (2,β)-Banach spaces

Let $\mathbb{R}$ be the set of real numbers. In this paper, we first introduce the notions of non-Archimedean $(2,\beta )$-normed spaces $(X,{6?,?6}_{?,\beta })$ and we will reformulate the fixed point theorem [10, Theorem 1] in this space, after it, we introduce and solve the radical quintic functional equation

$f\left(\sqrt[5]{x^5+y^5}\right)=f(x)+f(y),x,y\in \mathbb{R}.$

Also, under some weak natural assumptions on the function $\gamma :\mathbb{R}\times \mathbb{R}\times X\to [0,\infty )$, we show that this theorem is a very efficient and convenient tool for proving the hyperstability results when $f:\mathbb{R}\to X$ satisfy the following radical quintic inequality

${6f\left(\sqrt[5]{x^5+y^5}\right)?f(x)?f(y),z6}_{?,\beta }\le \gamma (x,y,z),x,y\in \mathbb{R}?\{0\},z\in X,$

with $x\ne ?y$.     

New a posteriori L∞(L2) and L2(L2)-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L^∞(J; L²Ω)-norm and L²(J; L²Ω)-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.     

Weyl’s theorem for algebraically wF(p, r, q) operators with p, r?>?0 and q?≥?1

If T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1 acting in an infinite-dimensional separable Hilbert space, then we prove that Weyl’s theorem holds for f(T) for any f ∈ Hol(σ(T)), where Hol(σ(T)) is the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is a wF(p, r, q) operator with p, r > 0 and q ≥ 1, then the a-Weyl’s theorem holds for f(T). In addition, if T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1, then we establish the spectral mapping theorems for the Weyl spectrum and for the essential approximate point spectrum of T for any f ∈ Hol(σ(T)), respectively. Finally, we examine the stability of Weyl’s theorem and the a-Weyl’s theorem under commutative perturbations by finite-rank operators.