Pub Date : 2022-01-01DOI: 10.30546/1683-6154.21.3.2022.246
{"title":"Nanosatellite Attitude Estimation in Sun and Eclipse Periods without Gyroscopes","authors":"","doi":"10.30546/1683-6154.21.3.2022.246","DOIUrl":"https://doi.org/10.30546/1683-6154.21.3.2022.246","url":null,"abstract":"","PeriodicalId":55503,"journal":{"name":"Applied and Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":10.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82003428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This document was an opportunity for us to measure the contributions of researchers on the asymptotic behavior of the extremes random variables. Beyond the available results, we have proposed an analysis of the behavior of the extremes of random variables of geometric type. We succeeded in determining a subsequence which allows us to establish a convergence in law of the extremes of this type of random variable while passing by the determination of a speed of convergence. We then exposed the limited law which results from it then we called upon the copulas of the extreme values to propose a joint limited law for two independent samples of random variables of geometric type. These results will allow us to analyze, in a document, not only the convergence in moment of order of the other extremes of the random variables of geometric type but also the general asymptotic behavior of the extremes of a serie of random variables with integer value. This document was an opportunity for us to measure the contributions of researchers on the asymptotic behavior of the extremes random variables. Beyond the available results, we have proposed an analysis of the behavior of the extremes of random variables of geometric type. We first made the case of the fact that the random variables of geometric type could be constructed from the random variables of exponential distribution and that they were not only integer variables but also that in general there were no sequences standards that allowed their extremes to converge. To do this, we first built a convergent ϕ(k) subsequence which we then used to define a geometric type Tϕ(k) subsequence of random variables. We have also proved the convergence in distribution of the extremes of the random variables Tϕ(k). We have also exhibited the resulting limit law. Finally, in this document, we have dealt with the multivariate case of random variables of geometric type. We considered two independent samples of random variables of geometric types. Using a copula of extreme values, in particular the logistic copula, we proposed a joint limit distribution of two independent samples of subsequences of geometric type random variables. We then exposed the limited law which results from it then we called upon the copulas of the extreme values to propose a joint limited law for two independent samples of random variables of geometric type.
{"title":"Asymptotic Behavior of Multivariate Extremes Geometric Type Random Variables","authors":"Frédéric Béré, Kpèbbèwèrè Cédric Somé, Remi Guillaume Bagré, Pierre Clovis Nitiéma","doi":"10.11648/J.ACM.20211006.12","DOIUrl":"https://doi.org/10.11648/J.ACM.20211006.12","url":null,"abstract":"This document was an opportunity for us to measure the contributions of researchers on the asymptotic behavior of the extremes random variables. Beyond the available results, we have proposed an analysis of the behavior of the extremes of random variables of geometric type. We succeeded in determining a subsequence which allows us to establish a convergence in law of the extremes of this type of random variable while passing by the determination of a speed of convergence. We then exposed the limited law which results from it then we called upon the copulas of the extreme values to propose a joint limited law for two independent samples of random variables of geometric type. These results will allow us to analyze, in a document, not only the convergence in moment of order of the other extremes of the random variables of geometric type but also the general asymptotic behavior of the extremes of a serie of random variables with integer value. This document was an opportunity for us to measure the contributions of researchers on the asymptotic behavior of the extremes random variables. Beyond the available results, we have proposed an analysis of the behavior of the extremes of random variables of geometric type. We first made the case of the fact that the random variables of geometric type could be constructed from the random variables of exponential distribution and that they were not only integer variables but also that in general there were no sequences standards that allowed their extremes to converge. To do this, we first built a convergent ϕ(k) subsequence which we then used to define a geometric type Tϕ(k) subsequence of random variables. We have also proved the convergence in distribution of the extremes of the random variables Tϕ(k). We have also exhibited the resulting limit law. Finally, in this document, we have dealt with the multivariate case of random variables of geometric type. We considered two independent samples of random variables of geometric types. Using a copula of extreme values, in particular the logistic copula, we proposed a joint limit distribution of two independent samples of subsequences of geometric type random variables. We then exposed the limited law which results from it then we called upon the copulas of the extreme values to propose a joint limited law for two independent samples of random variables of geometric type.","PeriodicalId":55503,"journal":{"name":"Applied and Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":10.0,"publicationDate":"2021-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80274848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-05DOI: 10.11648/J.ACM.20211006.11
L. Davis, F. Pahlevani, T. S. Rajan
The focus of this paper is the development, numerical simulation and parameter analysis of a model of the transcription of ribosomal RNA in highly transcribed genes. Inspired by the well-known classic Lighthill-Whitham-Richards (LWR) traffic flow model, a linear advection continuum model is used to describe the DNA transcription process. In this model, elongation velocity is assumed to be essentially constant as RNA polymerases move along the strand through different phases of gene transcription. One advantage of using the linear model is that it allows one to quantify how small perturbations in elongation velocity and inflow parameters affect important biology measures such as Average Transcription Time (ATT) for the gene. The ATT per polymerase is the amount of time an individual RNAP spends traveling through the DNA strand. The numerical treatment for model simulations includes introducing a low complexity and time accurate method by adding a simple linear time filter to the classic upwind scheme. This improved method is modular and requires a minimal modification of adding only one line of code resulting in increased accuracy without increased computational expense. In addition, it removes the overdamping of upwind. A stability condition for the new algorithm is derived, and numerical computations illustrate stability and convergence of the filtered scheme as well as improved ATT estimation.
{"title":"An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations","authors":"L. Davis, F. Pahlevani, T. S. Rajan","doi":"10.11648/J.ACM.20211006.11","DOIUrl":"https://doi.org/10.11648/J.ACM.20211006.11","url":null,"abstract":"The focus of this paper is the development, numerical simulation and parameter analysis of a model of the transcription of ribosomal RNA in highly transcribed genes. Inspired by the well-known classic Lighthill-Whitham-Richards (LWR) traffic flow model, a linear advection continuum model is used to describe the DNA transcription process. In this model, elongation velocity is assumed to be essentially constant as RNA polymerases move along the strand through different phases of gene transcription. One advantage of using the linear model is that it allows one to quantify how small perturbations in elongation velocity and inflow parameters affect important biology measures such as Average Transcription Time (ATT) for the gene. The ATT per polymerase is the amount of time an individual RNAP spends traveling through the DNA strand. The numerical treatment for model simulations includes introducing a low complexity and time accurate method by adding a simple linear time filter to the classic upwind scheme. This improved method is modular and requires a minimal modification of adding only one line of code resulting in increased accuracy without increased computational expense. In addition, it removes the overdamping of upwind. A stability condition for the new algorithm is derived, and numerical computations illustrate stability and convergence of the filtered scheme as well as improved ATT estimation.","PeriodicalId":55503,"journal":{"name":"Applied and Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":10.0,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79474215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-16DOI: 10.11648/J.ACM.20211005.12
Liu Weili, Huimin Lu
Knot theory is a branch of the geometric topology, the core question of knot theory is to explore the equivalence classification of knots; In other words, for a knot, how to determine whether the knot is an unknot; giving two knots, how to determine whether the two knots are equivalent. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation, but to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. Currently, scholars have also defined multiple knot invariants, but they also have certain limitations, and even more difficult to understand. In this paper, based on existing theoretical results, we define a knot invariant through the skein relation with two equations. To prove this knot invariant, we define a function f(L), and to prove f(L) to be a homology invariant of a non-directed link, we need to show that it remains constant under the Reideminster moves. This article first defines the fk(L), the property of f(L) is obtained by using the properties of fk(L). In the process of proof, the induction method has been used many times. The proof process is somewhat complicated, but it is easier to understand. And the common knot invariant is defined by one equation, which defining the knot invariant with two equations in this paper.
{"title":"A Knot Invariant Defined Based on the Skein Relation with Two Equations","authors":"Liu Weili, Huimin Lu","doi":"10.11648/J.ACM.20211005.12","DOIUrl":"https://doi.org/10.11648/J.ACM.20211005.12","url":null,"abstract":"Knot theory is a branch of the geometric topology, the core question of knot theory is to explore the equivalence classification of knots; In other words, for a knot, how to determine whether the knot is an unknot; giving two knots, how to determine whether the two knots are equivalent. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation, but to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. Currently, scholars have also defined multiple knot invariants, but they also have certain limitations, and even more difficult to understand. In this paper, based on existing theoretical results, we define a knot invariant through the skein relation with two equations. To prove this knot invariant, we define a function f(L), and to prove f(L) to be a homology invariant of a non-directed link, we need to show that it remains constant under the Reideminster moves. This article first defines the fk(L), the property of f(L) is obtained by using the properties of fk(L). In the process of proof, the induction method has been used many times. The proof process is somewhat complicated, but it is easier to understand. And the common knot invariant is defined by one equation, which defining the knot invariant with two equations in this paper.","PeriodicalId":55503,"journal":{"name":"Applied and Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":10.0,"publicationDate":"2021-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88746875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-09DOI: 10.11648/J.ACM.20211004.12
Wajahat Ali, Zhipeng Qiu
HIV spreads by cell-to-cell transfer and the release of cell-free particles. A slightly more effective method of retroviral transmission is the direct cell-to-cell transfer of HIV, according to recent reports. Intracellular interaction between unhealthy and healthy cells, in combination with cytokine discharged by the cells included, may affect the susceptibility of a target resting CD4+T cell to HIV infection and the formation of latent infection. We suggest a class of HIV latency mathematical model, integrating both cell-free virus transmission and direct cell-to-cell diffusion to improve the understanding of the dynamics of the latent reservoirs. We incorporate four components in our model: the uninfected T cells, the latently infected T cells, the active-infected T cells and the HIV viruses. We examine the latency model by introducing the basic reproduction number. We first establish the non-negativity and boundedness of the solutions of the system, and then we investigate the global stability of the steady states. The diseased-free equilibrium is globally stable when the basic reproduction number is less than 1 and if the basic reproduction number is greater than 1, the diseased equilibrium exists and is globally stable. Numerical simulations are executed to interpret the theoretical outcomes and evaluate the relative contribution of latency fractions in the virus production and the HIV latent reservoir by providing estimates.
{"title":"The Global Dynamics of HIV Latency Model Including Cell-to-Cell Viral Transmission","authors":"Wajahat Ali, Zhipeng Qiu","doi":"10.11648/J.ACM.20211004.12","DOIUrl":"https://doi.org/10.11648/J.ACM.20211004.12","url":null,"abstract":"HIV spreads by cell-to-cell transfer and the release of cell-free particles. A slightly more effective method of retroviral transmission is the direct cell-to-cell transfer of HIV, according to recent reports. Intracellular interaction between unhealthy and healthy cells, in combination with cytokine discharged by the cells included, may affect the susceptibility of a target resting CD4+T cell to HIV infection and the formation of latent infection. We suggest a class of HIV latency mathematical model, integrating both cell-free virus transmission and direct cell-to-cell diffusion to improve the understanding of the dynamics of the latent reservoirs. We incorporate four components in our model: the uninfected T cells, the latently infected T cells, the active-infected T cells and the HIV viruses. We examine the latency model by introducing the basic reproduction number. We first establish the non-negativity and boundedness of the solutions of the system, and then we investigate the global stability of the steady states. The diseased-free equilibrium is globally stable when the basic reproduction number is less than 1 and if the basic reproduction number is greater than 1, the diseased equilibrium exists and is globally stable. Numerical simulations are executed to interpret the theoretical outcomes and evaluate the relative contribution of latency fractions in the virus production and the HIV latent reservoir by providing estimates.","PeriodicalId":55503,"journal":{"name":"Applied and Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":10.0,"publicationDate":"2021-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85819239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-30DOI: 10.11648/j.acm.20211003.14
E. S. Shoukralla
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.
{"title":"Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind","authors":"E. S. Shoukralla","doi":"10.11648/j.acm.20211003.14","DOIUrl":"https://doi.org/10.11648/j.acm.20211003.14","url":null,"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.","PeriodicalId":55503,"journal":{"name":"Applied and Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":10.0,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88489621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-29DOI: 10.11648/j.acm.20211003.13
A. J. Ogunniran, Kayode S. Adekeye, J. Adewara, M. Adamu
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.
{"title":"A Review of Change Point Estimation Methods for Process Monitoring","authors":"A. J. Ogunniran, Kayode S. Adekeye, J. Adewara, M. Adamu","doi":"10.11648/j.acm.20211003.13","DOIUrl":"https://doi.org/10.11648/j.acm.20211003.13","url":null,"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.","PeriodicalId":55503,"journal":{"name":"Applied and Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":10.0,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77411836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-26DOI: 10.11648/j.acm.20211003.12
Malick Fall, I. Faye, Alassane Sy, D. Seck
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 under constraints volume. Finally, shape derivative of the functional is established by using Hadamard formula’s and an optimality condition is also given.
{"title":"On Shape Optimization Theory with Fractional Laplacian","authors":"Malick Fall, I. Faye, Alassane Sy, D. Seck","doi":"10.11648/j.acm.20211003.12","DOIUrl":"https://doi.org/10.11648/j.acm.20211003.12","url":null,"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 under constraints volume. Finally, shape derivative of the functional is established by using Hadamard formula’s and an optimality condition is also given.","PeriodicalId":55503,"journal":{"name":"Applied and Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":10.0,"publicationDate":"2021-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72538180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-22DOI: 10.11648/j.acm.20211003.11
Nicolae Popoviciu
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.
{"title":"Two Hypothesis on a Combinatorial Problem for Possible States on the Arrival Line for n Competitor Runners","authors":"Nicolae Popoviciu","doi":"10.11648/j.acm.20211003.11","DOIUrl":"https://doi.org/10.11648/j.acm.20211003.11","url":null,"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.","PeriodicalId":55503,"journal":{"name":"Applied and Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":10.0,"publicationDate":"2021-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74410475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-16DOI: 10.11648/j.acm.20211002.12
Ernesto Borges Batista, Luis Alberto Escalona Fernández, Kirelis Napoles Dominguez, Y. Sarmiento, Claudia del Carmen Pupo Marrero
Aims: Fractal for comparison of radiological imagery between morphologic and pathological elements confirms the behavior of the experimental information through dimension itself. The irregularity of the human body is its own characteristic. However, it has traditionally been measured with Euclidean metrics, by approximating its shapes to regular lines, areas and volumes. In response to this impossibility of making reliable measurements of this class of objects, fractal geometry is developed, which allows to adequately characterize the irregular shape of the human body. Method: they use the theoretic methods: Analysis synthesis, induction deduction and abstraction concretion. Processes of understanding, explanation and interpretation. Methods, procedures and mathematical algorithms, as well as information-technology professional programs are applicable. Come true quest of information about the application of dimension fractal in the diagnostic one belonging to diseases, based in radiological imagery. The diagnostic method fractal consists in the calculation of dimension for three cellular objects defined as: the nucleus, the cytoplasm without a nucleus and the entire cell. Results: Methods and procedures to ratify diseases, where the different authors yield a mathematical model, propose which themselves fractal for the comparison of histological and pathological elements confirms the behavior of the experimental data represented in radiological imagery, by means of dimension. About fractal geometry, the fractal dimension is obtained, which is a numerical measure that represents the degree of irregularity of an object. However, it has traditionally been measured with Euclidean metrics, by approximating its shapes to regular lines, areas and volumes. In response to this impossibility of making reliable measurements of this class of objects, fractal geometry is developed, which allows to adequately characterize the irregular shape of the human body. Conclusions: A methodology of work based in radiological imagery by comparison of histological and pathological elements to determine different diseases in patients becomes established.
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