The Conditions of Existence of the Extremal Element for the Problem of Finding the Distance Between Two Sets, the Unity of an Extremal Element for its Equivalent Problem, the Properties of the Function of the Distance

AbstractIn this paper, the problem of determining heat transfer from convecting-radiating fin of triangular and concave parabolic shapes is investigated.We consider one-dimensional, steady conduction in the fin and neglect radiative exchange between adjacent fins and between the fin and its primary surface. A novel intelligent computational approach is developed for searching the solution. In order to achieve this aim, the governing equation is transformed into an equivalent problem whose boundary conditions are such that they are convenient to apply reformed version of Chebyshev polynomials of the first kind. These Chebyshev polynomials based functions construct approximate series solution with unknown weights. The mathematical formulation of optimization problem consists of an unsupervised error which is minimized by tuning weights via interior point method. The trial approximate solution is validated by imposing tolerance constrained into optimization problem. Additionally, heat transfer rate and the fin efficiency are reported.

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TWO-LEVEL BURN-IN FOR RELIABILITY AND ECONOMY IN REPAIRABLE SERIES SYSTEMS HAVING INCOMPATIBILITY

International Journal of Reliability Quality and Safety Engineering ◽

10.1142/s0218539304001464 ◽

2004 ◽

Vol 11 (03) ◽

pp. 197-211 ◽

Cited By ~ 2

Author(s):

KYUNGMEE O. KIM ◽

WAY KUO

Keyword(s):

Nonlinear Programming ◽

Poisson Process ◽

Prediction Models ◽

Renewal Processes ◽

Assembly Process ◽

Equivalent Problem ◽

System Failures ◽

The Past ◽

And Performance ◽

Series Systems

When a system is assembled from components, incompatibility often occurs as a result of the assembly process. The ability to quantify incompatibility is very important for making burn-in decisions because the goal of system burn-in is to minimize the incompatibility factor. In the past, incompatibility has been only partially represented in the system prediction models because it was assumed that assembly had no effect on the components. This paper presents a more accurate model for system prediction by allowing for the possibility that, in some cases, assembly adversely affects the components. After applying a superposition of delayed renewal processes and a nonhomogeneous Poisson process for modeling times between system failures, we derive and analyze the effects of component and system burn-in on the system cost and performance. Examples are included to demonstrate how to determine optimal component and system burn-in times simultaneously based on an equivalent problem formation and nonlinear programming.

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A Systematic Approach to Evaluating Design Prompts in Supporting Experimental Design Research

Proceedings of the Design Society: International Conference on Engineering Design ◽

10.1017/dsi.2019.282 ◽

2019 ◽

Vol 1 (1) ◽

pp. 2755-2764 ◽

Cited By ~ 1

Author(s):

Apurva Patel ◽

Maria-Vittoria Elena ◽

Joshua Summers

Keyword(s):

Experimental Design ◽

Systematic Approach ◽

Design Research ◽

Problem Statement ◽

Equivalent Problem ◽

Step Process ◽

Initial Comparison ◽

Experiment Verification ◽

Multiple Variables ◽

The Given

AbstractExperiments that study engineering behavior in design often rely on participants responding to a given design prompt or a problem statement. Moreover, researchers often find themselves testing multiple variables with a relatively small participant pool. In such situations multiple design prompts may be used to boost replication by giving each participant an equivalent problem with a different experimental condition. This paper presents a systematic approach to compare given design prompts using a two-step process that allows an initial comparison of the prompts and a post-experiment verification of the similarity of the given prompts. Comparison metrics are provided which can be used to evaluate a level of similarity of existing prompts as well as develop similar problems. These metrics include complexity (size, coupling, and solvability), familiarity, and prompt structure. Statistical methods are discussed for post-experiment verification. Guidelines are provided for a post-experiment survey which may be used for an additional perspective of prompt similarity. The proposed approach is demonstrated using an experiment where two design prompts were used for within-subject replication.

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AUV-Method for a Class of Constrained Minimized Problems of Maximum Eigenvalue Functions

Journal of Function Spaces ◽

10.1155/2017/5309698 ◽

2017 ◽

Vol 2017 ◽

pp. 1-6

Author(s):

Wei Wang ◽

Ming Jin ◽

Shanghua Li ◽

Xinyu Cao

Keyword(s):

Exact Penalty ◽

Maximum Eigenvalue ◽

Convex Approximation ◽

Equivalent Problem ◽

Composite Function ◽

Primal Problem ◽

Approximate Problem ◽

Constrained Problem ◽

Constrained Minimization Problem ◽

Unconstrained Problem

In this paper, we apply theUV-algorithm to solve the constrained minimization problem of a maximum eigenvalue function which is the composite function of an affine matrix-valued mapping and its maximum eigenvalue. Here, we convert the constrained problem into its equivalent unconstrained problem by the exact penalty function. However, the equivalent problem involves the sum of two nonsmooth functions, which makes it difficult to applyUV-algorithm to get the solution of the problem. Hence, our strategy first applies the smooth convex approximation of maximum eigenvalue function to get the approximate problem of the equivalent problem. Then the approximate problem, the space decomposition, and theU-Lagrangian of the object function at a given point will be addressed particularly. Finally, theUV-algorithm will be presented to get the approximate solution of the primal problem by solving the approximate problem.

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Information Transfer in Solving Problems

Journal for Research in Mathematics Education ◽

10.5951/jresematheduc.10.2.0094 ◽

1979 ◽

Vol 10 (2) ◽

pp. 94-102

Author(s):

Gerald Kulm ◽

Harold Days

Keyword(s):

Information Transfer ◽

Problem Solution ◽

Equivalent Problem ◽

Algebraic Problem ◽

Special Case

The study investigated the transfer between problems having related structures. Problems were related to the Hens and Rabbits (algebraic) and the Missionaries and Cannibals (puzzle) problems in four ways: equivalent, similar, special case, and generalization. The results indicated that transfer occurred for both the algebraic and puzzle problems, and significant transfer resulted when the generalization was solved first. Solving an equivalent problem was effective for the puzzle but not the algebraic problem, whereas the opposite was true for solving a similar problem. Solution of related problems helps subjects to focus on relevant strategies, but distantly related structures or different contexts appear to interfere with transfer in solving some problems.

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On the Tensor Convolution and the Quantum Separability Problem

Open Systems & Information Dynamics ◽

10.1142/s1230161210000217 ◽

2010 ◽

Vol 17 (04) ◽

pp. 331-346

Author(s):

Gabriel Pietrzkowski

Keyword(s):

Abelian Group ◽

Tensor Product ◽

Hermitian Operator ◽

Product Formula ◽

Locally Compact Abelian Group ◽

Locally Compact ◽

Equivalent Problem ◽

Finite Dimensional ◽

Separability Problem ◽

Convolution Formula

We consider the problem of separability: decide whether a Hermitian operator on a finite dimensional Hilbert tensor product [Formula: see text] is separable or entangled. We show that the tensor convolution [Formula: see text] defined for mappings [Formula: see text] on an almost arbitrary locally compact abelian group G , gives rise to formulation of an equivalent problem to the separability one.

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Wavelets Galerkin Method for the Fractional Subdiffusion Equation

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4034391 ◽

2016 ◽

Vol 11 (6) ◽

Cited By ~ 3

Author(s):

M. H. Heydari

Keyword(s):

Galerkin Method ◽

Fractional Derivatives ◽

Operational Matrix ◽

Practical Importance ◽

Time Derivative ◽

Algebraic Equations ◽

Legendre Wavelets ◽

Equivalent Problem ◽

Subdiffusion Equation ◽

The Galerkin Method

The time fractional subdiffusion equation (FSDE) as a class of anomalous diffusive systems has obtained by replacing the time derivative in ordinary diffusion by a fractional derivative of order 0<α<1. Since analytically solving this problem is often impossible, proposing numerical methods for its solution has practical importance. In this paper, an efficient and accurate Galerkin method based on the Legendre wavelets (LWs) is proposed for solving this equation. The time fractional derivatives are described in the Riemann–Liouville sense. To do this, we first transform the original subdiffusion problem into an equivalent problem with fractional derivatives in the Caputo sense. The LWs and their fractional operational matrix (FOM) of integration together with the Galerkin method are used to transform the problem under consideration into the corresponding linear system of algebraic equations, which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account, automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

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Electrodynamics of charged scalar solitons

Canadian Journal of Physics ◽

10.1139/p79-296 ◽

1979 ◽

Vol 57 (12) ◽

pp. 2171-2177 ◽

Cited By ~ 3

Author(s):

T. F. Morris

Keyword(s):

Hamiltonian Formalism ◽

Finite Energy ◽

The Self ◽

Equivalent Problem ◽

Complex Scalar ◽

Charged Scalar ◽

Localized Solutions ◽

Interaction Solutions ◽

Complex Scalar Field ◽

Nonzero Charge

The electrodynamics of a nonlinear, complex scalar field is developed from the basis of a Hamiltonian formalism. By means of a canonical transformation, the equations for a stationary state are reduced to the consideration of an equivalent problem in static equilibrium. Localized solutions are defined. For specified conditions on the self-interaction, solutions of finite energy, and finite, nonzero charge, are localized.

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An Effective Algorithm for Globally Solving Sum of Linear Ratios Problems

Journal of Control Science and Engineering ◽

10.1155/2017/8138975 ◽

2017 ◽

Vol 2017 ◽

pp. 1-7

Author(s):

Hongwei Jiao ◽

Lei Cai ◽

Zhisong Hou ◽

Chunyang Bai

Keyword(s):

Programming Problem ◽

Computational Efficiency ◽

Nonconvex Programming ◽

Optimal Solution ◽

Space Region ◽

Effective Algorithm ◽

Equivalent Problem ◽

Solution Quality ◽

Space Partition ◽

Pruning Technique

In this study, we propose an effective algorithm for globally solving the sum of linear ratios problems. Firstly, by introducing new variables, we transform the initial problem into an equivalent nonconvex programming problem. Secondly, by utilizing direct relaxation, the linear relaxation programming problem of the equivalent problem can be constructed. Thirdly, in order to improve the computational efficiency of the algorithm, an out space pruning technique is derived, which offers a possibility of pruning a large part of the out space region which does not contain the optimal solution of the equivalent problem. Fourthly, based on out space partition, by combining bounding technique and pruning technique, a new out space branch-and-bound algorithm for globally solving the sum of linear ratios problems (SLRP) is designed. Finally, numerical experimental results are presented to demonstrate both computational efficiency and solution quality of the proposed algorithm.

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