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Two Hypothesis on a Combinatorial Problem for Possible States on the Arrival Line for n Competitor Runners
Issue:
Volume 10, Issue 3, June 2021
Pages:
46-55
Received:
9 April 2021
Accepted:
25 May 2021
Published:
22 June 2021
Abstract: In a very small t-time interval, several runners could occupy the same place on the arrival line (hypothesis 1). Each runner has his own name and a competition number (on the shirt). The number of runners is a natural number n. For each given n, the hypothesis creates a combinatorial problem having a lot of posible states. All notations are choose so that to indicate easily by name their meaning. The states are separated into two classes: non-nominal states and nominal states. The states are related with the place I, II, III etc on arrival line. It is necessary to generate the total number of non-nominal states (on arrival line) and the total number of nominal states. In order to generate the states the work uses some formulas and some specialised algorithms. For example, the consrtuction of all non-nominal states recommends that the string for the position I to use a decreasing string. The same rule is validly for position II, but for sub-strings etc. A lot of numerical examples ilustrate the states generation. An independent method verifies the correctitude of states generation. In order to continue the study of combinatorial problem, the work introduces two new notions in section 5. The notions of partial frequency and final frequency are defined for a nominal known runner in final classification, together with computational formulas. The section 6 constructs the random variables attached to final classification and the probability of each place on arrival line. Each runner receives a score (a number of points) related with his final classification. May be the runner is interested to know the probability to ocuppy the first place (place I) and to estimate the number of possible points. All the results could be written in a centralisation table (section 7). Section 8 contains several numerical examples with statistical computations. At the end of the work we replace hypothesis 1 by hypothesis 2: only one runner could ocuppay each place. All the above notions have a new specific form. The numerical examples ilustrates the theory.
Abstract: In a very small t-time interval, several runners could occupy the same place on the arrival line (hypothesis 1). Each runner has his own name and a competition number (on the shirt). The number of runners is a natural number n. For each given n, the hypothesis creates a combinatorial problem having a lot of posible states. All notations are choose ...
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On Shape Optimization Theory with Fractional Laplacian
Malick Fall,
Ibrahima Faye,
Alassane Sy,
Diaraf Seck
Issue:
Volume 10, Issue 3, June 2021
Pages:
56-68
Received:
17 April 2021
Accepted:
13 May 2021
Published:
26 June 2021
Abstract: The fractional Laplacian is a nonlocal operator that appears in biology, in physic, in fluids dynamic, in financial mathematics and probability. This paper deals with shape optimization problem associated to the fractional laplacian ∆s, 0 < s < 1. We focus on functional of the form J(Ω) = j(Ω,uΩ) where uΩ is solution to the fractional laplacian. A brief review of results related to fractional laplacian and fractional Sobolev spaces are first given. By a variational approach, we show the existence of a weak solution uΩbelonging to the fractional Sobolev spaces Ds, 2(Ω) of the boundary value problem considered. Then, we study the existence of an optimal shape of the functional J(Ω) on the class of admissible sets 
under constraints volume. Finally, shape derivative of the functional is established by using Hadamard formula’s and an optimality condition is also given.
Abstract: The fractional Laplacian is a nonlocal operator that appears in biology, in physic, in fluids dynamic, in financial mathematics and probability. This paper deals with shape optimization problem associated to the fractional laplacian ∆s, 0 < s < 1. We focus on functional of the form J(Ω) = j(Ω,uΩ) where uΩ is solution to the fractional laplacian. A brief review of results related to fractiona...
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A Review of Change Point Estimation Methods for Process Monitoring
Ademola John Ogunniran,
Kayode Samuel Adekeye,
Johnson Ademola Adewara,
Muminu Adamu
Issue:
Volume 10, Issue 3, June 2021
Pages:
69-75
Received:
15 March 2021
Accepted:
7 April 2021
Published:
29 June 2021
Abstract: When one or more observations fall outside the control limits, the chart signals the existence of a change in the process. Change point detection is helpful in modelling and prediction of time series and is found in broader areas of applications including process monitoring. Three approaches were proposed for estimating change point in process for the different types of changes in the literature. they are: Maximum Likelihood Estimator (MLE), the Cumulative Sum (CUSUM), and the Exponentially Weighted Moving Average (EWMA) approaches. This paper gives a synopsis of change point estimation, specifies, categorizes, and evaluates many of the methods that have been recommended for detecting change points in process monitoring. The change points articles in the literature were categorized broadly under five categories, namely: types of process, types of data, types of change, types of phase and methods of estimation. Aside the five broad categories, we also included the parameter involved. Furthermore, the use of control charts and other monitoring tools used to detect abrupt changes in processes were reviewed and the gaps for process monitoring/controlling were examined. A combination of different methods of estimation will be a valuable approach to finding the best estimates of change point models. Further research studies would include assessing the sensitivity of the various change point estimators to deviations in the underlying distributional assumptions.
Abstract: When one or more observations fall outside the control limits, the chart signals the existence of a change in the process. Change point detection is helpful in modelling and prediction of time series and is found in broader areas of applications including process monitoring. Three approaches were proposed for estimating change point in process for ...
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Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind
Issue:
Volume 10, Issue 3, June 2021
Pages:
76-85
Received:
15 June 2021
Accepted:
24 June 2021
Published:
30 June 2021
Abstract: We establish a new straightforward interpolation method for solving linear Volterra integral equations with weakly singular kernels. The proposed method is fundamentally different from all other published methods for solving this type of equations. We have modified some vector-matrix barycentric Lagrange interpolation formulas to be convenient for interpolating the kernel twice concerning the two variables of the kernel and introducing new ideas for selecting interpolation nodes that ensure isolation of the singularity of the kernel. We create two rules for selecting the distribution nodes of the two kernel variables that do not allow the denominator of the kernel to contain an imaginary value. We interpolate the unknown and data functions into the corresponding interpolant polynomial; each of the same degree via three matrices, one of which is a monomial. By applying the presented method based on the two created rules, we transformed the kernel into a double interpolant polynomial with a degree equal to that of the unknown function via five matrices, two of which are monomials. We substitute the interpolate unknown function twice; on the left side and on the right side of the integral equation to get an algebraic linear system without applying the collocation method. The solution of this system yields the unknown coefficients matrix that is necessary to find the interpolant solution. We solve three different examples for different values of the upper integration variable. The obtained results as shown in tables and figures prove that the obtained interpolate solutions are extraordinarily faster to converge to the exact ones using interpolants of lowest degrees and give better results than those obtained by other methods. This confirms the originality and the potential of the presented method.
Abstract: We establish a new straightforward interpolation method for solving linear Volterra integral equations with weakly singular kernels. The proposed method is fundamentally different from all other published methods for solving this type of equations. We have modified some vector-matrix barycentric Lagrange interpolation formulas to be convenient fo...
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