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Text Define χ ( n ) recursively by χ ( 1 ) = 1 and χ ( n ) = ϕ ( n ) + χ ( n / q ) for all integers n > 1 , where q is the least prime factor of n, and where ϕ is the Euler totient function. We show that χ ( n ) = ϕ ( d ) ( χ ( l ) − 1 ) + χ ( d ) , where n = d l and the prime factors of d are greater than the prime factors of l. We also show χ ( n m ) ≤ χ ( n ) χ ( m ) when n and m are coprime numbers. As an application, we show that for all primes p ≥ 11 , χ ( p 2 − p ) > χ ( p 2 − 1 ) . We discuss the interpretation of χ as the clique number of the power graph of a finite cyclic group and the significance of the inequality in this context. Video For a video summary of this paper, please visit https://youtu.be/p8finzAEJps .

In this paper a numerical approach combining the least squares method and the genetic algorithm (sequential and multi-core parallelization approach) is proposed for the determination of temperature in an inverse heat conduction problem (IHCP). Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data. Results show that an excellent estimation can be obtained by implementation sequential genetic algorithm within a CPU with clock speed 2.7 GHz, and parallel genetic algorithm within a 16-core CPU with clock speed 2.7 GHz for each core.

The problem of identifying the coefficient in a square porous medium is considered. It is shown that under certain conditions of data f , g , and for a properly specified class A of admissible coefficients, there exists at least one a ∈ A such that ( a , u ) is a solution of the corresponding inverse problem.

This study is intended to provide a numerical algorithm for solving a one-dimensional inverse heat conduction problem. The given heat conduction equation, the boundary conditions, and the initial condition are presented in a dimensionless form. The numerical approach is developed based on the use of the solution to the auxiliary problem as a basis function. To regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization method to obtain the stable numerical approximation to the solution.

In this paper, we consider an inverse heat conduction problem (IHCP). A set of temperature measurements at a single sensor location inside the heat conduction body is required. Using a transformation, the ill-posed IHCP becomes a Cauchy problem. Since the solution of Cauchy problem, exists and is unique but not always stable, the ill-posed problem is closely approximated by a well-posed problem. For this new well-posed problem, the existence, uniqueness, and stability of the solution are proved.

A numerical approach combining the use of the finite difference method with the solution of ordinary differential equation (ODE) is proposed for the determination of boundary conditions in an inverse parabolic problem. The least-squares method is adopted to modify the solution. Results show that an excellent estimation can be obtained within a couple of minutes CPU time at pentium IV-2.4 GHz PC.

This study is intended to provide a new numerical algorithm for solving a one-dimensional inverse nonlinear parabolic problem. At the beginning of the study, Taylor’s series expansion is employed to linearize nonlinear terms and then finite-difference method is used to discretize the problem domain. The present approach is to rearrange the matrix forms of the differential governing equations and estimate unknown functions. The least-squares method is adopted to find the solution. It is assumed that no prior information is available on the functional form of the unknown coefficient in the present study, thus, it is classified as the function estimation. Results show that an excellent estimation on the unknown functions of the inverse problem which can be obtained within a couple of minutes CPU time at Pentium IV-2.4 GHz PC.

In this paper, we consider an inverse heat conduction problem (IHCP). The given heat conduction equation, the boundary condition, and the initial condition are presented in a dimensionless form. A set of temperature measurements at a single sensor location inside the heat conduction body is required. Using a linear transformation, the ill-posed IHCP becomes a Cauchy problem. This problem will be solved by applying Legendre polynomials. Results show that an excellent estimation can be obtained within a couple of minutes CPU time at pentium IV-2.4 GHz PC.

In this paper, a numerical solution of an inverse non-dimensional heat conduction problem will be considered. By using a sensor located at a point inside the body and measuring the temperature at a point x = x 1 , 0 x 1